UNIVERSITY of CALIFORNIA, IRVINE Kinetic Studies Of

Home , Adhesion

UNIVERSITY OF CALIFORNIA, IRVINE

Kinetic Studies of Multivalent Nanoparticle Adhesion

DISSERTATION

submitted in partial satisfaction of the requirements for the degree of

DOCTOR OF PHILOSOPHY

in Biomedical Engineering

Mingqiu Wang

Dissertation Committee: Assistant Professor Jered Haun, Chair Associate Professor Jun Allard Professor Young Jik Kwon

2018

DEDICATION

To my parents, for their unconditional love and support.

ii TABLE OF CONTENTS

DEDICATION...... II

TABLE OF CONTENTS...... III

LIST OF FIGURES ...... V

LIST OF TABLES ...... VII

ACKNOWLEDGMENTS ...... VIII

CURRICULUM VITAE ...... X

ABSTRACT OF THE DISSERTATION ...... XI

1. INTRODUCTION ...... 1

1.1. TARGET NANOPARTICLE ADHESION ...... 1

1.2. NANO ADHESIVE DYNAMICS (NAD) SIMULATION AND KINETIC ANALYSIS ...... 2

2. BACKGROUND ...... 5

2.1. MULTIVALENT NANOPARTICLE ADHESION ...... 5

2.2. FLOW CHAMBER ASSAY ...... 6

2.3. ADHESION MODEL ...... 7

2.4. BROWNIAN DYNAMICS ...... 8

2.5. BIRTH AND DEATH MODEL ...... 9

3. MODEL DEVELOPMENT ...... 11

3.1. OVERVIEW OF NANO ADHESIVE DYNAMICS (NAD) SIMULATION ...... 11

3.2. NANOPARTICLE MOTION ...... 11

3.3. NANOPARTICLE ADHESION ...... 16

3.4. ADHESION MOLECULE CONFIGURATIONS ON SURFACE ...... 19

3.5. PARAMETERS...... 20

3.6. COMPUTATION DETAILS ...... 23

3.7. ATTACHMENT AND DETACHMENT RATES ...... 24

4. THE EVOLUTION OF PARTICLE DETACHMENT ...... 26

iii 4.1. INTRODUCTION ...... 26

4.2. METHODS ...... 27

4.3. GENERAL DYNAMICS OF NANOPARTICLES AND BONDS ...... 30

4.4. TRACKING BOND DISTRIBUTIONS ...... 36

4.5. MECHANICAL STATE DIAGRAM ...... 41

4.6. MEASURING � USING OPTICAL TWEEZERS ...... 45

4.7. FINAL FITTING OF EXPERIMENTAL DATA...... 46

4.8. DISCUSSION...... 54

5. HETEROGENEITY IN MULTIVALENT NANOPARTICLE ...... 64

5.1. INTRODUCTION ...... 64

5.2. METHODS ...... 64

5.3. BOND POTENTIAL (BP) DETACHMENT MODEL ...... 77

5.4. RELATING PARAMETERS FOR THE BP AND EMPIRICAL MODELS ...... 84

5.5. BOND STATE MODEL TO DETERMINE TRANSITION RATES...... 86

5.6. ESTIMATING NANOPARTICLE DETACHMENT RATE FROM TRANSITION RATES ...... 89

5.7. OBTAINING DETACHMENT RATES FOR ALL SUB-POPULATIONS ...... 90

5.8. DISCUSSION...... 116

6. NAD ATTACHMENT SIMULATIONS ...... 121

6.1. INTRODUCTION ...... 121

6.2. METHODS ...... 121

6.3. GENERAL DYNAMICS OF NANOPARTICLE ATTACHMENT ...... 124

6.4. INFLUENCE OF BOX HEIGHT AND ADHESION MOLECULE DENSITY ...... 125

7. CONCLUSION AND FUTURE DIRECTIONS ...... 127

REFERENCES ...... 130

iv LIST OF FIGURES

Figure 3.1 Size-scaled depiction of the adhesion molecule system employed...... 19 Figure 3.2 Optimizing simulations...... 24 Figure 4.1 Algorithm for NAD detachment simulations in which nanoparticles were initiated with a single bond...... 28 Figure 4.2 Nanoparticle and bond dynamics...... 31 Figure 4.3 Total bond numbers versus time and �...... 33 Figure 4.4 Bond biophysics and dynamics...... 35 Figure 4.5 Bond biophysics and reaction rates...... 36 Figure 4.6 Bond valency distributions and bond potential...... 38 Figure 4.7 Bond number dynamics...... 39 Figure 4.8 Corrected bond distributions to reveal bond potential...... 40 Figure 4.9 Mechanical state diagram...... 43 Figure 4.10 Dynamic force spectroscopy using optical tweezers...... 44 Figure 4.11 Final fitting of experiments for different ICAM-1 clustering conditions...... 48 Figure 4.12 Bond numbers for all ICAM-1 configurations...... 53 Figure 4.13 Mean bond potential for dimer and monomer cases...... 53 Figure 4.14 Single tether simulations and valency state-dependent detachment dynamics...... 57 Figure 4.15 Single tether simulations...... 58 Figure 5.1 Bond transition rate model fitting parameters...... 72 Figure 5.2 Full population BP model fits for two components using 2 and 3 degrees of freedom...... 78 Figure 5.3 BP model analysis of the base case...... 79 Figure 5.4 Individual BP detachment fits for cases 2-5...... 81 Figure 5.5 BP model analysis of cases 2-5...... 82 Figure 5.6 BP model analysis of cases 6-9...... 83 Figure 5.7 Individual BP detachment fits for cases 6-9...... 84 Figure 5.8 Bond transition rate modeling for the base case...... 88 Figure 5.9 Bond transition rate modeling for the cases 2-5...... 89

v Figure 5.10 Bond transition rate modeling for the base case using one component...... 90 Figure 5.11 Bond transition rate modeling for the cases 2-5...... 93 Figure 5.12 Final BP model fitting results...... 94 Figure 6.1 Nanoparticle attachment dynamics...... 125 Figure 6.2 Attachment rates at varied adhesion molecule densities and box heights...... 126

LIST OF TABLES

Table 3.1 Physical parameters used in NAD simulations...... 22 Table 4.1 Nanoparticle dynamics at � = 0.01 N/m across different � values...... 41 Table 4.2 Different � and � combinations resulting in nanoparticle dynamics that match experimental results...... 45 Table 4.3 Final simulation results across all valency conditions for � = 0.27 nm, � = 0.8 N/m, and the dimer configuration for ICAM-1...... 49 Table 4.4 Final simulation results across all valency conditions for � = 0.27 nm, � = 0.8 N/m, and the monomer configuration for ICAM-1...... 50 Table 4.5 Final simulation results across all valency conditions for � = 0.27 nm, � = 0.8 N/m, and the clustered dimer configuration for ICAM-1...... 52 Table 5.1 BP model parameters for two component fits assuming one does not detach...... 80 Table 5.2 Correlation between empirical and BP model parameters...... 86 Table 5.3 Mean first passage time calculations ...... 88 Table 5.4 Final fitting parameters using detachment fitting and mean first passage time criteria ...... 95 Table 5.5 BP model parameters for single component fits...... 96 Table 5.6 BP model parameters for two component fits using 2 and 3 degrees of freedom.97 Table 5.7 Pause time fitting results used to estimate bond number transition rates...... 98 Table 5.8 Bond transition rates from ODE model...... 107 Table 6.1 Nanoparticle attachment dynamics over varied box heights and adhesion molecule densities...... 126

vii ACKNOWLEDGMENTS

Getting to know and working with Prof. Jered Haun is the most precious part of my experience at UC Irvine. He has been supportive from the first day I started working on this project. He inspired me with his broad knowledge and experience in many fields, from biology to engineering, from wet lab experiments to computational modeling. I would not have finished this work without his valuable insights as an experimental scientist. He also encouraged me constantly with his humor, enthusiasm, patience, and persistence when in difficulties. I would always remember his thorough edits on my papers, correcting each of my typos and grammar mistakes from a non-native English speaker; remember his understanding, consideration, and support when I was upset and was struggling with our first paper. During the past five years, he not only imparted the knowledge of multivalent nanoparticle adhesion to me, but passed on me the ability and freedom of learning what I want to learn. His passion and love for his researches have enlightened those dark days during my Ph.D. study, and will always back me up in future.

I also want to thank Prof. Jun Allard. He started offering me mathematical guidance in the first hour we met at UCI CCBS retreat. Since then, his extensive knowledge in biophysics, statistics, and mathematics has directed me to the right paths every time I got stuck with my model. I could not have imaged this thesis being done without his help. I cannot express deeper appreciation to Prof. Jun Allard for making each of our discussion a delightful experience, teaching me knowledge, encouraging me by his optimism and kindness, and enabling a timid me to become confident in research through the past four years.

viii I would like to thank my committee member Prof. Young Jik Kwon for his precious suggestions and comments on my thesis. I would also like to thank my collaborators Prof.

Elliot Botvinick and Dr. Shreyas Ravindranath for valuable expertise on setup of optical tweezer spectroscopy. I am grateful to all their advice and guidance that I received through our fruitful discussions. I would like to thank Dr. Maha Rahim for helping me prepare

ICAM-1 CHO cells for optical tweezer experiments. I would like to thank Nianjiang Tan for the help with the model development of dynamical system analysis.

I would like to express my sincerest gratitude to all members of the Haun lab. You have filled my past five years with happiness and joy. I cherish the days we spent together, traveling to Tijuana, Tampa, Minneapolis; playing monopoly together; exchanging gifts from annually lab Secret Santa, and even more. You have made my Ph.D. life such a colorful experience!

I would like to acknowledge the American Chemical Society for providing me with permissions to include copyrighted materials as part of my dissertation.

In the end, I want to thank my parents. Thank you for your unconditional love and support on me all the time.

ix CURRICULUM VITAE

Mingqiu Wang

2012 - 2018 Ph.D. in Biomedical Engineering, University of California, Irvine

2012 - 2013 M.S. in Biomedical Engineering, University of California, Irvine

2008 - 2012 B.S. in Chemical Engineering, Southeast University

x ABSTRACT OF THE DISSERTATION

Kinetic Studies of Multivalent Nanoparticle Adhesion

Mingqiu Wang

Doctor of Philosophy in Biomedical Engineering

University of California, Irvine, 2018

Assistant Professor Jered Haun, Chair

Targeted delivery of functional nanoparticles (NPs) holds tremendous potential in diagnostics and therapeutics of vascular diseases and cancer. One of the major attributes is their ability of forming multiple bonds with target cells, thereby providing the flexibility for engineering desired adhesion strength. However, the multivalent binding of nanoparticles is still poorly understood, particularly from a dynamic perspective.

In this work, we first developed Nano adhesive dynamics (NAD) simulation to investigate the binding dynamics between an antibody-conjugated, 210-nm-diameter sphere and an ICAM-1-coated surface. In the NAD simulation, the particle motions are determined using the Langevin equation and the ICAM-1-Antibody bond association and dissociation are computed using Bell’s law. The NAD simulation replicated the time-decay in NP detachment rate as found in previous flow chamber assays. We found that the decreased detachment rate resulted from the heterogeneity of NPs in bonding ability, which is inherent in NP multivalence. NPs detached preferentially from those unable to quickly accumulate bonds. This could serve as a selection mechanism to eliminate NPs

xi confined to a lower bonding ability. In addition, we found that NP detachment was driven by mechanical forces due to NP thermal motions.

In order to understand the influence of NP bonding heterogeneity on adhesion dynamics, we further examined NP detachment behaviors at higher resolutions: at sub- population level, in which we grouped NPs with similar bonding ability as in one bond potential (BP) category; and further at bond level, in which we grouped NPs with the same bond number at a time point as in one bond state. At sub-population level, we found that the whole population detachment curve from NAD simulation data could be recovered via summing up fitted detachment curves of each BP category, fitted using mixture exponential distribution with two components. At bond level, we modeled NP transition between different bond states as a birth and death process (BD process), and the transition rate matrix of BD process was parameterized from NAD simulation data. Thereby we extracted detachment rates from the transition rate matrix using first passage times to reach the detached state. Combining different levels of analysis, we were able to provide a comprehensive study on multivalent nanoparticle adhesion dynamics.

xii 1. INTRODUCTION

1.1. Target nanoparticle adhesion

The targeted delivery of imaging or therapeutic agents to disease sites within the body still remains a major medical goal even after decades of research. Nanoparticle carriers have been shown to offer numerous advantages as a delivery platform, including high-loading capacity and protection of agents, facile attachment of affinity molecules, and favorable pharmacokinetics.[1-3] Another powerful attribute is the ability to form multiple bonds with target cells, thereby improving overall adhesion strength and internalization rate into cells.[4-13] However, our understanding of multivalent nanoparticle adhesion has primarily been based on thermodynamic behavior, limited by a lack of insight into key details such as bond formation rate, stability, and equilibrium number. For example, binding performance has typically been assessed after systems have reached equilibrium, and results were assessed in terms of an apparent affinity, also termed the avidity. Another issue is that it is nearly impossible to control for differences in context between different experimental systems. Thus, critical knowledge gaps remain in the field regarding the time- course by which nanoparticles evolve from initial capture via one or more bonds to the final multivalent state, and how this is influenced by system parameters. Such information would be extremely powerful for designing nanoparticle carriers that exhibit optimal targeting performance for different disease scenarios. The major challenge is that experimental systems cannot directly interrogate individual bonds, and thus interpretations must be made based on macroscopic nanoparticle behavior. Nearly all multivalent adhesion studies have assessed results only after reaching thermodynamic equilibrium, and therefore dynamic information from the early stages of adhesion were

1 neglected. Finally, multivalent adhesion is intrinsically linked to contextual factors including, but not limited to, nanoparticle size, adhesion molecule densities, and bond chemical and mechanical properties. Values for these factors are often unknown, and the individual and combined effects can be difficult to isolate and predict. The above limitations apply to experimental systems, as well as computational approaches that have sought to predict thermodynamic adhesion energy as a function of bond number.[14-20]

Thus, there is a need to investigate nanoparticle adhesion dynamics under the control of contextual parameters so that it will be possible to leverage multivalent interactions for disease targeting.

1.2. Nano Adhesive Dynamics (NAD) simulation and kinetic analysis

In this work, we used computation simulation to model the multivalent nanoparticle behaviors on an adhesive substrate, and studied the binding kinetics from simulation as well as dynamical system analysis. First, we develop nano adhesive dynamics (NAD) simulations to study the kinetics and biophysics of multivalent nanoparticle adhesion. We modeled nanoparticle adhesion using Langevin equation[21] and assessed bond formation/breakage using Bell’s model[22]. We also include biophysical components such as hydrodynamic fluid flow to replicate nanoparticle adhesion within a flow chamber. We started investigate nanoparticle stability after it initializes one bond with the substrate.

Thus, the NAD detachment simulations are initiated with the nanoparticle bound to a planar substrate via a single bond tether in order to focus on the temporal dynamics of nanoparticle detachment and multivalent bond formation. Results are compared to Haun’s experimental work[6] using a 210 nm polystyrene sphere coated with a monoclonal antibody and a glass substrate coated with ICAM-1, for which all model parameters are

2 known except bond mechanical properties (reactive compliance, �, spring constant, � and transitional spring constant, �ts). NAD simulations replicate time-dependent nanoparticle detachment behavior, consistent with experimental results at several different mechanical property combinations. We conclude it is mechanical work results in bonds rupture that is equivalent to the bond chemical energy. We used optical tweezers force spectroscopy experiments to determine that bond reactive compliance � equal to 0.27 nm, and parameterized spring constant of bond � equal to 0.8 N/m best fit to experiments across a broad range of adhesion molecule density conditions. The time-dependent nanoparticle detachment arises from the heterogeneity of nanoparticle bonding ability. Thus, detachment serves as a selection mechanism to evolve the remaining nanoparticle population towards higher valency and overall adhesion stability. Taken together, we conclude that our phenomenological detachment rate captures the two critical aspects of

0 valency selection. The magnitude parameter kD represents the combined detachment rate for the initial nanoparticle population across all valency potentials, while the temporal parameter describes the valency state-dependent rate at which nanoparticles are lost with time. We also conclude that average bond lifetime is extremely short, on the order of 0.1 s, due to applied mechanical forces in excess of 300 pN. Examining our simulation, we find that mechanical force primarily arises from Brownian motion of the nanoparticle, but we also note a contribution from bonds pulling on each other that increases with valency.

Secondly, we also develop dynamical system model to investigate the time-decaying detachment rate. The fundamental hypothesis is that categorizing nanoparticles based on a metric of bonding ability will result in subpopulations that exhibit simple and consistent detachment behavior. Specifically, we define the categorization criteria as the bond

3 potential (BP), and expect detachment for each BP category to follow first-order kinetics characterized by an exponential decay and constant detachment rate. Detachment profiles for each BP category will then sum in a manner that reconstructs the complex detachment profile of the full population. Using NAD simulation data, we show that mode bond number provides a good initial classification criterion for BP. However, non-first-order kinetics is still observed within BP sub-populations. We address this issue by assuming that each BP can contain two hidden components, which would both display first-order kinetics but with relatively faster and slower detachment rates. At this stage, we lack detachment information for the slow detaching components of the low BPs, and high BPs overall, and therefore we assume that these sub-populations do not detach at all. Under these assumptions, we show that the BP model can still match NAD simulation data very well. We then employ a survival analysis to formalize relationships between the parameters of the

0 new BP model and the original empirical model, which confirm that β and kD are fundamentally explained by nanoparticle population heterogeneity. Next, we seek to attain non-zero detachment rates for all BPs and hidden components, even those that did not detach on the observed time-scale for NAD simulations. Therefore, we develop a bond state model to determine all bond transition rates using a system of ordinary differential equations, and then estimate a composite nanoparticle detachment rate for each BP from the mean first passage time. Overall, we use the BP model and mean first passage time results to determine detachment rates for all BP and hidden component sub-populations.

4 2. BACKGROUND

2.1. Multivalent nanoparticle adhesion

The delivery of contrast or therapeutic agents to disease sites has been a major medical goal for decades, but success has been limited in human clinical settings. Site- specific targeting is generally achieved by adding an affinity molecule that binds to a distinct molecular determinant of the disease. In this manner, molecular imaging can be used to interrogate diagnostic, prognostic, or therapeutic indicators or drugs can be delivered directly to disease sites to maximize efficacy and reduce adverse side-effects.

Nanoparticle carriers offer numerous advantages as a targeted delivery platform, including high-loading capacity, protection of sensitive or insoluble agents, attractive pharmacokinetic and biodistribution properties, and facile attachment of affinity moieties for targeting diseases such as pathological vasculature and cancer.[1, 23-27] Another powerful attribute of nanocarriers is the ability to form multiple bonds with target cells, thereby improving overall adhesion strength and internalization rate into cells.[5-13, 27]

However, our understandings of the processes that govern multivalent binding remain limited. In part this is due to the fact that experimental and computational analyses have been based almost exclusively on thermodynamic behavior. For example, binding performance has typically been assessed after systems have reached equilibrium, with results reported as an apparent affinity or avidity. Moreover, numerous approaches have been developed to model multivalent binding, but nearly all have been based on correlating bond valency states with an overall thermodynamic free energy.[14-19, 28-30] Another issue is that differences in context across experimental systems have not been fully factored into the analyses. Finally, nanoparticle adhesion in a biological context is subject

5 to unique interfacial phenomena due to the nature of the macromolecules mediating binging. For example, the size scale of many molecular adhesion receptors can be quite large (10 nm antibodies, 20 nm targets such as ICAM-1) relative to the nanoparticle (50-

150 nm), and mechanical forces can modulate the influence of bond chemical energy. Thus, critical knowledge gaps remain in the field regarding the time-course by which nanoparticles evolve from initial capture via one or more bonds to the final multivalent state, and how this is influenced by various system parameters. Another aspect critical to many disease applications is the need to control targeting selectivity, since disease determinants are rarely unique but are instead upregulated above a basal expression level.

A prime example is the inflammatory molecule ICAM-1, which is present at ~150 molecules/μm2 on normal endothelium and then unregulated by a factor of ~10 during inflammation.[6, 24] These expression levels establish the inherent targeting selectivity, but we and others have shown that this can be eroded if nanoparticles are too reactive.[6,

12] Similarly, tumor cell targeting via HER2-specific affibody and transferrin ligand were both optimal at sub-maximal coating densities.[31] While maintaining disease selectivity is important, a major goal of the targeted delivery field is to achieve superselectivity. This involves adhesion that is exquisitely sensitivity to target expression level, with a switch- like change in binding efficiency. To date, superselectivity has only been observed in a computational model,[17] and experimental demonstration has remained elusive.

2.2. Flow chamber assay

Haun et al [6, 8, 11] developed a unique framework for assessing multivalent nanoparticle adhesion from a kinetic viewpoint. Specifically, the rates of attachment (kA) and detachment (kD) were determined for nanoparticles mediated by the interaction

6 between an antibody and ICAM-1 in flow chamber assays. kD was not a constant in time, but

0 rather decreased following a power law relationship that contained magnitude (kD ) and temporal (β) parameters. Furthermore, β was constant over a broad range of antibody and

ICAM-1 densities and particle sizes (40 nm to 1 µm).[6, 8, 11] However, β did vary for different types of binding interactions, such as recombinant single-chain antibodies and avidin/biotin. While the kinetic approach has provided unique insights into multivalent nanoparticle adhesion, there was no understanding of the underlying mechanisms behind the time-dependent detachment rate phenomenon, most notably the number and dynamic behavior of individual bonds.

2.3. Adhesion model

Adhesive Dynamics is a simulation framework originally developed to model kinetic and biophysical aspects of leukocyte rolling adhesion to inflamed endothelium.[36] The method employs a combination of deterministic equations of motion for the cell and a probabilistic treatment for bond formation and breakage. Bonds are modeled as Hookean springs and the Bell model is used to establish the effect of force on both bond rupture and formation rates. Within discrete time steps, all forces are vectorially summed, the cell is translated and rotated, and bond breakage and formation are assessed based on chemical and mechanical considerations using a Monte Carlo algorithm. Adhesive Dynamics simulations have achieved considerable success replicating rolling, weak, and firm binding behavior of neutrophils, protein-coated microbeads, and platelets.[33-37, 39, 40, 44] The general Adhesive Dynamics approach was also adapted to model the binding of human immunodeficiency virus (HIV) to cells. Brownian motion was introduced into the simulation due to the nano-scale size of HIV, around 100 nm, and thus the methodology

7 was termed Brownian Adhesive Dynamics (BRAD).[41-43] To date, Adhesive Dynamics simulations have not been used to model multivalent adhesion of targeted nanoparticles.

2.4. Brownian dynamics

Numerous computational approaches have been developed in an effort to understand multivalent binding phenomena. The most common approach has been to partition multivalent species into discrete bond valency states that are attributed an overall thermodynamic free energy.[28-30]In this manner, the Dormidontova group used

Monte Carlo simulations to investigate multivalent binding of polymer-coated nanoparticles under different bond density, energy, length, and clustering conditions to determine overall effects on binding free energy.[14-16] Martinez-Veracoechea et al. later presented a numerical simulation that calculated binding free energies using statistical mechanical functions, which led to the first prediction of superselective behavior.[17]

Although the above works offer useful insights into multivalency, they included little to no discrete bond detail beyond the chemical energy, notably lacking a role for mechanical forces. It is well established that applied forces accelerate rupture of non-covalent, biomolecular bonds by lowering the potential energy barrier.[22, 45-49] Decuzzi et al. incorporated bond mechanical considerations by modeling bonds as Hookean springs to determine bond force, and then using the Bell model to predict the effects of force on bond rupture rate.[18] A stochastic multivalent nanoparticle binding model was then used to predict adhesion strength to cells and the probability of endocytosis. Liu et al. later used a

Metropolis Monte Carlo simulation and a weighted histogram analysis to quantify the binding free energy of antibody-coated nanoparticles to endothelial cells via ICAM-1, which matched equilibrium adhesion data obtained from experiments. [19] Furthermore,

8 mechanical force predictions on the order of 200 pN were corroborated using atomic force microscopy. While these works have been illuminating, focus remained solely on equilibrium behavior. To date, temporal dynamics of multivalent nanoparticle adhesion has only been studied by Wang et al. using dissipative particle dynamics simulations.[20] The number of bonds and time constant required to reach equilibrium were studied at different overall binding strengths, but bond kinetics and biophysics were not evaluated. Thus, a simulation approach combining dynamic analysis of nanoparticle adhesion with discrete bond kinetic and mechanical properties has not yet been demonstrated.

2.5. Birth and death model

Given the inherent variability of biology materials[50], many biological reaction problems have been studied as stochastic processes, such as molecule diffusion, ligand- receptor adhesion, gene expression, and metabolism, etc. Among a variety of stochastic models, birth and death (BD) process has been widely applied for modeling complex biological and chemical reactions. A BD process is a stochastic process with Markov property in which transitions occur between neighbor states in a Markov chain, where

“birth” terms increment of state variable by one, while “death” terms decrement of state variable by one. A Poisson process is a pure birth process. It is defined on real line on where the time between successive occurrences of an event follows an exponential distribution. Moreover, if the real line is divided into equal finite intervals, the number of occurrences in each interval obeys a Poisson distribution. BD processes have many applications in biology, chemistry, epidemiology, and queueing theory. They may be used, for example to study the evolution of bacteria, the number of people with a disease within a population, or the number of customers in line at the supermarket. The time when a

9 stochastic process first hits a threshold is called first passage time. The threshold could be a boundary, an activation barrier, or a special state. BD models have wide applications in biology reaction via addressing problems which could be formulated into transition or state probabilities of the process, stationary distribution, mean, variance and distribution of times of the first entrance to a particular set of states, probabilities of extinction, the mean time of existence, etc.[51]

10 3. MODEL DEVELOPMENT

3.1. Overview of nano adhesive dynamics (NAD) simulation

We developed nano adhesive dynamics (NAD) simulation to model nanoparticles’ behaviors on an adhesive substrate. The NAD simulation incorporates both nanoparticles’ motion using Langevin equation and nanoparticles’ adhesion using Bell’s adhesion model

[22, 52, 53]. The NAD simulation was built upon the same principles as Adhesive Dynamics for modeling dynamic binding of leukocytes under shear flow and Brownian Adhesive

Dynamics for modeling HIV docking.[36, 41-43] For this study, we employed a 210 nm diameter sphere decorated with monoclonal antibodies and a planar substrate decorated with ICAM-1 proteins, as in Haun’s flow chamber experiments.[6]

3.2. Nanoparticle motion

Nanoparticle motion was determined using the Langevin equation, as in previous

BRAD simulations.[41, 42] Briefly, translational and rotational trajectories resulting from random thermal motion and all deterministic forces and torques, including hydrodynamic shear, bonding, and steric repulsion between the nanoparticle and substrate were solved numerically.[21, 53] Bond force was estimated by modeling bonds as Hookean springs:

�⃗ − �⃗ �⃗ = �(� − � ) � 3.1 where σ is the spring constant, λ is the length of the bond, λe is the equilibrium length of the bond, rr is the position vector originating from where the antibody is attached to the nanoparticle surface, and rl is the position vector originating from where ICAM-1 is attached to the substrate. For convenience, we will define the term δ = |λ - λe|, which determines the length that the bond is stretched or compressed from its equilibrium

11 length. Steric repulsion between the nanoparticle and substrate surfaces was determined based on compression of a surface protein layer using a nonspecific repulsion model.[52]

Finally, shear force acting on the nanoparticle was modeled using theoretical relationships from Goldman, Cox, and Brenner that are valid near the wall region.[54, 55] Note that since we are now using a nanoparticle, we employed the shear force/torque relationships valid for large h/R, where R is the particle radius and h is the distance of its center from the wall.

Nanoparticle translation and rotation

Nanoparticle movement was governed by the Langevin equations. For nanoparticle translation:

��⃗ = �⃗ �� 3.2 ��⃗ �⃗ = − + �⃗ + �⃗ �� where r is the position vector, v is the velocity vector, t is time, A is the vector of accelerations resulting from thermal collisions, and K is the vector of accelerations resulting from deterministic forces (bond force, repulsion force, and shear force; see below for detailed descriptions). tv is the viscous relaxation time, and is given by the Stokes–

Einstein relationship for a sphere:

� � = 3.3 6�� where µ is the solution viscosity, R is the radius of the nanoparticle, and m is the mass of the nanoparticle. For the nanoparticle system studied in this work, tv = 2.6 ns. Nanoparticle trajectories were generated by numerically solving Equations 3.2 and 3.3,[21, 53] as follows:

12 �⃗(� + ∆�) = �⃗(�) + ��⃗(�)∆� + ��⃗∆� + ��⃗

�⃗(� + ∆�) = ��⃗(�) + ��⃗∆� + ��⃗

� = exp (−Δ�/�) 3.4 (1 − �)� � = Δ�

(1 − �)� � = Δ� where ∂rG is a random position vector and ∂vG is a random velocity vector. The elements of these two random vectors were chosen in a position-velocity pairwise fashion from a bivariate Gaussian distribution, which was generated from a SIMD-oriented Fast Mersenne

Twister random number generator with seed obtained from system noise. The uniform distribution has a zero mean, with variances in position (�r) and velocity (�v) and correlation coefficient (crv) given by:

( ) ( ) �� 3 − 4 exp −∆�/�� + exp −2∆�/�� = ∆� 2 − � ∆�/��

�� = 1 − exp−2∆�/� 3.5 � �

�� = (1 − exp(−∆�/� )) � �

Similarly, for nanoparticle rotation:

��⃗ = �⃗ �� 3.6 ��⃗ �⃗ = − + +�⃗ + �⃗ �� ,

� 3.7 � = , 27��3

13 where ω is the vector of angular positions, θ is the vector of angular velocities, tv,rot is the rotational viscous relaxation time, Arot is the vector of angular accelerations caused by random torques resulting from thermal motion, Krot is the vector of angular accelerations resulting from deterministic torques, and I is the rotational inertia of the nanoparticle. The expressions in Equation 3.6 were solved in the same manner as Equation 3.2, with results analogous to Equation 3.4. Variances in angular position (�ω) and velocity (�θ) and correlation coefficient for rotational motion (cωθ) are defined identically to Equation 3.5, but with I substituted for m and tv,rot substituted for tv. For the nanoparticle system studied in this work, tv,rot = 0.8 ns. Antibody positions were updated after the nanoparticle was rotated using a three-dimensional rotation matrix.

The translational accelerations (K) and angular accelerations (Krot) resulting from deterministic forces are given by

∑ �⃗ + �⃗ + �⃗ �⃗ = 3.8 �

⃗ ∑ ⃗ �⃗ = 3.9

where ∑ �⃗ and ∑ �⃗ are vectorial summations of all bond forces and torques, respectively, Fr is the steric repulsion force between the nanoparticle and substrate, and Fs and Ts are the shear force and torque from hydrodynamic flow, respectively.

Force Calculations

Bond, shear, and steric repulsion forces were treated in a similar manner to previous Adhesive Dynamics and Brownian Adhesion Dynamics works.[36, 41-43] Bond force was determined by modeling the antibody/ICAM-1 interaction as a Hookean spring

14 (Equation 3.1). We assumed that molecules were rigidly attached to the surface. Each bond was then described by a time-varying vector (�⃗ − �⃗), from which the force and torque could be calculated as follows[36]:

�⃗ = (�⃗ − �⃗) × �⃗ 3.10 where r is the position vector pointing from the center of mass of the nanoparticle.

The nanoparticle and substrate were prevented from coming too close together due to their respective surface protein layers, as well as electrostatic forces and other long- range interactions. These factors result in a net repulsion force that scales with the separation distance. Equation 3.11 represents the mechanical work that must be performed against nonspecific repulsive forces to bring a unit area of adhesive substrate from an infinite separation distance to a separation distance of s, as given by Bell[22]:

� � Γ(�) = exp − 3.11 � � where s is the separation distance, ξ is the compressibility coefficient of surface proteins, and τ is a measure of the combined thickness of the protein layers. The repulsion force per area is the first order derivative of Equation 3.11 above, with contact area given by:

� = �(� − (� + � − �) ) 3.12

Therefore the repulsion force was calculated as follows:

1 1 � � = ��(� − (� + � − � )) + exp − 3.13 � ��

Once separation distance exceeds one bond length, the particle does not experience any repulsion force. As the nanoparticle moves closer to the surface, the repulsion forces

15 increases quickly, acting on the particle uniformly through its center of mass, so as not to generate torque.

Shear force and torque acting on the nanoparticle are modeled using theoretical relationships from Goldman, Cox, and Brenner that are valid near the wall region[54, 55]:

9 � � = 6��ℎ 1 + 16 ℎ 3.14

3 � 3.15 � = 4�� 1 − 16 ℎ where h is the distance from the particle center to the surface wall and S is the undisturbed shear rate. These equations are valid for large h/R. The length of ICAM-1/Ab bond

(41.1nm) is relatively large compared to the particle radius (105 nm), and thus h/R is approximately 1.4. The alternative is the near-wall condition, in which h/R is approximately 1. However, it should be noted that the difference between these two cases is small, with shear force and torque varying by 20 and 5%, respectively. Since fluid flow was designated along the x-axis, Fs has a none-zero component in the same direction and Ts has a none-zero component along the y-axis.

3.3. Nanoparticle adhesion

Non-covalent biomolecular bonds stochastically fluctuate between bound and unbound states based on intrinsic chemical energy, generally described as a thermally- assisted escape over a potential energy barrier. Applying an external force on the bond lowers this energy barrier, accelerating rupture.[18, 19, 22, 45-47, 49] We used the Bell model to capture the effect of force on the rate constant for bond rupture (kr):

16 ⃗ � � �� = � �� = � �� 3.16 ��

0 where kr is the intrinsic reverse reaction rate constant observed in the absence of force, γ is the bond reactive compliance that characterizes the sensitivity to force, kB is the

Boltzmann constant, and T is the absolute temperature. The term γσδ effectively defines the amount of mechanical work applied to the bond by force. Within each time-step, the cumulative probability of rupture (Pr) was calculated for each bond as follows:

� = 1 − exp (−� ∆�) 3.17

A uniformly distributed random number was then generated for each bond, and if the random number was less than Pr the bond was considered to have broken.

Bond formation is similarly influenced by the distance separating two adhesion molecules, which can consist of two steps: first the adhesion molecules must come sufficiently close to each other, and only then can the bond form at its intrinsic rate. The first step requires extension or compression of the unbound adhesion molecules, invoking an energetic penalty on bond formation that can again be captured using a Hookean spring model.[56] The rate constant for bond formation (kf) was thus:

−�(� − �) � = � �� 2�� 3.18

0 where kf is the intrinsic rate of bond formation and σts is the transition-state spring constant that applies to unbound adhesion molecules. Within each time-step, the cumulative probability of formation (Pf) was calculated for all possible interactions between unbound adhesion molecules as follows:

17 � = 1 − exp (−� ∆�) 3.19

A uniformly distributed random number was generated for each potential bonding interaction. If the random number was less than Pf, a bond was considered to have formed.

In order to save computational cost, potential bonds only included adhesion molecule pairs

−6 that were reasonably close to each other, which we defined as having Pf > 10 . Truncation of the probability density function in this manner introduced an error of < 0.01% to the cumulative probability of bond formation.

Intrinsic rate for bond formation and breakage were computed from macroscopic

m m kinetic rates. The macroscopic kinetic rates of bond formation (kf ) and rupture (kr ) were measured to be 1.6×105 M-1s-1 and 1.1×10-4 s-1, respectively, using surface plasmon resonance experiments[57]. These macroscopic rates were converted to intrinsic rates using the method presented by Bell[22]. First the diffusion-limited rate of formation, d+, and dissolution, d−, of the encounter complex were calculated as:

� = 4��

3.20 d = 3πD/R

where D is the diffusivity of soluble ICAM-1 and RAB is the encounter distance for the anti-

ICAM-1 antibody and ICAM-1 binding interaction. Using the Stokes-Einstein relation, the diffusion coefficient for soluble ICAM-1 is ~8.5 × 10−11 m2s-1. The value for the encounter distance was assumed to be 0.75 nm, as previously proposed for a hapten-antibody

0 0 system[22]. The intrinsic bond formation (kf ) and rupture (kr ) rates were then found by solving the following system of equations:

18 � = �� /(� + � ) 3.21 � = �� /(� + � )

0 5 -1 0 -1 The resulting intrinsic rates were kf = 1.6×10 s and kr = 1.1e-4 s .

3.4. Adhesion molecule configurations on surface

Both the antibody and ICAM-1 molecules were randomly immobilized on their respective surfaces as dimers to reflect experimental conditions (Figure 3.1). Antibodies consist of two Fc stalks, each of which connects to flexible binding arms (Fab). ICAM-1 protein was purchased as a chimera with human IgG1 Fc, which was reported to be a dimer by the manufacturer (R&D Systems) due to disulfide bonding within the Fc domain. To represent dimers, we randomly separated the two molecules by a distance ranging from 5-

10 nm for antibody Fab domains and 0.5-2 nm for ICAM-1/Fc. These values were chosen from published crystal structures or our best estimates. It should also be noted that ICAM-

1/Fc was linked to the surface using protein G, which contains two Fc binding domains. We initially assumed that only one of those binding domains was occupied with an ICAM-1 dimer, but it is possible that both could be occupied.

Figure 3.1 Size-scaled depiction of the adhesion molecule system employed. Images published in the Protein Data Bank: mouse IgG1 antibody (1IGY), ICAM-1 (combination of 1IAM AND 1P53), human IgG1 Fc (3D03), and protein G (3GB1).

19 To reduce computational burden, we did not track the motion of the free end of adhesion molecules about their fixed attachment points on the nanoparticle and substrate.

Instead, it was assumed that all molecules were oriented normal to its binding partner when bound or potential binding partners when unbound. This is a reasonable assumption for bonds under tension, and thus bond length was calculated based on the distance separating the attachment points on the nanoparticle and substrate. For unbound molecules, the assumption effectively implies that the molecules can fully sweep out their local area in search of a binding partner during the course of a single time-step. We did implement an algorithm to prevent a new bond from forming if it would intersect a current bond. This was determined based on the minimum distance that would separate the central axes of the potential and actual bond. If the separation distance was less than 2 nm, which was again chosen based on crystal structures and our best estimate, then the new bond was not allowed to form. This check to prevent bond crossing lowered total bond numbers by as much as 40%. Once bonds had formed, we did not assess whether they crossed each other as the nanoparticle underwent translational and rotational motion in an effort to save computational cost.

3.5. Parameters

All parameter values used in this study are given in Table 3.1. The macroscopic kinetic rates for the anti-ICAM-1 antibody (clone BBIG) and human ICAM-1 binding interaction were measured by surface plasmon resonance to be 1.6×105 M-1s-1 for formation and 1.1×10-4 s-1 for rupture.[57] These were converted to intrinsic rate constants

0 0 (kf , kr ) for individual bonds using the method described by Bell et al.[22] Antibody and

ICAM-1 densities were based on previous experiments, which reflect total ICAM-1 binding

20 sites (i.e. Fab domains) and ICAM-1 molecules.[6] Molecular lengths were estimated based on crystal structures, as previously presented.[11] The fluid was assumed to be water flowing with a wall shear rate of 100 s-1. The steric repulsion force between the nanoparticle and substrate was determined by estimating the surface protein layer thickness on the substrate (τ = 5 nm), and then tuning compressibility (ξ) to be the minimum value required to prevent the nanoparticle from penetrating into the adhesive substrate. The time-step (∆t) was 1 ns, similar to previous Adhesive Dynamics and BRAD simulations.[36, 41-43] ∆t is also very close to the viscous relaxation time, so it can be assumed that the particle has no inertia. The only unknown parameters in our model were related to bond mechanics (γ, σ, σts). To reduce our parameter investigation, we assumed that σ and σts had similar values, which mechanistically means that the spring constant for both bound and unbound states was the same. While this assumption is reasonable on its own merit, we have also mitigated the importance of σts by immediately assessing for bond breakage following each formation event. Thus, all new bonds must survive at least one rupture challenge before it can exert a force and/or torque on the nanoparticle.[32]

21 Table 3.1 Physical parameters used in NAD simulations.

Parameter Parameter Value Dimension

R Nanoparticle radius 105 nm

ρ Nanoparticle density 1.05 g/cm3

λe Equilibrium bond length 41.1 nm

Antibody length 11.4 nm

ICAM-1 length 29.7 nm

Antibody density 410/1080/3400 μm-2

ICAM-1 density 21/41/134 μm-2

5 0 1.6x10 -1 kf Intrinsic bond formation rate s

-4 0 1.1x10 -1 kr Intrinsic bond breakage rate s

μ Viscosity 0.001 g/nm/s

T Temperature 300 K

∆t Time step 1 ns

S Fluid shear rate 100 s-1

τ Polymer thickness 5 nm

ξ Compressibility 0.03 pN

22 3.6. Computation details

To ensure accuracy and statistical significance in determining nanoparticle

0 detachment parameters (β and kD ) based on Equation 3.24, we tested different nanoparticle ensemble sizes and simulation times. These studies were performed using � =

-2 0.1 N/m, �ts = 0.1 N/m, and � = 0.98 nm at low antibody (410 µm ) and low ICAM-1 (21

-2 0 µm ) densities. For a 30 s simulation, we found that both β and kD began to converge around ensemble sizes of 150 nanoparticles (Figure 3.2A). Using a 200 nanoparticle

0 ensemble, both β and kD converged around 20 s (Figure 3.2B). Therefore, we selected to use 200 nanoparticle ensembles and 30 s simulation times for all studies. We also tested nanoparticle dynamics at different initial separation distances between the nanoparticle and surface, from 40.3 to 41.9 nm, which was the full range of bond extension or compression lengths allowed while maintaining the bond breakage probability (Pr,

−6 Equation 3.17) at < 10 for this set of conditions (� = 0.1 N/m, �ts = 0.1 N/m, and � = 0.98 nm). We found that nanoparticle detachment dynamics were not significantly affected by initial bond length (Figure 3.2C). Finally, we investigated whether new bonds would physically intersect with a current bond before we allowed them to form. We estimated that the diameter of an antibody/ICAM-1 bond was ~2 nm. If a potential bond were to form within a distance less than 2 nm from an existing bond, we didn’t allow bond formation regardless of the result of the stochastic algorithm. We found this checking for bond intersection reduced mean bond number by 40%, from 3.1 to 2.2, for the set of conditions described above.

Figure 3.2 Optimizing simulations. (A,B) Nanoparticle detachment profiles were fit using different (A) ensemble numbers and 0 (B) simulation times. Both the β and kD detachment rate parameters converged for ensembles larger than 150 nanoparticles and simulation times longer than 20 s. (C) Nanoparticle detachment dynamics were unaffected by the initial separation distance between the nanoparticle and surface, within the limits with which a bond would actually −6 form (i.e. conditions with Pr < 10 ).

3.7. Attachment and detachment rates

Nanoparticle detachment rate was defined in a similar manner to classic kinetic treatments of molecular binding:

�� = −� � �� 3.22 where B is the bound particle number. An empirical power law equation was used to capture time-dependent detachment rate behavior, as Haun et al previously defined[6]:

� � = (�/�) 3.23

0 where kD and β are magnitude and temporal parameters, respectively, and tref is a reference time. tref simply maintains unit consistency, and we will use a value of 1 s as in previous work. Substituting Equation 3.22 into Equation 3.23 and integrating yields:

� � = exp (� ) � � − 1 3.24 where B0 is the initial particle number in the ensemble and t is the simulation time.

24 Nanoparticle attachment rate is defined as:

�� = k C − � � dt 3.25 where kA is the nanoparticle attachment rate and Cw is the particle concentration at the wall.

To ensure statistical significance, we determined detachment parameters for

0 different ensemble numbers and total simulation times. We found that both β and kD converged for ensemble numbers larger than 150 and simulation times longer than 20 s

(Figure 3.2). All parameter fits were therefore performed using 200 ensembles and 30 s simulation times. Fitting errors were estimated using the Bootstrap method.

4. THE EVOLUTION OF PARTICLE DETACHMENT

4.1. Introduction

We started from NAD detachment simulations to investigate the nanoparticle binding stability and study the bond accumulation process in order to understanding the time-decay kD phenomena as observed in Haun’s flow chamber assay[6]. Haun et al[6, 8,

11] found that kD was complex, appearing to decrease over the time-scale of minutes

0 following an empirical power law with magnitude (kD ) and temporal (β) parameters.

Based on NAD detachment simulation results, we developed a multi-scale, biophysical simulation called Nano Adhesive Dynamics (NAD) to further investigate the time- dependent detachment phenomenon.[32] NAD simulations included both fluid flow and

Brownian motion to model nanoparticle adhesion within flow chambers, and thus combined elements from previous Adhesive Dynamics works modeling leukocytes,[33-38] platelets,[39, 40] and viruses.[41-43] Biophysical details such as mass transport, mechanical forces, and bonding properties were also included to directly reconstitute experimental systems and results. NAD simulations successfully recapitulated time-

0 dependent detachment behavior, providing the same β and very similar kD values across many different experimental conditions. We concluded that long-term, minute-scale changes in nanoparticle detachment rate were not caused by bond accumulation. Instead, the nanoparticle population was heterogeneous with respect to the number of bonds that can ultimately be formed in a given substrate location and nanoparticle orientation. Over time, nanoparticles restricted to lower bond numbers were removed by detachment, and through this selection mechanism the remaining nanoparticle population evolved with

26 time in terms of bond number and overall adhesion stability. With this new insight, we have postulated that the empirical detachment rate model was successful because it captured this selection/evolution mechanism. Most importantly, we now understand that a population of nanoparticles cannot be represented by a single multivalent detachment rate, and for that matter, affinity/avidity. Instead, a series of detachment rates co-exist together, each corresponding to sub-populations with different bond numbers and dynamics.

4.2. Methods

Overview of NAD detachment simulation

The NAD process flow is shown in Figure 4.1. The nanoparticle and substrate were first defined, followed by random distribution of adhesion molecules at specific total densities. A single bond was then placed between the nanoparticle and substrate at its equilibrium length. This initial condition was used because our primary goal was to monitor multivalent bond formation and time-dependent nanoparticle detachment

0 dynamics (β and kD parameters), and this is similar to a previous study that used the completed double layer boundary integral equation method (CDL-BIEM), with Brownian motion introduced as an external force, to investigate the effect of Brownian motion on platelet adhesion via GPIb-α.41 We also found that the initial bond length did not significantly bias results (Figure 3.2). Simulations were then started using a defined time- step (∆t). Within each time-step, bonds were assessed for potential breakage, unbound adhesion molecules were examined for potential bond formation, forces were vectorially summed, and the nanoparticle was translated and rotated. We note that tethering of a

27 nanoparticle from the bulk could readily be studied using the methods described, in similar manner to previous BRAD simulations[41-43].

Figure 4.1 Algorithm for NAD detachment simulations in which nanoparticles were initiated with a single bond.

Nanoparticle detachment rates

To recreate nanoparticle detachment profiles from experiments, we combined the results from an ensemble of NAD simulations. As discussed above, nanoparticles were initially bound to the substrate via a single bond at its equilibrium length. Simulations were then performed until a predefined simulation time was reached or the nanoparticle translated from its initial location by a distance that was at least 2.5 times its diameter. The latter condition was designed to replicate nanoparticle-tracking experiments, where significant movement of the nanoparticle would appear as a new binding event. Movement

28 could result from the loss of all bonds, reentry into the bulk fluid, and motion downstream with the bulk fluid or a net diffusive motion while still remaining bound to the substrate.

Detachment profiles were constructed based on the number of nanoparticles that remained bound as a function of time throughout the simulation.

Measurement of bond mechanical properties using optical tweezers

Biotinylated anti-human ICAM-1 monoclonal antibody (clone BBIG) was purchased from R&D Systems (Minneapolis, MN). Streptavidin-coated, 3 µm polystyrene beads were purchased from Spherotech (Lakeforest, IL). Antibody conjugates were prepared by first washing beads (50 µL) three times by centrifugation at 1000 x g for 5 minutes and resuspending in 1 mL of PBS containing 1% BSA (PBS+). Beads were then incubated with 5

µg/mL biotinylated antibody for 1 hr at room temperature and washed three 3 times by centrifugation. Normal and ICAM-1 expressing CHO-K1 cells were obtained from ATCC and cultured as recommended. Prior to experiments, cells were plated on 35 mm x 10 mm cell culture dishes containing glass coverslips on the bottom (MatTek Corporation, Ashland,

MA) and cultured overnight. The dishes were then mounted on a motorized piezo-stage with nanometer precision (Physik Instrumente GmbH & Co. KG, Karlsruhe, Germany). All experiments were conducted at 25°C using CO2 Independent Medium (Thermo Fisher

Scientific, Waltham, MA).

Optical tweezers experiments were performed to determine bond rupture force at different force loading rates, similar to previous work using biomembrane force probe and atomic force microscopy.[19, 49, 58-60] Anti-ICAM-1 antibody-coated beads were trapped in a custom-built optical tweezers system described previously.[61, 62] A cell was translated using the piezo-stage to the laser-trapped bead, held in contact for a few

29 milliseconds, and then retracted at constant velocity. During each retraction cycle, interactions between the bead and cell resulted in the bead being pulled from the center of the laser trap, which in turn exerted an equal and opposite force on the bead in proportion to the displacement. Thus, bead displacement increased the restoring force of the laser trap until the strength of the interaction was overcome, and then the bead returned to the center of the laser trap. Several hundred contact and release cycles were performed per condition, and bead displacement was tracked using a quadrant photodiode and converted to force based on a calibration of the optical trap, as described.[61, 62] Three different retraction velocities of the piezo-stage were tested: 4, 24, and 56 µm/s. These velocities corresponded to loading rates of 213.9, 1544.5, and 2995.8 pN/s, respectively. For each velocity condition, rupture forces measured in each cycle were plotted as normalized histograms, also known as rupture force spectra. Distinct force modes were identified and characterized as arising from non-specific, single bond, or multiple bonding interactions.

Finally, single bond rupture force was plotted versus the logarithm of the force loading rate to determine �, in accordance with the Bell model.[19, 49] Control experiments utilized streptavidin-coated beads and normal CHO-K1 cells.

4.3. General dynamics of nanoparticles and bonds

As a starting point for an exploration into antibody/ICAM-1 bond mechanical properties, we used 0.1 N/m for both bond spring constants (� and �ts), similar to previous

BRAD simulations of HIV.[41] We also chose the lowest ICAM-1 coating density (21 µm-2) on the substrate and two different antibody coating densities on the nanoparticle (410 µm-2 and 3400 µm-2) from experiments.[6] We then identified � values that resulted in nanoparticle detachment that varied from stable to unstable regimes. At low antibody

30 density, varying � from 0.72 to 1.08 nm resulted in continuous nanoparticle detachment throughout the duration of 30 s simulations, with the total number of detached particles ranging from 18% to 99% (Figure 4.2A).

Figure 4.2 Nanoparticle and bond dynamics. (A,C) Nanoparticle detachment profiles obtained for � = 0.1 N/m, � = 0.72 to 1.08 nm, and low ICAM-1 density at (A) low and (C) high antibody density. Time-dependent behavior can clearly be seen at high �, with an initial rapid decline that transitioned to a more stable regime around ~5 s. (B,D) Mean bond number increased over time for both (B) low and (D) high antibody density conditions following a similar temporal pattern as nanoparticle detachment. Bond number increased and became more stochastic as fewer nanoparticles remained bound. (E,F) Detachment profiles were fit using Equation 3.23 to obtain (E) β and 0 (F) kD parameters. (E) The temporal parameter β increased with � until saturating at 0.75, which was the value measured in experiments. There was a slight shift to higher � as 0 antibody density increased. (F) The magnitude parameter kD progressively increased with � and decreased with antibody density regardless of β, reflecting overall nanoparticle stability.

The rate of detachment and total number of detached nanoparticles generally increased with �. The only exception was a slight crossing over of the detachment curves for � = 0.78 and 0.82 nm, which we attributed to the stochastic nature of the simulation.

Time-varying detachment behavior was clearly evident at high �, which can characterized

31 as a rapid decrease during the first 5 s before flattening out beyond 10 s. Mean bond number per bound nanoparticle increased with time (Figure 4.2B), mirroring the corresponding nanoparticle detachment profile. Surprisingly, final mean bond numbers increased as nanoparticle stability decreased. However, as expected the total number of bonds for the system of 200 nanoparticles decreased with time and correlated with stability (Figure 4.3). Similar nanoparticle and bond dynamics were observed at high antibody density, but with higher bound nanoparticle percentages and lower bond numbers for each value of � (Figure 4.2C and D). Using Equation 3.24, we fit the

0 detachment curves to obtain the β and kD parameters. We found that β initially increased with � before saturating at 0.75 (Figure 4.2E), which remarkably was the β value we found in experiments. The trend for β was similar for both cases, but shifted to higher � with

0 increased antibody density. For kD , values correlated with nanoparticle detachment, decreasing with antibody density and increasing steadily with � (Figure 4.2F). The latter held true even if β remained constant at 0.75. The simulation results at � = 0.92 nm

0 matched experimental findings quite well in terms of both β (~0.75) and kD . At low

0 -1 antibody density, kD was nearly identical between simulation and experiment (93 ms ), but at high density the difference was nearly 5-fold (10 ms-1).

Figure 4.3 Total bond numbers versus time and �. (A,B) Total bonds across all bound nanoparticles decreased over time for low (A) and high (B) antibody density conditions. Total bonds also decreased at each time point as nanoparticle stability decreased. (C,D) Same data represented only during the first 0.5 s to show that there was an initial increase in bond number that lasted at most 0.1 s, and then decreased with time. (E) Total bond numbers at the end of simulations (30 s) decreased as adhesion became less stable (increasing �, lower antibody density).

Focusing at the bond level, force played a significant role in destabilizing bonds at high �. This can be seen in the average bond force at rupture, FB,R, which increased with � until saturating at around 95 pN (Figure 4.4A). At the condition best matching experiments (� = 0.92 nm), average FB,R was slightly less than 95 pN, with a distribution ranging from 50 to 140 pN (Figure 4.5). Average bond force, FB, was significantly lower however, around 16 pN for the same condition (Figure 4.5). The mechanical work applied to bonds at rupture, which is the product �FB,R, increased steadily with � (Figure 4.4B).

Interestingly, mechanical work was very close to the bond chemical energy of 87 pN*nm at

� = 0.92 nm, and then continued to increase further at higher � values. The fact that FB,R increased as bonds became more sensitive to force (increasing �) may seem counterintuitive, but this was expected because bond extension (or compression) length at

33 rupture, δR, should also increase with � (Figure 4.4C). In fact, δR remained very close to � until reaching a maximum at 0.9 nm. It is interesting that FB,R, δR, and β all show similar saturation behavior starting at � = 0.9 nm. As for bond stability, we observed that average lifetime decreased with � from second to millisecond time-scales (Figure 4.4D). Though very rare, a few bonds were able to persist the full duration of the simulation (Figure 4.5).

Bond formation rate increased with � (Figure 4.4E), but this was likely a secondary effect driven by decreasing bond stability, as the same bonds could continually break and reform.

All bond force and rate metrics shown in Figure 4.4A-E were identical at high and low antibody density. We did observe that bond formation rate was slightly greater at high antibody density during very early stages of the simulation (Figure 4.5). Final mean bond number at 30 s also varied with antibody density (Figure 4.4F), ranging between 2.1 and

3.7 bonds per nanoparticle.

Figure 4.4 Bond biophysics and dynamics. (A) Bond rupture force (FB,R) increased with � before saturating around 95 pN. (B) Mechanical work at bond rupture (�FB,R) increased steadily with �, surpassing the bond chemical energy (dashed line) around 0.9 nm. (C) Bond extension or compression length at rupture (δR) was slightly greater than � until saturating around 0.9 nm. (D,E) Average bond (D) lifetime and (E) formation rate exhibited opposing trends as bonding became more dynamic with increased �. (F) Mean bond number at the end of simulation (30 s) increased as adhesion became less stable, both in terms of increasing � and decreasing antibody density. Error bars represent the standard error from 200 simulations.

Figure 4.5 Bond biophysics and reaction rates. (A) Bond rupture force (FB,R) distributions for � = 0.92 nm for the low and high antibody densities, showing a peak at slightly less than 100 pN. (B) Bond force (FB) distributions for � = 0.92 nm, showing that bonds were typically exposed to forces that were significantly less than FB,R. (C) Mean FB traces at � = 0.92 nm fluctuated around an average value of 16 pN. (D,E) Bond lifetime distributions for � = 0.92 nm, showing that only a few bonds persisted longer than a few seconds. The only difference between panels D and E is the scaling of the y-axis. (F) Bond formation rate, evaluated only within the first 0.1 s of the simulation, was slightly elevated for the high antibody density case.

4.4. Tracking bond distributions

To further investigate the inverse correlation between mean bond number and nanoparticle stability (Figure 4.2B and D, Figure 4.4F), we constructed histograms of final mean bond number for all conditions (Figure 4.6A and B). We found that very few particles were bound via a single bond at the end of simulations for any condition and as � increased, fewer particles remained bound via two or even three bonds. This effect was more pronounced at low antibody density, which can best be seen for the di- and trivalent states. At four bonds and above, adhesion appeared to be insensitive to �, although sample sizes were small. These findings suggest that � did not affect the inherent bond distribution

36 attained for a nanoparticle population, only the ability for nanoparticles to successfully remain bound at the lower valencies. Moreover, as the low valency nanoparticles detached, the bond distribution for the population shifted to a higher mean value. Since this effect is reminiscent of a survival of the fittest scenario, we shall refer to it as valency selection. At high antibody density, the extent of valency selection was diminished for a given value of �, suppressing the shift in bond number. To account for valency selection, we investigated bond numbers early in the simulation and found that most bonds had already formed within the first 0.1 s (Figure 4.7). This can also be seen by tracking the total bond numbers across the entire system of nanoparticles, which only increased during the first 0.1 s

(Figure 4.3). Using bond numbers from this early snapshot, when detachment events were minimal, we could eliminate the influence of valency selection and predict the inherent bond distribution available to each nanoparticle population, which were indeed similar regardless of � (Figure 4.8). Corrected mean bond number, or mean bond potential, results are shown in Figure 4.6C and D as a function of time, and confirm that the bond steady state had already been established by the 0.1 s time-point. Finally, mean bond potential did increase with antibody density from approximately 1.7 to 1.9 (Table 4.1). Actual mean bond numbers increased over time from these inherent bond potentials exclusively due to valency selection, as the remaining population comprised a steadily increasing number of bonds per nanoparticle.

Figure 4.6 Bond valency distributions and bond potential. (A,B) Bond number histograms obtained at the end of simulations (30 s) for � = 0.1 N/m, � = 0.72 to 1.08 nm, and low ICAM-1 density at (A) low and (B) high antibody density. Detached nanoparticles were categorized under 0 bonds. (C,D) Mean bond potential values as a function of time at (C) low and (D) high antibody density. Mean bond potential represents the mean bond number determined right after the bond steady state was achieved (0.1 s), and after correcting for nanoparticles that had detached. Mean bond potential did not vary with �, but shifted from ~1.7 to ~1.9 with increased antibody density.

Figure 4.7 Bond number dynamics. Instantaneous and mean bond numbers during the first 0.5 s of the simulation for � = 0.1 N/m, low antibody density, and � values of (A) 0.72 nm (B) 0.78 nm (C) 0.82 nm (D) 0.86 nm (E) 0.92 nm (F) 0.98 nm (G) 1.02 nm and (H) 1.08 nm. For all conditions, bond number increased predominantly within the first 0.1 s of the simulation, and then continued to slowly increase with time. Detached nanoparticles are indicated by red dots placed at zero bonds. After the 0.1 s time-point, the slow increase in mean bond number correlates with the frequency of detachment events. These results are consistent with the true bond steady state being reached before 0.1 s, and then valency selection leading to the subsequent increases in mean bond number for the remaining population over time.

Figure 4.8 Corrected bond distributions to reveal bond potential. (A,B) Bond number histograms were corrected for valency selection using bond numbers seen at the 0.1 s time-point of the simulation for the low (C) and high (D) antibody densities. If a nanoparticle had detached prior to that time point, it was assigned its maximum bond number. (C,D) Bond numbers were corrected for valency selection using either final bond number for nanoparticles that remained bound throughout 30 s simulations or the value noted at 0.1 s prior to detaching for those that were lost. Results are presented at low (A) and high (B) antibody densities. Both correction methods yielded similar results, indicating that the true bond potentials likely had been captured. Bond potential distributions did not vary with �, but shifted to higher valency with increased antibody density. Note that a significant number of nanoparticles were restricted to a single bond for both cases.

40 Table 4.1 Nanoparticle dynamics at � = 0.01 N/m across different � values. The conditions highlighted in yellow (� = 0.92nm) best match experimental results in 0 terms of the β and kD fitting parameters.

0 � Bound NP kD Final Mean Bond Mean Bond Antibody β (nm) (%) (ms-1) Number Potential

0.72 72 0.11 ± 0.11 15 ± 4 2.2 1.7

0.78 64 0.45 ± 0.06 46 ± 6 2.3 1.7

0.82 62 0.69 ± 0.04 52 ± 7 2.5 1.9

0.86 54 0.76 ± 0.03 66 ± 7 2.4 1.8 Low Density (410 µm-2) 0.92 37 0.74 ± 0.03 106 ± 11 2.6 1.8

0.98 18 0.77 ± 0.03 187 ± 21 2.9 1.9

1.02 4 0.73 ± 0.03 351 ± 34 3.1 1.9

1.08 2 0.79 ± 0.04 441 ± 44 3.7 2.0

0.72 93 0.10 ± 0.39 3 ± 2 2.1 1.9

0.78 89 0.005 ± 0.271 4 ± 2 2.1 1.8

0.82 82 0.48 ± 0.08 19 ± 3 2.2 1.9

0.86 73 0.40 ± 0.06 34 ± 4 2.4 1.9 High Density (3400 µm-2) 0.92 66 0.76 ± 0.03 48 ± 6 2.5 1.9

0.98 48 0.76 ± 0.03 79 ± 8 2.4 1.7

1.02 39 0.74 ± 0.03 102 ± 9 2.8 1.9

1.08 14 0.68 ± 0.03 213 ± 15 3.2 2.0

4.5. Mechanical state diagram

Next we sought to characterize nanoparticle and bond dynamics across a broad range of mechanical parameter space. Therefore we independently varied � from 0.001 to

1 N/m and � from 0.1 to 10 nm, and categorized nanoparticle detachment behavior into one of three states: static, dynamic, or transient. The static and transient states were

41 defined by >95% and <5% of all nanoparticles remaining bound after 5 s, respectively. All other cases were designated as dynamic. Again using the low antibody density, but now paired with medium ICAM-1 density, we constructed the mechanical state diagram shown in Figure 4.9A. The dynamic regime spanned a continuum of � and � combinations, but with a stronger sensitivity to �. We then identified combinations within the dynamic

0 regime that yielded β and kD values matching experiments, which are listed in Table 4.2 and depicted as red circles in Figure 4.9A. For each of these matching cases, bond lifetimes were ~0.1 s and final mean bond numbers were ~3. We did observe differences in FB,R, ranging from 10 to 400 pN, but �FB,R was consistently close to the bond chemical energy of

87 pN*nm (Table 4.2). Thus, experimental nanoparticle detachment dynamics, both in terms of temporal behavior and magnitude, could be replicated by matching bond mechanical work at rupture to bond chemical energy. We again observed that δR was close in value to � for most conditions (Figure 4.9B), but there appeared to be an inflection point around 50 pN. Since we could not identify a unique �-� combination from simulations, we measured the adhesion strength of the antibody/ICAM-1 interaction using optical tweezers-based force spectroscopy (Figure 4.10). Antibody-coated, 3 µm beads were brought into contact with CHO-K1 cells expressing human ICAM-1, and rupture force histograms were obtained at force loading rates of 200, 1500, and 3000 pN/s. The resulting single bond rupture forces were 5.0 ± 2.9, 17.4 ± 4.3, and 52.0 ± 8.9 pN, respectively, which corresponded to � = 0.27 nm. Using this result, the best-fit � value was ~0.8 N/m, and this combination is listed in Table 4.2 and depicted as teal circles in Figure 4.9.

Figure 4.9 Mechanical state diagram. Nanoparticle detachment dynamics at low antibody and medium ICAM-1 densities, assessed across a large range of � and � values. The transient regime (blue) corresponds to highly unstable adhesion, defined as <5% of nanoparticles remaining bound after 5 s. The static regime (brown) corresponds to highly stable adhesion, with >95% remaining bound after 5 s. The dynamic regime (red) lies in between, and the red circles indicate the mechanical property combinations that precisely matched experiments. (B) Bond rupture length (δR) was slightly less than � at low FB,R, but became increasingly larger after FB,R exceeded ~95 pN. Teal squares denote the matching condition using � measured with optical tweezers force spectroscopy experiments (0.27 nm) and the best fit � (0.8 N/m).

Figure 4.10 Dynamic force spectroscopy using optical tweezers. (A) Micrograph of approach and retraction cycle for an antibody-coated 3 µm bead to an ICAM-1 expressing CHO cell. (B) Continuous force waveform obtained over the course of several approach and retraction cycles during a representative experiment. Each cycle was ~1 s in duration, and peak forces were used to construct rupture force spectra. (C-E) Rupture force spectra at (C) 214, (D) 1544, and (E) 2995 pN/s loading rates. Rupture modes were fit to a Gaussian distribution, and rupture forces are presented as the mean ± standard deviation in the legend. (F) Rupture force spectra of specific antibody/ICAM-1 interaction and control interaction at a loading rate of ~1500 pN/s. A specific rupture mode is clearly seen, centered at ~17.5 pN, whereas the control only has a low force mode (<12 pN). Inlay shows the full spectra. (G) Single bond rupture force versus natural logarithm of the loading rate. Red line represents the fit used to determine � = 0.27 nm.

44 Table 4.2 Different � and � combinations resulting in nanoparticle dynamics that match experimental results.

0 � � kD FB,R δR Rupture work β (N/m) (nm) (ms-1) (pN) (nm) (pN•nm)

0.001 9.5 0.77 ± 0.02 140 ± 10 8.6 8.6 81.7

0.005 4.4 0.75 ± 0.03 140 ± 10 19.9 4.0 87.6

0.01 3 0.81 ± 0.02 90 ± 10 28.4 2.8 85.2

0.03 1.7 0.78 ± 0.03 90 ± 10 49.5 1.7 84.2

0.07 1.1 0.75 ± 0.03 100 ± 10 76.4 1.1 84.0

0.1 0.92 0.79 ± 0.02 110 ± 10 92.3 0.9 84.9

0.5 0.38 0.74 ± 0.03 150 ± 10 226.4 0.5 86.0

0.8 0.27 0.78 ± 0.04 47 ± 6 324.4 0.4 87.6

1 0.24 0.75 ± 0.02 120 ± 10 393.7 0.4 94.5

4.6. Measuring � using optical tweezers

Since we could not identify a unique �-� combination from our mechanical state diagram, but did observe differences in bond biophysical behavior with respect to δR, we measured the adhesion strength of the antibody/ICAM-1 interaction using optical tweezers-based force spectroscopy. An adherent, live CHO-K1 cell expressing human

ICAM-1 was brought into contact with a laser trapped, BBIG antibody coated bead using a motorized piezo-stage with nanometer precision (Figure 4.10A). After allowing time for bond formation (<0.1 s), the stage was translated away at a specific velocity to achieve force driven bond dissociation. Bead position was monitored throughout approach and retraction phases (Figure 4.10B), and was later converted to force using a calibration of the optical trap. Rupture force histograms obtained at force loading rates of approximately

45 200, 1500, and 3000 pN/s are shown in Figure 4.10C-E. The largest peak in each plot, appearing at low force, corresponded to non-specific interactions (Figure 4.10F). Specific antibody/ICAM-1 binding interactions contained additional rupture modes at higher force.

Results at 1500 pN/s were easiest to interpret, with only a single high force mode centered at 17.4 ± 4.3 pN that correlated to 17.9% of all interactions. Two high force rupture modes were observed at the other loading rates. At 200 pN/s, the second rupture mode overlapped significantly with the non-specific mode, while the third mode was very broad and centered over a similar force range seen at 1500 pN/s. We believe that both of these force modes corresponded to specific antibody/ICAM-1 binding, the first under monovalent and the second under multivalent contexts. The monovalent force mode was centered at 5.0 ± 2.9 pN. At 3000 pN/s, we again observed two high force modes. However, in this case we attributed the second mode centered at 28 pN to non-specific interactions because the frequency was far greater than would be expected from a specific binding interaction. Furthermore, it is well known that high force loading rates result in substantially larger standard deviations in rupture force, which is more consistent with the third mode with rupture force centered at 52.0 ± 8.9 pN. Based on these results, we plotted rupture force versus the logarithm of the loading rate (Figure 4.10G), and determined that

� was approximately 0.27 nm for our antibody/ICAM-1 interaction based on a linear fit.

4.7. Final fitting of experimental data

As a final exercise, we simulated all antibody and ICAM-1 density conditions using the �-� combination identified in the previous section. β values all ranged between 0.74

0 and 0.81, very close to 0.75 from experiments (Table 4.3). kD values generally matched experiments as well (Figure 4.11A), remaining within a factor of 2 for most conditions.

46 However, no detachment was observed at high ICAM-1 density. Increasing � had a weak effect at high ICAM-1, but led to significant deviations for the lower ICAM-1 density cases.

This led us to postulate that each protein G molecule on the glass substrate may have bound two ICAM-1 dimers. We tested this by modifying our ICAM-1 distribution algorithm to randomly apply protein G molecules at a density equal to one-fourth the high ICAM-1 density (33.5 µm-2). ICAM-1/Fc dimers were then randomly assigned at the appropriate total density. Simulation results under the new clustered ICAM-1 dimer arrangement are

0 listed in Table 4.5, and kD values are shown Figure 4.11A. The low and medium ICAM-1 densities did not change significantly, since most ICAM-1 was still dispersed as single dimers. At high ICAM-1 density though, all ICAM-1 molecules were now clustered in groups of 4, which lowered binding stability such that detachment was now observed at all density conditions. β values were greater than 0.75 from experiments, but were also associated

0 with high error due to the small number of detachment events. Fitting kD with β = 0.75 resulted in values that were lower than experiments, but within an order of magnitude

(Table 4.5). Since ICAM-1 clustering significantly influenced nanoparticle dynamics, we also performed simulations in which ICAM-1 was distributed entirely as single molecules

(Table 4.4). Nanoparticle adhesion was most stable for monomers (Figure 4.11A), with no detachment observed at high ICAM-1 or the medium ICAM-1/high antibody combination. β decreased slightly, ranging from 0.65-0.75.

Figure 4.11 Final fitting of experiments for different ICAM-1 clustering conditions. 0 (A) Comparison of kD across all antibody and ICAM-1 densities between experiments and NAD simulations conducted using the final mechanical conditions (� = 0.27 nm, � = 0.8 N/m). ICAM-1 was presented in three different configurations: dimers, clustered dimers, and monomers. Clustering of ICAM-1 decreased nanoparticle stability, particularly at high ICAM-1 density. (B) Mean bond potential was highest for ICAM-1 monomers. Dimer configurations were similar at low and medium ICAM-1, but the clustered dimer surprisingly had elevated mean bond potentials at high ICAM-1. (C) Bond potential histograms for the clustered dimer case. Note the large number of nanoparticles restricted to 1 or 2 bonds at low and medium ICAM-1 densities. (D) Mean bond potential versus time for the clustered dimer case, shown only at early time points to illustrate that the bond steady state was reached before 0.1 s at low and medium ICAM-1 density. At high ICAM-1 density, most bonds formed prior to 0.1 s, but a second, slower bond accumulation phase 0 was also observed out to 0.5 s. (E) kD and mean bond potential closely followed an exponential relationship for all molecular density and ICAM-1 clustering conditions.

48 Table 4.3 Final simulation results across all valency conditions for � = 0.27 nm, � = 0.8 N/m, and the dimer configuration for ICAM-1. Bond Rupture Final ICAM-1 Ab k 0 Bound F δ Bond β D lifetime B,R work R bond (µm-2) (µm-2) (ms-1) NP (%) (pN) (nm) potential (s) (pN•nm) number

0.75 100 ± 21 410 ± 52 0.17 307.2 84.2 0.38 2.5 1.4 10 0.03

0.75 21 1080 ± 67 ± 8 65 0.18 306.2 83.9 0.38 2.4 1.7 0.03

0.74 21 3400 ± 38 ± 6 79 0.18 305.3 83.7 0.38 2.3 2.0 0.03

0.78 41 410 ± 47 ± 6 73 0.08 324.4 88.9 0.41 3.0 2.4 0.04

0.81 41 1080 ± 30 ± 4 83 0.14 316.9 86.8 0.40 2.9 2.6 0.03

0.74 41 3400 ± 12 ± 3 93 0.15 312.7 85.7 0.39 2.9 2.9 0.06

0.80 134 410 ± 2 ± 1 99 0.01 344.1 94.3 0.43 4.7 4.7 0.90

134 1080 NA 0 100 0.05 347.0 95.1 0.43 5.0 5.0

134 3400 NA 0 100 0.06 345.0 94.5 0.43 5.1 5.1

49 Table 4.4 Final simulation results across all valency conditions for � = 0.27 nm, � = 0.8 N/m, and the monomer configuration for ICAM-1. Bond Rupture Final ICAM-1 Ab k 0 Bound F δ Bond β D lifetime B,R work R bond (µm-2) (µm-2) (ms-1) NP (%) (pN) (nm) potential (s) (pN•nm) number

21 410 0.70 65 56 0.13 318.0 87.1 0.40 3.1 2.0

21 1080 0.74 34 73 0.15 313.8 86.0 0.39 3.0 2.4

21 3400 0.68 17 86 0.15 314.6 86.2 0.39 2.9 2.9

41 410 0.62 14 88 0.09 333.1 91.3 0.42 4.0 4.0

41 1080 0.71 3 97 0.09 332.1 91.0 0.42 4.1 4.1

41 3400 NA 0 100 0.09 330.3 90.5 0.41 4.1 4.1

134 410 NA 0 100 0.03 360.0 98.6 0.45 6.2 6.2

134 1080 NA 0 100 0.02 376.0 103.0 0.47 7.2 7.2

134 3400 NA 0 100 0.03 368.0 100.8 0.46 7.1 7.1

For all conditions simulated, FB,R exceeded 300 pN, increasing modestly with ICAM-1 density and degree of clustering but remaining insensitive to antibody density (Table 4.5,

Table 4.1, and Table 4.3). δR exceeded � by ~50% for most cases, only deviating for the clustered dimer configuration at high ICAM-1 and antibody densities (150%), likely because FB,R was so high at 540 pN (Table 4.5). Bond lifetimes were all less than 0.2 s, following the same general trend as FB,R. Final mean bond numbers ranged from ~2.5 to 7, while again exhibiting evidence of valency selection since very few nanoparticles remained bound via a single bond state (Figure 4.12). Investigating bond numbers at early time points, we determined that the bond steady state was still attained within 0.1 s for all cases except the clustered dimer configuration at high ICAM-1 density (Figure 4.12). For this case, it appeared that there was a second, slower phase of bond accumulation that delayed the steady state to around 0.5 s. After correcting for valency selection, we determined mean

50 bond potentials for all conditions (Figure 4.11B). As expected, mean bond potentials were highest for the monomer configuration, but the clustered dimer was slightly greater than the dimer even though nanoparticle adhesion was generally less stable. Looking at the bond potential histograms (Figure 4.13), it was clear that the clustered dimer configuration had a broader bond distribution, with more nanoparticles associated with the single bond state. Bond potential histograms also illustrate how bond distributions shifted to higher mean values as antibody and ICAM-1 density increased. Plotting mean bond potential versus time confirmed that bonds did not accumulate after the bond steady state was reached within the first 0.1 to 0.5 s of the simulation (Figure 4.11D and Figure

0 4.13). Finally, we observed that kD and mean bond potential closely followed an exponential relationship across all densities and ICAM-1 configurations (Figure 4.11E).

51 Table 4.5 Final simulation results across all valency conditions for � = 0.27 nm, � = 0.8 N/m, and the clustered dimer configuration for ICAM-1.

Bond Rupture Final ICAM-1 Ab k 0 Bound F δ Bond β D lifetime B,R work R bond (µm-2) (µm-2) (ms-1) NP (%) (pN) (nm) potential (s) (pN•nm) number

0.80 ± 102 ± 21 410 39 0.15 311.7 85.4 0.39 2.8 1.8 0.02 10

0.77 ± 21 1080 69 ± 8 54 0.15 314.8 86.3 0.39 2.8 1.8 0.03

0.68 ± 21 3400 55 ± 7 56 0.16 312.9 85.7 0.39 2.5 2.0 0.03

0.82 ± 41 410 52 ± 6 64 0.10 327.0 89.6 0.41 3.5 2.5 0.03

0.83 ± 41 1080 31 ± 5 73 0.11 328.2 89.9 0.41 3.3 2.6 0.02

0.75 ± 41 3400 28 ± 5 79 0.10 329.6 90.3 0.41 3.3 2.7 0.03

0.94 ± 134 410 3 ± 1* 98 0.03 369.1 101.1 0.46 5.5 5.3 0.68

0.93 ± 134 1080 3 ± 2* 97 0.02 373.5 102.3 0.47 5.9 5.7 0.55

0.99 ± 134 3400 1 ± 1* 99 0.02 542.1 148.5 0.68 6.1 6.0 0.44

0 *Fitting of kD was performed using β = 0.75

Figure 4.12 Bond numbers for all ICAM-1 configurations. (A) Bond number histograms for ICAM-1 presented as clustered dimers. (B-C) Mean bond number versus time over the first 0.5 s of simulations for ICAM-1 arranged as (B) clustered dimers, (C) dimers, and (D) monomers.

Figure 4.13 Mean bond potential for dimer and monomer cases. (A,C) Bond potential histograms for ICAM-1 arranged as (A) dimers and (C) monomers. (B,D) Traces of mean bond potential for ICAM-1 arranged as (B) dimers and (D) monomers. The bond steady state was attained within 0.1 s at all adhesion molecule densities for both clustering cases.

4.8. Discussion

We have applied the Adhesive Dynamics simulation framework to investigate how multivalent nanoparticle adhesion to a surface via specific biomolecular interactions stabilizes over time. NAD simulations accurately reproduced the multivalent nanoparticle detachment behavior that we observed in experiments for an antibody/ICAM-1 binding pair. Specifically, detachment rate (kD) continually decreased over time following a power

0 law relationship with temporal (β) and magnitude (kD ) fitting parameters (Equation 3.23).

The β parameter was insensitive to adhesion molecule densities on the nanoparticle and substrate surfaces, as we observed in experiments. Instead, β was strongly influenced by individual bond stability, with the experimentally observed value (β ~ 0.75) emerging whenever bonds were subjected to an amount of mechanical work that was equal to the

0 bond chemical energy. The kD parameter varied with adhesion molecule densities, bond mechanical properties, and ICAM-1 clustering, generally tracking with nanoparticle stability. While mean bond number per bound nanoparticle did increase in a manner consistent with the decrease in nanoparticle detachment rate, we found that this was only due to bond accumulation during the first 0.1-0.5 s of simulations. Mean bond number continued to slowly increase at longer times because nanoparticle populations were heterogeneous with respect to bond potential, and detachment primarily occurred for nanoparticles that were restricted to one or two bonds. This valency selection mechanism manifested in the evolution of the remaining nanoparticle population due to a classic

‘survival of the fittest’ scenario.

54 Our results conclusively demonstrated that nanoparticle adhesion was strongly influenced by mechanical forces. Experiments were matched if bonds ruptured under high force (FB,R > 300 pN), or more accurately, mechanical work (�FB,R ~ bond chemical energy).

Bonds were not consistently subjected to this level of force, but rather with a frequency such that bonds persisted for ~0.1 s. To put this into context, the half-life for the

0 -4 -1 antibody/ICAM-1 interaction in the absence of force (kr = 1.1x10 s ) is ~100 min. To further investigate the origin of mechanical force, we conducted a new simulation in which bonds were not allowed to form over time (i.e. kf = 0). Remarkably, all nanoparticles held by a single tether detached in less than 1 s (Figure 4.14A). Fitting the detachment profile using a simple exponential decay resulted in a rate constant of 3.6 s-1, almost four orders of

0 magnitude higher than kr . This finding was not affected by insufficient temporal resolution, as results were identical when we decreased the time-step to 0.1 ns (Figure

4.15). Bond force (FB) and rupture force (FB,R) distributions for single tether simulations were similar to the multivalent cases (Figure 4.14B and C). Average FB,R and bond lifetime were ~290 pN and ~0.25 s, respectively, both very close to the low valency conditions

(Table 4.5, Table 4.4, and Table 4.3). We found that Brownian motion of the nanoparticle was the only significant source of mechanical force in the single tether simulation. Shear force was very low at 0.036 pN, and simulation results were not affected by removal of fluid flow nor increasing shear rate by a factor of 100 (Figure 4.15). Note that the latter finding is consistent with our previous experiments in which nanoparticle detachment rate remained constant after varying shear rate up to 1000 s-1.[6] As expected, the effects of

Brownian motion on detachment correlated with nanoparticle size, further accelerating detachment rate as size decreased (Figure 4.15). Likewise, increasing size mitigated

55 Brownian effects, particularly since the influence of shear was also rapidly rising. Shear force only began to become significant for micron-sized particles, but Brownian motion remained the primary source of force inducing bond rupture (FB,R > 200 pN). This is in contrast to a previous study of platelet adhesion, where Brownian motion was determined to have minimal effect.[44] While there are differences in implementation of Brownian motion and fluid shear between these works, Adhesive Dynamics here versus CDL-BIEM for the platelet case, the key factor is likely the difference in how bonds were modeled.

Platelet adhesion was mediated by GPIb-α, which had a much higher � (0.71 nm) and was not modeled directly as a spring with a set spring constant. Regarding the mechanism by which Brownian motion induces bond rupture, Liu et al. [19] previously investigated the adhesion of a 100 nm diameter sphere mediated by an antibody/mouse ICAM-1 interaction using a thermodynamic model that included an entropic penalty for tethering the nanoparticle’s motion. The Bell model was then used to provide an estimate for FB,R that was as high as 230 pN, but no justification was given for the loading rate rates that were employed. While this maximum rupture force is similar to the 290 pN value that we found using the single tether simulation, an important distinction is that our result emerged without having to assume the loading rate.

Figure 4.14 Single tether simulations and valency state-dependent detachment dynamics. (A) Nanoparticles held by a single tether all detached within 1 s, with a profile that closely resembled the initial phase of rapid detachment observed for multivalent cases. (B) Bond force distribution for the single tether simulation was nearly identical to the multivalent cases. (C) Bond rupture force distributions were similar between the single tether and low ICAM-1 density cases, but rupture force shifted to higher values with increased valency. (D- F) Valency state dependent detachment dynamics. Mean bond number (black line) is shown over time at low antibody density and either (D) low, (E) medium, or (F) high ICAM- 1 density. All detachment events are included in the plot, and color-coded based on maximum bond number achieved: 1 bond (green), 2 bonds (orange), or 3 bonds (purple). The point of detachment is indicated by the triangle (△), and lines then trace back up to the time point at which that nanoparticle was at its maximum bond number, which is indicated by an upside down triangle (▽). Nanoparticles restricted to a single bond detached rapidly, most within the first few seconds. Nanoparticles that detached from the second and third bond states persisted longer, and quickly dropped all the way to zero bonds, typically within 0.1 s, limiting the chance for bonds to re-form.

Figure 4.15 Single tether simulations. Nanoparticle detachment was simulated using only a single bond, with no bond formation allowed. (A) Single tether detachment profiles were identical after reducing the simulation time-step from 1 ns down to 0.5 and 0.1 ns. (B) Results were also unchanged when fluid shear flow was removed, as well as when shear rate was increased up to a factor of 100 (10,000 s-1). Significant shear force effects were observed starting at 50,000 s-1 shear rate, which continued to become stronger at 100,000 s-1. (D-F) Effect of nanoparticle size. (C,D) At 100 s-1 shear rate, particle adhesion via a single tether becomes increasingly more stable as radius increases, which can be seen based on (C) delayed detachent profiles and (D) lower bond rupture forces. Inset in (C) displays results from fitting detachment profiles using a single exponential decay to obtain an effective bond detachment rate (kr) and mean FB,R. No significant different is observed between particles with 500 and 1000 nm radii, likely due to counter-effects of shear force. (E,F) Without flow, there is minimal change in single tether (E) detachment rate or (F) bond rupture force histogram for a 105 nm radius particle. However, adhesion of a 1000 nm radius particle is considerably more stable than when shear flow is present due to lower forces being experienced at rupture.

We have also uncovered a second source of mechanical force, which became more pronounced as bond valency increased and ICAM-1 molecules were clustered (Table 4.5,

Table 4.4, and Table 4.3). We presume that this force arose from bonds pulling on each other. For most cases, inter-bond pulling was modest, increasing FB,R by only 10-30% relative to the result of the single tether simulation, but was as high as 100%. We should note that these estimates assume that the contribution of force from nanoparticle

58 Brownian motion remained constant, which may not have been the case. Regardless, it was clear that multivalent nanoparticle adhesion was not stabilized by reducing mechanical force loads on individual bonds, but rather by the fact that the rate of bond formation was simply much faster than rupture for all conditions. Although the mechanically stressed bonds would generally be considered very unstable, persisting only ~0.1 s, we found that the full bond steady state could be attained on this same time-scale for nearly all conditions

(Figure 4.11D and Figure 4.13). Based on the sum of these findings, we conclude that nanoparticles quickly populated and were maintained near their highest valency state, which was likely determined by the local availability of free adhesion molecules. This is supported by the fact that most nanoparticles that detached were only able to attain a maximum of one or two bonds (Figure 4.14D-F). Specifically, single bond restricted particles were all lost within ~1 s, similar to the single bond tether simulation (Figure

4.14A), but some did persist out to 10 s due to re-binding. For nanoparticles that reached bond valencies of two and even three, detachment tended to occur through the successive loss of all bonds within 0.1 to 0.2 s, thus minimizing the chance for bonds to re-form. It should be noted that since only single bond-restricted nanoparticles were lost at high

ICAM-1 density, it is certainly possible that some nanoparticles detached before reaching their true bond potential.

Based on the above findings, we believe that our phenomenological time-dependent detachment rate (Equation 3.23) captures the two critical aspects of valency

0 selection/evolution. The parameter kD provides a metric for the combined detachment rate of the entire nanoparticle population, across the full distribution of bond valencies that are possible, including those nanoparticles that will ultimately detach. The β parameter

59 then modulates this inherent detachment probability as nanoparticles do detach, starting with the initial rapid loss of those restricted to a single bond and then transitioning to the lower multivalent states (Figure 4.14). To illustrate these concepts, imagine that the nanoparticle population was categorized into i states based on valency potential, and then

0 0 each state was assigned a different kD value (i.e. kD ,i) and initial nanoparticle number (i.e.

Ni). In this scenario, only the Ni values would change with time, decaying at a constant rate

0 defined by the respective kD ,i values. Formalization of these valency state-dependent relationships, including connection to β, will be a focus of future work. A key implication of this interpretation for time-dependent detachment is that the effects will continue beyond the minute time-scale observed in experiments, lasting until only nanoparticles in the highest attainable valency state remain. Our power law equation is not ultimately bounded in this manner, and thus we cannot extrapolate out to time scales longer than minutes. NAD simulations cannot provide this insight either because they are too computationally expensive. Hence, the best approach for predicting long-term behavior will be to develop a deterministic model with rate equations defined for each valency state, similar to previous work.[28-30] While this requires a large number of parameters to be defined, our simulation results should greatly simplify the process by providing single bond rupture rate, scaling of FB,R with valency (Table 4.5), bond potential distributions (Figure 4.11C), and bonding dynamics during the approach to the steady state (Figure 4.11D). These insights highlight the advantage of our kinetic approach to understanding multivalent nanoparticle adhesion. Our results also suggest that thermodynamic equilibrium concepts such affinity or avidity cannot be used to understand multivalent nanoparticle adhesion in a straightforward manner, as the population is heterogeneous and will be changing with

60 time. Our work indicates that the ideal time point to define the equilibrium is likely within the first second after binding, but this cannot be assessed experimentally, and even at this time-point the system will have likely already been influenced significantly by valency selection.

Regarding bond biophysics, we found that bonds ruptured after being separated (or compressed) by a length (δR) very close to �. This was generally to be expected, as � has mechanistically been described as the length scale for adhesive interactions within the binding pocket.[49] However, δR is also a length scale related to stretching/compression of the antibody and ICAM-1 molecules, and thus future work will seek to add the necessary molecular detail to distinguish between these different length contributions. Nevertheless, the end result was that over much of the mechanical property parameter space that we investigated, bonds ruptured under a mechanical work that was ~��2. This result is reminiscent of the bond spring energy used in previous Adhesive Dynamics simulation

2 works, which was defined as �δ .[36, 38] However, it is important to clarify that our result was only valid at rupture, with ��δ prevailing at shorter extension/compression lengths (δ

2 < �). We did attempt to use the �δ relationship in early simulations, but could not match

0 experimental results in terms of β and kD . To uniquely identify the bond mechanical properties for our antibody/ICAM-1 pair, we performed force spectroscopy studies using optical tweezers and found � = 0.27 nm. This is close to the value measured for an antibody/murine ICAM-1 interaction,[19] and in general is representative of a moderate bond strength.[49] Extrapolating out to a rupture force of 300 pN would equate to a force loading rate of ~108 nN/s, but both of these values are well outside of the range that we used for optical tweezers experiments (maximum ~50 pN). Thus, it is possible that a

61 different energy landscape barrier could be governing bond mechanics, which would equate to a larger �.[49] This force regime will be examined experimentally in future studies. Using � = 0.27 nm, we found that � = 0.8 N/m provided the best fit to experiments, placing δR at ~0.4 nm, or approximately 50% larger than �. Our � value was larger than previous works using HIV or nanoparticles,[19, 41-43] but was not so high as to introduce artifacts into the simulation. This was determined based on the spring time � = 2��/� , where m is the mass of the particle.[63] The value of � was ~16 ns, which is an order of magnitude greater than the time-step used for simulations and 5-fold greater than the viscous relaxation time (2ns).

Using our final mechanical parameters, we achieved excellent correlation between

0 NAD simulations and experiments in terms of both β and kD kinetic parameters at most, but not all, adhesion molecule densities. Since no detachment was observed at high ICAM-1 density, we chose to cluster ICAM-1 dimers in a manner that was consistent with our experimental set-up using protein G molecules to bind ICAM-1/Fc. This change successfully destabilized results at high ICAM-1 without negatively affecting the other cases. If we consider the valency state concepts introduced in the preceding paragraphs, then fine-

0 tuning � primarily modulated the inherent detachment rates for each valency state (i.e. kD ,i values), while clustering ICAM-1 primarily redistributed the initial number of nanoparticles in each valency state (i.e. Ni values, see Figure 4.11C and Figure 4.13, for examples). Even though the final kinetic consistency between NAD simulations and experiments was very strong, deviations still remained. It is possible that fitting could be improved by further adjusting � or the molecular configurations (i.e. clustering, discrete physical detail). Alternatively, simulations may be overestimating bond formation rate,

62 which could be adjusted by decoupling �ts from � or by tracking the orientation of unbound adhesion molecules to account for the actual separation distance between binding domains.

Another related phenomenon is that we didn’t prevent bonds from occupying the same physical space as nanoparticles translated and rotated, which would be expected have a larger effect at higher bond valencies. We did prevent new bonds from forming if they were within 2 nm of another bond, but this estimate may have been too conservative. Finally, discrepancies may simply be related to measurement errors within the experiments.

63 5. HETEROGENEITY IN MULTIVALENT NANOPARTICLE

5.1. Introduction

In the previous chapter, we used NAD detachment simulation to model the bond accumulation process and assessed the nanoparticle stability after it forms one bond. We found the reason behind time-decaying detachment rate kD, that the nanoparticle population is heterogonous in bonding ability, and nanoparticles preferentially detach from those with lower bonding ability. We terms the bonding ability as bond potential. In order to further understand the heterogeneity of nanoparticle population, in this chapter we applied dynamical system analysis to provide a more theoretical inspection on particle detachment dynamics.

5.2. Methods

Empirical Detachment Model

In previous work, we modeled nanoparticle detachment in a similar manner to classic kinetic treatments of molecular binding. Nanoparticle detachment following initial tethering can be expressed as in Equation 3.22. We observed that kD decreased over time, which was captured using an empirical power law as in Equation 3.23. We originally treated tref as a variable of convenience to maintain unit consistency,[6] and chose 1 s.

However, NAD simulations indicated that the steady state for multivalent bond formation

was achieved at approximately 0.1 s, which would be a more systematic choice for tref. This would mean that at approximately the time when bond steady state has been achieved, the

0 value for kD was equal to kD . For this part, we re-analyzed previously reported experimental and simulation data using Equation 3.24 with tref = 0.1 s.

64 Bond Potential Classification

NAD simulations revealed considerable heterogeneity in steady state bond number for a given population of nanoparticles. This was due to the fact that adhesion molecules were distributed randomly on both the nanoparticle and substrate, and thus the availability of adhesion molecules within the contact zone varied for different nanoparticles. We developed the term bond potential (BP) to characterize this phenomenon. For this work, we have chosen to define BP based on the mode bond number observed during NAD simulations. For example, a nanoparticle with BP = 2 predominantly had two bonds, even if there were significant periods of time in which there was 1, 3, or more bonds. We further assume that nanoparticles within the same BP category generally share the same adhesion properties, including kinetic rates and thermodynamic energy. However, we do note that this is a broad categorization, and thus variability may still exist. We will account for to this possibility by allowing for the presence of two or more hidden components within each BP.

We concluded that this approach was more intuitive than defining BPs with non-integer values, as bond number will always be a whole number at any given time. The hidden components will further sub-categorize BPs into distinct groups that display significantly different adhesion properties.

BP Detachment Model Development

Within each BP category, we hypothesized that detachment would follow Equation

3.22, but now subject to a constant kD value that was unique to each BP, as follows:

�� = −� � �� , 5.1 where i designates BP and kD,i is the detachment rate for BP i. This gives the integral form:

� (�) = ��, 5.2

Thus, each BP will follow a simple exponential decay, and the results for each BP can be summed to recreate the detachment profile for the entire population.

, �(�) = � (�) = � �

5.3 � = �

0 where Bi is the initial number of nanoparticles in BP i and m is the maximum BP observed for the population.

As discussed, it is possible that BP categories greater than 1 may exhibit significant bonding heterogeneity, resulting in hidden components. Under this assumption, Equation

5.2 can be generalized to include terms for each relevant hidden component:

() () , �(�) = � � �

5.4 () 1 = �

(j) where j designates hidden component number, a i is the relative number of BP i

(j) nanoparticles in component j, k D,i is the detachment rate for BP i nanoparticles in component j, and k is the total number of hidden components. If a BP only contains a single component, then Equation 5.4 reduces back to Equation 5.2. As described in the results section, we found that NAD simulation data was well characterized using two hidden

66 components, with the detachment rate for the slower detaching component equal to zero, which simplifies Equation 5.4 to:

� , = �� + (1 − �) � 5.5

Survival Analysis to Relate BP and Empirical Detachment Models

We wish to find the relationship between the BP model in the previous section with the empirical (population-scale) detachment model in Equation 3.22. To do this, we use survival analysis. We first defined a hazard function, H(t), to describe the instantaneous detachment rate, as follows:

1 �� (�) �(�) = − = − ln �(�) �� 5.6

We again assumed that each BP can be divided into hidden components with relatively faster and slower detachment rates. Specifically, we assumed there were two hidden components, one of which did not detach over the time-scale of interest. Using

Equation 5.4 and 5.5, we can thus approximate B(t), denoted as � (�), as follows:

� (�) � � ( ( ) ) ( ) = 1 − � 1 − � � exp − � + � 1 − � � � �, = (1 − �)� + �

where, for convenience, we defined 5.7

� = �(1 − �)� , � � � = �,

0 and wi is the initial ratio of nanoparticles in BP i relative to the initial full population (i.e.

0 0 Bi /B ). Combining Equation 5.6 and 5.7, we can approximate H(t), denoted as � (�), as:

67 1 1 � (�) = ⋅ � � / 5.8 1 + 1 − � �

We can also approximate a hazard function, denoted H*(t), for the empirical model using Equation 3.23:

∗ � � (�) = (�/�) 5.9

This would imply the following bond survival probability:

� � ∗ ∗ () � (�) = exp − � (�′)��′ = exp − ⋅ (�/�) 1 − � 5.10

Given the two hazard function approximations above, we imposed equality at specific time points. We first chose tref, the bond steady state. At t = tref, assuming � (�) =

∗ � (�) implies

1 1 � = ⋅ � � / 5.11 1 + 1 − � �

We also used an arbitrary later time that we defined as tmax. At t = tmax, assuming

∗ � (�) = � (�) implies

� / 1 + 1 − � � ln � 1 + �/ 1 − � 5.12 � = � ln �

0 where Equation 5.11 was used to substitute for kD . Thus, Equation 5.11 and 5.12 provide

0 approximate relationships between the empirical parameters kD and β and the BP

0 expansion parameters kD,i, wi , and ai.

68 Bond State Model Development

The bond state model requires bond formation and rupture rates for all bond state transitions. To estimate these, we tracked instantaneous bond numbers within each BP category throughout the simulations. We defined the relative number of nanoparticles in

BP i with bond number j as Si,j. At any given time, the number of nanoparticles in each BP category is equal to the sum of all bond number states:

� (�) = � (�) � , 5.13

where N is the maximum bond number observed for BP i. Si,j is further defined based on the rates of bond formation (kf) and rupture (kr) for all possible state transitions, as dictated by the following master equation:

�� , = −� � − � � + � � + � � �� ,, , ,, , ,, , ,, ,

�� , = � � �� , 5.14 or:

�� = � � �� where Ki is transition matrix for BP i containing all formation and rupture rates and Si is a vector of all bond states (Si,j). The terms Si,d and kdiff are the relative number of detached nanoparticles and the rate at which a nanoparticle with zero bonds translates 1 diameter from the contact area, respectively, and were included to account for the fact that nanoparticle tracking experiments cannot detect a detachment event until a nanoparticle has translated far enough from the site of attachment. As indicated by Equation 5.14, the

69 rate of change for nanoparticles with j bonds is determined by four components: the rate of acquiring an additional bond (kf,i,j+1), the rate of losing a bond (kr,i,j), the rate of nanoparticles with j-1 bonds acquiring an additional bond (kf,i,j), and the rate of nanoparticles with j+1 bonds losing a bond (kr,i,j+1).

If we allow for the possibility of two hidden components within each BP, we can modify the master equation (Equation 5.14) as follows:

��() , = � (�() �() + �() �() ) �� ,, ,, , ,, ,

() () () () +�,,(�,,�, + �,,�,)

() () −�,,�,

��() , = (1 − � )(�() �() + �() �() ) �� ,, ,, , ,, ,

() () () () 5.15 +�,,(�,,�, + �,,�,)

() () −�,,�,

�� , = � (�() + �()) �� , ,

or:

�� = � � ��

() () where � , and � , correspond to hidden components 1 and 2, respectively, �,,(∈

[0,1]) is the relative number of nanoparticles transitioning from state j-1 to j for component

70 1, and �,,(∈ [0,1]) is the relative number of nanoparticles transitioning from state j to j-1 for component 1. Under the two-component scenario, we assume that one component will have lower bonding energy, and thus be analogous to the fast detaching component in the

BP model, but we did attempt to enforce this as a constraint. Furthermore, we assumed that a nanoparticle could change configuration with time, and thereby cross from one component to the other. This effectively means that the system of nanoparticles does not retain memory. Based on these assumptions, the system of differential equations must be solved simultaneously and components 1 and 2 cannot be interpreted in isolation from each other.

Since there are a large number of transition rates in Equations 5.14 and 5.15, we employed two simplification strategies before solving the system of differential equations.

First, we determined exact values for kr,1,1 and kdiff using simplified NAD simulations in which the nanoparticle was only attached via a single tether and additional bonds were not allowed to form (Figure 5.1). The second simplification was to estimate the remaining transition rates (kf and kr) from full NAD simulations using the average pause times between bond number transitions, and further details are described in Figure 5.1. NAD simulation data was fit using Equation 5.13 and either Equations 5.14 and 5.15 using the

“minimize” function in Python LMFIT library, with kr,1,1 and kdiff values set as constants and the pause time kf and kr values employed as initial conditions.

Figure 5.1 Bond transition rate model fitting parameters. (A,B) Single bond tether simulations to determine values for (A) kr1,1 and (B) kdiff. (C,D) Representative example of simulation pause time fitting to determine initial transition rate values for ODE model fits using (C) one and (D) two components.

We constructed a transition matrix for each BP, Ki, by writing the bond number state master equation (Equation 5.14) in vector form, as follows:

−� − � , � , 0, 0, 0, ⎡ ,, ,, ,, ⎤ � , 0, 0, … , ⎢ �,,, −�,, − �,,, ,, ⎥ ⎢ 0, � , −�,, − �,,, �,,, 0, ⎥ = ,, … , ⎢ 0, 0, �,,, −�,, − �,,, �,,, ⎥ 5.16 ⎢ … , … , … , … , … , ⎥ ⎣ �,,, 0, 0, 0, 0, … ,⎦

S = [�,, �,, �,, �,, … , �,,�]

T = [�,, �,, �,, �,, … , �, ]

We then parameterized Ki using the transition rates determined from fitting the

NAD simulation data, and determined the average time to detachment (Ti) using Equation

5.18. The vector Ti contained elements that were all very similar in value. For example, T1 of the base case (medium ICAM-1 and medium antibody, ICAM-1 monomers) had values of

1.47 and 1.76 s for bond states 0 and 1, respectively, and T2 values were 54.06, 54.06,

54.11, 54.11, and 54.12 s for bond states 0-4, respectively. Clearly state 0 of BPs 1 and 2 should not have such disparate detachment times, as both involve unbound nanoparticles simply diffusing away. Thus, Ti must be thought of as the time to detachment from the average state for each BP system. Thus, we averaged all elements of Ti to obtain a single

(M) detachment time, which we then inverted to determine k D,i (Equation 5.19).

Similarly, if we allow for the possibility of two hidden components within each BP, the transition matrix for each BP (Ki) can be modified as follows:

5.17

The vector Ti contains elements that were still all very similar in value. For example,

T2 values were 20.50, 19.15, 21.15, 20.45, 20.95, 21.19, 20.59, 20.60, 20.52, 20.60s for bond states 0-4 (2 components in each state).

Estimating a Detachment Rate from the Bond Transition Matrix

We can use the bond state model from of the previous section to estimate a detachment rate from the transition matrix (Ki) using the mean first passage time.[64] We defined Ti,j as the first passage time for a nanoparticle in BP i and bond state j, and Ti is a

73 vector containing the first passage times for all N bond states. The vector Ti can be

� = −� � � � 5.18 calculated directly from the transition matrix, as follows: where E is a vector with N+2 elements that are all equal to unity. The elements of vector Ti can then be averaged to determine the mean first passage time, and inverted to obtain a

(M) detachment rate, which we will refer to as k D,i :

�() = �/ � , , 5.19

Estimating bond number state transition rates from NAD simulation pause times

Forward and reverse bond state transition rates were estimated using the waiting times between bond state transition observed from NAD simulations. Results are shown for the base case, bond state transition from 4 to 5, and BP 5 assuming a single population for

Figure 5.1C or two sub-components in Figure 5.1D. The two sub-component result exhibited better alignment with the pause time data, which again confirmed the presence of bonding heterogeneity even within BPs. However, for simplicity, we chose to use single exponential fits as initial values for solving the system of ODEs in Equations 5.15 and 5.16.

74 Combining the BP Model and Mean First Passage Time Estimates

Simulation data was fit a second time using the BP model (Equation 5.4) with two

(M) hidden components, but now using k D,i as a constraint. Specially, we used three fitting

() () () parameters: detachment rates �, and �, as well as the weight � , as follows:

( ) () ( ) () , , �(�) = � [� � + (1 − � )� ]

such that: 5.20 () () � 1 − � � , = () + () �, �,

In this manner, we achieved results that are consistent with both macroscopic

(nanoparticle detachment) and microscopic (bonding) phenomena. Furthermore, we

() obtained non-zero detachment rates for the slow detaching component (�, ), which improved longer-term predictions of detachment behavior.

Measuring kr,1,1 and kdiff using single bond NAD simulations

Simplified NAD simulations in which nanoparticles were initially tethered via a single bond, but no additional bonds were not allowed to form, were used to determine the rupture rate from 1 to 0 bonds (kr,1,1) and the diffusion rate of unbound nanoparticles at least 2.5 diameters away from the initial position (kdiff). This was accomplished by separately tracking the times to rupture and diffusion for an ensemble of 200 nanoparticles, resulting in the profiles shown in Figure 5.1. The time distributions were then fit using an exponential decay to obtain the respective rates. This resulted in kdiff = 66

-1 0 5 - s under all parameter conditions. For the standard set of bond properties (kf = 1.5x10 s

1 0 -4 -1 -1 , kr = 1.1x10 s , s = 0.8 N/m, and g = 0.274 nm), we obtained kr,1,1 = 3.8 s . For the new

75 0 -4 -1 0 -3 -1 bond property conditions g = 0.29 nm, g = 0.3 nm, kr = 5x10 s , and kr = 1.1x10 s , we

-1 obtained kr,1,1 values of 8.9, 17.7, 14.8, and 19.8 s , respectively.

Bond formation and rupture rate estimates from simulation pause times

For each forward and reverse bond state transition, we will define the waiting time,

�, and assume that the distribution of waiting times for an ensemble of transitions follows an exponential distribution:

�(�) = �exp (−��) 5.21

where k is the transition rate, which can be either describing bond formation (kf) or rupture (kr). We can then estimate k as the inverse of mean pause times:

� = 1/�̅ 5.22

Univariate fitting was implemented using maximum-likelihood estimation in R’s MASS library.

Alternatively, we assumed the distribution of waiting times follows a mixture of two components:

�(�) = �� exp(−��) + (1 − �)� exp(−��) 5.23

where a is the relative number of nanoparticles in component 1 and � and � are the component rates. Bivariate fitting was implemented using the expectation-maximization

(EM) algorithm in R’s RENEXT library.

76 5.3. Bond Potential (BP) Detachment Model

We postulated that appropriately categorizing nanoparticles based on bond potential (BP) would result in a series of detachment profiles that followed first-order kinetics and could then be combined to reconstruct the complex profile of the full population (Equation 5.1, 5.2, and 5.3). As a test case, we used NAD simulation results at medium antibody density (1080 nm-2) on the nanoparticle and medium ICAM-1 density (41 nm-2) on the substrate. We first assigned each nanoparticle to a BP based on mode bond number. This resulted in six BP categories that were distributed as shown in Figure 5.3A.

Also indicated in Figure 5.3A is the number of nanoparticles that detached by the end of the simulation, which varied from 100% for BP1 to 30% for BP2, 2% for BP3, and 0% for

BP4 and above. Detachment profiles for BPs 1-3 are shown separately in Figure 5.3B, along with exponential decay fits that were obtained using Equation 5.2,. These fits matched the simulation data well for BP 1, but the deviations seen for BPs 2 and 3 clearly indicated a significant deviation from first-order kinetics, suggesting significant bonding heterogeneity was still present within these sub-populations. Therefore, we assumed that hidden components were present, and using Equation 5.4 to fit the simulation data we found that two hidden components was sufficient. This led us to conclude that bonding heterogeneity was still present in each BP, which we hypothesized could be captured using multiple hidden components, as described by Equation 5.4. Fitting results using 2 hidden components are shown in Figure 5.2 for the base case, and generally match NAD simulation data well. Fitting parameters, including kD,i values for both components and the relative number of component 1 (ai), are listed in Table 5.1. Upon inspection, kD,i values for component 2 were much less than component 1, and in most cases were 0. Thus, we

77 chose to simplify two component detaching fitting by assuming that the second component does not detach at all (kD,i = 0). As shown in Figure 5.2, this simplification led to similar fitting accuracy.

Figure 5.2 Full population BP model fits for two components using 2 and 3 degrees of freedom. Two degrees of freedom (2 DOF) assumes that one component does not detach while three degrees of freedom (3 DOF) assumes that both can detach, and the fits were identical. Results shown are for the base case (medium antibody and ICAM-1 density, g = 0.274 nm, 0 -4 -1 kr = 1.1x10 s ).

We also observed that fitting results were similar if we assumed that the second

(2) component did not detach (i.e. k D,i = 0, see Figure 5.2), as described by Equation 5.5.

Detachment rate constants (kD,i) and the relative contribution of the fast-detaching component (ai) are listed in Table 5.1. As expected, both kD,i and ai decreased significantly with BP due to greater adhesion stability. The detachment profile for the full nanoparticle population is shown in Figure 5.3C, along with the reconstructed profile obtained from all

BPs using Equation 5.3 for both one and two component scenarios. The empirical model was also used to fit the simulation data using Equation 3.23. The two-component BP model matched the NAD simulation data best, as the other fits diverged at either short or long times.

Figure 5.3 BP model analysis of the base case. 0 -4 -1 (medium antibody and ICAM-1 density, g = 0.274 nm, kr = 1.1x10 s ). (A) Histogram showing how nanoparticle population was distributed across BPs 1-6. The number of nanoparticles that detached during 30 s simulations is shown in red. (B) Individual detachment profiles for (i) BP 1, (ii) BP 2, and (iii) BP 3 and fits performed using single and dual components. (C) Detachment profile for the full nanoparticle population and fits performed using the BP model with single and dual components, as well as the empirical model.

79 Table 5.1 BP model parameters for two component fits assuming one does not detach.

kD,1 a1 kD,2 a2 kD,3 a3 kD,4 a4 kD,5 a5 kD,6 a Case 6 (s-1) (s-1) (s-1) (s-1) (s-1) (s-1)

1. Base 0.57 1 0.19 0.30 0.01 0.05

0 2. Low Antibody 3.17 1 0.42 0.03 0.12 .06

3. Low ICAM-1 0.57 1 0.06 0.33

4. ICAM-1 dimer 0.81 1 0.08 0.23 0.05 0.08

5. Clustered 0.82 1 0.07 0.20 dimer

0 6. � = 0.29 nm 2.18 1 0.10 0.02 0.08 .55

0 7. � = 0.3 nm 8.48 1 0.15 0.09 0.17 0.08 0.08 0.075 0.04 .75

0 -4 -1 8. kr = 5x10 s 7.10 1 0.10 0.55 0.07 0.15 0.05 0.08

0 -3 -1 9. kr = 1x10 s 18.22 1 0.20 0.67 0.05 0.40 0.05 0.10 0.04 0.12 0.04 0.12

Next we followed the same approach using four more conditions, each of which varied only one parameter: lower antibody density (410 µm-2), lower ICAM-1 density (21

µm-2), ICAM-1 arranged as dimers, and ICAM-1 arranged as clustered dimers of 4 molecules. Bond potential distributions are shown in Figure 5.5A-D, and generally shifted to lower values compared to the base case in Figure 5.3A. The downward trend in bond number was also associated with higher levels of nanoparticle detachment, although most still detached from BPs 1 and 2. Detachment fits for each BP can be found for both one

(Equation 5.2) and two component (Equation 5.5) scenarios in Figure 5.4.

Figure 5.4 Individual BP detachment fits for cases 2-5. Base case parameters were modified to (A) low antibody density, (B) low ICAM-1 density, (C) ICAM-1 dimers, and (D) clustered ICAM-1 dimers. Detachment profiles and fits performed using single and dual components for (i) BP 1, (ii) BP 2, and (iii) BP 3.

Full population fits using the BP model with one and two components and the empirical model (Equation 3.23) are given in Figure 5.5E-H. Two component BP model fits again matched simulation data best across all conditions. Detachment rate constants (kD,i) and relative numbers (ai) for the first component of each BP are listed in Table 5.1. The

81 most striking differences relative to the base case were in kD,1, which increased for all but the lower ICAM-1 condition. Since bond rupture rate for BP1 (kr,1,1) should be the same for these conditions, equal to the value observed in single bond simulations (3.8 s-1, see Figure

5.1), these changes most likely reflected a decrease in bond formation rate (kf,1,1). All kD,2 values were lower than the base case, but a2 varied both higher (lower antibody and ICAM-

1 density) and lower (ICAM-1 clustering). BP3 was difficult to evaluate due to very low ai values.

Figure 5.5 BP model analysis of cases 2-5. Base case parameters were modified to (A,E) low antibody density, (B,F) low ICAM-1 density, (C,G) ICAM-1 dimers, and (D,H) clustered ICAM-1 dimers. (A-D) Histograms showing how nanoparticle population was distributed across BPs 1-6. The number of nanoparticles that detached during 30 s simulations is shown in red. (E-H) Full population detachment profiles and fits performed using the BP model with single and dual components, as well as the empirical model.

We then explored detachment from higher BPs by performing new NAD simulations using the base case, but destabilizing individual bonds via increased � (0.29 and 0.3 nm) or

0 -4 -3 -1 kr (5x10 and 10 s ). BP distributions are shown in Figure 5.6A-D, and were all very similar to the base case in Figure 5.3A. However, poor individual bond stability resulted in

82 a higher percentage of nanoparticles detaching from BPs 2 and 3, and significant

0 detachment was now seen from BP4. Detachment was even observed from BP 6 for kr =

10-3 s-1. Detachment fits for each BP category can be found for both one (Equation 5.2) and two component (Equation 5.5) scenarios in Figure 5.7. Full population fits using the BP model with one and two components and the empirical model (Equation 3.23) are given in

Figure 5.6E-H. Even though detachment was now higher in magnitude and had contributions from more BPs, the two component BP model still matched the NAD simulation data very well. Detachment rate constants (kD,i) and relative numbers (ai) for the first component of each BP are listed inTable 5.1. The largest differences were again

0 related to kD,1, which increased progressively with both � and kr due to elevated kr,1,1 ( see

Figure 5.1). Differences in kD,2 from the base case were modest, but a2 increased substantially. Both kD,i and ai values continued to decrease with BP, but only modestly, indicating little change in the fast detaching component.

Figure 5.6 BP model analysis of cases 6-9. 0 -4 Base case parameters were modified to (A,E) g = 0.29 nm, (B,F) g = 0.3 nm, (C,G) kr = 5x10 -1 0 -3 -1 s , and (D,H) kr = 10 s . (A-D) Histograms showing how nanoparticle population was distributed across BPs 1-6. The number of nanoparticles that detached during 30 s simulations is shown in red. (E-H) Full population detachment profiles and fits performed using the BP model with single and dual components, as well as the empirical model.

Figure 5.7 Individual BP detachment fits for cases 6-9. 0 -4 -1 Base case parameters were modified to (A) g = 0.29 nm, (B) g = 0.3 nm, (C) kr = 5x10 s , 0 -3 -1 and (D) kr = 10 s . Detachment profiles and fits performed using single and dual components for (i) BP 1, (ii) BP 2, (iii) BP 3, (iv) BP 4, (v) BP 5, and (vi) BP 6.

5.4. Relating Parameters for the BP and Empirical Models

We next performed a survival analysis to derive a full population-scale detachment rate (kD) for the BP model based on the individual BP detachment rates (kD,i) and weights

(ai). We again assumed that two components were present for each BP, allowing us to use the results in Table 5.1. We then defined a hazard function (H) to characterize an instantaneous detachment rate, and used the corresponding instantaneous nanoparticle survival probability (�), given by Equation 5.8. The hazard function contained magnitude and time-dependent elements, similar to our original empirical detachment rate (Equation

3.23). The magnitude term was based on the sum of the fast detaching component of each

BP weighted by their respective kD,i values, which we defined as the parameter �̃.

Specifically, the magnitude parameter was 1/�̃, which corresponds to an initial detachment

84 rate for the fast detaching component. The time-dependent term was complex, scaling with

1/(1+�/), and also including the parameter �, which is simply the sum of the non- detaching components. In contrast, the empirical detachment rate scaled with a power law,

1/tβ, and thus we cannot directly compare the magnitude components. Instead, we evaluated the hazard functions at specific time points, the time to reach multivalent bonding, tref, and an arbitrary later time, tmax. At tref, the time-dependent element of the

0 empirical model vanished, allowing kD to be expressed explicitly in terms of BP model parameters �̃ and � (Equation 5.11). At tmax, we solved explicitly for β and substituted for

0 0 kD using Equation 5.11, resulting in Equation 5.12. Values for kD and β determined using the empirical and the BP models are listed in Table 5.2, along with the BP parameters �̃

0 and �. The respective kD values were remarkably similar, matching precisely for many

0 -1 cases and varying by at most 10%. Values for kD ranged from 0.03 s for the base case up

-1 0 0 -3 -1 to 0.18 s for the clustered dimer. We note that kD was relatively low for kr = 10 s even though it had the highest level of detachment, which was likely due to the strong influence of �̃. Before we could evaluate β, we still needed to select a specific value for tmax. First, we observed that the BP model results matched well for most of the 9 parameter cases at tmax =

5.5 s. Second, we optimized tmax to match β, which resulted in values ranging from 4 s for

0 the clustered dimer to 12 s for the highest kr . Interestingly, the optimized tmax values were

0 -3 -1 all approximately 3-fold greater than �̃. Values for β ranged from 0.45 for kr = 10 s to

0.81 for the clustered dimer, and we observed a strong correlation with 1/�̃.

85 Table 5.2 Correlation between empirical and BP model parameters.

Empirical BP Model

Case 0 0 kD kD tmax -1 β � �̃ (s) -1 β (s ) (s ) (s)

1. Base 0.03 0.73 0.94 1.8 0.03 0.72 5.5

0 2. Low Antibody 0.04 0.55 2.1 0.05 0.54 6.5 .87

3. Low ICAM-1 0.12 0.67 0.77 2.7 0.10 0.66 6.5

4. ICAM-1 dimer 0.15 0.73 0.77 1.7 0.13 0.75 5.5

5. Clustered dimer 0.19 0.82 0.74 1.2 0.21 0.81 4.0

1 6. � = 0.29 nm 0.05 0.69 0.88 0.06 0.66 5.5 .9

1 7. � = 0.3 nm 0.14 0.65 0.73 0.14 0.64 5.5 .9

0 -4 -1 8. kr = 5x10 s 0.09 0.67 0.82 1.9 0.09 0.68 5.5

0 -3 -1 9. kr = 1x10 s 0.08 0.45 0.65 4.8 0.07 0.44 12.0

5.5. Bond State Model to Determine Transition Rates

Detachment modeling can only provide a metric of nanoparticle adhesion stability for the BPs and sub-components that contribute to detachment. To obtain more information about the full population, we sought to determine the rates of bond formation and rupture for all state transitions. This required that we further distinguish nanoparticles within each BP (i) based on instantaneous bond number (j), and we designated bond number state as Si,j. We then defined a system of ordinary differential equations to describe all bond number states and transition rates, including bond formation (kf,i,j) and rupture

(kr,i,j), as given by Equation 5.14. The system of ODEs was used to fit NAD simulation data to

-1 -1 determine kf,i,j and kr,i,j . We used exact values for kr,1,1 (3.8 s ) and kdiff (66 s ) that were

86 determined from single bond NAD simulations, and initial estimates for all other bond transition rates determined from pause times observed during full NAD simulations (Table

5.7). Bond number state fitting results for the base case (medium antibody and ICAM-1

0 -4 -1 density, g = 0.274 nm, kr = 1.1x10 s ) are presented in Figure 5.7, and in general did not capture key features of the simulation data. We therefore assumed the presence of bonding heterogeneity within each BP, and modified the ODE master equation to include two

(1) (2) separate, but interacting components (S i,j and S i,j), as indicated by Equation 5.15. Bond number state fitting results for two components are presented for the base case in Figure

5.8A, and matched the NAD simulation data well for all BPs. State transitions rates (kf,i,j and kr,i,j) and the relative number of nanoparticles in component 1 for formation and rupture

(Pf,i,j and Pr,i,j) are listed in the Table 5.8. We then used the results in Table 5.3 to reconstruct nanoparticle detachment curves for each BP (Figure 5.8B) and the entire population (Figure 5.8C), which closely matched the NAD simulation data. We performed the same exercise for the other eight parameter conditions, which provided similar results

( see Figure 5.9 and Table 5.8).

Figure 5.8 Bond transition rate modeling for the base case. 0 -4 -1 (medium antibody and ICAM-1 density, g = 0.274 nm, kr = 1.1x10 s ) using two components. (A-C) Simulation profiles and fits for nanoparticles within each bond number state for (A) BP 1, (B) BP 2, and (C) BP 3. The detachment profile is also overlaid in black for each BP category. (D,E) Detachment curves from the simulation and fits for (D) each BP category shown relative to the initial distribution and (E) the full population.

Table 5.3 Mean first passage time calculations

(M) -1 k D,i (s )

Case BP 1 BP 2 BP 3 BP 4 BP 5 BP 6

1. Base 0.70 0.06 4.6x10-6

2. Low Antibody 3.30 0.01 2.4x10-3

3. Low ICAM-1 0.62 0.01 2.0x10-5

4. ICAM-1 dimer 0.92 8.5x10-3 2.0x10-3

5. Clustered dimer 0.96 7.2x10-3

6. � = 0.29 nm 3.57 0.02 3.8x10-7

7. � = 0.3 nm 5.25 0.05 7.3x10-3 2.7x10-3 7.5x10-8

0 -4 -1 -3 8. kr = 5x10 s 5.12 0.03 0.01 1.6x10

0 -3 -1 -3 9. kr = 1x10 s 22.75 0.04 0.01 2.1x10

Figure 5.9 Bond transition rate modeling for the cases 2-5. Base case parameters were modified to (A) low antibody density, (B) low ICAM-1 density, (C) ICAM-1 dimers, and (D) clustered ICAM-1 dimers. (i-iv) Simulation profiles and fits for nanoparticles within each bond number state for (i) BP 1, (i) BP 2, and (i) BP 3. The detachment profile is also overlaid in black for each BP category. (iv-v) Detachment curves from the simulation and fits for (iv) each BP category shown relative to the initial distribution and (v) the full population.

5.6. Estimating Nanoparticle Detachment Rate from Transition Rates

Next we used the bond transition, or microscopic, rates (kr,i,j, kf,i,j, kdiff) to estimate a nanoparticle-scale detachment, or macroscopic, rate (kD,i) based on the mean first passage time. The time expected for detachment from each possible bond state, or first passage time (Ti,j), was calculated for each BP (i) and bond state number (j) from the transition matrix (K), as described by Equation 5.18, using the transition rates listed in Table 5.8. It should be noted that since we chose to allow nanoparticles to pass between the components in our model, we could only determine a single Ti,j that corresponded to the

89 effects of both components. Interestingly, we observed that first passage times were similar within each BP, regardless of bond number state and component. We therefore averaged

(M) (M) Ti,j over state j and to obtain k D,i , as described in Equation 5.19. k D,i values are listed in

Table 5.3 for each BP of all 9 case studies. Comparing to the respective kD,i values

(M) determined from BP model fitting listed in Table 5.1, k D,i values were consistently lower because they included contributions from both components. Conversely, kD,i only included the fast detaching component, as the second component was assumed to not detach.

Figure 5.10 Bond transition rate modeling for the base case using one component. (A-C) Simulation profiles and fits for nanoparticles within each bond number state for (A) BP 1, (B) BP 2, and (C) BP 3. The detachment profile is also overlaid in black for each BP category. (D,E) Detachment curves from the simulation and fits for (D) each BP category shown relative to the initial distribution and (E) the full population.

5.7. Obtaining Detachment Rates For All Sub-Populations

As a final exercise, we sought to obtain detachment rate constants for all BP categories and sub-components by using the BP model and k(M)D,i. Therefore, we again fit nanoparticle detachment data using Equation 5.4, but now using as an additional constraint k(M)D,i , as shown in Equation 5.20. Full population fits are shown in Figure 5.12, and in

90 general match the simulation data in a similar manner to the initial BP model. The only exception was the base case, for which more nanoparticles detached than was observed in the NAD simulation. Inspection of the profile in Figure 5.12 indicated that detachment abruptly stopped after 5 seconds, which was likely due to a sampling issue since this condition had very low detachment numbers. Final detachment rates and weights for both components are listed in Table 5.4.

Figure 5.11 Bond transition rate modeling for the cases 2-5. 0 -4 -1 Base case parameters were modified to (A) g = 0.29 nm, (B) g = 0.3 nm, (C) kr = 5x10 s , 0 -3 -1 and (D) kr = 10 s . (i-iv) Simulation profiles and fits for nanoparticles within each bond number state for (i) BP 1, (i) BP 2, (i) BP 3, (iv) BP 4, (v) BP 5, and (vi) BP 6. The detachment profile is also overlaid in black for each BP category. (vii-viii) Detachment curves from the simulation and fits for (vii) each BP category shown relative to the initial distribution and (viii) the full population.

Figure 5.12 Final BP model fitting results. Results are shown for all 9 cases including (A) base case, (B) low antibody density, (C) low ICAM-1 density, (D) ICAM-1 dimers, (E) clustered ICAM-1 dimers, (F) g = 0.29 nm, (G) g = 0 -4 -1 0 -3 -1 0.3 nm, (H) kr = 5x10 s , and (I) kr = 10 s . Utilizing the mean first passage time criterion improves long-term predictions of the BP model relative to the initial results obtained assuming no detachment of the second component. Empirical fits are also included for comparison.

94 Table 5.4 Final fitting parameters using detachment fitting and mean first passage time criteria

, a )

, 1 - ) 1 s - (

s i (

D, i D, (1) k (2) k

1 2 3 4 5 6

BP BP BP BP BP BP Case

4 6 - -

x10 x10

Base

0.70 0.70 0.07 0.06 0.06 0.08 1.00 1. 1.5 4.6

5 4 4 -

- -

x10 x10 x10 Low

0.62 0.28 0.01 0.12 0.05 0.10 0 2. 4.0 Antibody 3.1 2. -

3 3 3 - - -

ICAM

x10 x10 x10 1

0.63 3.30 0.97 0.53 0.01 0.02 2 4 4 Low 2. 2. 2.

1 3 3 3 - - - -

x10 x10 x10

1.00 0.93 0.93 0.09 0.46 0.05 ICAM 1 dimer

8.5 2.3 2. 4.

3 -

x10

0.02 0.25 0.96 0.05 2.40 2 dimer Clustered

7. 5.

3 3 3 - - -

0.29

x10 x10 x10 =

0.04 0.04 1.00 0.02 0.01 0.02 0.01 2.20 1 1 � 22.80

2. 2. 2.8 6.

3 3 nm - -

x10 x10 0.3

5.12 0.03 0.58 0.63 0.10 0.23 = 0.20 0.04 1.00 37.40

�

5.0 1.6 7.

4 -

3 3 3 3 7 3 4 8 4 ------

x10

5 1

- x10 x10 x10 x10 x10 x10 x10 x10 x10 = s

7.10 1.00 0.05 0.05 0.02 5.25 0 3 7 3 5 3 3 4 r k 2.7 2. 7. 1. 1.5 7. 1. 7. 5.

3 -

7 4 4 5 - - - - 15

x10

1 1 10

- x10 x10 x10 x10

= s

0.26 0.02 0.14 3.57 1.00 0.53 0.02 0.64 0.06 1.00 0.02 0.61 0 6 0 18.22 1x r . k 3.8 1. 7.3 6 2.

Table 5.5 BP model parameters for single component fits.

-1 -1 -1 -1 -1 Case kD,1 (s ) kD,2 (s ) kD,3 (s ) kD,4 (s ) kD,5 (s )

1. Base 0.57 0.02 1.5x10-4

2. Low Antibody 3.17 0.02 3.1x10-3

3. Low ICAM-1 0.57 1.2x10-2

4. ICAM-1 dimer 0.81 0.01 2.3x10-3

5. Clustered dimer 0.82 8.0x10-3

6. � = 0.29 nm 2.18 0.03 1.7x10-3 3.3x10-3 2.0x10-3

7. � = 0.3 nm 8.48 0.06 7.2x10-3 2.2x10-3

0 -4 -1 -3 -3 -4 8. kr = 5x10 s 7.10 0.03 5.5x10 3.1x10 7.3x10

0 -3 -1 -2 9. kr = 1x10 s 18.22 0.05 1.3x10

96 Table 5.6 BP model parameters for two component fits using 2 and 3 degrees of freedom. Two degrees of freedom assumes that one component does not detach while three degrees of freedom assumes that both can detach.

(1) -1 (2) -1 Degrees k D,i, (s ), k D,i, (s ), αi Case of freedom BP 1 BP 2 BP 3

2 0.57 1.00 0.23 0 0.30 1.5x10-4 0 1.00 1. Base 3 0.57 1.00 0.23 0 0.30 1.5x10-4 1.5x10-4 0.00 2 3.17 1.00 0.12 0 0.32 5.2x10-2 0 0.10 2. Low ICAM-1 3 3.17 1.00 0.12 0 0.32 5.2x10-2 0 0.10 2 0.57 1.00 0.03 0 0.53 3. Low Antibody 3 0.57 1.00 0.44 1.0x10-2 0.03

2 0.81 1.00 0.12 0 0.20 2.3x10-3 0 1.00 4. ICAM-1 Dimer 3 0.81 1.00 0.15 1.4x10-3 0.17 2.3x10-3 2.3x10-3 0.00 2 0.82 1.00 0.07 0 0.20 5. Clustered Dimer 3 0.82 1.00 0.10 1.9x10-3 0.14

97 Table 5.7 Pause time fitting results used to estimate bond number transition rates. Case 1: Base

Bond Number State Parameter BP 0 1 2 3 4 5 6

1 66.0 3.0 0.0 0.0 0.0 0.0 0.0 2 66.0 3.8 0.1 9.3 6.8 255.1 0.0 3 66.0 0.8 11.4 13.7 24.5 48.5 0.0 (1) -1 k r,i,j (s ) 4 66.0 3.5 15.1 21.5 24.9 46.4 74.8 5 66.0 3.8 12.0 18.2 36.4 54.7 78.6 6 66.0 0.0 0.0 1101.9 49.6 60.8 62.3 1 66.0 3.1 0.0 0.0 0.0 0.0 0.0 2 66.0 3.8 10.5 25.5 76.7 267.3 0.0 3 66.0 290.8 357.6 27.6 57.0 159.0 0.0 (2) -1 k r,i,j (s ) 4 66.0 3.8 597.6 748.3 58.3 132.0 192.6 5 66.0 3.8 12.1 655.6 846.5 206.4 322.8 6 66.0 0.0 0.0 1101.9 1800.2 254.2 223.9 1 1.0 0.5 1.0 1.0 1.0 1.0 1.0 2 1.0 1.0 0.1 1.0 0.6 0.0 1.0 3 1.0 0.1 0.9 0.4 0.6 0.6 1.0 Pr,i,j 4 1.0 0.9 0.8 0.9 0.5 0.6 0.7 5 1.0 1.0 0.5 0.9 0.9 0.7 0.6 6 1.0 1.0 1.0 1.0 0.8 0.6 0.5 1 0.0 6650.9 0.0 0.0 0.0 0.0 0.0 2 0.0 53.0 2197.1 14.3 5.7 15474.6 1.0 3 0.0 94.9 251.0 160.0 98.5 408.3 0.0 (1) -1 k f,i,j (s ) 4 0.0 692.5 125.1 136.2 192.7 187.9 211.7 5 0.0 52.5 413.7 37.8 232.7 488.1 731.0 6 0.0 0.0 547.8 1305.0 520.5 516.7 952.0 1 0.0 204551.0 0.0 0.0 0.0 0.0 0.0 2 0.0 20969.5 6759.6 7403.3 6450.7 8305.5 0.0 3 0.0 9505.3 7517.9 7284.2 7381.6 8695.6 0.0 (2) -1 k f,i,j (s ) 4 0.0 8739.7 5990.6 7053.9 7568.0 7981.7 8273.3 5 0.0 52.5 6354.8 5691.6 7099.7 8729.7 12429.9 6 0.0 0.0 547.8 174609.0 11749.6 11841.9 13491.8 1 1.0 0.7 1.0 1.0 1.0 1.0 1.0 2 1.0 0.9 0.1 0.9 0.8 0.8 1.0 3 1.0 0.1 0.2 0.2 0.1 0.1 1.0 Pf,i,j 4 1.0 0.6 0.2 0.1 0.1 0.1 0.1 5 1.0 1.0 0.2 0.1 0.1 0.1 0.1 6 1.0 1.0 1.0 0.9 0.1 0.1 0.1

98 Case 2: Low antibody

Bond Number State Parameter BP 0 1 2 3 4 5 6 7 8 1 66.0 8.3 10.3 0.0 0.0 0.0 0.0 0.0 0.0 2 66.0 65.6 8.1 14.5 30.4 49.8 0.0 0.0 0.0 3 66.0 34.5 11.2 16.5 23.7 57.4 86.8 0.0 0.0 (1) -1 k r,i,j (s ) 4 66.0 5.6 11.6 26.5 32.2 52.9 141.9 0.0 0.0 5 66.0 392.3 7.9 16.3 39.5 57.3 149.9 0.0 0.0 6 66.0 0.0 0.0 18.2 50.4 76.7 147.4 1942.1 30376.8 7 66.0 233.7 9.6 126.6 312.5 84.6 114.3 240.1 0.0 1 66.0 8.3 10.3 0.0 0.0 0.0 0.0 0.0 0.0 2 66.0 5913.3 13.8 28.4 85.2 140.8 0.0 0.0 0.0 3 66.0 932.1 194.1 33.5 54.3 178.1 87.9 0.0 0.0 (2) -1 k r,i,j (s ) 4 66.0 1522.8 2042.7 569.4 104.9 207.2 892.8 0.0 0.0 5 66.0 392.3 70.2 353.8 422.5 372.1 2341.6 0.0 0.0 6 66.0 0.0 0.0 3324.9 764.1 591.3 2863.4 33169.8 520853.0 7 66.0 233.7 74.1 126.6 318.0 1384.5 368.7 926.2 0.0 1 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2 1.0 0.8 0.3 0.7 0.6 0.6 1.0 1.0 1.0 3 1.0 0.5 0.8 0.5 0.5 0.6 0.5 1.0 1.0

Pr,i,j 4 1.0 0.9 1.0 0.8 0.7 0.6 0.4 1.0 1.0 5 1.0 1.0 0.8 0.8 0.8 0.7 0.5 1.0 1.0 6 1.0 1.0 1.0 0.8 0.6 0.8 0.7 0.3 1.0 7 1.0 1.0 0.9 1.0 0.2 0.6 0.4 0.5 1.0 1 0.0 775.1 31.1 0.0 0.0 0.0 0.0 0.0 0.0 2 0.0 762.2 180.6 51.8 146.5 289.6 0.0 0.0 0.0 3 0.0 654.2 153.4 159.7 133.2 134.2 5525.1 0.0 0.0 (1) -1 k f,i,j (s ) 4 0.0 166.5 205.7 155.2 265.0 307.7 206.1 0.0 0.0 5 0.0 367.5 111.4 118.9 282.4 517.6 924.3 0.0 0.0 6 0.0 0.0 1204.5 126.0 879.6 571.2 715.9 1000.3 12274.8 7 0.0 2374.3 244.9 1449.4 1988.0 599.8 1254.8 2723.0 0.0 1 0.0 775.1 31.1 0.0 0.0 0.0 0.0 0.0 0.0 2 0.0 21341.0 3738.8 3554.2 3018.6 4861.6 0.0 0.0 0.0 3 0.0 7992.4 4010.7 4188.4 4173.6 4733.2 49814.5 0.0 0.0 (2) -1 k f,i,j (s ) 4 0.0 5127.9 2895.0 4422.7 4733.0 5024.6 3678.6 0.0 0.0 5 0.0 367.5 2276.5 5002.3 5108.7 6342.1 14928.4 0.0 0.0 6 0.0 0.0 1204.5 3361.2 9055.8 9024.4 13113.0 34148.3 47994.9 7 0.0 2374.3 6152.8 1449.4 22320.5 7266.9 15198.6 34249.5 0.0 1 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2 1.0 0.5 0.2 0.3 0.2 0.2 1.0 1.0 1.0 3 1.0 0.7 0.3 0.2 0.2 0.2 0.6 1.0 1.0

Pf,i,j 4 1.0 0.3 0.3 0.2 0.2 0.2 0.0 1.0 1.0 5 1.0 1.0 0.3 0.2 0.2 0.2 0.5 1.0 1.0 6 1.0 1.0 1.0 0.2 0.4 0.3 0.2 0.1 0.7 7 1.0 1.0 0.5 1.0 0.4 0.2 0.4 0.1 1.0

99 Case 3: Low ICAM-1

Bond Number State Parameter BP 0 1 2 3 4 5 6

1 66.0 3.2 3.7 0.0 0.0 0.0 0.0 2 66.0 17.7 7.9 11.8 14.9 0.0 0.0 3 66.0 2.8 13.7 14.4 22.3 28.3 0.0 (1) -1 k r,i,j (s ) 4 66.0 4.3 419.4 23.0 25.3 43.5 0.0 5 66.0 9.3 0.0 0.0 61.8 61.8 38.8 6 66.0 0.0 4.8 0.0 75.1 63.9 53.9 1 66.0 25.3 9.9 0.0 0.0 0.0 0.0 2 66.0 486.2 13.2 20.4 26.9 0.0 0.0 3 66.0 92.6 426.9 29.1 52.9 28.4 0.0 (2) -1 k r,i,j (s ) 4 66.0 4.3 3797.8 216.0 62.1 112.4 0.0 5 66.0 9.3 0.0 0.0 1857.5 234.4 112.2 6 66.0 0.0 4.8 0.0 169.6 2378.1 179.5 1 1.0 0.8 0.1 1.0 1.0 1.0 1.0 2 1.0 0.5 0.4 0.4 1.0 1.0 1.0 3 1.0 0.4 0.9 0.4 0.5 0.5 1.0 Pr,i,j 4 1.0 1.0 0.6 0.7 0.5 0.6 1.0 5 1.0 1.0 1.0 1.0 0.6 0.6 0.0 6 1.0 1.0 1.0 1.0 0.7 0.7 0.6 1 0.0 291.2 31.5 0.0 0.0 0.0 0.0 2 0.0 261.7 164.3 56.2 119.8 0.0 0.0 3 (1) -1 0.0 842.5 81.7 111.0 100.4 58.7 0.0 k f,i,j (s ) 4 0.0 675.7 48.0 272.6 230.9 180.6 0.0 5 0.0 254.8 4296.3 2128.4 903.4 757.0 232.1 6 0.0 0.0 271.2 81.0 2389.0 1819.0 1258.8 1 0.0 6458.8 4297.7 0.0 0.0 0.0 0.0 2 0.0 9863.9 6872.8 7417.2 6617.6 0.0 0.0 3 0.0 65455.0 5949.0 6964.9 7924.2 9001.7 0.0 (2) -1 k f,i,j (s ) 4 0.0 675.7 3009.2 7611.6 8048.2 8200.3 0.0 5 0.0 254.8 4296.3 2128.4 12195.8 10675.6 14430.4 6 0.0 0.0 698919.0 81.0 287045.0 13450.5 14458.1 1 1.0 0.2 0.2 1.0 1.0 1.0 1.0 2 1.0 0.4 0.2 0.3 0.3 1.0 1.0 3 1.0 0.6 0.1 0.1 0.2 0.3 1.0 Pf,i,j 4 1.0 1.0 0.2 0.1 0.1 0.1 1.0 5 1.0 1.0 1.0 1.0 0.2 0.2 0.1 6 1.0 1.0 0.8 1.0 0.9 0.3 0.3

100 Case 4: ICAM-1 dimer

Bond Number State Parameter BP 0 1 2 3 4 5 6

1 66.0 4.2 6.6 0.0 0.0 0.0 66.0 2 66.0 7.4 6.5 9.2 30.9 0.0 66.0 (1) -1 k r,i,j (s ) 3 66.0 7.4 10.9 13.3 19.9 0.0 66.0 4 66.0 5.7 3.8 20.1 28.5 41.2 66.0 5 66.0 0.0 11.9 4561.4 79.5 70.2 66.0 1 66.0 13.5 6.7 0.0 0.0 0.0 66.0 2 66.0 74.1 11.7 17.9 135.8 0.0 66.0 (2) -1 k r,i,j (s ) 3 66.0 198.6 420.0 25.7 49.7 0.0 66.0 4 66.0 161.3 14.2 153.3 74.1 105.5 66.0 5 66.0 0.0 11.9 4561.4 1822.9 195.9 66.0 1 1.0 0.9 0.5 1.0 1.0 1.0 1.0 2 1.0 0.4 0.2 0.2 0.8 1.0 1.0

Pr,i,j 3 1.0 0.7 0.9 0.4 0.5 1.0 1.0 4 1.0 0.8 0.1 0.8 0.6 0.6 1.0 5 1.0 1.0 1.0 1.0 0.3 0.6 1.0 1 0.0 276.1 31.6 0.0 0.0 0.0 0.0 2 0.0 360.1 123.0 52.5 369.4 0.0 0.0 (1) -1 k f,i,j (s ) 3 0.0 185.5 136.1 97.9 124.3 0.0 0.0 4 0.0 277.5 69.2 198.2 231.1 245.6 0.0 5 0.0 0.0 147.3 11536.0 2084.6 924.0 0.0 1 0.0 8798.5 10702.3 0.0 0.0 0.0 0.0 2 0.0 26690.7 10475.3 14689.1 14522.3 0.0 0.0 (2) -1 k f,i,j (s ) 3 0.0 7503.1 9902.6 11909.5 15083.1 0.0 0.0 4 0.0 3568.4 9088.9 13286.9 13753.9 15140.6 0.0 5 0.0 0.0 5747130.0 11536.0 18997.5 13763.9 0.0 1 1.0 0.2 0.5 1.0 1.0 1.0 1.0 2 1.0 0.3 0.2 0.3 0.1 1.0 1.0

Pf,i,j 3 1.0 0.2 0.2 0.1 0.1 1.0 1.0 4 1.0 0.4 0.2 0.1 0.1 0.1 1.0 5 1.0 1.0 0.9 1.0 0.2 0.1 1.0

101 Case 5: Clustered ICAM-1 dimer

Bond Number State Parameter BP 0 1 2 3 4 5 6

1 66.0 3.8 0.1 0.0 0.0 0.0 0.0 2 66.0 3.8 2.1 0.3 41.4 0.0 0.0 3 66.0 3.6 10.6 13.8 22.3 44.1 0.0 (1) -1 k r,i,j (s ) 4 66.0 0.0 10.6 23.5 30.4 50.0 0.0 5 66.0 2.3 8.4 63.5 50.1 58.9 75.5 6 66.0 0.0 0.0 0.0 0.0 161.4 92.2 1 66.0 1.0 0.0 0.0 0.0 0.0 0.0 2 66.0 2.7 1.5 0.2 29.7 0.0 0.0 3 66.0 150.4 191.7 26.9 69.8 149.2 0.0 (2) -1 k r,i,j (s ) 4 66.0 0.0 583.8 247.0 86.9 194.5 0.0 100000000 5 66.0 2.3 8.4 0.0 3067.5 260.0 221.5 6 66.0 0.0 0.0 0.0 0.0 2408.6 853.3 1 1.0 1.0 0.7 1.0 1.0 1.0 1.0 2 1.0 1.0 0.9 0.5 0.4 1.0 1.0 3 1.0 0.9 0.7 0.3 0.5 0.8 1.0 Pr,i,j 4 1.0 1.0 0.9 0.8 0.6 0.5 1.0 5 1.0 1.0 1.0 0.9 0.9 0.6 0.5 6 1.0 1.0 1.0 1.0 1.0 0.2 0.7 1 0.0 94.1 0.2 0.0 0.0 0.0 0.0 2 0.0 264.9 781.2 0.1 189.7 0.0 0.0 3 0.0 498.2 91.2 189.0 107.5 507.1 42.5 (1) -1 k f,i,j (s ) 4 0.0 0.0 88.4 93.2 286.1 321.4 0.0 5 0.0 181.7 693.1 1844.2 780.5 1129.8 920.2 6 0.0 0.0 6226.3 39377.8 506.4 3677.7 2828.4 1 0.0 18.8 0.0 0.0 0.0 0.0 0.0 2 0.0 4.8 14.3 0.0 3.5 0.0 0.0 3 0.0 142679.0 10954.7 12479.4 18129.1 17681.0 42.5 (2) -1 k f,i,j (s ) 4 0.0 0.0 6585.1 12203.1 13536.3 16470.3 0.0 5 0.0 181.7 132742.0 19392.0 13687.8 18459.4 23556.6 6 0.0 0.0 6226.3 39377.8 506.4 39605.0 28111.1 1 1.0 1.0 0.7 1.0 1.0 1.0 1.0 2 1.0 1.0 0.9 0.5 0.4 1.0 1.0 3 1.0 0.8 0.2 0.1 0.1 0.1 1.0 Pf,i,j 4 1.0 1.0 0.2 0.1 0.1 0.1 1.0 5 1.0 1.0 0.8 0.4 0.1 0.2 0.1 6 1.0 1.0 1.0 1.0 1.0 0.3 0.3

102 Case 6: � = 0.29 nm

Bond Number State Parameter BP 0 1 2 3 4 5 6 7 8 1 66.0 9.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2 66.0 26.1 33.1 39.5 106.7 110.9 0.0 0.0 0.0 3 66.0 137.7 39.3 53.3 74.2 152.6 308.5 0.0 0.0 (1) -1 k r,i,j (s ) 4 66.0 12.7 34.9 78.8 100.7 152.3 209.2 0.0 0.0 5 66.0 0.0 55.1 51.1 116.8 183.7 311.0 188.4 0.0 6 66.0 79.9 73.6 51.1 134.8 233.6 324.7 350.6 0.0 7 66.0 0.0 0.0 0.0 250.3 237.9 296.0 287.1 0.0 1 66.0 9.9 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2 66.0 533.8 86.6 80.4 3342.3 258.9 0.0 0.0 0.0 3 66.0 6549.5 519.2 129.6 184.4 630.3 1530.8 0.0 0.0 (2) -1 k r,i,j (s ) 4 66.0 106.9 887.8 2306.2 298.0 474.4 383.3 0.0 0.0 5 66.0 0.0 1170.1 1758.2 2794.2 1223.6 1495.8 574.8 0.0 6 66.0 79.9 3456.1 1189.4 3132.8 1470.6 2000.3 2657.8 0.0 7 66.0 0.0 0.0 0.0 3959.8 4409.6 1508.4 2736.4 0.0 1 1.0 0.5 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2 1.0 0.3 0.8 0.5 0.8 0.2 1.0 1.0 1.0 3 1.0 0.7 0.6 0.6 0.5 0.6 0.6 1.0 1.0

Pr,i,j 4 1.0 0.4 0.6 0.9 0.6 0.6 0.9 1.0 1.0 5 1.0 1.0 0.8 0.9 0.8 0.7 0.6 0.2 1.0 6 1.0 1.0 0.8 0.8 0.8 0.7 0.6 0.6 1.0 7 1.0 1.0 1.0 1.0 0.4 0.6 0.6 0.6 1.0 1 0.0 143.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2 0.0 575.8 348.3 174.8 532.5 942.2 0.0 0.0 0.0 3 0.0 292.2 332.1 305.8 251.0 445.8 2276.2 0.0 0.0 (1) -1 k f,i,j (s ) 4 0.0 351.9 238.3 429.7 543.1 659.6 771.4 0.0 0.0 5 0.0 0.0 704.3 215.0 529.3 1162.4 2017.2 601.7 0.0 6 0.0 9364.1 262.1 178.6 617.4 1693.9 2330.9 3765.0 0.0 7 0.0 0.0 4832.3 707.7 1285.9 1047.8 2322.0 3244.1 633.6 1 0.0 13928.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2 0.0 14002.5 7143.3 7592.8 8401.6 10974.8 0.0 0.0 0.0 3 0.0 5768.5 7839.6 8263.1 8596.7 10094.4 20055.4 0.0 0.0 (2) -1 k f,i,j (s ) 4 0.0 49334.6 6516.2 9063.1 9114.6 9869.6 13800.0 0.0 0.0 5 0.0 0.0 7764.5 7361.5 9335.9 12449.1 18350.1 34439.1 0.0 6 0.0 9364.1 14980.5 7169.1 9279.0 14652.4 20669.2 32670.2 0.0 7 0.0 0.0 4832.3 707.7 9642.1 12840.1 19803.0 33471.0 633.6 1 1.0 0.1 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2 1.0 0.5 0.2 0.2 0.1 0.1 1.0 1.0 1.0 3 1.0 0.2 0.2 0.1 0.1 0.1 0.3 1.0 1.0

Pf,i,j 4 1.0 0.7 0.2 0.1 0.1 0.1 0.2 1.0 1.0 5 1.0 1.0 0.3 0.2 0.1 0.2 0.3 0.1 1.0 6 1.0 1.0 0.5 0.2 0.1 0.2 0.3 0.2 1.0 7 1.0 1.0 1.0 1.0 0.2 0.2 0.3 0.1 1.0

103 Case 7: � = 0.3 nm

Bond Number State Parameter BP 0 1 2 3 4 5 6 7 8 1 66.0 29.1 31.8 0.0 0.0 0.0 0.0 0.0 0.0 2 66.0 94.9 80.3 86.3 153.4 0.0 0.0 0.0 0.0 3 66.0 57.2 140.3 125.1 174.8 347.5 579.4 0.0 0.0 (1) -1 k r,i,j (s ) 4 66.0 50.1 106.2 137.7 239.2 343.7 516.3 667.8 0.0 5 66.0 83.6 64.6 165.7 256.9 397.3 652.4 740.4 0.0 6 66.0 0.0 4633.8 152.1 392.1 472.5 645.9 955.4 0.0 7 66.0 0.0 0.0 126.1 309.3 476.4 600.6 744.4 0.0 1 66.0 29.2 31.9 0.0 0.0 0.0 0.0 0.0 0.0 2 66.0 1753.3 904.2 165.0 315.9 0.0 0.0 0.0 0.0 3 66.0 847.6 2210.4 317.6 397.4 1437.3 2176.4 0.0 0.0 (2) -1 k r,i,j (s ) 4 66.0 6261.4 1537.8 3032.6 1620.4 1267.1 2212.5 862813.0 0.0 5 66.0 5022.2 430.5 3393.0 2677.1 3261.9 3088.2 4855.3 0.0 6 66.0 0.0 4633.8 8009.1 4398.1 2229.4 4249.5 7355.8 0.0 7 66.0 0.0 0.0 590.3 9143.1 5607.2 3828.0 5847.9 0.0 1 1.0 0.7 0.5 1.0 1.0 1.0 1.0 1.0 1.0 2 1.0 0.5 0.9 0.4 0.5 1.0 1.0 1.0 1.0 3 1.0 0.3 0.5 0.6 0.5 0.7 0.6 1.0 1.0

Pr,i,j 4 1.0 0.5 0.8 0.8 0.7 0.6 0.6 1.0 1.0 5 1.0 0.6 0.8 0.8 0.8 0.7 0.6 0.7 1.0 6 1.0 1.0 1.0 0.6 0.4 0.7 0.7 0.6 1.0 7 1.0 1.0 1.0 0.1 0.9 0.4 0.7 0.6 1.0 1 0.0 146.1 26.8 0.0 0.0 0.0 0.0 0.0 0.0 2 0.0 704.5 340.4 281.1 405.8 0.0 0.0 0.0 0.0 3 0.0 531.8 635.0 648.3 545.9 829.3 2614.3 0.0 0.0 (1) -1 k f,i,j (s ) 4 0.0 820.9 889.6 399.2 828.7 1424.0 2176.6 592.9 0.0 5 0.0 1467.1 367.1 467.4 974.1 996.5 2860.9 1682.3 0.0 6 0.0 0.0 307.2 1707.9 2127.3 2187.9 2536.5 5348.2 0.0 7 0.0 0.0 5526.6 796.5 1860.5 2389.7 3331.9 4454.0 170823.0 1 0.0 14031.5 141.3 0.0 0.0 0.0 0.0 0.0 0.0 2 0.0 11388.3 7277.5 8155.0 8876.6 0.0 0.0 0.0 0.0 3 0.0 36017.2 10269.6 8947.4 9308.5 11994.2 19260.1 0.0 0.0 (2) -1 k f,i,j (s ) 4 0.0 8936.1 9404.1 9126.5 10590.3 12740.6 17689.1 10482.4 0.0 5 0.0 4662.6 8237.3 9625.2 10907.7 12435.4 22771.2 29979.5 0.0 6 0.0 0.0 6133.0 23138.7 15906.6 16063.6 20100.7 35698.6 0.0 7 0.0 0.0 5526.6 8464.8 18210.3 18931.6 23042.0 33404.5 170823.0 1 1.0 0.4 0.3 1.0 1.0 1.0 1.0 1.0 1.0 2 1.0 0.4 0.2 0.2 0.1 1.0 1.0 1.0 1.0 3 1.0 0.6 0.2 0.2 0.2 0.1 0.2 1.0 1.0

Pf,i,j 4 1.0 0.4 0.2 0.2 0.1 0.2 0.3 0.0 1.0 5 1.0 0.4 0.2 0.1 0.2 0.1 0.3 0.1 1.0 6 1.0 1.0 0.1 0.3 0.2 0.3 0.2 0.2 1.0 7 1.0 1.0 1.0 0.3 0.4 0.2 0.3 0.2 1.0

104 0 -4 -1 Case 8: kr = 5x10 s

Bond Number State Parameter BP 0 1 2 3 4 5 6

1 66.0 18.7 52.1 0.0 0.0 0.0 0.0 2 66.0 115.7 51.4 66.2 77.5 182.2 0.0 3 66.0 77.9 61.5 75.9 104.8 176.1 277.8 (1) -1 k r,i,j (s ) 4 66.0 38.9 63.5 95.4 131.1 204.8 303.0 5 66.0 0.0 137.5 142.4 158.7 222.8 370.9 6 66.0 69.5 31.2 75.8 165.5 292.9 440.9 1 66.0 36.0 52.1 0.0 0.0 0.0 0.0 2 66.0 4311.7 342.3 130.5 169.4 182.9 0.0 3 66.0 2695.9 2201.3 177.8 222.8 767.9 963.7 (2) -1 k r,i,j (s ) 4 66.0 2518.6 1693.7 2458.0 405.7 645.9 851.1 5 66.0 0.0 2050.9 3134.7 3889.8 1128.0 1700.3 6 66.0 69.5 92.2 388.0 3317.3 3158.2 3264.0 1 1.0 0.6 1.0 1.0 1.0 1.0 1.0 2 1.0 0.7 0.9 0.7 0.5 0.5 1.0 3 1.0 0.5 0.9 0.5 0.5 0.7 0.6 Pr,i,j 4 1.0 0.7 0.7 0.9 0.6 0.6 0.5 5 1.0 1.0 0.2 0.7 0.9 0.7 0.7 6 1.0 1.0 0.1 0.9 0.8 0.8 0.7 1 0.0 17112.2 108.8 0.0 0.0 0.0 0.0 2 0.0 726.4 336.4 222.1 210.4 369.9 0.0 3 (1) -1 0.0 259.5 628.8 375.6 336.5 657.7 2706.5 k f,i,j (s ) 4 0.0 673.0 412.3 458.7 497.3 907.7 925.8 5 0.0 0.0 484.0 771.1 535.4 871.0 2281.6 6 0.0 5415.3 1091.7 317.1 558.0 1713.6 2144.2 1 0.0 17112.2 165317.0 0.0 0.0 0.0 0.0 2 0.0 15060.9 7137.5 7940.5 7321.2 7954.4 0.0 3 0.0 17147.7 8146.4 8462.0 8600.3 8943.4 21847.2 (2) -1 k f,i,j (s ) 4 0.0 11244.1 8030.7 9063.7 9206.1 11542.7 11654.8 5 0.0 0.0 6529.1 9583.1 9965.3 12025.0 19913.9 6 0.0 5415.3 8931.5 9286.3 11281.1 14144.4 18959.9 1 1.0 1.0 0.9 1.0 1.0 1.0 1.0 2 1.0 0.4 0.1 0.2 0.2 0.1 1.0 3 1.0 0.3 0.1 0.1 0.2 0.2 0.3 Pf,i,j 4 1.0 0.4 0.2 0.2 0.1 0.2 0.1 5 1.0 1.0 0.2 0.1 0.1 0.1 0.3 6 1.0 1.0 0.2 0.2 0.1 0.2 0.2

105 0 -3 -1 Case 9: kr = 10 s

Bond Number State Parameter BP 0 1 2 3 4 5 6

1 66.0 136.9 104.5 0.0 0.0 0.0 0.0 2 66.0 133.1 113.6 131.2 222.6 0.0 0.0 3 66.0 61.5 142.4 201.0 232.0 372.2 0.0 (1) -1 k r,i,j (s ) 4 66.0 70.3 157.2 226.2 324.0 457.9 676.5 5 66.0 208.8 164.5 174.2 318.0 492.9 863.7 6 66.0 4249.1 98.0 170.0 481.0 476.5 687.1 1 66.0 136.9 104.5 0.0 0.0 0.0 0.0 2 66.0 2978.2 1665.5 225.8 551.6 0.0 0.0 3 66.0 1092.2 2725.0 2508.7 522.0 865.8 0.0 (2) -1 k r,i,j (s ) 4 66.0 1076.9 3487.9 4021.6 2910.9 1630.1 2684.4 5 66.0 210.1 4864.4 2697.0 1546.2 4036.2 4593.6 6 66.0 4249.1 417.7 3974.9 3923.0 2670.7 3626.4 1 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2 1.0 0.6 0.8 0.4 0.8 1.0 1.0 3 1.0 0.3 0.6 0.8 0.5 0.5 1.0 Pr,i,j 4 1.0 0.3 0.5 0.8 0.8 0.6 0.6 5 1.0 0.5 0.9 0.6 0.8 0.7 0.6 6 1.0 1.0 0.8 0.7 0.4 0.7 0.6 1 0.0 1614.6 58.2 0.0 0.0 0.0 0.0 2 0.0 359.4 489.7 395.0 515.3 0.0 0.0 3 (1) -1 0.0 396.5 551.7 616.1 739.7 717.2 0.0 k f,i,j (s ) 4 0.0 448.5 363.3 667.3 988.2 1714.2 2598.7 5 0.0 689.7 709.1 396.1 1221.7 1058.9 3563.0 6 0.0 8879.7 1401.8 238.1 1165.5 2361.1 2894.6 1 0.0 1614.6 58.2 0.0 0.0 0.0 0.0 2 0.0 9744.5 8310.9 8598.7 10111.7 0.0 0.0 3 0.0 13342.6 9286.4 9394.8 9968.2 10235.2 0.0 (2) -1 k f,i,j (s ) 4 0.0 9621.0 9428.5 9382.5 11100.8 14133.9 20962.2 5 0.0 14124.3 10470.3 9803.5 11140.8 12928.3 27177.9 6 0.0 8879.7 15486.3 10082.2 12669.5 15937.7 21784.7 1 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2 1.0 0.2 0.2 0.2 0.2 1.0 1.0 3 1.0 0.4 0.2 0.2 0.2 0.2 1.0 Pf,i,j 4 1.0 0.2 0.1 0.1 0.2 0.2 0.3 5 1.0 0.2 0.2 0.1 0.2 0.2 0.3 6 1.0 1.0 0.4 0.2 0.1 0.3 0.3

106 Table 5.8 Bond transition rates from ODE model.

Case 1: Base

Bond Number State Parameter BP 0 1 2 3 4 5 6

1 66.0 3.8 0.0 0.0 0.0 0.0 0.0 2 66.0 3.8 0.2 2.9 2.6 255.2 0.0 3 66.0 3.8 3.4 8.2 4.7 167.2 0.0 (1) -1 k r,i,j (s ) 4 66.0 3.5 15.1 21.5 24.9 46.4 74.8 5 66.0 3.8 12.0 18.2 36.4 54.7 78.6 6 66.0 0.0 0.0 1,101.9 49.6 60.8 62.3 1 66.0 1.1 0.0 0.0 0.0 0.0 0.0 2 66.0 22.1 1.0 16.6 15.3 1,483.3 0.0 3 66.0 0.2 0.1 0.3 0.2 7.0 0.0 (2) -1 k r,i,j (s ) 4 66.0 3.8 597.6 748.3 58.3 132.0 192.6 5 66.0 3.8 12.1 655.6 846.5 206.4 322.8 6 66.0 0.0 0.0 1,101.9 1,800.2 254.2 223.9 1 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2 1.0 1.0 0.1 1.0 0.6 0.1 1.0 3 1.0 1.0 0.9 0.4 0.6 0.7 1.0 Pr,i,j 4 1.0 0.9 0.8 0.9 0.5 0.6 0.7 5 1.0 1.0 0.5 0.9 0.9 0.7 0.6 6 1.0 1.0 1.0 1.0 0.8 0.6 0.5 1 0.0 283.2 0.0 0.0 0.0 0.0 0.0 2 0.0 53.5 1,746.1 3.4 4.4 10,239.3 1.0 3 0.0 55.3 6,828.6 358.4 2.8 82.3 0.0 (1) -1 k f,i,j (s ) 4 0.0 692.5 125.1 136.2 192.7 187.9 211.7 5 0.0 52.5 413.7 37.8 232.7 488.1 731.0 6 0.0 0.0 547.8 1,305.0 520.5 516.7 952.0 1 0.0 176.8 0.0 0.0 0.0 0.0 0.0 2 0.0 4.9 160.8 0.3 0.4 943.0 0.0 3 0.0 0.6 75.1 3.9 0.0 0.9 0.0 (2) -1 k f,i,j (s ) 4 0.0 8,739.7 5,990.6 7,053.9 7,568.0 7,981.7 8,273.3 5 0.0 52.5 6,354.8 5,691.6 7,099.7 8,729.7 12,429.9 6 0.0 0.0 547.8 174,609.0 11,749.6 11,841.9 13,491.8 1 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2 1.0 1.0 0.1 1.0 0.6 0.1 1.0 3 1.0 1.0 0.9 0.4 0.6 0.7 1.0 Pf,i,j 4 1.0 0.6 0.2 0.1 0.1 0.1 0.1 5 1.0 1.0 0.2 0.1 0.1 0.1 0.1 6 1.0 1.0 1.0 0.9 0.1 0.1 0.1

107 Case 2: Low antibody

Bond Number State Parameter BP 0 1 2 3 4 5 6 7 8 1 66.0 3.7 27.3 0.0 0.0 0.0 0.0 0.0 0.0 2 66.0 3.8 0.2 0.6 77.1 60.4 0.0 0.0 0.0 3 66.0 3.8 42.3 6.2 1.4 39.6 182.9 0.0 0.0 (1) -1 k r,i,j (s ) 4 66.0 3.6 11.6 26.5 32.2 52.9 141.9 0.0 0.0 5 66.0 3.9 7.9 16.3 39.5 57.3 149.9 0.0 0.0 6 66.0 0.0 0.0 18.2 50.4 76.7 147.4 1,942.1 30,376.8 7 66.0 233.7 9.6 126.6 312.5 84.6 114.3 240.1 0.0 1 66.0 26.7 194.4 0.0 0.0 0.0 0.0 0.0 0.0 2 66.0 7.5 0.3 1.1 152.3 119.3 0.0 0.0 0.0 3 66.0 0.3 3.2 0.5 0.1 3.0 13.7 0.0 0.0 (2) -1 k r,i,j (s ) 4 66.0 1.5 2,042.7 569.4 104.9 207.2 892.8 0.0 0.0 5 66.0 3.9 70.2 353.8 422.5 372.1 2,341.6 0.0 0.0 6 66.0 0.0 0.0 3,324.9 764.1 591.3 2,863.4 33,169.8 520,853.0 7 66.0 233.7 74.1 126.6 318.0 1,384.5 368.7 926.2 0.0 1 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2 1.0 1.0 0.3 1.0 0.6 0.5 1.0 1.0 1.0 3 1.0 1.0 0.8 0.5 0.5 0.6 0.5 1.0 1.0

Pr,i,j 4 1.0 0.9 1.0 0.8 0.7 0.6 0.4 1.0 1.0 5 1.0 1.0 0.8 0.8 0.8 0.7 0.5 1.0 1.0 6 1.0 1.0 1.0 0.8 0.6 0.8 0.7 0.3 1.0 7 1.0 1.0 0.9 1.0 0.2 0.6 0.4 0.5 1.0 1 0.0 23.8 14.2 0.0 0.0 0.0 0.0 0.0 0.0 2 0.0 54.4 1,014.0 1.7 283.3 85.3 0.0 0.0 0.0 3 0.0 0.0 1,212.0 50.1 1.2 0.1 5,880.3 0.0 0.0 (1) -1 k f,i,j (s ) 4 0.0 16,652.2 2,056.9 155.2 265.0 307.7 206.1 0.0 0.0 5 0.0 367.5 111.4 118.9 282.4 517.6 924.3 0.0 0.0 6 0.0 0.0 1,204.5 126.0 879.6 571.2 715.9 1,000.3 12,274.8 7 0.0 2,374.3 244.9 1,449.4 1,988.0 599.8 1,254.8 2,723.0 0.0 1 0.0 1.2 0.7 0.0 0.0 0.0 0.0 0.0 0.0 2 0.0 1.3 23.3 0.0 6.5 2.0 0.0 0.0 0.0 3 0.0 0.0 39.7 1.6 0.0 0.0 192.7 0.0 0.0 (2) -1 k f,i,j (s ) 4 0.0 5,127.9 2,895.0 4,422.7 4,733.0 5,024.6 3,678.6 0.0 0.0 5 0.0 367.5 2,276.5 5,002.3 5,108.7 6,342.1 14,928.4 0.0 0.0 6 0.0 0.0 1,204.5 3,361.2 9,055.8 9,024.4 13,113.0 34,148.3 47,994.9 7 0.0 2,374.3 6,152.8 1,449.4 22,320.5 7,266.9 15,198.6 34,249.5 0.0 1 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2 1.0 1.0 0.3 1.0 0.6 0.5 1.0 1.0 1.0 3 1.0 1.0 0.8 0.5 0.5 0.6 0.5 1.0 1.0

Pf,i,j 4 1.0 0.3 0.3 0.2 0.2 0.2 0.0 1.0 1.0 5 1.0 1.0 0.3 0.2 0.2 0.2 0.5 1.0 1.0 6 1.0 1.0 1.0 0.2 0.4 0.3 0.2 0.1 0.7 7 1.0 1.0 0.5 1.0 0.4 0.2 0.4 0.1 1.0

108 Case 3: Low ICAM-1

Bond Number State Parameter BP 0 1 2 3 4 5 6

1 66.0 3.4 1.1 0.0 0.0 0.0 0.0 2 66.0 4.0 4.1 0.5 281.7 0.0 0.0 3 66.0 2.8 13.7 14.4 22.3 28.3 0.0 (1) -1 k r,i,j (s ) 4 66.0 4.3 419.4 23.0 25.3 43.5 0.0 5 66.0 9.3 0.0 0.0 61.8 61.8 38.8 6 66.0 0.0 4.8 0.0 75.1 63.9 53.9 1 66.0 7.9 2.4 0.0 0.0 0.0 0.0 2 66.0 0.9 0.9 0.1 60.4 0.0 0.0 3 66.0 92.6 426.9 29.1 52.9 28.4 0.0 (2) -1 k r,i,j (s ) 4 66.0 4.3 3797.8 216.0 62.1 112.4 0.0 5 66.0 9.3 0.0 0.0 1857.5 234.4 112.2 6 66.0 0.0 4.8 0.0 169.6 2378.1 179.5 1 1.0 1.0 0.0 1.0 1.0 1.0 1.0 2 1.0 1.0 0.7 0.1 0.7 1.0 1.0 3 1.0 0.4 0.9 0.4 0.5 0.5 1.0 Pr,i,j 4 1.0 1.0 0.6 0.7 0.5 0.6 1.0 5 1.0 1.0 1.0 1.0 0.6 0.6 0.0 6 1.0 1.0 1.0 1.0 0.7 0.7 0.6 1 0.0 302.7 0.8 0.0 0.0 0.0 0.0 2 0.0 19.4 1594.5 0.1 141.0 0.0 0.0 3 (1) -1 0.0 842.5 81.7 111.0 100.4 58.7 0.0 k f,i,j (s ) 4 0.0 675.7 48.0 272.6 230.9 180.6 0.0 5 0.0 254.8 4296.3 2128.4 903.4 757.0 232.1 6 0.0 0.0 271.2 81.0 2389.0 1819.0 1258.8 1 0.0 1.0 0.0 0.0 0.0 0.0 0.0 2 0.0 0.4 32.0 0.0 2.8 0.0 0.0 3 0.0 65455.0 5949.0 6964.9 7924.2 9001.7 0.0 (2) -1 k f,i,j (s ) 4 0.0 675.7 3009.2 7611.6 8048.2 8200.3 0.0 5 0.0 254.8 4296.3 2128.4 12195.8 10675.6 14430.4 6 0.0 0.0 698919.0 81.0 287045.0 13450.5 14458.1 1 1.0 1.0 0.0 1.0 1.0 1.0 1.0 2 1.0 1.0 0.7 0.1 0.7 1.0 1.0 3 1.0 0.6 0.1 0.1 0.2 0.3 1.0 Pf,i,j 4 1.0 1.0 0.2 0.1 0.1 0.1 1.0 5 1.0 1.0 1.0 1.0 0.2 0.2 0.1 6 1.0 1.0 0.8 1.0 0.9 0.3 0.3

109 Case 4: ICAM-1 dimer

Bond Number State Parameter BP 0 1 2 3 4 5

1 66.0 3.8 19.9 0.0 0.0 0.0 2 66.0 3.4 0.1 20.5 37.9 0.0 (1) -1 k r,i,j (s ) 3 66.0 3.8 58.6 6.1 3.0 0.0 4 66.0 5.7 3.8 20.1 28.5 41.2 5 66.0 0.0 11.9 4,561.4 79.5 70.2 1 66.0 12.3 64.5 0.0 0.0 0.0 2 66.0 1.7 0.1 10.5 19.4 0.0 (2) -1 k r,i,j (s ) 3 66.0 0.2 3.0 0.3 0.2 0.0 4 66.0 161.3 14.2 153.3 74.1 105.5 5 66.0 0.0 11.9 4,561.4 1,822.9 195.9 1 1.0 1.0 0.5 1.0 1.0 1.0 2 1.0 1.0 0.2 1.0 0.8 1.0

Pr,i,j 3 1.0 1.0 0.8 0.5 0.4 1.0 4 1.0 0.8 0.1 0.8 0.6 0.6 5 1.0 1.0 1.0 1.0 0.3 0.6 1 0.0 299.4 2.6 0.0 0.0 0.0 2 0.0 8.7 283.4 6.3 4.0 0.0 (1) -1 k f,i,j (s ) 3 0.0 0.0 1,987.3 59.6 0.4 0.0 4 0.0 277.5 69.2 198.2 231.1 245.6 5 0.0 0.0 147.3 11,536.0 2,084.6 924.0 1 0.0 47.8 0.4 0.0 0.0 0.0 2 0.0 0.4 11.6 0.3 0.2 0.0 (2) -1 k f,i,j (s ) 3 0.0 0.0 30.8 0.9 0.0 0.0 4 0.0 3,568.4 9,088.9 13,286.9 13,753.9 15,140.6 5 0.0 0.0 5,747,130.0 11,536.0 18,997.5 13,763.9 1 1.0 1.0 0.5 1.0 1.0 1.0 2 1.0 1.0 0.2 1.0 0.8 1.0

Pf,i,j 3 1.0 1.0 0.8 0.5 0.4 1.0 4 1.0 0.4 0.2 0.1 0.1 0.1 5 1.0 1.0 0.9 1.0 0.2 0.1

110 Case 5: Clustered ICAM-1 dimer

Bond Number State Parameter BP 0 1 2 3 4 5 6

1 66.0 3.8 6.9 0.0 0.0 0.0 0.0 2 66.0 3.7 9.1 0.3 75.7 0.0 0.0 3 66.0 3.6 10.6 13.8 22.3 44.1 0.0 (1) -1 k r,i,j (s ) 4 66.0 0.0 10.6 23.5 30.4 50.0 0.0 5 66.0 2.3 8.4 63.5 50.1 58.9 75.5 6 66.0 0.0 0.0 0.0 0.0 161.4 92.2 1 66.0 4.6 8.4 0.0 0.0 0.0 0.0 2 66.0 3.3 8.2 0.2 67.8 0.0 0.0 3 66.0 150.4 191.7 26.9 69.8 149.2 0.0 (2) -1 k r,i,j (s ) 4 66.0 0.0 583.8 247.0 86.9 194.5 0.0 1,000,000,0 5 66.0 2.3 8.4 00.0 3,067.5 260.0 221.5 6 66.0 0.0 0.0 0.0 0.0 2,408.6 853.3 1 1.0 1.0 0.8 1.0 1.0 1.0 1.0 2 1.0 1.0 1.0 0.4 0.4 1.0 1.0 3 1.0 0.9 0.7 0.3 0.5 0.8 1.0 Pr,i,j 4 1.0 1.0 0.9 0.8 0.6 0.5 1.0 5 1.0 1.0 1.0 0.9 0.9 0.6 0.5 6 1.0 1.0 1.0 1.0 1.0 0.2 0.7 1 0.0 196.4 1.3 0.0 0.0 0.0 0.0 2 0.0 324.9 1,036.2 0.1 935.5 0.0 0.0 3 0.0 498.2 91.2 189.0 107.5 507.1 42.5 (1) -1 k f,i,j (s ) 4 0.0 0.0 88.4 93.2 286.1 321.4 0.0 5 0.0 181.7 693.1 1,844.2 780.5 1,129.8 920.2 6 0.0 0.0 6,226.3 39,377.8 506.4 3,677.7 2,828.4 1 0.0 25.3 0.2 0.0 0.0 0.0 0.0 2 0.0 2.7 8.7 0.0 7.8 0.0 0.0 3 0.0 142,679.0 10,954.7 12,479.4 18,129.1 17,681.0 42.5 (2) -1 k f,i,j (s ) 4 0.0 0.0 6,585.1 12,203.1 13,536.3 16,470.3 0.0 5 0.0 181.7 132,742.0 19,392.0 13,687.8 18,459.4 23,556.6 6 0.0 0.0 6,226.3 39,377.8 506.4 39,605.0 28,111.1 1 1.0 1.0 0.8 1.0 1.0 1.0 1.0 2 1.0 1.0 1.0 0.4 0.4 1.0 1.0 3 1.0 0.8 0.2 0.1 0.1 0.1 1.0 Pf,i,j 4 1.0 1.0 0.2 0.1 0.1 0.1 1.0 5 1.0 1.0 0.8 0.4 0.1 0.2 0.1 6 1.0 1.0 1.0 1.0 1.0 0.3 0.3

111 Case 6: � = 0.29 nm

Bond Number State Parameter BP 0 1 2 3 4 5 6 7 8 1 66.0 17.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2 66.0 17.7 39.6 4.5 131.4 89.6 0.0 0.0 0.0 3 66.0 17.7 14.3 100.6 75.8 50.9 259.0 0.0 0.0 (1) -1 k r,i,j (s ) 4 66.0 12.7 34.9 78.8 100.7 152.3 209.2 0.0 0.0 5 66.0 0.0 55.1 51.1 116.8 183.7 311.0 188.4 0.0 6 66.0 79.9 73.6 51.1 134.8 233.6 324.7 350.6 0.0 7 66.0 0.0 0.0 0.0 250.3 237.9 296.0 287.1 0.0 1 66.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2 66.0 0.3 0.6 0.1 2.1 1.4 0.0 0.0 0.0 3 66.0 0.0 0.0 0.2 0.2 0.1 0.5 0.0 0.0 (2) -1 k r,i,j (s ) 4 66.0 106.9 887.8 2,306.2 298.0 474.4 383.3 0.0 0.0 5 66.0 0.0 1,170.1 1,758.2 2,794.2 1,223.6 1,495.8 574.8 0.0 6 66.0 79.9 3,456.1 1,189.4 3,132.8 1,470.6 2,000.3 2,657.8 0.0 7 66.0 0.0 0.0 0.0 3,959.8 4,409.6 1,508.4 2,736.4 0.0 1 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2 1.0 1.0 0.9 0.5 0.9 0.2 1.0 1.0 1.0 3 1.0 1.0 0.8 0.6 0.5 0.4 0.8 1.0 1.0

Pr,i,j 4 1.0 0.4 0.6 0.9 0.6 0.6 0.9 1.0 1.0 5 1.0 1.0 0.8 0.9 0.8 0.7 0.6 0.2 1.0 6 1.0 1.0 0.8 0.8 0.8 0.7 0.6 0.6 1.0 7 1.0 1.0 1.0 1.0 0.4 0.6 0.6 0.6 1.0 1 0.0 10.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2 0.0 218.7 449.1 1.4 0.1 1,037.4 0.0 0.0 0.0 3 0.0 721.7 11,472.5 1,421.2 82.6 5.0 396.7 0.0 0.0 (1) -1 k f,i,j (s ) 4 0.0 351.9 238.3 429.7 543.1 659.6 771.4 0.0 0.0 5 0.0 0.0 704.3 215.0 529.3 1,162.4 2,017.2 601.7 0.0 6 0.0 9,364.1 262.1 178.6 617.4 1,693.9 2,330.9 3,765.0 0.0 7 0.0 0.0 4,832.3 707.7 1,285.9 1,047.8 2,322.0 3,244.1 633.6 1 0.0 1.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2 0.0 17.4 35.7 0.1 0.0 82.4 0.0 0.0 0.0 3 0.0 1.0 16.1 2.0 0.1 0.0 0.6 0.0 0.0 (2) -1 k f,i,j (s ) 4 0.0 49,334.6 6,516.2 9,063.1 9,114.6 9,869.6 13,800.0 0.0 0.0 5 0.0 0.0 7,764.5 7,361.5 9,335.9 12,449.1 18,350.1 34,439.1 0.0 6 0.0 9,364.1 14,980.5 7,169.1 9,279.0 14,652.4 20,669.2 32,670.2 0.0 7 0.0 0.0 4,832.3 707.7 9,642.1 12,840.1 19,803.0 33,471.0 633.6 1 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2 1.0 1.0 0.9 0.5 0.9 0.2 1.0 1.0 1.0 3 1.0 1.0 0.8 0.6 0.5 0.4 0.8 1.0 1.0

Pf,i,j 4 1.0 0.7 0.2 0.1 0.1 0.1 0.2 1.0 1.0 5 1.0 1.0 0.3 0.2 0.1 0.2 0.3 0.1 1.0 6 1.0 1.0 0.5 0.2 0.1 0.2 0.3 0.2 1.0 7 1.0 1.0 1.0 1.0 0.2 0.2 0.3 0.1 1.0

112 Case 7: � = 0.3 nm

Bond Number State Parameter BP 0 1 2 3 4 5 6 7 8 1 66.0 8.9 2.7 0.0 0.0 0.0 0.0 0.0 0.0 2 66.0 8.9 140.3 1.2 213.5 0.0 0.0 0.0 0.0 3 66.0 8.9 171.6 210.3 31.3 344.8 447.3 0.0 0.0 (1) -1 k r,i,j (s ) 4 66.0 8.9 111.1 111.0 243.9 351.4 467.5 499.8 0.0 5 66.0 8.9 8.4 70.4 91.6 366.0 393.9 499.7 0.0 6 66.0 0.0 4,633.8 152.1 392.1 472.5 645.9 955.4 0.0 7 66.0 0.0 0.0 126.1 309.3 476.4 600.6 744.4 0.0 1 66.0 151.7 45.8 0.0 0.0 0.0 0.0 0.0 0.0 2 66.0 0.1 2.2 0.0 3.3 0.0 0.0 0.0 0.0 3 66.0 0.4 7.4 9.1 1.4 14.9 19.3 0.0 0.0 (2) -1 k r,i,j (s ) 4 66.0 0.1 0.7 0.7 1.6 2.3 3.0 3.2 0.0 5 66.0 0.0 0.0 0.1 0.2 0.7 0.8 1.0 0.0 6 66.0 0.0 4,633.8 8,009.1 4,398.1 2,229.4 4,249.5 7,355.8 0.0 7 66.0 0.0 0.0 590.3 9,143.1 5,607.2 3,828.0 5,847.9 0.0 1 1.0 1.0 0.5 1.0 1.0 1.0 1.0 1.0 1.0 2 1.0 1.0 1.0 0.3 0.5 1.0 1.0 1.0 1.0 3 1.0 1.0 0.5 0.5 0.6 0.7 0.9 1.0 1.0

Pr,i,j 4 1.0 1.0 0.8 0.8 0.6 0.0 0.6 1.0 1.0 5 1.0 1.0 0.8 0.8 0.7 0.6 0.4 1.0 1.0 6 1.0 1.0 1.0 0.6 0.4 0.7 0.7 0.6 1.0 7 1.0 1.0 1.0 0.1 0.9 0.4 0.7 0.6 1.0 1 0.0 21.5 4.9 0.0 0.0 0.0 0.0 0.0 0.0 2 0.0 301.3 627.6 0.5 4.9 0.0 0.0 0.0 0.0 3 0.0 0.1 889.5 5,228.4 51.7 4.6 2,170.9 0.0 0.0 (1) -1 k f,i,j (s ) 4 0.0 16.7 717.3 274.6 80.2 103.3 70.1 295.4 0.0 5 0.0 1,699.7 5,675.6 186.2 339.1 291.5 398.8 172.5 0.0 6 0.0 0.0 307.2 1,707.9 2,127.3 2,187.9 2,536.5 5,348.2 0.0 7 0.0 0.0 5,526.6 796.5 1,860.5 2,389.7 3,331.9 4,454.0 170,823.0 1 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2 0.0 37.1 77.4 0.1 0.6 0.0 0.0 0.0 0.0 3 0.0 0.0 18.3 107.3 1.1 0.1 44.6 0.0 0.0 (2) -1 k f,i,j (s ) 4 0.0 0.8 32.5 12.5 3.6 4.7 3.2 13.4 0.0 5 0.0 5.5 18.3 0.6 1.1 0.9 1.3 0.6 0.0 6 0.0 0.0 6,133.0 23,138.7 15,906.6 16,063.6 20,100.7 35,698.6 0.0 7 0.0 0.0 5,526.6 8,464.8 18,210.3 18,931.6 23,042.0 33,404.5 170,823.0 1 1.0 1.0 0.5 1.0 1.0 1.0 1.0 1.0 1.0 2 1.0 1.0 1.0 0.3 0.5 1.0 1.0 1.0 1.0 3 1.0 1.0 0.5 0.5 0.6 0.7 0.9 1.0 1.0

Pf,i,j 4 1.0 1.0 0.8 0.8 0.6 0.0 0.6 1.0 1.0 5 1.0 1.0 0.8 0.8 0.7 0.6 0.4 1.0 1.0 6 1.0 1.0 0.1 0.3 0.2 0.3 0.2 0.2 1.0 7 1.0 1.0 1.0 0.3 0.4 0.2 0.3 0.2 1.0

113 0 -4 -1 Case 8: kr = 5x10 s

Bond Number State Parameter BP 0 1 2 3 4 5 6

1 66.0 21.8 12.4 0.0 0.0 0.0 0.0 2 66.0 19.4 5.7 2.8 53.4 169.2 0.0 3 66.0 20.7 270.0 43.8 37.2 231.3 277.4 (1) -1 k r,i,j (s ) 4 66.0 19.4 37.0 80.4 41.9 206.5 315.7 5 66.0 0.0 137.5 142.4 158.7 222.8 370.9 6 66.0 69.5 31.2 75.8 165.5 292.9 440.9 1 66.0 15.2 8.7 0.0 0.0 0.0 0.0 2 66.0 1.6 0.5 0.2 4.3 13.6 0.0 3 66.0 0.3 3.4 0.5 0.5 2.9 3.5 (2) -1 k r,i,j (s ) 4 66.0 0.2 0.4 0.9 0.5 2.4 3.6 5 66.0 0.0 2,050.9 3,134.7 3,889.8 1,128.0 1,700.3 6 66.0 69.5 92.2 388.0 3,317.3 3,158.2 3,264.0 1 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2 1.0 1.0 0.5 0.5 0.2 0.5 1.0 3 1.0 1.0 1.0 0.6 0.5 0.3 0.5 Pr,i,j 4 1.0 1.0 0.7 0.9 0.7 0.6 0.5 5 1.0 1.0 0.2 0.7 0.9 0.7 0.7 6 1.0 1.0 0.1 0.9 0.8 0.8 0.7 1 0.0 53.8 33.0 0.0 0.0 0.0 0.0 2 0.0 4.0 604.0 7.1 20.7 27.1 0.0 3 (1) -1 0.0 20.9 5,060.1 83.7 38.4 19.4 248.5 k f,i,j (s ) 4 0.0 5.0 1,191.4 357.9 47.3 47.9 7.6 5 0.0 0.0 484.0 771.1 535.4 871.0 2,281.6 6 0.0 5,415.3 1,091.7 317.1 558.0 1,713.6 2,144.2 1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2 0.0 0.1 8.0 0.1 0.3 0.4 0.0 3 0.0 0.2 44.7 0.7 0.3 0.2 2.2 (2) -1 k f,i,j (s ) 4 0.0 0.2 45.1 13.5 1.8 1.8 0.3 5 0.0 0.0 6,529.1 9,583.1 9,965.3 12,025.0 19,913.9 6 0.0 5,415.3 8,931.5 9,286.3 11,281.1 14,144.4 18,959.9 1 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2 1.0 1.0 0.5 0.5 0.2 0.5 1.0 3 1.0 1.0 1.0 0.6 0.5 0.3 0.5 Pf,i,j 4 1.0 1.0 0.7 0.9 0.7 0.6 0.5 5 1.0 1.0 0.2 0.1 0.1 0.1 0.3 6 1.0 1.0 0.2 0.2 0.1 0.2 0.2

114 0 -3 -1 Case 9: kr = 10 s

Bond Number State Parameter BP 0 1 2 3 4 5 6

1 66.0 16.3 385.7 0.0 0.0 0.0 0.0 2 66.0 14.2 407.8 3.7 118.1 0.0 0.0 3 66.0 14.5 131.7 185.6 232.6 362.4 0.0 (1) -1 k r,i,j (s ) 4 66.0 15.8 162.3 181.9 295.7 258.7 500.0 5 66.0 14.8 203.5 155.9 304.8 485.5 500.0 6 66.0 14.7 68.9 164.4 399.8 488.8 465.3 1 66.0 79.2 1,877.8 0.0 0.0 0.0 0.0 2 66.0 0.2 6.4 0.1 1.8 0.0 0.0 3 66.0 0.8 7.1 10.0 12.6 19.6 0.0 (2) -1 k r,i,j (s ) 4 66.0 0.5 5.2 5.9 9.5 8.3 16.1 5 66.0 0.3 4.0 3.0 5.9 9.5 9.7 6 66.0 1.2 5.7 13.5 32.8 40.1 38.2 1 1.0 1.0 0.1 1.0 1.0 1.0 1.0 2 1.0 1.0 1.0 0.4 0.6 1.0 1.0 3 1.0 1.0 0.7 0.8 0.5 0.5 1.0 Pr,i,j 4 1.0 1.0 0.5 0.6 0.6 0.6 0.5 5 1.0 1.0 0.5 0.6 0.8 0.6 0.6 6 1.0 1.0 0.8 0.8 0.4 0.8 0.6 1 0.0 68.6 108.6 0.0 0.0 0.0 0.0 2 0.0 258.6 2,200.6 1.3 6.6 0.0 0.0 3 (1) -1 0.0 21.0 947.0 827.9 255.9 17.7 0.0 k f,i,j (s ) 4 0.0 0.9 3,179.0 4,125.3 3,021.2 1,099.1 55.5 5 0.0 29.3 3,090.1 1,864.3 1,233.9 249.7 405.3 6 0.0 12,604.9 166.8 46.7 11,098.8 8,022.9 326.7 1 0.0 3.6 5.7 0.0 0.0 0.0 0.0 2 0.0 5.9 49.9 0.0 0.1 0.0 0.0 3 0.0 2.1 92.6 80.9 25.0 1.7 0.0 (2) -1 k f,i,j (s ) 4 0.0 0.0 15.5 20.1 14.7 5.3 0.3 5 0.0 0.6 68.4 41.2 27.3 5.5 9.0 6 0.0 155.3 2.1 0.6 136.8 98.9 4.0 1 1.0 1.0 0.1 1.0 1.0 1.0 1.0 2 1.0 1.0 1.0 0.4 0.6 1.0 1.0 3 1.0 1.0 0.7 0.8 0.5 0.5 1.0 Pf,i,j 4 1.0 1.0 0.5 0.6 0.6 0.6 0.5 5 1.0 1.0 0.5 0.6 0.8 0.6 0.6 6 1.0 1.0 0.8 0.8 0.4 0.8 0.6

115 5.8. Discussion

The goal of this work was to create a new model of multivalent nanoparticle detachment that addressed bonding heterogeneity within a population. This would provide a set of detachment rates, one for each sub-population, that combine to give the population’s empirically-observed behavior. A critical aspect was the categorization criteria, which we defined as the BP and for which we selected to use the mode bond number observed during NAD simulations. While this approach was generally successful, the need to utilize two hidden components to accurately fit detachment data indicated that significant bonding heterogeneity was still present with BPs. This suggests that nanoparticles tending to possess the same number of bonds can still display significantly different adhesion energies. Allowing for two hidden components provided for sub- categorization based on relatively faster or slower detachment rates, or lower and higher bond energies. We believe that these hidden components likely arose from a physical mechanism with respect to the way in which adhesion molecules and/or bonds were distributed within the contact zone between the nanoparticle and substrate. For example, certain bond configurations could lead to higher levels of strain on one or more neighboring bonds, accelerating rupture. Alternatively, excess unbound receptors or ligands could drive bond re-formation after a rupture event. We would expect that a configuration-based mechanism would be most prominent at lower BPs, where variation in adhesion molecule/bond distribution would be highest, and this can in fact be seen in

Table 5.1, Table 5.3 and Table 5.4 by the decrease in ai or Pf,i, P,r,I with BP.

Using the two-hidden component interpretation, we fit NAD simulation detachment data for each BP category (Equation 5.4). However, it was clear that there was not enough

116 information to characterize the slower detaching component, which didn’t contribute appreciably to detachment during the 30 s simulations. Therefore, we simplified the fitting

(2) process by assuming that the slow detaching species did not detach at all (k D,i = 0). It should be noted that this assumption is only strictly valid over the time frame observed, as all nanoparticles should possess non-zero kD,i values. The resulting BP model fits matched simulation data as well or better than our original empirical model with time-dependent detachment rate (Equation 3.23). We then performed a survival analysis to estimate an instantaneous population detachment rate (kD) for the BP model (Equation 5.8), which scaled with time following 1/(1+�/), a functional form sometimes called a logistic decay.

Comparing the empirical model kD that used a power law (Equation 3.23), it was quite remarkable that both models could capture temporal changes with such precision using two very different time functions. By evaluating kD for the two models at two important

0 time points, we then developed relationships for the empirical model parameters (kD and

β) in terms of the BP model parameters (wi, kD,i, a,i), as shown in Equation 5.11 and 5.12. By

0 definition, the empirical model reduces to kD at tref, the multivalent bonding steady state.

For the BP model at tref, the exponential term reduces to ~1 for all cases, leaving two scaling factors: 1/�̃ and 1/(1+�/(1 − �)). The parameter �̃ is the sum of the fast detaching component within each BP, weighted by 1/kD,i, and thus can be considered a characteristic time for detachment. The parameter � is the sum of the slow detaching population component of each BP, and thus the term �/(1 − �) is simply the ratio of the slow to fast detaching components. In the absence of a slow component, this term would reduce to 0

0 0 and kD would simply be equal to 1/�̃. As � increases, kD also increases because kD is essentially decaying to a larger non-zero value. From the population perspective, 1/�̃ can

117 be thought of as a characteristic detachment rate that dictates the period of time required for the population kD to stabilize to its long-term value, which is non-zero because of �. To assess β, we defined kD in terms of an arbitrary time point tmax, and discovered that the BP model best matched the empirical model at tmax ~ 3�̃. While we do not know the reason for this specific relationship, it does make intuitive sense that β depends on �̃, as both dictate how long it will take the population kD to stabilize. But we note that tmax is not just a threshold that ensures the system time has exceeded �̃, as further increasing tmax causes β to diverge from the empirical model. While �̃ does appear within the numerator of Equation

5.12, this was not the reason that there was such a strong connection with β, as the exponential terms reduced to constants for all cases studied in this work. Instead, �̃ affected the denominator through its indirect connection to tmax. Ultimately, we found that Equation

5.11 and 5.12 were very accurate in relating BP and empirical model parameters (Table

5.2). This implies that both models share the same basic assumption that the nanoparticle population contained sub-components that could never detach during the time-scale that was observed during experiments or simulations. As a consequence, it is unclear how well either model can predict what would happen to the remaining nanoparticle population on longer timescales.

Since direct information about the slow detaching population could not be obtained from nanoparticle detachment (macroscopic) data, we turned to the bonding (microscopic) data. Using a system of ordinary differential equations (Equation 5.14 and 5.15), we determined bond state transition (forward and reverse) rate values for each BP. We then used the bond transition rates and mean first passage time of detachment (Equation 5.18 and 5.19) to estimate a nanoparticle detachment rate. We still assumed that two hidden

118 components were present for each BP, but we chose to allow them to interact, such that a nanoparticle could crossover between components at any time. We felt that this interpretation was more realistic than fully separating the components, as nanoparticle rotation and translation could reveal new adhesion molecule configurations within the

(M) contact zone. As a result, we could only calculate a single detachment rate (k D,i ) that applied to both components. To determine individual detachment rates, we again fit the simulation detachment data (Equation 5.4), but now optimized those fits using all three

(1) (2) (M) parameters (k D,i, k D,i, ai) while also utilizing k D,i as an additional constraint (Equation

(2) (M) 5.20). Fitting results for the initial (k D,i = 0) and final (k D,i ) iterations of the BP model were similar during the 30 s simulated time. We note that beyond 30 s, the initial fits would predict that no more nanoparticles would detach, which clearly is not accurate. Thus, the final fits with both macroscopic and microscopic consistency would provide superior predictions for long-term detachment behavior. This could be important because NAD simulations are computationally expensive, requiring days of real time to simulate seconds of adhesion.

Based on the combined findings of this work, we conclude that BP represents a true bonding classification of multivalent nanoparticles. As such, the characteristic parameter kD,i represents both a macroscopic, kinetic property of the nanoparticle and a microscopic, thermodynamic construct representative of the individual bonds. To determine the thermodynamic energy of the nanoparticle, often termed the avidity, we must also consider the kinetic rate of nanoparticle attachment (kA). We anticipate that the kA of a population of nanoparticles should be less heterogeneous, and possibly even uniform, because it only accounts for the formation of the first bond at a given location on the substrate. However, it

119 is possible that kA and BP could be associated. For example, attachment would be favored for regions of the substrate with higher than average adhesion molecule density, which would favor a higher ultimate BP after multivalent bonding has reached equilibrium. If such an effect were significant, it would lead another selection mechanism driving nanoparticles to higher BP and overall stability, which will be studied in future work. After initial attachment however, kD,i captures all context relevant detail for a given BP, including reattachment of the nanoparticle to the same location from 0 bonds.

120 6. NAD ATTACHMENT SIMULATIONS

6.1. Introduction

In previous chapters, we focused on assessing the stability of nanoparticle adhesion and bond accumulation process after initial bindings. In this chapter, we expand our NAD simulations to include tethering from the bulk fluid to model nanoparticle attachment rate

(kA). This together with previous NAD detachment simulations enables full kinetic evaluation of experimental binding data. The attachment rate of nanoparticle was defined in Equation 3.25. Particle attachment data obtained during the binding period do not reflect detachment; thus, the attachment rate (kACw) can be determined by the slope of bound particle number versus time by setting kD = 0 in Equation 3.25. Besides, we could use this attachment simulation result to parameterize σts, the last unknown mechanical property of ICAM-1-Antibody bond, which, for computational simplicity, was assumed to be the same as σ in previous detachment simulation. We can use NAD attachment simulation to independently fit σts since kA only depends on the rate constant for bond formation (kf) as in Equation 3.18. By the end of this chapter, we will have a complete computational tool for investigate multivalent nanoparticle adhesion process.

6.2. Methods

Overview of NAD attachment simulation

Similar to NAD detachment simulations, we employed a 210-nm-diameter sphere decorated with monoclonal antibody and a planar substrate decorated with ICAM-1 proteins. The substrate was first initialized by randomly distributing ICAM-1 dimers with effective ICAM-1 molecule density nL. A free particle was then randomly initialized in a

121 cuboid box with its bottom surface on the substrate. Antibody dimers were distributed on the surface of particle with effective Fab molecule density nR. Unlike our previous detachment simulation, no prior bond was formed attaching particle to the substrate.

Attachment simulation started using a predefined time step (Dt). In each time step, the particle was translated and rotated based on the forces vectorially summed on the particle; if the particle was one equilibrium bond length close to the substrate, unbound adhesion molecules within the contact area were assessed for potential bond formation. If a bond was formed, we recorded this time as its binding time, and fed a new free particle into the box, with its height uniformly distributed between 0 and box height h. If no bond was formed, we checked if the particle moved out from the top of the box; if so, we re-put the particle into the box with its height uniformly distributed between 0 and h. Then the simulation proceeded onto next time step until a predefined maximum time step was reached. We used periodic boundary condition (PBC) on four sides of the box, so that once the particle flowed out of the box from one side, it was re-put into the box from its opposite side. Due to PBC, the size of box base and the initial position of particle projected onto box base (x and y axis) do not influence simulation results.

Nanoparticle attachment rate

Nanoparticle attachment rate kA was computed using Equation 6.1.

� = h/� 6.1

where h is the box height, and tl (with the unit s) is the rate parameter of the distribution of waiting times for free particles to attach. Here we assumed particle attachment follows a

122 Possion process, so the waiting times obey an exponential distribution. The parameter estimation for exponential distributed was done by Python "scipy.stats" package.

Damkohler number for attachment scales the kinetic rates to the diffusive transport rate, given by:

� ℎ � = � 6.2

where h is the box height, and D is particle diffusivity, 2.09´10-12 m2/s computed using

Stokes-Einstein equation for a 210nm diameter spherical nanoparticle in water at 300K.

Based on the transport-reaction model in Haun et al [6], particle concentration within the flow chamber is given by the two dimensional transient convective-diffusion equation. The flow within the chamber is assumed fully developed and the laminar flow is parabolic. Particle concentration at the inlet is equal to the starting concentration. The heterogeneous reaction at the ligand-coated chamber surface matches the reaction rate to the flux of particles to the wall. In the reaction-limited regime (δA << 1), where particle diffusion is much faster than attachment, the substrate experiences a constant probability density of particles to be 1/h, which provides an effective local concentration. Following the assumptions of our NAD simulation, and also of many other biochemical processes, the derivative of particle concentration with respect to box height diffusion is then proportional to kA and particle concentration[65], thus tl has the relationship:

h � ∝ �� 6.3

We therefore defined a fundamental attachment rate κ, which is independent of adhesion molecule density (nR and nL) and box height (h).

123 k ℎ κ = = �� 6.4

6.3. General dynamics of nanoparticle attachment

We started NAD attachment simulations with the configuration of low antibody density (410 nm-2) on the nanoparticle, medium ICAM-1 density (41 nm-2) on the substrate, spring constant (σ) and transitional spring constant (σts) 0.8 N/m, reactive compliance (γ)

0.274 nm, ICAM-1 in dimer configuration as in our previous NAD detachment simulation.

We used an ensemble number of 50 (boxes), and maximum time for each box was 50s. We chose a box height as 2000nm (10 times larger than the particle diameter). There were 628 particles attached through the simulation. The number of attached particles increased linearly with time as in Figure 6.1A. The rate parameter tl of this system was estimated as

0.358s (Figure 6.1B), and the kA based on equation 6.1 was then computed as 5586 nm/s, corresponding to a Damkohler number (dA) 5.35. Compared with the kA 7.6 nm/s, and dA

0.92 in Haun’s experiment, the high simulation attachment rate indicated particles are over reactive. We then sought to tune down the reactivity by increasing σts.

124

Figure 6.1 Nanoparticle attachment dynamics. (A) Nanoparticle attachment profile. (B) Histogram of waiting times between particle initialization and attachment. The data was collected from the nanoparticle population � = 0.8 N/m, �ts = 0.8 N/m, � = 0.274 nm, h = 2000nm, and medium ICAM-1/low antibody density.

6.4. Influence of box height and adhesion molecule density

In order to match the simulation to experiment, we compared experimental κ with simulation κ, computed from Equation 6.4. Experimental κ was fitted to be 827nm5 s-1 from

Haun’s flow chamber assay result, as the slope of experimental kA verse the product of adhesion molecule density nRnL.[6] Before fitting κ value, we needed to confirm the particle systems were in the reaction-limited regime, where Equations 6.3 and 6.4 hold. We first assessed the influence of the box height h, and adhesion molecule densities nL and nL on attachment rate kA, which is characterized by the rate parameter tl by Equation 6.1. Next we sought to remove the influence of h, nL and nL by converting tl into κ. Finally, we compared simulation κ with experimental κ to decide the parameterization of �ts.

In the initial experiment, we ran a particle system with medium ICAM-1 density and low antibody density in a 2000nm-height box and found δA is 5.35, indicating this system in the diffusion-limited region. Base on the equation 6.3, the rate parameter tl is

125 proportionate to h and inversely proportionate to the product of the antibody density nL and the ICAM-1 density nR at reaction-limited regime. In order to bring the particle system

-2 -2 to the reaction-limited regime, we decreased nL from 410 nm to 10-50 nm , and nR from

-2 41 nm in six systems as in Table 6.1. δA of all six systems are below 1. Figure 6.2 shows

the linearity between tl and 1/nLnR(A) and between tl and h. This confirmed reaction- limited. However, in all six conditions, κ are two orders of magnitude larger than experimental κ. In future work, our goal is to make reaction less favorable by tuning up σts.

Figure 6.2 Attachment rates at varied adhesion molecule densities and box heights. (A) tl versus inverse adhesion molecule densities at h = 2000nm. (B) tl versus inverse h at -2 -2 nL = 10 µm and nR = 5 µm .

Table 6.1 Nanoparticle attachment dynamics over varied box heights and adhesion molecule densities.

Antibody ICAM-1 5 -1 density nL density nR h (nm) tl (s) kA (nm/s) δA κ (nm s ) (µm-2) (µm-2) 50 5 2000 5.78 346.02 3.31E-01 1.38E+06 10 10 2000 12.28 162.87 1.56E-01 1.63E+06 10 5 2000 22.57 88.61 8.48E-02 1.77E+06 10 1 2000 78.5 25.48 2.44E-02 2.55E+06 10 5 1500 16.8 89.29 6.41E-02 1.79E+06 10 5 1000 11.08 90.25 4.32E-02 1.81E+06

126

7. CONCLUSION AND FUTURE DIRECTIONS

The targeted delivery of nanoparticle carriers holds tremendous potential to transform the detection and treatment of diseases. A major attribute of nanoparticles is the ability to form multiple bonds with target cells, which greatly improves adhesion strength.

However, multivalent binding of nanoparticles is still poorly understood, particularly from a dynamic perspective. In previous flow chamber assay, Haun et al[6] studied the kinetics of nanoparticle adhesion and found that the rate of detachment decreased over time. Here, we have applied the Adhesive Dynamics simulation framework to investigate binding dynamics between an antibody-conjugated, 210 nm diameter sphere and ICAM-1 on a surface at the scale of individual bonds. We found that Nano Adhesive Dynamics (NAD) simulations could replicate the time-varying nanoparticle detachment behavior that we observed in experiments. As expected, this was correlated with a steady increase in mean bond number with time, but this was only attributed to bond accumulation during the first second that nanoparticles were bound. Longer-term increases in bond number instead manifested from nanoparticle detachment serving as a selection mechanism to eliminate nanoparticles that had randomly been confined to lower bond valencies. Thus, time- dependent nanoparticle detachment reflects an evolution of the remaining nanoparticle population towards higher overall bond valency. We also found that NAD simulations precisely matched experiments whenever mechanical force loads on bonds was high enough to directly induce rupture. These mechanical forces were in excess of 300 pN and primarily arose from the Brownian motion of the nanoparticle, but we also identified a valency-dependent contribution from bonds pulling on each other. In summary, we have

127 achieved excellent kinetic consistency between NAD simulations and experiments, which has revealed new insights into the dynamics and biophysics of multivalent nanoparticle adhesion, allowing us to uncover the mechanisms underlying time-dependent nanoparticle detachment and identify a key role for mechanical forces. Based on the NAD simulation results, we analyzed the complex detachment dynamics of heterogeneous multivalent nanoparticle populations in an effort to obtain detachment rates for each sub-population.

We first classified nanoparticles based on mode bond number, which we termed the bond potential (BP). Our BP detachment model matched simulation data well, provided we assume that each BP category contained hidden components and that the stable sub- populations did not detach over the timescale observed during simulations. We then formalized relationships between the BP model and an empirical model that we developed in previous work to capture time-dependent effects, demonstrating that both models fundamentally describe nanoparticle population heterogeneity effects. We obtained detachment rates for all sub-populations by combining the BP model with bond transition rate information obtained using a system of ordinary differential equations and mean first passage time analysis. The BP detachment model will enable analysis of multivalent nanoparticle adhesion heterogeneity and provide nanoparticle detachment rates and avidities for the various sub-populations. Finally, we also simulated nanoparticle attachment process, enabling full kinetic evaluation of experimental binding data. Based on our NAD simulation and dynamical system analysis tool, we anticipate new and powerful strategies will be uncovered that will enable greater control over nanoparticle adhesion and maximize targeting performance. Our unique kinetic approach may even make it possible to direct early stage behavior toward different final states.

128 In future work, we will also incorporate discrete physical detail for receptor and ligand molecules to increase accuracy and better represent bond biophysics. It will also be extremely valuable to explore a broader range of parameter space across experiments and simulations, including nanoparticle size and shape as well as bond kinetics and mechanics.

Regarding potential targeted delivery applications, we will begin to evaluate adhesive behavior in a cellular context. This will require that target ligands be allowed to diffuse laterally within the plasma membrane, and potentially be translated in response to force.

Motion of the ligand within the substrate would follow similar treatments used here for nanoparticles, as demonstrated in previous work.[41-43] Furthermore, we will continue to use the BP model to help extract key information from NAD simulations, such as the detachment rates and affinities for all sub-populations. We are also interested in utilizing the BP and empirical models to directly analyze naïve experimental nanoparticle adhesion data without the need for computationally costly NAD simulations. In this scenario, the BP model would be limited to a simplified interpretation in which all slow detaching BPs and sub-components would be binned together into a non-detaching group. However, it may be possible to estimate key insight into BP distributions and detachment rates for sub- populations that did contribute to detachment during the experiment. This would greatly expedite analysis, and in general make the work far more accessible to other researchers in the field. Finally, we will explore the use of the BP model to predict behavior over longer time periods than are not accessible for NAD simulations.

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