Molecular Electromechanics: Modeling
Electrostatic Forces Between GAG Molecules
by
Delphine Marguerite Denise Dean
Submitted to the Department of Electrical Engineering and Computer
Science
in partial fulfillment of the requirements for the degrees of
Bachelor of Science in Electrical Engineering and Computer Science
and
Master of Engineering in Electrical Engineering and Computer Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May 2001
Copyright 2001 Delphine Marguerite Denise Dean. All rights reserved.
The author hereby grants to M.I.T. permission to reproduce and
distribute publicly paper and electronic copies of this thesis and to
MAS T
grant others the right to do so.
S NOLOGY
JUL 112001
LIBRARIES
Author ..........................................
Department oj.Electrical Engineering and Computer Science
May 23, 2001
Certified by .......
Professor
Accepted by...
.............
.
...... ...........
Alan J. Grodzinsky
Electric ,echanicl,
and Bioengineering
ThesisSupervisor
.....................
Arthur C. Smith
Chairman, Department Committee on Graduate Students
2
Molecular Electromechanics: Modeling Electrostatic Forces
Between GAG Molecules
by
Delphine Marguerite Denise Dean
Submitted to the Department of Electrical Engineering and Computer Science
on May 23, 2001, in partial fulfillment of the
requirements for the degrees of
Bachelor of Science in Electrical Engineering and Computer Science
and
Master of Engineering in Electrical Engineering and Computer Science
Abstract
Electrostatic repulsions between glycosaminoglycan (GAG) molecules are the principle contributors to the compressive strength of articular cartilage tissue. These
molecular-level repulsion forces can be measured in aqueous solutions under various
conditions with a molecular force probe (MFP). In order to determine the contribution of electrostatic repulsion and to understand the significance of the components
of the total force, accurate models need to be developed.
In this thesis, three different models of the electrostatic interactions are investigated. The first models a GAG "brush"-like layer as a smooth flat surface charge.
This model has a simple analytical solution in certain limits and provides a qualitative comparison of ionic strength dependent trends. The second model treats a GAG
layer as a uniform volume charge density. Since the length of the GAG molecule is
comparable to the distance at which the MFP measures forces, this volume charge
model is a more appropriate model than the flat surface model for force calculations.
The third is a novel model that represents the GAG molecules as rods of uniform
charge density, which better describes the geometry of the system.
Because the equations relating the electrostatic force to the known system parameters (charge density and ionic strength of the buffer) are nonlinear, there are no
known analytical solutions for the last two models. Therefore, numerical methods
were developed and their feasibility and results are reported.
Thesis Supervisor: Alan J. Grodzinsky
Title: Professor of Electrical, Mechanical, and Bioengineering
3
4
Acknowledgments
This thesis would not have been possible without the help of many people. I would
like to thank Professor Alan Grodzinsky, my thesis and academic advisor, for all
the insight, support, and encouragement he has given me during the course of my
studies. I would also like to acknowlege the crucial work of Joonil Seog in determining
the chemistry for attaching the GAG to gold, running all the MFP experiments, and
basically getting this project off the ground. Professor Christine Ortiz provided useful
advice and references, and helped introduce me to fields in which I had little prior
experience.
I would like to thank Dr. Anna Plaas and Shirley Wong-Palms at the Shriners
Hospital in Tampa, who provided the GAG chains and helped with the attachment
chemistry. I also would like to thank Professor Laibinis and Ivan Lee for their insights
on surface chemistry. To everyone in Prof. Grodzinsky's lab: thanks for your friendship and for making the lab such a nice place to work. N6ra Szisz helped me work
out my theory derivations, read through my thesis and in general provided lots of
moral support and friendship. I want to credit Moonsoo Jin for correctly predicting
that my numerical simulation would take at least half a year to develop, and also for
his valuable suggestions along the way. I've enjoyed the company of my office mate
Han-Hwa Huang, who was always willing to lend a hand in lab-related matters. Linda
Bragman provided assistance with several administrative problems and supplied my
inbox with many humorous emails. Thanks also to Jiang-Ti Kong, who started the
work on this project. I also want to thank everyone in Prof. Ortiz's lab for all the
great Monday meetings and discussions, especially Monica Rixman who put up with
listening to me rant about current publications on electrostatics.
Professor Jacob White taught a great numerical methods class and helped me
with the Jacobian-free Newton method. Jonathan Blanford, Mathew Cain, Shannon Cheng, Timothy Garnett, Douglas Heimburger, May Lim, Michelle Nadermann,
James Paris, Natalia Toro, Qian Wang, and Boriz Zbarsky are all very nice people who
gladly donated computer processor time to run my program in exchange for chocolate
5
cake. Justin Cave, Anne Gaumond, Raffi Krikorian, Julie Park and Michael Spitznagel are also very nice and not only did they run my program on their computers
but they also proofread this thesis and found all (hopefully!) of my errors. Benjamin
Yoder is especially nice and read over my thesis, helped me debug my program and
ran it on his "little" computer farm in the Media Lab.
I want to thank Cecile Le Cocq, my sister, who let me take naps at her dorm on
campus during those long writing sessions, and who gave me lots of moral support.
Catherine and Christian Le Cocq, my parents, who are all-around really cool and
super parents, ran my program, proofread this document, listened to my late night
rants about GAGs and still managed to stay sane! Finally, I would like to thank
Brian Dean, my husband, who helped me debug, read my thesis, discussed my ideas,
made me dinner every night for the last three months, and was always there for me
when I stressed out. And to you the reader, thanks for reading!
6
Contents
1
Introduction
17
1.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
1.2
O bjective
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
1.3
O verview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2 Articular Cartilage
2.1
2.2
3
19
Cartilage Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.1.1
Extracellular Matrix . . . . . . . . . . . . . . . . . . . . . . .
19
Proteoglycans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.2.1
21
Glycosaminoglycan . . . . . . . . . . . . . . . . . . . . . . . .
Molecular Level Forces
23
3.1
Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3.1.1
The Poisson-Boltzmann Equation . . . . . . . . . . . . . . . .
23
3.1.2
Electrostatic Force
. . . . . . . . . . . . . . . . . . . . . . . .
26
Non-Electrostatic Forces . . . . . . . . . . . . . . . . . . . . . . . . .
28
3.2
4 Experimental Methods
5
29
4.1
High Resolution Force Spectroscopy . . . . . . . . . . . . . . . . . . .
29
4.2
Experimental Setup and Results . . . . . . . . . . . . . . . . . . . . .
30
Modeling of Electrostatic Force
35
5.1
Smooth Surface Charge Model . . . . . . . . . . . . . . . . . . . . . .
35
5.1.1
35
Analytical Solution to Linearized Flat Surface Charge Model
7
.
5.1.2
5.2
37
. . . . . . . . . . . . . . . . . . . . . . . . . .
39
. . . . . . . . .
41
. . . . . . . . . . . . . . . . . . . . . . . . . . .
43
Numerical Method for Solving Charged Rod Model . . . . . .
44
Volume Charge Model
5.2.1
5.3
Numerical Method for Flat Surface Charge Model . . . . . . .
Numerical Method for Volume Charge Model
Charged Rod M odel
5.3.1
51
6 Results and Discussions
6.1
Smooth Surface Charge Model . . . . . . . . . . . . . . . . . . . . . .
51
6.2
Volume Charge Model
. . . . . . . . . . . . . . . . . . . . . . . . . .
55
6.3
Charged Rod Model
. . . . . . . . . . . . . . . . . . . . . . . . . . .
62
6.4
Comparison of Results . . . . . . . . . . . . . . . . . . . . . . . . . .
67
71
7 Conclusions
7.1
Sum m ary
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
7.2
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
A Electrostatic Interactions Between MFP Tip and Substrate: Generalized 2-D Models for "Neutral" and Charged Surfaces
73
. . . . . . . . . . . . . . . . . . . . . . . . .
75
A.2 Electrostatic Force from Induced Charge . . . . . . . . . . . . . . . .
76
A.1 Induced Surface Charge
79
B Detailed Experimental Methods
8
List of Figures
2-1
The structure of cartilage (not drawn to scale).
. . . . . . . . . . . .
20
2-2
Schematic drawing of aggrecan, a proteoglycan.
. . . . . . . . . . . .
21
2-3
(a) The structure of GAG; (b) Molecular structure of Chondroitin-6sulfate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
3-1
Schematic of some of the forces felt by GAG molecules. . . . . . . . .
24
4-1
The molecular force probe (MFP).
. . . . . . . . . . . . . . . . . . .
30
4-2
(a) An example of deflection curve; (b) An example of a force curve. .
31
4-3
Schematics of the experiments: (a) GAG coated substrate versus monolayer coated tip; (b) GAG coated substrate versus GAG coated tip. .
4-4
Experimental data of force between a charged monolayer coated tip
and a GAG coated substrate at various concentrations of NaCl.
4-5
. . .
. . . . . . . . .
36
Potential between two charged surfaces. The control box is shown with
the dashed line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-3
33
(a) Two approaching planes with charged surfaces; (b) Charged hemisphere tip approaching a flat charged planar surface.
5-2
32
Experimental data of force between a neutral monolayer coated tip and
a GAG coated substrate at various concentrations of NaCl. . . . . . .
5-1
31
39
(a) Flat surface charge over a fixed charge volume; (b) Two approaching surface with fixed charge volumes. . . . . . . . . . . . . . . . . . .
9
40
5-4
(a) A flat surface charge over a fixed charge volume a distance D >
brush height apart; (b) A flat surface charge over a fixed charge volume
. . . . . . . . . . . . . . . . . . . . . .
42
5-5
Approximated hemisphere of charge over a fixed volume of charge. . .
44
5-6
Two parallel plates with charged rods.
. . . . . . . . . . . . . . . . .
44
5-7
Repeating unit in the parallel charged rod plate model. D is the dis-
a distance D' < brush height.
tance between the square base plates. This unit is discretized into
cubic elements of size 6, by Jy by 62.
6-1
. . . . . . . . . . . . . . . . . .
A comparison of the force predicted by the surface charge model using
the full solution and the linearized approximation. . . . . . . . . . . .
6-2
54
.....................................
Force between a smooth volume of fixed charge and a flat charged
surface (hnm x 1nm) at 0.01M. . . . . . . . . . . . . . . . . . . . . .
6-4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
Force between rods of fixed charge and a flat charged surface (1nm x
1nm ) at 0.01M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-9
59
The volume charge model of a GAG-coated tip approaching a GAGcoated surface.
6-8
58
The volume charge model compared to experimental data for GAG
surface versus neutral monolayer-coated tip. . . . . . . . . . . . . . .
6-7
57
The volume charge model compared to experimental data for GAG
surface versus charged monolayer-coated tip. . . . . . . . . . . . . . .
6-6
56
Force between a smooth volume of fixed charge and a flat charged
hemisphere (radius=25nm) at 0.01M. . . . . . . . . . . . . . . . . . .
6-5
53
Surface charge model (nonlinear and linearized) versus experimental
data........
6-3
45
63
The charged rod model compared to the experimental results of a
charged monolayer-coated tip approaching GAG surface.
. . . . . . .
64
6-10 The charged rod model compared to experimental data for a neutral
monolayer-coated tip approaching a GAG surface. . . . . . . . . . . .
10
65
6-11 Force between two approaching 1nm x Inm surfaces with charged rods
at 0.01M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
6-12 The charged rod model of a GAG-coated tip approaching GAG surface. 66
6-13 Charged monolayer versus GAG force at 0.01M NaCl compared to 2
models with added approximated steric effect. . . . . . . . . . . . . .
67
6-14 Neutral monolayer versus GAG experimental results at 3M NaCl compared to the de Gennes steric model.
. . . . . . . . . . . . . . . . . .
68
6-15 Comparison of the volume charge and charged rod models of GAGcoated tip approaching a GAG-coated substrate. . . . . . . . . . . . .
A-i Model of tip approaching a charged surface.
70
. . . . . . . . . . . . . .
74
A-2 General interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
B-1 Schematic of the chemistry to graft GAG onto substrate surface. . . .
80
11
12
List of Tables
6.1
The Debye length at various NaCL concentrations. . . . . . . . . . . .
52
6.2
The flat surface charge model parameters.
. . . . . . . . . . . . . . .
52
6.3
The smooth volume charge model parameters. . . . . . . . . . . . . .
56
6.4
The rod model parameters . . . . . . . . . . . . . . . . . . . . . . . .
62
13
14
List of Symbols
AFM
atomic force microscope
CS
chondroitin sulfate
ECM
extracellular matrix
GAG
glycosaminoglycan
HRFS
high resolution force spectroscopy
MFP
molecular force probe
PB
Poisson-Boltzmann
PG
proteoglycan
SAM
self-assembled monolayer
r1
Debye length
<D)
electrostatic potential
E
electric field
F
force
W
energy
T
temperature
CO
bulk salt concentration
p
volume charge density
a
surface charge density
15
fo
permittivity of free space
8.85 x 10-1
22
cf
permittivity of fluid
6.923 x 10-
N2
R
gas constant
Nm
8.314 molK
F
Faraday constant
C
96484.6 mol
k otmanconstant
1.381 x -1 0 3N
16
Chapter 1
Introduction
1.1
Background
Articular cartilage is the load bearing tissue found in joints. Its strength in response to different loads is highly dependent on the composition of its extracellular
matrix. It has been shown with macroscopic tissue-level experiments that much of
its compressive strength is due to highly charged polyelectrolyte molecules known
as glycosaminoglycans (GAGs).
The GAGs are packed very closely in the tissue
(with spacing approximately 2 - 4nm) and the electrostatic repulsion between them
is thought to contribute 50-75% of the tissue's equilibrium modulus in compression.
However, molecular level measurements of these forces have been only recently reported [24, 23, 20].
The development of high resolution force spectroscopy (HRFS) instruments, like
the atomic force microscope (AFM), has made it possible to measure the nanoNewton scale interactions of molecules [14, 3]. In HRFS, the force is measured by moving
a small cantilever or spring with an attached tip of known geometry towards a flat
surface. As the tip approaches the surface, the interaction forces will deflect the cantilever. From this deflection measurement, one can calculate the force of interaction
between the tip and the surface. Using a new type of HRFS known as a molecular
force probe (MFP), the interactions between GAGs can be measured directly.
17
1.2
Objective
The force between chondroitin-sulfate glycosaminoglycans (CS-GAGs) attached to a
substrate and a charged cantilever tip was measured using a molecular force probe.
To interpret these forces, models are needed which accurately represent the system.
The goal of this thesis is to develop models for the electrostatic component of the
force. By comparing these to the experimental results, one can determine which
factors play the most important role in the interaction between GAGs.
There has been some previous work done on modeling brush layers of polyelectrolytes [3, 19] although not specifically for the case of GAG molecules. These previous
models are simple and lend themselves to efficient numerical solutions. However, their
simplifications often do not reflect well the actual molecular geometry of the GAGGAG electrostatic interactions. In this thesis, a new and potentially more accurate
model is described, and a different numerical technique is employed for its solution.
These models are compared and evaluated with respect to initial experimental data
in order to determine the optimal approach for modeling GAG-GAG interactions.
1.3
Overview
This thesis is structured as follows. Chapter 2 gives background on the biological
properties of cartilage. Chapter 3 describes the theory of some of the molecular level
forces. In chapter 4, experimental methods and results are discussed. Chapter 5
describes in detail the three models which were implemented and tested. Chapter 6
gives the results of these models and compares them to some preliminary experimental
results. Finally, chapter 7 summarizes the findings and touches on directions for future
research.
18
Chapter 2
Articular Cartilage
Articular cartilage is the load bearing connective tissue found on the surface of movable joints. It normally sustains high compressive loads, 10 - 20MPa, without damage. The biomechanical properties of cartilage, such as its high compressive resistance,
are directly related to the molecular structure of extracellular macromolecules.
2.1
Cartilage Biology
Cartilage is a very complex avascular and alymphatic tissue (Figure 2-1). Articular
cartilage tissue is composed of 70-80% water by weight and contains only 20 - 40
thousand cells, called chondrocytes, per cubic millimeter [16, 17]. However, this low
density of cells maintains the extracellular matrix (ECM) under normal conditions.
2.1.1
Extracellular Matrix
The extracellular matrix provides most of the strength of cartilage tissue. The ECM
is produced and maintained by the chondrocytes. While it is composed mostly of
water, it is surprisingly strong mechanically.
The different macromolecular structures of the ECM account for different mechanical properties. The collagen molecules contribute to the tissues' shear and tensile
strength, while the highly charged proteoglycan molecules provide most of the com19
Hyaluronate
Agggrecan
ollagen
Chondrocyte
GAG chain
~~
Figure 2-1: The structure of cartilage (not drawn to scale).
pressive strength.
2.2
Proteoglycans
Proteoglycans make up 5-10% of the cartilage wet weight (35% by dry weight) [17, 25].
They are composed of a long core protein to which one or more glycosaminoglycan
(GAG) chains are covalently bound. Aggrecan is the most abundant proteoglycan
in cartilage (Figure 2-2). The core protein of aggrecan contains three globular domains. The first, GI, is found near the amino-terminal and uses a link protein to
attach to the binding region of hyaluronan, a glycosaminoglycan consisting of several
thousand repeating disaccharide units. The second globular domain, G2, is found
further down on the core protein and the third globular domain, G3, is found near
the carboxyl-terminal of the core protein. Between G2 and G3 is a highly charged glycosaminoglycan rich region. The core protein is several hundred nanometers long and
~ 300kDa. Proteoglycans, such as aggrecans, form large aggregates with hyaluronan
(Figure 2-2).
20
CS
/
N-linked oligosaccharide
Hyaluronate
binding region
0-linked oligosaccharide
Link protein
Hyaluronate
H a-r
CS = Chondroitin sulfate
KS-rich region
KS = Keratan sulfate
Figure 2-2: Schematic drawing of aggrecan, a proteoglycan.
2.2.1
Glycosaminoglycan
Aggrecan contains three major types of glycosaminoglycans (GAG): chondroitin-6sulfate, chondroitin-4-sulfate, and keratan-sulfate. The keratan-sulfate chains, which
are
-
5kDa, are the shortest consisting of about 10 repeating disaccharides. These
chains are located mainly near the G2 region of the core protein. The chondroitinsulfate (CS) chains occupy most of remaining GAG-rich region of aggrecan and comprises 95% of the molecular weight of the entire proteoglycan molecule (Figure 2-3).
The chondroitin-sulfate chains are 30 - 40nm long and are spaced approximately
2 - 4nm apart on the core protein. One end of the CS is covalently linked to the core
protein while the reducing end points into the intra-tissue space (Figure 2-3a). The
CS-GAG chain is composed of alternating glucuronic acid and N-acetyl-6(or 4)-sulfate
galactosamine (Figure 2-3b). Under normal physiological conditions, the carboxylic
acid and the sulfate groups are negatively charged.
It has been found that the sulfation of GAG chains, and therefore its charge, is
decreased in cartilage disease, such as osteoarthritis [22].
Studies have shown with
macroscopic measurements [16, 2] that the high negative charge of the GAG molecules
is the major determinant of cartilage compressive loading properties, responsible for
21
Ser
Xyl
=
00-
Serine
H2OSO-
HH
= Xylose
= Galactose
Gal
= Glucuronate
GluA
GlcNAc = N-Acetyl-galactosamine
\
HH
H
HH
H
H====
H
Sr-O-Xyl-Gal-Gal--fO-GluA-O-GlcNAc -n
HH
Glucuronate
Chondroitin sulfate
H
NHCOCH3
N-Acetyl-galactosamine-6-sulfate
(b)
(a)
Figure 2-3: (a) The structure of GAG; (b) Molecular structure of Chondroitin-6sulfate.
50-75% of the equilibrium modulus in compression.
22
Chapter 3
Molecular Level Forces
Intermolecular interactions are complex and must be modeled differently depending
on the distance and charge states of the molecules. Far from trying to give a general
model that takes every kind of interaction into consideration, this short description
will concentrate on the forces that are of interest for the modeling of the cartilage
GAG molecules (Figure 3-1).
3.1
3.1.1
Electrostatics
The Poisson-Boltzmann Equation
The Poisson-Boltzmann equation gives an expression for the potential in space in an
electrolyte solution [26, 10, 25]. This potential can then be used to compute the total
electrostatic force. There are several simplifying assumptions made: the ions in solution are treated as point charges and therefore take up no volume. The permittivity,
C, is everywhere the same as that of the bulk solution, ef, and is independent of any
electric field.
Start with Gauss's and Faraday's laws from Maxwell's equations,
V-cfE
V xE
p
=
-
23
(3.1)
at
(pH)
(3.2)
Electrostatic force between opposing GAGs Electrostatic force between interdigitating GAGs
Electrostatic force between neighboring GAGs
Steric force
Figure 3-1: Schematic of some of the forces felt by GAG molecules.
where E is the electric field, p is the volume charge density, p is the magnetic permeability, and H is the magnetic field. If time varying magnetic fields are assumed to
be negligible then the curl of the electric field is zero and a scalar potential, <b) (volts),
can then be defined. This is known as the electro quasi- static approximation:
V x E ~ 0, E =_- V D.
(3.3)
From 3.1 and 3.3, the potential and space charge distribution are then related by
Poisson's equation:
V2(b =
(3.4)
For biological tissues, the volume charge density will be due to the mobile ion con-
centrations in solution, pi plus any added fixed charge density pfi., associated with
macromolecules of the ECM. When the system is in thermodynamic equilibrium, pi
can be related to the potential by the Boltzmann distribution, which represents the
balance between ionic diffusion and electrical migration.
Fick's first law describes the relationship between the flux of the solute due to
24
diffusion and its concentration gradient [9, 27]. Also, if the solute is charged, it will
move in the presence of an electric field. The component of the flux due to this
electrical migration is proportional to the concentration, ci. The total flux of species
i, Fi, is due to both electrical migration and diffusion:
zi
-i
(3.5)
-piciE - DjVcj
where zi is the valence, pi is the electrical mobility, Di is the ionic diffusion coefficient
and ci is the concentration of species i.
In equilibrium, the flux of each ionic species is zero, and so the electrical migration
and diffusion fluxes must balance:
z.
piciE = DjVc.
(3.6)
lzil
As stated above, the electric field can be written as the gradient of a potential:
zi
piciv4
=-DiVci,
lzil
and this equation can be rearranged then integrated to get an expression for the
concentration profile of species i, ci :
i
=
jV4)
-Di( 1)Vci
Ci
l~il
piV_
=
-DjV(ln(c ))
Pi4 + Constant
=
-Di ln(ci)
_
lzil
lzil
using the Einstein relation,
D
=
RT:
zjFD + Constant
=
-RT ln(ci)
=i
(Constant)e-RT a
z=F
25
When D = 0, the concentration of i must be the bulk concentration, so the concentration profile of species i at steady state will be:
(3.7)
,T
Ci = Cioe~
where cio is the bulk concentration of species i.
Using 3.7, the charge density can be expressed as:
ziFci + pix
p =
S ziFc
0 6tY
±TPfix
(3.8)
where zi is the valence and ci(x) is the concentration profile, cio is the bulk concentration of ion i, F is Faraday's constant, R is the gas constant, T is the temperature,
and pfix is the fixed volume charge density.
For a solution of monovalent salt such as NaCl having bulk concentration Co,
equations 3.4 and 3.8, combine to give the Poisson-Boltzmann equation:
2
= 2FIC0
-1s .
2
F(D
)h
pf ix
Pf
(3.9)
The Debye length, r,-, is defined to be:
6f RT
r
2F 2 C,
(3.10)
This is the characteristic length over which the potential will decay. As shown in the
equation, it varies inversely with the square root of salt concentration.
3.1.2
Electrostatic Force
The total force on a surface can be calculated one of two ways: either by balancing
the forces on a control volume, or by differentiating the total energy with respect to
26
the direction (the energy method).
The force will be due to two terms: the osmotic term due to the ion concentrations
gradient, and the Maxwell stress term due to the force of the electric field acting on
the ion charge. The osmotic pressure due to the ion concentration relative to the
bulk concentration at a distance r from the charged surface, where (D
(=(r),for a
monovalent salt solution is given by van't Hoff's law:
Posm - P
(ci(r) - ci (rrf))
= RT
-';,
The outward swelling pressure is:
P.Smotic = RT(Coe-
"+ Coe+-T) - RTCO = 2RTCO (cosh
(
.
The Maxwell stress tensor, defined as acting outward from the surface, is:
Tig = EEJ
Tie
-c~jj
j
1
Tzz
-64
-EE,
2
=
1~ ~x
EkEk)
2
6 ij (-I
1
2 E(E,2+
EY)
The Maxwell pressure (defined as acting in on the surface) is the normal force that a
charge feels due to the electric field:
PMaxwell
2(V
-
2
2.
Thus, the total electrostatic force, Fe, per unit area in the z-direction will be the sum
of these two pressure terms:
Fe
Area
- 2RTCo
cosh F
\RT
- 1) +
2
(V))2 .(3.11)
Force, as always, can be written as the derivative of the energy, We:
a
Fe =
-We
Fe)z
27
(312)
The energy will be the sum of the terms due to the fixed charges, the osmotic pressure
due to the ion concentration gradient, and the Maxwell stress:
W = Jf
dS +
p101
-
-dV
2RTCo(cosh
T)
-
1) +
'f(V4)2)
dV
(3.13)
where the first term is a surface integral and the other two terms are volume integrals.
3.2
Non-Electrostatic Forces
There are of course other types of forces between molecules besides the electrostatic
forces, including steric and other short range forces. Steric forces are associated with
molecules that are being bent or stretched. Steric force may, however, be hard to
seperate from electrostatic components because they can also be caused by intramolecular repulsion. Therefore, the steric component is sometimes hard to isolate.
The repulsive force per area between two uncharged brushes has been previously
described. According to the theory by de Gennes [6], the force per unit area on an
uncharged brush of thickness h at separation D is:
Ps(D) =
F
Area
~
S3
D
_
2h
,h)D<2h
(3.14)
where the brush is composed of end grafted polymers and the polymers are modeled
as chains of s-sized sections. The first term is due to the non-ionic osmotic repulsion
between the brushes while the second term comes from the elastic interactions of the
chains. Since the GAG is a charged polymer, the total force could be approximated
as the sum of the electrostatic force and this uncharged brush steric force.
As molecules get closer and closer, their structure becomes of utmost importance.
Their geometry, their charge distribution and their mechanical degrees of freedom are
all affected. There is no general tool to describe very close interaction of long chain
charged polymers. Close polymer interaction is a vast field of study in itself [6, 19].
28
Chapter 4
Experimental Methods
4.1
High Resolution Force Spectroscopy
With the development of high-resolution force spectroscopy (HRFS) instruments like
the molecular force probe (MFP), it is possible to measure the nanoscale interaction
forces between molecules [14, 7, 11, 24, 23].
AFMs have been popular tools for
imaging surfaces, and recent work has shown that they are reliable tools for force
measurements as well [1, 5]. The MFP (Figure 4-1) is very similar to the AFM but
has been optimized specifically for the measurement of force. It overcomes some of
the optical interference problems inherent to standard AFMs and has slightly better
resolution.
In HRFS, a fixed surface is slowly approached by a small cantilever or spring with
a tip of known geometry and size (in these experiments, a hemisphere of 50nm in
diameter) held by a piezoelectric ceramic translator. As the tip feels repulsive or
attractive forces, the cantilever bends and its deflection is measured by a laser. Thus,
a cantilever deflection versus piezoelectric actuator distance curve (deflection curve)
is obtained. The cantilever spring constant is measured by analyzing its resonant frequencies. The data is then converted into a force versus distance curve (force curve)
using the known cantilever spring constant (Figure 4-2). These measurement techniques are particularly useful for biological applications because they can be carried
out in an aqueous environment without harming the sample. Therefore, using these
29
HEAD
15 rnLD
IR laser
mirror
laser
focusing te
tis
ho
dr
I tical
and
cariener
UP
ationt
Xra
lquid meniscus
m-pl--
---
stage
X
o.ptical
BASE
microscope
0
objective
Figure 4-1: The molecular force probe (MFP).
Thsterasaientswroeb
onlSo
2,2,
0
n
ealdepaaino
techniques, it is now possible to make molecular level measurements of the repulsion
forces between the GAG molecules, leading to a better understaning of how cartilage
behaves in a variety of conditions.
4.2
Experimental Setup and Results
These experiments were done by Joonil Seog [24, 23, 20] and a detailed explanation of
these methods is given in Appendix B. Glycosaminoglycan molecules (approximately
32nm in length) from rat cells were end grafted to a gold substrate (Figure 4-3). The
density of GAG on the surface was approximately one chain per 6nm x 6nm area
and the charge per GAG chain was approximately 1.097 x 10-17C. A gold coated
AFM tip was coated with either a charged monolayer or a neutral monolayer (Figure
4-3a). One substrate was also coated with charged self-assembled monolayer (SAM).
The force between it and the charged monolayer coated tip was measured in order to
determine the monolayer surface charge density. It was found to be approximately
30
0
tip touching surface force
1-)
Z
42
.,.
.. ......... .. ,
repul ive force
0
piezoelectric distance (nm)
(a)
0
0
no force
"
tip to surface distance (nm)
(b)
Figure 4-2: (a) An example of deflection curve; (b) An example of a force curve.
Gold
GAr
D
monolayer
backfill SA
t-GAG7
Gold(b)
(a)
Figure 4-3: Schematics of the experiments: (a) GAG coated substrate versus monolayer coated tip; (b) GAG coated substrate versus GAG coated tip.
31
......
Charged tip versus GAG surface
2.5\
2.
1.0M
1.5
0.01 M
0.001 M
0.0001 M
8
0.5
\-
0
50
100
150
distance (nm)
Figure 4-4: Experimental data of force between a charged monolayer coated tip and
a GAG coated substrate at various concentrations of NaCl.
-8.16 x 10-
. The forces between the modified tips and the GAG coated surface
were measured at different NaCl concentrations (from 0.0001M to 3.OM) using a
molecular force probe. Future experiments will be carried out with the tip also coated
with GAG (Figure 4-3b).
The measured force between the charged tip and GAG surface is shown in Figure
4-4. The majority of the force seems to be electrostatic in origin as the force decreases
with increasing ionic strength. The force between the neutral tip and GAG surface
is shown in Figure 4-5. The forces observed are smaller than those with the charged
monolayer-coated tip. However, it is interesting to note that there still appears to be
a long range force of electrostatic origin. Except for 0.0001M, the force gets smaller
as the ionic strength increases. The forces measured at 0.0001M are slightly smaller
32
Neutral tip versus GAG surface
2
1.8
1.6
1.4
-3.OM
1.2
-0.1M
0.01 M
S ..
001 M
2
0.0001 M-
-
0.8
-
0.6
--
0.4
--
0.2
---
0
10
20
40
30
50
60
70
distance (nm)
Figure 4-5: Experimental data of force between a neutral monolayer coated tip and
a GAG coated substrate at various concentrations of NaCl.
than those at 0.001M possibly because there is an offset of the curve due to an
incompressible layer of GAGs. This would result in the 0 distance being offset by the
thickness of this incompressible layer. The 0.1M-3M results are very close together
because the forces are only significant at a range when steric and non-electrostatic
effects occur. Therefore, the change in ionic strength only produces a small change in
the total force. The origin of this electrostatic force is discussed in detail in Appendix
A.
33
34
Chapter 5
Modeling of Electrostatic Force
To better understand the forces measured in the high resolution force spectroscopy
experiments, models were developed to more accurately represent the system [23].
Because the electrostatic forces are related by a nonlinear differential equation, these
models do not usually lead to a closed form analytical solution and, therefore, numerical techniques need to be employed to solve them.
5.1
Smooth Surface Charge Model
The simplest model is one in which the glycosaminoglycan (GAG) coated surfaces are
modeled as smooth, infinite planar surface charges (Figure 5-1a). Due to symmetry,
this system is essentially a one-dimensional problem.
5.1.1
Analytical Solution to Linearized Flat Surface Charge
Model
The Poisson-Boltzmann equation (Equation 3.9) is difficult to solve because it is
nonlinear. However, one can obtain an analytical solution to this problem by first
taking a linear approximation as follows [21, 3]:
,24
2=2FCo sinh
Ef
(
FG\
RT
35
2F 2 C
"I = rC
K
cf RT
(5.1)
r
"D
-_
---
- - --
--------
(a)
-
G,
0-
-
--
(b)
Figure 5-1: (a) Two approaching planes with charged surfaces; (b) Charged hemisphere tip approaching a flat charged planar surface.
where r,-' is the Debye length at the salt concentration Co. This is a good approximation if F'I is small.
The solutions to this linearized equation has the form 4D = Ae"z + Be-". Since
there is a surface charge density on both of plates (Figure 5-1a), the boundary conditions at z = 0 and z = D are
<-2 = -12 and
b =El respectively. Using
these boundary conditions and the linearized equation, the solution for the potential
between two infinite sheets of charge (Figure 5-1a) at a distance D apart is:
<(Z) =
92(cs
D)cosh(rz) - sinh(Kz) l
cosh(rz)
cfsinh(rD)
g
)
cf r, sinh(KD)
(5.2)
The force per unit area at any position z acting in the z-direction is equal to
the sum of two terms: the osmotic pressure due to the ion concentrations that are
distributed according to Boltzmann statistics and the Maxwell electric field stress
(Equation 3.11). The calculation for the force becomes trivial once the potential is
known and a ground (reference potential) is well defined. Here, the charged surfaces
are infinite in the x and y-direction but are finite in the z-direction. At z -+ 00,
the potential and the electric field are zero since this is very far from the charge
(see Appendix A). Therefore, the full solution for the linearized approximation of the
36
pressure, force per area, between infinite sheets of charge is [21]:
+
SFfl2t
PfAlat - Area
2cr1
j
cosh(iD) + aO
2 (2
2cf sinh
sinh2(rD)
2
(5.3)
Using the above equation, one can integrate this force on small circular cross-sections
and get an expression for a hemisphere tip of radius r (Figure 5-1b) [3]. If the surface
charge on the tip and on the substrate are of the same order and KD is small, then
some terms may be dropped and the resulting force is:
2w
where
j
Pflatrdr =
Fhemisphere
4wuou E2 r
-D
efK
Jo
(5.4)
-1 is the charge on the tip and o 2 is the effective surface charge of the GAG
modified substrate.
The solution to the nonlinear Poisson-Boltzmann equation for this geometry can
be solved numerically.
5.1.2
Numerical Method for Flat Surface Charge Model
Since this geometry is simple, one can use a Newton method on finite differences to
solve the nonlinear Poisson-Boltzmann equation. There are only boundary conditions
at z = 0 and z = D as discussed above.
Since the problem is one dimensional, the potential in space can be represented as
a one-dimensional matrix or vector, <D, where each entry in the vector is the potential
at evenly spaced points in the z-direction. The derivative in the z-direction can be
written as differences between neighboring discretizations.
a
41jontk
point
k
-+
([k + 1] -
Oz
02
19Z2
<D[k]
6Z
[k + 1] +
point
[k - 1] - 2D[k]
6en
+2
(5.5)
(5.6)
where 6z is the distance between points k and k + 1.
The Poisson-Boltzmann equation (Equation 3.9) for each discrete entry plus the
37
boundary conditions leads to a set of N nonlinear equations, where N is the number
of discretizations, all equal to zero if the potential at each point is correct. These can
be rearranged and represented as a vector-valued function A as follows:
<D[1]
- (1[2] + 6x L
0
2D[2] - o[1] - (D[3] + 62(2FO sinh(F"42]))
A (D
2i]
-Di_1]-_
'14i + 1] +
-1
2(D[N - 1] - ([N - 2] - (<[N] +
<D[N]
62 (2Fyo
6 2(2FCo
0
sinh(FEi]))
sinh(F[N-1]))
0
- (D[N - 1] - 6Ol
(5.7)
If a close enough initial guess for the value of the potential at all points is given,
then that guess can be refined using a Taylor series expansion. This is repeated until
the change in potential at each step is small enough. This algorithm is known as a
Newton method for solving multidimensional systems (here the dimension refers to
the number of elements in the vector, N).
Newton algorithm
where
(Dk
1.
(Do = Initial Guess, k = 0
2.
Compute A((Dk) and the Jacobian matrix of A,
3.
A
4.
(Dk+1 =
5.
k =k + 1
6.
Error
7.
Loop back to 2 until Error is below some threshold.
=
(
OA
(A(D
=
k + A
JIA1
2
is the vector of the calculated potential values at each of the discretized
points at the kth iteration.
The potential is then converted to a force by taking a bounding box with one
surface at point i where the derivative of the potential is zero and the other surface
38
.- . - --.
-
----
--
. ------
.
I
E
I
I
z
=
US1
S1
Z=0
z =O
iz=D
The control box is shown with
Figure 5-2:IFPotential between two charged surfaces.
F
the dashed line.
at infinity where the potential and electric fields are zero. The force on the enclosed
plate is then the osmotic pressure at i (Figure 5-2):
= 2RTCO cosh
<I[i]
(RT
Area
(5.8)
The hemisphere tip geometry is approximated by using the force results between the
flat surfaces and summing up the force on appropriately sized concentric cylinders. In
effect, this is the numerical version of the integral done for the closed form hemisphere
tip solution.
This numerical method was run in Matlab. Space was discretized to 200 increments (i.e. N=200) and the maximum error threshold was 10-10.
5.2
Volume Charge Model
Since GAG molecules are approximately 30nm long, the "brush"-like layer takes up
some volume above the gold substrate. Without taking into account the shape of
the separate molecules, this region can be modeled as a fixed uniform volume charge
density [19]. This new geometry should be a significant improvement over the the
39
AD
z
1
h
(I)
P
(I)
h =h
brush
(II)
brush
height
0
x
(I)
height
0
(a)
(b)
Figure 5-3: (a) Flat surface charge over a fixed charge volume; (b) Two approaching
surface with fixed charge volumes.
previous flat surface model because the flat surface model is only appropriate if the
separation distance D is much greater than the dimension of the charged molecules
grafted to the substrate (e.g., the length of the GAG chain) but the forces were
measured at forces at distances on the order of the GAG length. Therefore, the
experiments where the tip is coated with sulfate, the charged monolayer, can be
modeled as shown in Figure 5-3a and the experiments where the substrate and tip
are both coated with GAG molecules can be modeled as shown in Figure 5-3(b).
The model has two different regions. In the region outside the fixed volume charge
(I), the Poisson-Boltzmann equation has the same form as in the previous model. In
the region inside the fixed volume charge (II), the Poisson-Boltzmann equation has
an extra term added to take into account the fixed volume density.
I)
V24
= 2F0o sinh
Ef
II)
V2 D =
2F0O sinh
Ef
(I)
(RT
(I)
RT
(5.9)
(5.10)
-
Ef
The boundary conditions are at the two surfaces and at the edge of the volume. At
the surfaces, the boundary conditions are the same as before: the derivative of the
40
potential is proportional to the surface charge density. At the edge of the volume
charge, there is a continuity condition so that the potential and its derivative must
be continuous at that point.
Of course, since the Poisson-Boltzmann equation is nonlinear, these equations
are very difficult to solve analytically. However, due to symmetry, the problem is
one-dimensional and thus it can be solved numerically using a Newton method.
5.2.1
Numerical Method for Volume Charge Model
For the model shown in Figure 5-3a, there are three different boundary regions. At z
= D, there is a surface charge density so the condition is -24 az -2-.Ef
At z = 0, there
0 (see Appendix C for a more general
is no surface charge so the condition is
discussion of this boundary condition). Finally, at the boundary between region (I)
and (II) at z = h, the potential and electric fields must be continuous. The model in
Figure 5-3b has one extra condition on the edge of the second volume but overall the
equations look the same.
Since the problem is one dimensional because of its symmetry, the potential in
space can be represented as a vector (Figure 5-4a), 4D, where each entry is the potential
at evenly spaced points in the z-direction, as in the previous model. This again leads
to a set of N nonlinear equations, where N is the number of discretizations. These can
be rearranged to define a vector valued function A as follows (where the kth point
41
D
"z
D'<h
(I)
y
NIX = (Pflx) (h/D)
D>h
h=
brush
height
NPix
(a)
(b)
Figure 5-4: (a) A flat surface charge over a fixed charge volume a distance D >
brush height apart; (b) A flat surface charge over a fixed charge volume a distance
D' < brush height.
corresponds to the point on the edge of the volume charge):
P[1] - D[2] +
2D[2] - 1[1] - 1[3] +
21[i]
-
4[i - 1]
-
A(D) =
-
Ef
'1[i + 1] + 6
6
0
Pf-)
6 (2"O sinh( F"d2]) -
- 2] - 1[k] +
2-[k -1]-
sinh(F",l])
5 (2"O
RT
Pfi2)
Ef
)
(2FCo sinh(ETE) -
2(2FCO
sinh(F11[k-1])
_
___
0
24)[k] - 4)[k - 1] - D[k + 1]
21[k + 1] - 4[k] - 4[k + 2] + 62(2FCO sinh(F
21[i]
-
_ 1]
-[i
-
1[i + 1] +
21[N - 1] - 1[N - 2] -
[N] +
62(2F00
6
2(2FCo
4[N] - P[N - 1] - 6x
[k+l]))
sinh(ETE))
sinh(FlN-1]))
0
If the plates are close enough together and the volume charge takes up the whole
space, then there are only two boundary conditions and the system simplifies to the
42
following (Figure 5-4b):
1[1] - (D[2] +
21[2]
A(f) =
2[i] -
-
[i-
62(
D[1] - D[3] +
1]
-
X
6
sinh(F("]
RT
Ef
2 (2Fo
XEf
1 [i + 1] +
24[N - 1] - @[N - 2] - P[N] +
sinh( F"2
T
62(2FCo
6
-]
-
PfE
sinh(!f E) -
2 ( 2FCo
X
0
P-ix
Ef
)
sinh( F4D[N-1)
Ef
4[N] - 4[N - 1] - 6x(Ewhere p'
= pfix-
0
0TE
0
is the adjusted volume charge density now that the plates are
a distance D' < h apart. These two systems of equations can both be solved by a
Newton method (as in previous section). The solution to the linearized equation is
used to obtain a good initial guess at the first distance. The solution for the potential
at each distance is used as the initial guess for the potential at the next distance.
The potential is used to compute a force by taking a Gaussian bounding box with
one surface in region (I) where the derivative of the potential is zero and the other
surface at infinity where the potential and electric fields are zero. Then, the force on
the enclosed surface is then due only to the osmotic pressure term at the first surface
in region (I).
Finally, the force due to the hemisphere tip geometry is approximated by summing
the forces due to appropriately sized concentric cylinders (Figure 5-5).
This numerical method was implemented in Matlab. Space was discretized into
200 increments (N=200) and the maximum error threshold was 10-10
5.3
Charged Rod Model
Since the chondroitin sulfate chains on our surface are approximately 30nm long but
only about Inm wide [2], the long chain polymeric shape might have an effect on
the force profile that is not taken into account by the smooth volume charge model
above. To account for certain aspects of molecular shape, the GAG coated surfaces are
43
(I)
D
h
(II)
Pfix
Figure 5-5: Approximated hemisphere of charge over a fixed volume of charge.
Figure 5-6: Two parallel plates with charged rods.
modeled as two parallel plates with charged rods (Figure 5-6). Once more, because
the Poisson-Boltzmann equation is nonlinear, this can only be solved numerically.
However, due to of its lack of symmetry, the model of two approaching brush surfaces
(Figure 5-6) is now a three-dimensional problem. This must be solved by a slightly
modified method, as the matrices that would result from a normal Newton method
would be too large (trillions or more entries) to be manipulated and stored efficiently
on normal modern computers.
5.3.1
Numerical Method for Solving Charged Rod Model
The surfaces can be broken down into small repeating units (Figure 5-7). The space
potential of one unit is solved numerically and the force corresponding to that potential is calculated. The rods are modeled as volumes with a fixed charge density
44
charged
rod
~3 nm
Figure 5-7: Repeating unit in the parallel charged rod plate model. D is the distance
between the square base plates. This unit is discretized into cubic elements of size 6o,
by 6o, by 62.
and the substrates onto which they are attached are modeled as flat surfaces with no
surface charge. The dimensions of the unit are determined by the coating density of
the GAG on the surface and the distance between the tip and surface. The total force
for the two surfaces at the current spacing, D, will be the force for one unit times the
number of units needed to represent the surface. The hemisphere geometry of the
tip can be approximated by numerically integrating in the same manner as with the
previous -models.
The units are discretized into small elements with dimensions 6x, 6g, and
6 2.
The fixed volume charge and potential for each cubic element are stored in a threedimensional matrix. As before, the continuous derivatives in Cartesian coordinates
45
are discretized to finite differences:
a2
02
92
OZ24
([i + 1, J, k] + 4)[1 - 1, j, k] - 24)[i, j, k]
VD point(ij,k)
62
P[i, j + 1, k] +
+
[ij
-
1, k] - 2 [i, j, k]
62
y
[i, j, k + 1] + 1[i, j, k - 1] - 2)[i, j, k]
62
z
Then, the continuous Poisson-Boltzmann equation can be written in discrete form
(where pfix[x, y, z] is the volume charge for the discrete volume at (x, y, Z)):
f (4[i, j, k])
f (a)
P[i-7j,k]
= P[i, j, k], where
=
1
2a(
x
+
1
) + A sinh(Ba)
z
=(1~i±1,i~kl±1[i-1,jik])(-
k] +
+([i,j+±1,
1
[i, j - 1, k])
y
+(1[i,j,k + 1] + 1[i,j, k - 1])(62)
z
Pf ix [i7j, k]
For each element that is not a boundary, this equation is used to update the potential. Boundary conditions determine which values of 4[x, y, z] and which differences,
P[x, y, z], are fixed:
a
ai
-,a- where
i is the direction of the boundary.
(5.11)
For example, if there is a charged surface between the element at (i, j, k) and the ones
at (i + 1,] , k) and (i, j + 1, k), Equation 5.11 leads to the following two equations for
46
the potential at point (i, j, k):
TD[i + 1, j, k] - (1[i,j, k]
_-
1[i, j + 1, k] - 1[i, j, k]
o-
If the equation is not satisfied, then (D[i, j, k] must be changed by some amount, A,
approximated using the Taylor series approximation as follows:
f (4b +,A)
~f (,D) + A f '(<D)
f(14i,j,k]+A)
P[i, j, k]
=
~
P[i, j, k]
f((D[i, j, k]) + Af'((b[i, j, k])
P[i, j, k] - f (4b[i, , k])
f'(1b[i, j, k])
For the boundary conditioned elements, A is taken to be the average of the A calculated for each of the equations. Therefore, for the example boundary above, A can
be defined by these equations:
A,
=
A2
A
D[i + 1,jk] -
<[ij,
<b[Zjj+1,k]-
<[ijk]+
Al
+ A2
2
The full algorithm is as follows:
47
k]+
Ef
Ef
6x
6y
1.
Initialize D (set everything to 0 for example)
2.
For all (i, j, k), calculate A[i, j, k] based on current 1[i,j, k]; check if it is a boundary
and calculate A[i, j, k] appropriately if it is.
3.
D <- 4+ A
4.
Error =
5.
Loop back to 2 until
il 2
a. a certain fixed number of iterations is reached
b. until Error < MAXERROR
6.
Calculate force from 1 at current distance D
7.
D = D - d (decrease the distance between the two plates by d)
8.
Loop back to 2 until D = 0
For these experiments, the maximum number of iterations was 10000 and the MAXERROR was 10'0.
This is a Newton method like the one used in the volume charge model above
but in this case the Jacobian matrix is too large to store in memory. Therefore, this
method employs a "matrix-free" Newton method approach.
Because the potential needs to be solved for several values of D in order to get
a full force curve, the solution of the potential at D is used as the initial condition
for the potential D - d, the next distance. If d is made small enough, the potential
at D will be very close to the potential at D - d and therefore the algorithm should
converge fairly rapidly at each iteration.
The energy method is used to calculate the force in this model. The total energy
at distance D is calculated by numerical integration. The energy contribution of one
48
element that is not a boundary is:
j, k]
=
(pfix[ii , k]f[i, j, k] )6x6y64
w[ij,k]
=
-(2RTCo(cosh(
Wf [i,
M
. e
wmz,,k
k]
[i)
=
TI[i,, k]) - 1) 6X6 6z
[i
-+ 1') k,
2
k]
6X
+ 4D[i, j + 1, k] - D[i, j, k]
6Y
+
w[ij,k]
[i,j,k + 1] -
[, j, k])
6 6
wf +wo+m
where Wf is the energy contribution due to the fixed charge density, wo is the energy
contribution due to the osmotic pressure, and wm is the energy contribution due to
the electric field stress.
The energy contribution of an element with boundary conditions is the average
of the potential at (i, j, k) and the potential on the other side of the boundary multiplied by the surface charge density and by the surface area. Hence, for the example
boundary element described above, the contribution to the energy will be:
. .
w[i, j, k] =-
((D[iI j, k] _+(D[z + 1, j, k]) 6j +or(4)[Z, j, k] + (1[t., j + 1, k])
2
+
66
The total energy at distance D will be the sum of all these contributions.
Wtotal =
w[i, j, k]
Z: Z
x
y
z
The force is the derivative of the energy in the z-direction, which is approximated by
finding a second order polynomial of best fit for the local energy. The force at point
z
=
D is approximated as the derivative of the quadratic fit of the energies calculated
at distances D-( < z < D+( evaluated at z = D, where ( is 1.5nm unless otherwise
49
stated.
The numerical method to solve for the potential and the energy calculation were
implemented in C and run in parallel on several machines so as to calculate the energy
at every increment of one nanometer. There were two machines allocated per ionic
strength run. For example, there were two machines calculating the energies at 0.01M
NaCl starting at 100nm down to 2nm where the first machine was calculating at even
number distances and the second one at the odd distances. It was determined that
discretizing by 50 increments in the x- and y-directions and 100 in the z-direction was
enough to ensure convergence since more discretizations produced the same results.
At this discretization, it takes about 24 - 48 hours to run the program at a single
concentration for all the distances on one 1GHz PentiumIII. Therefore, doing the
calculations in parallel, getting the results for all the concentrations at all the distances
takes about 24 hours. The conversion of the energy to force was done in Matlab.
This model assumes the rods are stiff so that they do not move until there is contact. After contact, the rods maintain the same volume by getting shorter and wider
and maintaining their total charge. If the GAG molecules are not really completely
rods and move as the surfaces approach one another, the rod behavior as the surfaces
approach could be altered.
Of course, the description of this simulation by no means models all that is happening at the surfaces. There are other interaction forces besides the electrostatics.
However, because GAG molecules are very highly charged, the contribution of these
forces to the total force should be small, especially at distances greater than the
length of the GAG chain.
50
Chapter 6
Results and Discussions
Measurements were taken using the charged and neutral monolayer-coated tips. The
tip radii were measured to be approximately 25nm. In this chapter, the different
models are compared to the experimental data (see Chapter 4). Because of difficulties
in coating the tip with GAG chains, the experiments of GAG versus GAG are not
yet completed. However, it is interesting to see what the different models predict for
these experiments since modeling GAG-GAG interactions is the ultimate goal of the
project.
6.1
Smooth Surface Charge Model
The smooth surface charge model is a useful tool as it is relatively easy to solve and
there is a closed form analytical solution to it in the linearized approximation limit
(see Chapter 5). However, although the model predicts the correct trend, it does not
fit the data well.
The Debye length at various salt (NaCl) concentrations is summarized in Table
6.1. The values calculated using the model are compared to the experimental results
when all model parameters are fixed at the values determined experimentally or from
theory (see Table 6.1).
The Debye length is the length scale over which the force
should decay away when two charged surfaces approach each other. However, since
the GAG "brush"-like layer takes up a volume, is not flat, and the forces are acting
51
NaCl concentration (M) Ir (nm)
3M
0.18
iM
0.30
0.1M
0.96
0.01M
3.04
0.001M
9.61
Table 6.1: The Debye length at various NaCl concentrations.
Parameter
NaCI Concentration
Debye Length
Value
1-0.001M
See Table 6.1
Tip Radius
Tip Surface Charge
Density
Effective GAG Surface Charge Density
25nm
-8.16 x 10 4 6
Method for determining
Known quantity of salt added
Theoretical value from NaCl concentration
(see Chapter 3)
Measured from SEM picture
From previous experiments (see Chapter 4)
-0.18964
Calculated from experimental values of
chain length, number of repeating charge
units and the GAG coating density on the
surface
Table 6.2: The flat surface charge model parameters.
52
x 10-4
Comparison of nonlinear surface charge model to its linearized approximation
1
-
0.9
-
I
-
0.01 nonlinear
- M linear
0.01M linear
---
0)
C.)
M noniinear
0.M nonlinear
0.8
0. 1M linear
0.7
-M linear
0.6
0.01M nonlinear
0.5
0.1M nonlinear
0.1M linear
0.01M linear
0
1M nonlinear
0.4
0.3
0.2
- .
0.1
.........
..
0
0
10
20
30
40
50
60
70
distance (nm)
Figure 6-1: A comparison of the force predicted by the surface charge model using
the full solution and the linearized approximation.
53
Comparison of nonlinear and linearized surface charge models to experimental data
I
F
1M I UMnled[
- - a.M nonlinear
0.01 M nonlineai_
1 M data
0.1
0.09-
- M linear
0.1M data
0.08
0.1M linear
-
-
0.01M
-
data
0.07-
0.01M linear
1 M data
0.1M data
Q0.1M data
0.01M linear
0.06 k
0'
0.05
0.1M nonline ar
0
0.04
0.1 M nonlinea r
0.03
- M linear and
1 M nonlinear
0.02
0.01
0
0
0
10
20
30.
40
10
20
30
40
50
60
50
60
distance (nm)
80.... 90.100
70......
70
80
90
100
Figure 6-2: Surface charge model (nonlinear and linearized) versus experimental data.
54
on the length scale of the GAG molecules, the force measured does not decay in the
same way as it would for flat charged surfaces (Figure 6-2).
The nonlinear model
does not fit as well as the linearized approximation model because the linearized
approximation overpredicts what the force should be between two charged flat sheets
(Figure 6-1). However, since in this case the GAG molecules take up a volume and
the forces observed are larger than those for flat surface charges, the error in the
linearized approximation is in the right direction and therefore the linearized model
is closer to the data then the nonlinear model (Figure 6-2).
Of course, some of the model parameters can be changed so that the data and
the surface charge model results are closer. For example, one can change the model
Debye length in order for it to fit the data more closely. However, the discrepencies
between this model and the experimental data are largely due to the geometry of the
model not reflecting the geometry of the experimental setup. Artificially changing
the Debye length to fit the data reveals no new information and can be misleading.
Although the flat surface charge model predicts the right trends, the GAG surface is
not well modeled by a flat surface charge density and so the model results are not
expected to be close to the experimental data.
6.2
Volume Charge Model
The model parameters for the volume charge model were fixed to the previously
known values (see Table 6.2). The volume charge model fits the data reasonably well.
The model results for the volume of charge approaching a flat charged surface are
shown in Figure 6-3. There is a cusp at the point where the flat surface touches the
top of the volume (32nm). However, when the tip is made to be a hemisphere, the
cusp is not as conspicuous as it is smoothed out in the numerical integration (Figure
6-4).
The model tends to underpredict the experimental results when compared to
the GAG versus charged monolayer data (Figure 6-5).
This is probably because
modeling the GAG surface as a volume charge does not take into account the discrete
55
Method for determining
Value
Parameter
Known quantity of salt added
Theoretical value from NaCl concentration
1-0.001M
See Table 6.1
NaCl Concentration
Debye Length
Tip Radius
Tip Surface Charge
Density
Monolayer
Neutral
(see Chapter 3)
Measured from SEM picture
From previous experiments (see Chapter 4)
25nm
-8.16 x 10-4
From literature [13, 18]
1-2 times Eo
Permittivity
GAG Volume Charge
Density
-5.925 * 106
Calculated from experimental values of
chain length, number of repeating charge
units and the GAG coating density on the
C
surface
Charge
Height
Determined from GAG length
32nm
Volume
Table 6.3: The smooth volume charge model parameters.
1.
X 10'
Volume charge model of flat surface approaching volume charge
5
Volume height
=
32nm
Force at 0.01 M
0.5
0
10
20
30
40
distance (em)
50
60
70
Figure 6-3: Force between a smooth volume of fixed charge and a flat charged surface
(1nm x 1nm) at 0.01M.
56
Volume charge model
of
charged hemisphure approaching volume charge
0.5
0.450.4-
Force at 0.01 M
0.35-
0.34
-
Volume height
32nm
a 0.25
0.2-
0.15-
0.1
0.05-
0
10
20
30
40
distance (cm)
50
60
70
Figure 6-4: Force between a smooth volume of fixed charge and a flat charged hemisphere (radius=25nm) at 0.01M.
nature of the GAG chains. The discrepancies are most apparent at distances less than
10nm. This is because at distances smaller than 10nm, the steric and nonelectrostatic
components of the force also tend to be more significant. Therefore, any model which
only considers electrostatic forces should deviate from experimental results at small
distances.
The smooth volume charge model can be made to fit the data more closely by
increasing the surface charge density parameter from the value previously calculated.
Although this will make the lower concentrations fit better, it will not change the fit
at the higher concentrations notably since the distance on which the surface charge
makes significant differences to the force is very small (see Table 6.1).
The model tends to overpredict the force for the neutral monolayer-coated tip
(Figure 6-6) at distances greater than 10nm. This discrepancy in the force is most
likely due to the modeling of the tip monolayer as a perfect insulator; in reality, the
layer is not a perfect insulator and has a finite non-zero conductance. Therefore, the
actual induced surface charge on the surface will be smaller than that predicted by
the model. This in turn leads to a smaller force. For distances less than 10nm, the
model predicts forces which are smaller than the experimental value. This is because
57
Comparison of volume charge model to charged monolayer vs GAG experimental data
0.5
- M model
0.1M model
0.01M model
0.45
S1M data
- 0.1M data
- 0.01M data
1 M data
0.4
0.1M data
0.35
0.01 M data
0.3
-M model
Volume height =32nm
C
0.25
1M model
0
0.2
0.01M
model
0.15
0.1
0.05
C
0
5
10
15
25
20
30
35
40
45
50
distance (nm)
Figure 6-5: The volume charge model compared to experimental data for GAG surface
versus charged monolayer-coated tip.
58
Comparison of volume charge model to neutral monolayer versus GAG experimental data
0.5
omMcidel
1 M model
0.45
0.1M model
3M data
0.01M model
- 3M data
S1M data
- 0.1M data
1 M data
0.4
0.1M data
0.351
0.01 M data
''-
0.3
Volume height
3M model
=
32nm
(D
C.) 0.25
1 M model
0
0.2
0.1 M model
0.15
0.01M model
0.1
0.05
0
0
' ''' '
5
10
15
20
25
30
35
40
45
50
distance (nm)
Figure 6-6: The volume charge model compared to experimental data for GAG surface
versus neutral monolayer-coated tip.
59
at small distances nonelectrostatic effects become very significant. This is especially
true at the higher ionic strengths where the electrostatic component of the force will
be relatively small. There appears to be a certain nonelectrostatic component of the
force which is independent of ionic strengths and begins at about 10nm. This is
evident since the experimental data at 1-3M for distances less than 10nm lie rather
close. The electrostatic component of the force for these ionic strength should be
small. The model, which overpredicts the actual electrostatic component, predicts
forces which are about 5-10 times smaller than the experimental results at IM and
3M NaCl. Therefore, most of the force measured is due to steric and nonelectrostatic
effects.
For the volume charge versus volume charge (which models the GAG versus GAG
surface), the cusp is at 64nm when the two parallel volumes (i.e. flat tip versus a
parallel flat substrate) approach each other since that is when the two volumes touch.
Of course, the cusp is smoothed out when the model assumes a hemisphere tip. This
model assumes that the GAG brushes do not interpenetrate at all. Once the fixed
charge volumes touch, the fixed charge density is constant everywhere between the
tip and the bottom of the substrate. The results are illustrated in Figure 6-7.
Because of difficulties with the GAG tip modification chemistry, there are not yet
any experimental data to which these model results can be compared. Nonetheless,
a few interesting remarks can be made about these results. The force at 32nm and
0.1M NaCl is about 0.03nN. This translates roughly to a pressure of 0.1MPa. This
predicted force can be compared to the macroscopic measurements of cartilage as the
volume charge density at that distance is equivalent to a GAG-GAG spacing of 3nm,
which is approximately physiological conditions. The value reported in literature for
the measured modulus of cartilage tissue is about 0.7MPa and it is thought that
electrostatic interactions of GAG account for about 0.29MPa [8, 15, 2].
60
Volume charge model of GAG-tip approaching GAG surface
0I
0.
I
3M
iM
0.1M
-
0.01 M
0.5
32nm
64nm
0.01M
0.4
U.IM
0
I vi
0.3 -
3M
0.2 -
0.1
0
0
-
10
I
-
20
30
40
50
60
70
80
90
100
distance (nm)
Figure 6-7: The volume charge model of a GAG-coated tip approaching a GAG-coated
surface.
61
Parameter
NaCl Concentration
Debye Length
Value
1-0.001M
See Table 6.1
Tip Radius
Tip Surface Charge
25nm
-8.16 x 10-4c
Method for determining
Known quantity of salt added
Theoretical value from NaCl concentration
(see Chapter 3)
Measured from SEM picture
From previous experiments (see Chapter 4)
Density
Monolayer
Neutral
Permittivity
GAG Volume Charge
Density
Charge Rod Height
Charge Rod Spacing
Charge Rod Width
From literature [13, 18]
1-2 times co
106__
Calculated from experimental values of
chain length and number of repeating
charge units
From GAG length
From GAG coating density
Approximated from GAG chain diameter
in literature
-5.925 * 106
32nm
6nm
1nm
Table 6.4: The rod model parameters.
6.3
Charged Rod Model
The model parameters for the volume charge model were fixed to the previously
known values (see Table 6.3). The model results for charged rods approaching a flat
charged surface are shown in Figure 6-8. Similar to the charged volume model, there
is a cusp where the rods first touch the flat surface (32nm) but when the tip is made to
be a hemisphere this cusp is smoothed out in the numerical integration. The charged
rod model predicts higher forces than the smooth volume model because the same
amount of charge is in a much smaller volume. It overpredicts the force at 0.1M but
fits the data at the other concentrations better than the volume charge model (see
Figure 6-9). The model results predict smaller forces than the data at distances less
than 10nm as that seems to be the range where steric and nonelectrostatic forces are
more significant. Also, the model tends to slightly underpredict the force at
-
32nm.
This is probably due to polydispersity, which means that not all the GAG chains
are 32nm long but have an average length of 32nm with some standard deviation.
Therefore, the longer chains will cause the force to increase sooner than predicted.
62
I-
Charge rod model of flat charged surface approaching a surface with charged rods
0.9 -
0.8 -
0.7 Rod height = 32nm
0.6 Force at 0.01 M
0.50.4 -
0.3-
0.2-
0.1 -
0
10
20
30
40
distance (nm)
50
60
70
Figure 6-8: Force between rods of fixed charge and a flat charged surface (Inm x 1nm)
at 0.01M.
There might also be some initial resistance that is not electrostatic in nature when
the tip hits the top of the GAG chains that cannot be taken into account by the
charged rod model.
This model also tends to overpredict the force for the neutral monolayer-coated
tip (Figure 6-10). This is expected, as explained in the previous section, since the
monolayer is not a perfect insulator. For distances less than 10nm, the model predicts
forces which are less then the experimental value. Again, this is because, at those short
distances, non-electrostatic components of the force become much more significant.
Because of difficulties with the GAG tip modification chemistry, there are not yet
any experimental data to which these model results can be compared. However, these
data can be compared to macroscopic modulus measurements [8, 15, 2]. The force
at 32nm and 0.1M NaCl is about 0.03nN or about 0.1MPa (Figures 6-11 and 6-12).
The value reported in the literature for the measured modulus of cartilage tissue is
about 0.7MPa and the electrostatic interactions of GAG is about 0.29MPa of the
total modulus. This is essentially the same as the result for the volume charge model
since the two models' result curves cross at approximately 32nm.
63
Comparison of rod model to charged monolayer vs GAG experimental data
1 model
0.1 M model
--
lM data
0.45
S
0.1M data
. 0.01M model
.Mdata
0.1 M data
0.4
-
0.01M data
0.35
- M model
0.3
-
>0.25
rod height = 32nm
-
model
-0.1M
0
0.2
0.01M model
-. -.
0. 15
0.1
0.051
- -
- -.
0
0
5
10
-
15
20
30
25
distance (nm)
35
40
45
50
Figure 6-9: The charged rod model compared to the experimental results of a charged
monolayer-coated tip approaching GAG surface.
64
Comparison of charged rod model to neutral monolayer versus GAG experimental data
0.5
- M model
0.1M model
0.01M model
S1M data
0.1 M data
0.01 M data
0.45|
1 M data
0.4F
0.1M
0.35F
data
0.01 M data
0.31 M model
Rod height= 32nm
0.25
0. 1M
0.2
model
0.01 M model
0.150.1
- *
'.
0.05
0
4
8
12
16
20
24
28
distance (nm)
32
36
40
44
48
Figure 6-10: The charged rod model compared to experimental data for a neutral
monolayer-coated tip approaching a GAG surface.
65
1
Charge rod model of two approaching surfaces with charge rods
x 10'
0.9
0.8
0.7
Rod height
=
32nm
2x Rod height
o-
64nm
0.6
0.5
Force at 0.01 M
-
0.4
0.3
0.2
0.1
0
10
20
40
30
50
distance (nm)
70
60
80
100
90
Figure 6-11: Force between two approaching 1nm x 1nm surfaces with charged rods
at 0.01M.
0. ID I
I"
I
Rod charge model of GAG-tip approaching GAG surface
I
:
iM
--
0.5
411
32nm
0.4
0.1MA
0.01 M
64nm
001M
i.M
0.3
U
1M
0.2
0.1
01
0
10
20
30
40
50
distance (nm)
60
70
80
90
100
Figure 6-12: The charged rod model of a GAG-coated tip approaching GAG surface.
66
Force between charged monolayer and GAG compared to models with added steric component
1
-
__
0.9
0.01M data
Steric model (de Gennes)
Volume charge model + StE c
Charged rod model + steric
0.8
0.01M data
0.70.6
0
0.5
charge rod model + steric mod el
L
volume charge model + steric nodel
0.4
steric model
0.30.2 0.1 U. I
0
10
20
30
distance (nm)
40
50
60
Figure 6-13: Charged monolayer versus GAG force at 0.01M NaCl compared to 2
models with added approximated steric effect.
6.4
Comparison of Results
All the models predict forces smaller than the experimental results at short distances
(less than 10nm) because no steric or other non-electrostatic effects were taken into
account. Using the equation in Chapter 3 for the force between uncharged brush layers, the non-electrostatic component of the force can be approximated. As previously
mentioned in Chapter 3, the steric component is hard to isolate from the electrostatic
effects in theory and therefore it is not surprising that although the new models with
added steric effects fit the data better they still show large discrepancies at the smaller
distances. The de Gennes steric model [6] does not take into account intra-GAG electrostatic repulsion and therefore is not really appropriate to use for these experiments
for anything more than rough approximations. This is also noticeable if the predicted
67
0.3
Neutral monolayer tip versus GAG at 3M NaCl experimental data compared to de Gennes steric theory
.
- 3M experimental dat6
-de Gennes theory
0.25
0.2experimental data
z
)0.
15
-
P
0
0.1
steric theory
0.05-
0
0
10
30
20
40
50
60
distance (nm)
Figure 6-14: Neutral monolayer versus GAG experimental results at 3M NaCl compared to the de Gennes steric model.
steric force is compared to the experimental results of the neutral monolayer coated
tip and GAG surface at 3M NaCl. At this ionic strength the inter-GAG electrostatic
forces should be very small since the Debye length is only about 0.2nm and therefore
the steric and nonelectrostatic effects should dominate. However, as shown in Figure
6-14, the de Gennes steric model does not exhibit the behavior that would match the
data. From the very poor fit of the flat surface charge model to the experimental data
for the charged monolayer-coated tip versus GAG surface in comparison to the other
models, it can be concluded that the space that the GAG molecules occupy is very
important in determining interaction forces. This is reasonable considering that the
length of the GAG chain is of the same order as the distance between the tip and the
surface. It is not yet clear which of the two other models, the volume charge or the
68
charged rod, is more appropriate for the GAG surface. It seems that, in the range
where electrostatic forces are dominant, the charge volume model provides a lower
bound while the charged rod model provideds an upper bound on the experimental
data. Since the way the charged rods move as they are pressed on by the tip can make
significant changes in the resulting force, it might be that the GAG chains are best
modeled by rods which lie down instead of compress as the two surfaces approach.
Experimental data for the GAG versus GAG has not yet been obtained as there
were difficulties in coating the GAG onto the tip surface. However, observations can
be made about each of the different models' behavior. As can be seen in Figure
6-15, there are significant differences in the behavior of the predicted curves for the
volume charge and the charged rod models. The main reason is that the volume
charge model assumes that there is no interdigitation of GAG molecules whereas the
charge rod model assumes that there is. The predicted force at 32nm in O.1M NaCl
is about the same in both models which means they compare about the same way
to the macroscopic modulus measurements. Until some measurements are done on
the molecular level, there is no clear indication of precisely how the GAG molecules
behave at physiological ionic strength.
69
Comparison of 2 models of a GAG-coated tip approaching a GAG-coated surface
0.7
-
0.6-
0.5
m
1M volume model
0.1M volume model
0.01 M volume model
- rod model
1M
M rod model
0.o1e
0.01M rod model
0.01M rod model
-
0.01M volume model
0.4 -
0. 1M rod model
C
0
0. 1M volume model
0.3 r
1 M rod model
0.2-
0.1
1 M volume model
-
....
...
.
0
10
20
40
30
50
60
70
distance (nm)
Figure 6-15: Comparison of the volume charge and charged rod models of GAG- boated
tip approaching a GAG-coated substrate.
70
Chapter 7
Conclusions
7.1
Summary
Articular cartilage is the load bearing tissue found on joint surfaces. Macroscopic
tissue measurements have shown that its ability to withstand large compressive forces
is primarily due to the highly charged glycosaminoglycan (GAG) molecules found in
its extracellular matrix. With tools such as the molecular force probe (MFP), the
molecular forces between thesis GAG molecules can be measured [24, 23, 201.
In this thesis, different models for the electrostatic repulsion between GAG "brush"like layers were developed and compared to experimental MFP force data for a charged
monolayer coated tip and a GAG coated surface. It was shown that modeling the
surface coated with a "brush" -like layer of GAG as a flat surface charge predicted the
trends in the data but greatly underestimates the actual force measured. Modeling
the "brush" as a smoothed volume of fixed charge only slightly underestimated the
actual experimental results. This second model is a useful tool in that it fits the data
relatively well, since the resultant force from the volume model has a shape much
closer to the experimental data than the flat surface charge model, and it is an easy
model to implement. However, modeling the "brush" as a surface covered with rods
of fixed charge predicts a force which was even closer to the experimental results.
This model, however, tends to slightly overestimate the force in the region where
electrostatic forces are dominant. This would seem to indicate that the "brush"-like
71
layer behaves like a surface covered with rods that are not completely rigid.
It was also shown that there can be an electrostatic component of the force when
the MFP tip is coated with a neutral monolayer. The monolayer can be modeled
as a perfect insulator to predict the forces.
However, it appears that this is too
strong a condition and that incoporating a conductance into the model would be
more appropriate. It was noted, however, that the electrostatic component of the
force only dominates at distances greater than 10nm. At smaller distances, there
appears to be a large steric and non-electrostatic component of the force.
Since there are not yet any MFP results for a GAG-coated tip interacting with a
GAG coated surface, both the volume and the rod charge models were compared to
macroscopic results. They both found forces which were of the same order as these
experimental results.
7.2
Future Work
Because the rod model has more parameters and is more general than the smooth
volume model (a smooth volume is just one big rod), it can be better modified to
fit experimental results and give more insight into the behavior of the GAG chains.
Since the behavior of the rods as the distance gets smaller than their height greatly
determines the shape of the force curve, more studies need to be carried out to fully
understand how these GAG molecules behave in tissue.
Experiments need to be
carried out to measure the forces between GAG "brush"-like layers in order to relate
these to previous macroscopic measurements.
In this thesis, the rod model has been shown to be a feasible alternative to the
simpler models previously discussed in the literature [19, 3]. Accurate models give
good insight into the behavior of GAG chains in physiological conditions. Therefore,
the continued development of more realistic models will ultimately provide much
needed insight into the microscopic and macroscopic properties of articular cartilage.
72
Appendix A
Electrostatic Interactions Between
MFP Tip and Substrate:
Generalized 2-D Models for
"Neutral" and Charged Surfaces
In this appendix, a model is developed to describe the experiments where the tip
was coated with a neutral polymer. Even though the polymer may be neutral, the
charge from the GAG on the substrate may induce a surface charge on the tip. Thus,
electrostatic forces may still be induced and measured.
The tip is modeled as a grounded metal surface coated with an insulator that may
also contain surface fixed charge groups (Figure A-1). The substrate surface charge
is given as o-s; the charge at the insulator electrolyte interface,
U-2,
and the charge
on the insulator metal interface, 9s3, are both unknown. There are three regions in
this model: region 1 between the charged surface and the tip, which is filled with salt
solution, region 2 inside the insulator, which has no volumetric charge density, and
the region 3 inside the grounded metal, in which the electric fields must be zero.
The electric field and potential in each of the three regions are described by the
laws of electroquasistatics (Gauss, Faraday, and conservation of charge, respectively)
73
1
P
.7(D=
+
3
2
'
E=0
+
-, f
3
I
_-
+
+
0
z
z
D
zD+d
Figure A-1: Model of tip approaching a charged surface.
(a)
Ea
(b) nfb
Fb
a
E
Ea
interface,S
Figure A-2: General interface.
and the interface boundary conditions associated with each respective law (Figure
V - cE =p
n - (,EaE
V x E =0
n x (Ea - Eb)
V -J =
atp
n - (JaE,
- CbEb) = Us
-
= 0 (or <1a = <b*)
UbEb) =
at Os
* in the absence of an infinitesimal thin double layer.
These equations lead to three boundary conditions for the model, one at each of the
74
interfaces:
n
o1s1
'a D
=
-VD
-
n - (E2E2 - cf E1)
at z = 0
at z
O-s2,
=
,
-
D
0, at z = D + d
(A.1)
(A.2)
(A.3)
The electric field inside the insulator (region 2) will be constant since there is no
volume charge. From this, an expression for the induced surface charge density at
z = D can be derived.
A.1
Induced Surface Charge
The problem is in steady state so
2 =
at
0. Therefore, due to charge conservation,
- n - (9 2 E 2 - o-E
yUs2 =0
1)
where -1 and or2 are the conductivities of region 1 and 2 respectively. But since region
2 is an insulator, -2 ~ 0, and therefore,
E12 = 0
(A.4)
where E1 2 is the magnitude of the electric field in the z-direction inside region 1.
Gauss's law states that discontinuities in the electric fields at the boundary are
due to a surface charge:
0
2=
n - (f 2 E 2
-
f E1)
The electric field in region 1 at z = D- is zero as shown above. So this simplifies to:
Os2
-
-E2E22
75
where E 2, is the magnitude of the electric field in the z-direction inside region 2.
The electric field inside region 2 can be obtained by finding the slope of the
potential:
(z = D + d) - 4(z = D)
D+d-D
2-
on the insulator will be:
Therefore, the surface chargeE~z62
(A.5)
f24)2
-'
d
and similarly, the surface charge on the insulator at z
=s3
-
2E22
=-
=
D + d will be:
f2
dd
E2E22=
'
which means that:
O-s3
(A-6)
= -Us2.
Therefore, the net charge on the insulator is 0, as expected. However, there is a
non-zero surface charge density on the insulator at z
=
D.
Notice that the surface charge induced on the insulator at z
=
D is of the same
sign as the potential on its surface. The potential will be positive when a., > 0 and
negative when o-i < 0. Therefore, the induced charge on the tip will have the same
sign as that on the charged substrate. This will lead to electrostatic repulsive forces
even though the tip may not have had charge to begin with.
A.2
Electrostatic Force from Induced Charge
The potential in region 1 can be calculated numerically in the same manner as for the
flat surface charge model described in 5.1. The only difference is that the boundary
76
condition at z = D is (<D
= 0 as show above.
Once the potential is obtained inside region 1, the force can be calculated by using
the same balance of force method used in the calculation of force for the charged tip
versus charged substrate models. A Gaussian bounding box is defined where one
surface is at z = D_, where -Philz=D_
= 0, and the other surface is at infinity,
where the potential and the electric field are 0. Then the force on the tip enclosed
inside the box will be due only to the osmotic pressure at z = D.
This logic can be used if the surface substrate model is changed from a flat surface
charge to a volume charge, or even to charged rods.
77
78
Appendix B
Detailed Experimental Methods
These experiments were carried out by Joonil Seog at MIT [24, 23, 20].
Silicon wafers (Recticon Enterprises Inc., Pottstown, PA; test grade) and Si 3 N4
cantilevers (Thermomicroscope, Sunnyvale, CA) were coated with 2nm of chromium
to promote the adhesion of gold, followed by a 100nm of gold deposited using thermal
evaporator at 1.5). Gold-coated silicon wafer was cleaned using piranha solution (1:3
concentrated H2 SO 4 and H2 0 2 (30%)) for five minutes just before further modification
with CS-GAG.
Metabolically radiolabeled
3 5S-aggrecan
was obtained from rat chondrosarcoma
cell cultures and digested with proteinase K. The resulting amino-acid-terminated
35
S-CS-GAG chains were precipitated with ethanol, purified on superose-6, and re-
suspended in 0.01M phosphate buffer. The
35
S-CS-GAG chains with their terminal
reactive amine groups were treated with an excess of dithiobis[sulfosuccinimidyl propionate] (DTSSP, Pierce), and the terminal disulfide bond was reduced to a thiol
group using an excess of dithiothreitol (DTT, Pierce). After removal of excess reactants, 5pl aliquots of the thiol-terminated
3
chains were placed on cleaned
1S-CS-GAG
1cm x 1cm gold-coated silicon wafers. The slides were left to react at room temperature for 72 hours. After rinsing, the wafers were placed in 1 mM solution of C1 2H 25-SH
for 30 min. This process known as backfilling forms a neutral SAM (self-assembly
monolayer) on that part of the gold which was not modified with GAG. GAG surface
density on the wafer was assessed by removal of the
79
3
1S-CS-GAG
after AFM tests
HH
-NH
COOH
H
OH
H
H
OH
2
+
so;
o
H
CH
H
SOjNa4
H
0DTSSP0
H
NHCOCH,
&--10-50
0
0H
0
DTT, 1 h
0
-P
PBS buffer, 1 hr
Cs
H0
backfill with
methyl-terminated
$AM : C1 "1-5S"
''u"
s
HN
H
0
2S
H
0
Cs
24 hrs
~~ SO,-Na
HN
0
s§ s
Cs
end-gmfted CS-G AG
H
0
=0
S
=100 nm on silicon chip
backfill with
methyl-terminated
SAM:
Figure B-1: Schematic of the chemistry to graft GAG onto substrate surface.
by sonication, and measuments by scintillation counting. The parking density was
about 1 GAG chain per 6.5nm x 6.5nm (Figure B-1).
The gold coated tip was chemically modified by forming monolayers in one of
two ways.
It was immersed for 24 hours in either a 5mM ethanol solution of 2-
mercaptoethanesulfonic acid (Aldrich) to create a charged tip or a 5mM ethanol
solution of 11-mercaptoundecanol (from Prof. Laibinis lab at MIT) to create a neutral tip. The tip modified with 2-mercaptoethansulfonic acid was backfilled with
ethanethiol (Aldrich). Modified tips were rinsed in ethanol and dried under N2 just
prior to mounting them.
Various ionic strength solutions were prepared by mixing NaCl and dionized H2 0.
The pH of the solutions was about 5.6. Repulsive forces between the CS-GAG chains
and a chemically modified tip (Figure 4-3a) were then measured in the NaCl solutions
ranging in concentration from 0.0001M to 3.0M, using the Molecular Force Probe
(MFP; Asylum Res., Santa Barbara, CA). The standard silicon nitride tip had a
spring constant of 0.012). The spring constant of triangular, 320pam long silicone
nitride cantilever was determined using thermal method [4, 12].
80
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