402 CAPSTONE THESIS

Ning Shu
BIOEN402
SENIOR YEAR CAPSTONE THESIS
Designing An Aqueous Environment Adhesive Derived From E. Coli Fimbriae: A
Computational Approach
By Ning Shu
Principle Investigator/Mentor: Wendy E. Thomas
Department of Bioengineering, University of Washington

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Table of Contents
1. BACKGROUND & SIGNFICANCE .................................................................................................3
1.1 Introduction..................................................................................................................................................................3
1.2 Literature Review & Previous Work..............................................................................................................3
1.2.1 Current Adhesion Landscape...........................................................................................................................3
1.2.2 Previous Work on E. coli Fimbrial Adhesion............................................................................................5
1.3 Financial Support......................................................................................................................................................8
1.4 Consequenceof Success.........................................................................................................................................9
1.5 Economic Significance............................................................................................................................................9
1.6 Ethical and Legal Implications...........................................................................................................................9
2. METHODS & DESIGN APPROACH.............................................................................................10
2.1 Overview of Equations, Experimental Set-Up and Design Approach.......................................10
2.2 Data Acquisition......................................................................................................................................................12
2.2.1 Fimbrial Mesh Set-Up.......................................................................................................................................12
2.2.2 AFM Driver............................................................................................................................................................12
2.2.3 Global Parameters Input.................................................................................................................................12
2.2.4 Simulation Core Code.......................................................................................................................................13
2.2.5 Outputs & MATLAB...........................................................................................................................................14
2.2.6 VMD Visualization of the AFM experiment.............................................................................................14
2.2.7 AFM pull experiments: trajectory analysis.............................................................................................14
3. RESULTS & DISCUSSIONS .........................................................................................................16
3.1 Research Data & Design Process...................................................................................................................16
3.1.1 Figure 8: Effectof PullingSpeed on Maximum Rupture Forceand Workof Adhesion.
Using default k12 type I fimbria parameters....................................................................................................18
3.1.2 Figure 9: Effect of varying k1,2 on-rate and k2,1 off-rate.....................................................................19
3.1.3. Figure 10: Effects of Varying Coiling Parameter Ka,b.......................................................................21
3.1.4 Figure 11: Effects of Varying Coiling Parameter kb,a ..........................................................................22
3.1.5: Final Design & Improvements in Adhesiveness..................................................................................23
3.1.6 Statistical Analysis.............................................................................................................................................23
CONCLUSION & FUTURE DIRECTIONS .........................................................................................27
KEY PERSONNEL & RESOURCES ...................................................................................................28
Key Personnel...................................................................................................................................................................28
Resources............................................................................................................................................................................28
Works Cited ................................................................................................................................29

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1. BACKGROUND & SIGNFICANCE
1.1 Introduction
The biophysics of how organisms adhere to surfaces is of great interest to the field of
biomimetic and bioengineering, as the process of evolution has already provided many novel and
effective biological adhesive mechanisms. The operating conditions for these adhesive
mechanisms vary vastly in terms of the magnitude of force as well as the function it performs—
some adhesives are designed to release under certain conditions, such is the case for E. coli
adhesion, while some are meant to become death grips such as in the case of marine mussels.
Consequently, these tools provided by nature would have to be studied and applied in the context
of trade-offs between adhesive strength, adhesion reversibility, and the target chemical
environment.
The current adhesive design landscape is filled with options in terms of getting creative,
but designs generally fall into the categories of either dry or wet adhesives. The gecko is a
famous model organism for adhesive design, however as a dry adhesive it is not suited for
aqueous environments, a particular environment of great interest to bioengineers as it pertains to
organs and tissues of a biological system. Wet adhesives, on the other hand, are by definition
able to perform well within aqueous environments, particularly in cases of tissue adhesion inside
a body or organ. Two promising models exist for the wet adhesive landscape: the marine mussel
and the intestinal bacteria E. coli. E. coli is unique in its ability to not only form strong bonds
with a mannose surface, but also forming bonds that are completely reversible—in other words,
maintaining adhesive strengths after repeated attachment and detachment cycles to mannose
without leaving behind residues that could reduce the adhesiveness. However, the magnitude of
force associated with E. coli adhesion is only within the nano-scale, and efforts in replicating the
strong adhesion in the micro-scale has been unsuccessful, which poses a unique design
question—what could be done to increase the adhesiveness, but at a bigger scale and yet still
retaining the reversibility?
In terms of applications, drug delivery systems frequently target the mucous parts of the
body such as the intestines and nose for its accessibility and blood supply, but without proper
securement to the epithelia the delivery system would not have enough time to release its content
before becoming displaced by the shear flow of mucous layer. Likewise, diagnostic procedures
such as endoscopy has reached a stage involving less invasive techniques such as robotic
capsules, but the success of this method depends on the mobility of the capsule in shear flow
environments, and bio-adhesives that work in such an environment can greatly enhance its
effectiveness if it is able to bind strongly and reversibly. Therefore, understanding the binding
mechanism of E. coli could well be the solution to the creation of a smart adhesive designated for
use in aqueous application.
1.2 Literature Review & Previous Work
1.2.1 Current Adhesion Landscape
As mentioned briefly before, the gecko’s ability to adhere to vertical walls is well known
and this particular adhesion mechanism dominates the dry adhesive design landscape. The
gecko’s foot is composed of hair-like structures called setae that are capable of standing up to

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~200 μN of force each in non-aqueous environments, which amounts to a staggering support
load of 100 N per gecko foot [1]. Taking advantage of this amazing property, synthetic versions
of the gecko adhesives have been developed and refined, the most prominent product being the
Waalbot, a synthetic gecko-inspired wall-climbing robot that could climb steep walls at 90-
degree angles with great agility [2]. Yet despite the gecko adhesive’s impressive strength, it is
restricted from widespread biological applications as it fails in aqueous environments unless
modified.
On the wet adhesives landscape, the mechanisms of focus are primarily that of the marine
mussel and the E. coli bacteria. It is known that E. coli type I fimbria binds to a mannose
molecule or a mannose glycoprotein via a receptor-ligand mechanism and that shear forces
actually enhance the binding strength. This anti-intuitive behavior is proposed by Wendy et al. to
be the result of allosterically regulated catch bonds, where the E. coli receptor FimH undergoes
conformational change to a strong affinity state upon experiencing a certain level of shear force,
which results in the formation of bond that is longer-lived [3]. The implication of the allosteric
model is that when the shear force reverts to a lower level, the high affinity state of the receptor
actually switches back to the low affinity state, effectively “releasing” the ligand and creates a
new opportunity for binding—essentially, the catch bond behavior not only smartly increase
adhesion when needed but also has the potential for a completely reversible adhesive. However,
one of the biggest obstacles for an E. coli-based adhesive is scaling, as Atomic Force
Spectroscopy (AFM) experiments with FimH films demonstrate that adhesive strength is not
scaling proportionally higher as the contact area increases, which seriously hinders application
on the macroscale [4]. Fortunately, it has been discovered that the inclusion of the fimbriae
linkage portion FimA into the FimH films effectively increased bond lifetimes and the overall
adhesiveness, which suggests that the FimA structure has a significant role in buffering the
forces exerted on the E. coli fimbria [4]. In fact, Forero et al proposed that FimA is capable of
uncoiling its tertiary structure to elongate and buffer the forces applied to the fimbria [5]. The
mechanisms in which FimA and FimH affects fimbrial adhesion will be elaborated on in further
detail throughout this paper.
Aside from E. coli, the marine mussel is another organism that utilizes wet adhesive
mechanisms, specifically through the unique modified amino acid 3,4-dihydroxy-L-
phenylalanine (DOPA). DOPA is unique in that the dissociation force between its adhesive
bonds is strong, in the range of hundreds of pico-Newtons [3]. It has been observed that through
oxidation, DOPA allows the mussel to bind reversibly to a metallic surface but irreversibly to
organic, aqueous surfaces [4]. Although the DOPA amino acid exhibits extraordinary binding
strength and also provides unique flexibility in terms of binding to both organic and inorganic
surfaces, which are all useful properties, its lack of reversible binding to organic surfaces would
limit its range of applications unless this limitation could be overcome through synthetic
processes.
Most interestingly, there have been recent efforts in developing a hybrid gecko-mussel
adhesive and they were able to successfully develop the “geckel” adhesive, which retains the
adhesive’s binding strength under water (which normally would have been lost) by coating a
synthetic gecko-inspired micropillar adhesive with a mussel-inspired DOPA infused polymer
layer [5]. However, the geckel still performs far better in dry environments than in aqueous

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environments: the adhesive strength being higher for gecko in air than in water, and the
reversibility is higher in air than water as well.
A key distinction between the E. coli adhesive and the geckel adhesive is that the geckel
binds reversibly only to an inorganic surface whereas the E. coli adhesive is capable of reversibly
binding to organic, glycosylated surfaces. Therefore these two mechanisms are not the same and
should have different applications completely. In addition, it is not known if the geckel’s
adhesive efficacy can be scaled to a bigger area than the current half a square centimeter surface
area, which interestingly is a similar concern faced by E. coli adhesive design as scaling E. coli
adhesion to the macro scale has also proven to be difficult [5] [7].
1.2.2 Previous Work on E. coli Fimbrial Adhesion
Overview
As mentioned earlier, E. coli readily adhere to tissues in wet conditions using receptor-
ligand bonds. This particular mechanism is advantageous for three reasons: first, receptor-ligand
bonds are optimal for an aqueous environment; second, the ligand mannose is present in
mannose glycoproteins that exist on most human and tissues; third, previous work on FimH and
mannose sugar binding has shown that not only is FimH-mannose binding strong and reversible
on the nano-scale, it is also non-fouling, a term in adhesion meaning being able to retain bond
strengths after numerous repeated interactions between FimH and mannose without leaving
residues on the binding surface [4].
The study of these adhesion mechanisms is largely based on simulations coupled with
atomic force microscopy (AFM) and parallel-plate flow chamber experiments. Previous work
began on leukocyte microvilli adhesion, which has similarities to type I fimbrial adhesion as both
structures involve a “tether” linker structure with a receptor structure at its tip that identifies
carbohydrate molecules [3]. Briefly, leukocyte adhesion involves short and stubby microvilli
with selectin cells, whereas fimbria is comprised of the long coiled rope-like structure FimA and
the receptor FimH at the outer tip of the FimA. To reiterate a point made earlier, the adhesion
behavior of the fimbria is the combination of FimH catch-bond formation and FimA uncoiling
behavior. The structure of the fimbria is presented in Figure 1.
Figure 1: Forero et al.,
2006. Schematic of the
Fimbria Structure within
an AFM experiment.
Here an AFM cantilever
coated with mannose (red
dots) is pulling on a
fimbria that is uncoiling
at the base (FimA) while
attached to the receptor
FimH on top.

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Catch-Bonds
Catch-bond formation between the receptor FimH and mannose is shear-enhanced and
allosterically regulated. Specifically, the FimH adhesion receptor protein has two distinct shear-
force activated conformations, one low affinity state and one high affinity state, that result in two
distinct unbinding rates that are inversely related to the shear-force. In other words, since
unbinding rate is inversely proportional to the bond lifetime, FimH anti-intuitively forms a
longer lived, stronger bond with mannose at a high shear force, but a short lived and weaker
bond at low shear force [4]. Therefore, the allosteric catch bond model explains how E. coli
bacteria are able to achieve adhesion at higher shear flow, by activating the higher bond strength
conformation of FimH receptors. An implication of this behavior is that receptor-ligand bond
based adhesives are self-cleaning and reversible, since at low shear flow, the FimH receptor
returns to the weaker conformation and releases the ligand, therefore the target tissue (ligand) is
not being harmed through adhesion, leaving no residue of binding, and the FimH is ready to
bond strongly again when the shear increases.
However, data from flow-chamber experiments in the Wendy Thomas lab also shows that
between stationary adhesion at high shear and free floatation at low shear exists a “stop-and-go”
phase called “shear-stabilized rolling” that which the previous allosteric catch bond model could
not explain entirely, indicating that catch-bond mechanism is not the entire story [9]. The
implication of the discovery of this shear-stabilized rolling behavior of E. coli is that the E. coli
adhesion mechanism is not just about catch-bonds between FimH and mannose, but also highly
dependent on the physical behavior of the FimA “tether” segment that connects the E. coli body
with its FimH receptors.
Uncoiling
Indeed, FimA, (the fimbriae “tethers”) proved to be absolutely essential in the force
loading capability of E. coli in any force effects, not just shear forces, as observed in AFM
experiments. The fimA segments serve as shock absorbers for the FimH receptors at the tip when
the FimH-mannose bond is subject to high force. AFM experiments involving FimH films (FimH
without FimA) and fimbriae (FimH and FimA intact) films indicate that FimH receptors
arranged into a film is much weaker than films made out of whole fimbriae where the FimA is
present, and this is particularly true as the target mannose surface increased in size and contact
area increased to the micro-scale [4]. Further investigation on the fimbria properties indicates
that a fimbria is nonlinearly elastic, being extremely stiff at low shear force but highly elastic at
high shear force and even exhibits an uncoiling mechanism that buffers the force loading process
over a specific threshold shear force and equalizes the load across bonds like shock absorbers
[5]. In addition, this finding validates previous literature suggesting that pili-like structures like
fimbria can have cooperative binding of multiple pili that leads to the distribution of load and
improved bond strength such that two pili can take on more forces than allowed by each pili on
its own [9]. (see Figure 2)
This cooperativity of pili structures is in turn determined by the intrinsic qualities of the
individual pilus involved. Because real life adhesives are on a much larger size and time scale, it
is vital to consider incorporation of fimbriae properties into the simulation and design of the wet

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adhesive, since it is inherent that using mere FimH-mannose binding alone is not strong enough
to create a strong adhesive.
Figure 2: Elongation from single pillus and two pili combined
Source: Multipili attachment of bacteria with helixlike pili exposed to stress. [9]
In order to effectively design a reversible wet adhesive, the Wendy Thomas’ laboratory’s
approach is to perform research experiments on FimH and mannose bonding using AFM
experiments, and then use computational model to fit these experimental data so that the
simulations can then be used to determine optimum design parameters for the wet adhesive. The
fundamental framework of the initial Thomas lab fimbrial simulation by doctorate student
Matthew Whitfield was first derived from Adhesive Dynamics by Hammer and Apte, where the
approach indicated in the work served as the backbone structure of the computer codes [10]. E.
coli is modeled as a spherical object with many fimbrial structures surrounding it, and the entire
object is acted upon by numerous parameters such as but not limited to the ones indicated in
Figure 3.
Figure 3: E. Coli modeling diagram, forces and structural features involved in simulation.

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Source: Simulations of cell rolling and adhesion on surfaces in shear flow: general results
and analysis of selectin-mediated neutrophil adhesion. Hammer, D and Apte, S. 1992,
Biophysical Journal, Vol. 63.
A sophisticated foundation simulation protocol written in C++ programming language has
already been established, along with a set of automated MATLAB scripts that read and generate
graphs and other visualizations from the output of the simulation program. The basic principles
behind the simulations are: 1) uses Monte Carlo principles to simulate the randomness present in
a real biological system 2) treating the coiled fimbria as a Hookean spring that uncoils into a
worm-like chain and 3) sums the forces of fluid flow with fimbrial interactions with the mannose
surface to calculate the state of the bond. The coding can be readily changed to account for any
new parameters that arise and running the simulation can provide insights to the next direction
for wet lab experiments, and vice versa, fitting the simulation to new experimental data is always
crucial to acquire a more realistic model of the system.
The Fimbrial Simulation has two experiment modes, AFM or FLOW representing AFM
or flow chamber experimental set ups, respectively. The simulation code for the AFM mode has
been modified extensively by graduate student Keith Johnson to reflect realistically the
orientation of mannose and fimbriae surfaces. Specifically, the AFM experiment from the
previous version of the simulation has a bead covered with radially-spread fimbriae moving
towards a mannosylated surface, but in reality most of the AFM experiments performed, such as
the representative AFM experiment pulls from the publication “A Nanoadhesive Composed of
Receptor Ligand Bonds” has a mannosylated bead moving towards a fimbriae-covered surface.
The orientation mismatch has resulted in incongruent data and in response, the code was
modified to accommodate the addition of a fimbriae mesh set up, where the fimbriae are
arranged in a haystack manner to more realistically represent a fimbriated surface. The bead is
changed to a simple mannose bead similar to the kind that is attached to an AFM cantilever. The
protocol to achieve the fimbrial mesh is specified further in the methods section.
1.3 Financial Support
The study and designing of reversible wet adhesive at the Wendy E. Thomas laboratory is
supported by the National Science Foundation’s CAREER award. The mission statement of the
National Science Foundation is to promote the progress of science; to advance the national
health, prosperity, and welfare; [and] to secure the national defense" by “keeping close track of
research around the United States and the world, maintaining constant contact with the research
community to identify ever-moving horizons of inquiry, monitoring which areas are most likely
to result in spectacular progress and choosing the most promising people to conduct the research
[12].” The Career Award specifically awards junior faculty who demonstrates “leadership in
integrating education and research” [12]. This funding has provided not only research funding
for the development of a wet adhesive that would have an impact on healthcare but also provided
valuable research opportunities to both undergraduate and graduate students.
Generally, any research that can implicate the optimization of a drug delivery process is
of high interest to pharmaceutical companies, as successful delivery of dosage is often crucial to
obtain the statistically significant differences in drug trials, and passing drug trials opens the door

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to the launch of highly profitable products to market. In addition, while E. coli adhesion to the
intestinal wall is normally harmless, significant urinary tract infection could result from mutated,
virulent E. coli strains that display a different glycoprotein binding behavior. Knowledge of E.
coli adhesion could therefore lead to the development of anti-adhesive drugs against pathogenic
E. coli strains, effectively eliminating reliance on antibiotics for treatment of infections.
Furthermore, bioinstrumentation companies would also be likely to fund such research because
instruments often require strong adhesion to tissues and cells to function properly, but the
adhesive should not tarnish the tissue it binds to after the instrument is removed, therefore
requiring the reversible characteristic of the smart adhesive.
1.4 Consequence of Success
The immediate application of a successful E. coli-derived reversible wet adhesive would
most likely result in the drastic improvement of endoscopic micro-robots used for endoscopy
diagnostic technology. The adhesive would be certainly appropriate in terms of biocompatibility
since the material is E. coli derived and E. coli is naturally found within the intestinal walls. The
wet adhesive would allow the micro-robots to firmly adhere to the gastro-intestinal walls but at
the same time the reversible quality would also allow the micro-robots to release the adhesion
and move freely, while being less damaging to the GI tract walls compared to traditional
endoscopic micro-robots. Improved GI tract diagnostics can greatly benefit patients who are
suffering from Celiac disease, Crohn’s disease as well as colon cancer, as endoscopy capsules
are less invasive and can serve as a more regular diagnostic tool for their conditions. In addition,
the E. coli-derived reversible wet adhesive can be applied to biomedical instruments or drug
delivery systems in general. When applied to robotic arms or fingers, the adhesive can provide a
stronger grip but also mechanical endurance and dexterity, as the robotic device would have the
ability to let go as commanded and repeat a certain motion indefinitely.
1.5 Economic Significance
The cost of a capsule endoscopy procedure is extremely high, with the recorder costing 25,000
dollars while the capsule itself anywhere from 250 dollars to 450 dollars each [13]. Although
improved endoscopy methods would not necessarily cause a direct decrease in the cost of the
procedure, it can improve better data acquisition while the capsule is within the patient and this
in turn will reduce the number of repeats of the procedure, saving the patient money indirectly.
According to a recent study on capsule endoscopy, given the low compliance rate for
colonoscopy, colon capsule endoscopy might be a more effective and cheaper alternative for
patients, saving the economy 16,165 dollars to 29,244 dollars per life-year saved [14]”.
1.6 Ethical and Legal Implications
Because a wet reversible adhesive can have such wide range of potential areas of
application, the ethical implications of this design is far-reaching and complex. Although this
technology can obviously do inherent good through the improvements of medical instruments,
helpful robots and drug delivery, it is not unreasonable to assume the very same technology can
be used for the improvements of weapons and agents of biological warfare. As mentioned in one
particular lecture given by Dr. Paul Yager, chairman of the department of Bioengineering at the
University of Washington, the ability to achieve a strong grip can result in the improvement of

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handguns that might not serve the social interest even though the technology itself is quite
benign. The development of the smart reversible adhesive has no immediate legal or economic
implications, but due to the flexibility of this technology, there can be legal or economic impacts
that are out of the control of the original designer as a result of others applying this technology to
advance their own means. Certainly, if the adhesive is being used for refining biomedical or drug
delivery processes and robotics, one could expect increase in productivity leading to more
consumptions and jobs within at least the biomedical industry. However, it is possible, as with
any advances in the field of biology, that the technology could be used toward weapons of
biological warfare. A specific scenario could be the development of a mutant strain of E. coli that
is optimized adhesively to be the most virulent strain, which could devastate human population
should it contaminate food or water sources.
2. METHODS & DESIGN APPROACH
2.1 Overview of Equations, Experimental Set-Up and Design Approach
The simulation characterizes the adhesive behavior of a particular AFM experiment through a
series of parameters. The parameters can be roughly categorized into three categories,
experimental, state rate constants, and coiling rate constants. The values of these parameters are
set to the default values derived from previous experiments unless specifically altered for the
purpose of experimentations. Table 1 exhibits some of the experimental/simulation parameters.
The state rate parameters are k0,1, k0,2, k1,2 and k2,1, representing transition rate from unbound to
weak state, unbound to strong state, weak to strong state and strong to weak state respectively
(See Figure 5).
Figure 4: 2-state model of fimbrial adhesion binding. [3]
These bond transition are related to the force acting upon the fimbria as modeled by the
following equation:







Tk
Fx
kk
B
bijo
ijij exp
Equation 1: bond transition equation relating bond state, transition distance, and force where
i is 1 or 2, and j is 0, 1 or 2 and x are the transition distances. [3]

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Lastly, another major contributor to the adhesive behavior is the uncoiling phenomenon,
characterized by segmenting the FimA into three distinct segments. Section A is modeled as a
coiled spring, while section B and C are modeled as worm-like chains. As the FimA uncoils, the
effective distance of the three segments changes.
Equations 2 and 3: Force dynamics of the three uncoiling segments
The transition rates between states A and B are given by
the transition rates between state B and C:
One very important equation relates the velocity of extension to the force experienced by the
bond (uncoiling velocity and uncoiling force)
where Ku is Kab while Kc is Kba, ΔXu is Xab and ΔXc is Xbc. ΔL is 5.0 nm. [5]
Equation 8: Extension Velocity as a function of the Force on the Fimbria
kUðf Þ¼ Aexp
DUU þ f DxU
kbT
¼ k0
Uexp
f DxU
kbT
ð1Þ
Here, A isthe Arhenius frequency factor, kbT isthe thermal
energy, and k0
U is the uncoiling rate in the absence of force.
There will be an analogous rate equation for kC( f) for coiling
depending on the energy DUC and distance DxC differences
between the uncoiled and transition states. From the single
transition state assumption, we get that DxC ¼ DxU DL and
DUC ¼ DUU DU, where DU is the energy cost to uncoil a
monomer and DL is the extension due to one subunit
uncoiling.
With this model, we can relate the average extension
velocity V to the applied force
Vðf Þ¼ DL kUðf Þ kCðf Þ½ Š
¼ DL k0
Uexp
f DxU
kbT
k0
Cexp
f DxC
kbT
ð2Þ
This equation should hold whether the AFM tip pulls the
fimbrial tip at a constant velocity while recording the force
(Figures 1, 2, and 4A), or the tip maintains a constant force
while the position is recorded (Figures 3B and S1). A
consequence of Equation 2 is that there is a force, fbal, at
which the average velocity is zero and the uncoiling and
coiling rates balance each other [kbal ¼kU( fbal) ¼kC( fbal)]. We
can then rewrite Equation 2 as
range at which
Thus, the patte
shown in Figure
quaternary unco
Discussion
When force is
terminal end o
contract revers
constant velocity
force region wh
force increases
fimbriae are fair
at a velocity of
between 60 and
this range. Whe
within this int
contraction even
ably close to th
single FimA sub
orienting thems
nm individual e
confirm that th
elongation is re
uncoiling of th
dynamics of unc
experimental fin
29kbT, and a fo
events occur at
fimbriae can ge
and flow cell m
fimbriated FimH
intermediate co
in length but is
significant coilin
An explanatio
of type I fimbria
that they have c
This potentially
sequence of Fim
Coiling and unc
a number of w
derived here by
change under p
could then mod
ligand complex
bond lifetimes a
here, fimbriae c
Figure 5. FimH–1M Bond Lifetimes in a Flow Chamber
The binding of 1M beads to fimbriae-coated surfaces was monitored in
flow chambers at two well-defined flow rates that create drag forces of
2.5 pN (open triangles) and 37 pN (solid triangles) on the beads. Beads
moving in the field of view alternately paused (bound to the surface in a
stationary manner for a period of time) and moved at the hydrodynamic
velocity of the fluid flow. The lifetime of the pauses was measured and
assumed to correspond to single-bond lifetimes. The lifetime of these
pauses is plotted here as the fraction of all pauses that survive to the
time indicated on the x-axis. There are two distinct decay rates under
each condition, suggesting a weak and a strong conformational state,
each corresponding to short-lived and long-lived binding, respectively.
The fraction of pauses lasting more than 1 s (arrows) was calculated
directly from raw data like these and used in Figure 4B. This fraction
increased with force, consistent with the notion that FimH forms a catch
bond with mannose because of a force-induced switch to a long-lived
state. The lines correspond to the slopes of the exponential fits of the
lifetimes in the first 500 ms or between 2 and 20 s, as described in the
Materials and Methods section.
DOI: 10.1371/journal.pbio.0040298.g005

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Table 1 Simulation parameters
Symbol Variable Value
kB Boltzmann constant 1.38 10-23
J/K
T Temperature 298° K
n Number of fimbriae 186
ka Fimbrial stiffness 2.06 pN/nm
Lavg Average fimbrial length 0.572 μm
E Fimbrial elasticity 2 GPa
ρ Cell density 1300 kg/m3
η Fluid viscosity 0.001 Pa*s
r Cell radius Variable*
τ Time step 500 ns
2.2 Data Acquisition
2.2.1 Fimbrial Mesh Set-Up
Graduate student Keith Johnson has implemented a new scheme for the AFM experiment
mode, where fimbriae are arranged into a haystack-like orientation through a MATLAB script
called Fim_Fall. The script simulates a realistic fimbriated surface by letting the fimbriae “fall”
onto a surface and on top of one another in a randomized fashion, taking in fimbrial density,
fimbrial surface area and equilibrium fimbrial length as parameters. The output of this script is a
set of six coordinate files (x_start, x_end, y_start, y_end, z_start, z_end, respectively) that will
serve as inputs for the execution of the core simulation code. See Figure 6 for a visual example
of the fimbrial mesh. For all experiments performed, the fimbrial density used is 240 fimH per
square micron of area and the equilibrium fimbrial length is the literature value of 0.572 microns
[4].
2.2.2 AFM Driver
The simulated AFM experiment requires a driver input file that serves as the trajectory of
the AFM cantilever. The script driver.m is called to generate three driver files, one for each of
the axis. The inputs to the program are arrays xstr, ystr and zstr, which each represent an array
dictating the movement of in the specific axis. For example, xstr = {0, ‘t’, 20} would signify a
movement of 0 microns/s, with time variable t, for 20 seconds.
2.2.3 Global Parameters Input
As aforementioned, the global parameters are saved inside a text file that is read in by the
main simulation code upon execution (See Table 3).
Table 2. Coiling parameters
Parameter Value
kAB 0.069 s-1
kBA 4.3x106 s-1
xAB 0.461 nm
xBA -1.9 nm
keqBC 2.63x105 s-1
xeqBC 0.64 nm
LpB 3.35 nm
LpC 0.516 nm

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Table 3: List of global parameters and their respective values according to strain
PARAMETER DEFINITION UNITS/VALUE Strain:kb91 Strain: k12
TYPE type of experiment AFM or FLOW
TMAX runtime of simulation seconds(s)
Fast amount of outputs 0=all, 1 = subset
TIMESTEP time step of simulation seconds(s)
PPS times per second output times per second (s-1)
Numfim total number of fimbriae 5654
fluid_shear fluid shear rate times per second (s-1) 5000 5000
cell_density density of cell kg/m3 1300 1300
cell_radius radius of cell meters (m) 1.5*10^-6 1.5*10^-6
cellE Young's Modulus for cell pascals (Pa)
viscosity viscosity of fluid Pascal second (Pa*s)
eq.fim_length mean fimbrial length meters (m) 0.572 10^-6 0.572*10^-6
fim_stiffness effective fimbrial stiffness Newton/meter (N/m) 0.00875 0.00875
fimbrialE Young's Modulus for Fim pascals (Pa) 2*10^9 2.00E+09
kbT Thermal energy Joules (J) 4.21E-21 4.21E-21
k01 state 1 (weak) on-rate times per second (s-1) 3 3
k10 state 1 (weak) off-rate times per second (s-1) 6 1.37
k12 transition from weak to strong times per second (s-1) 1.25E-03 3.00E-05
k20 state 2 (strong) off-rate times per second (s-1) 5.00E-06 5.00E-06
k21 transition rate from strong to weak times per second (s-1) 0.025 0.11
ΔL10 transition distance (state 1 unbinding) meters (m) 4.52E-10 2.80E-10
ΔL12transition distance from weak to strong meters (m) 1.20E-10 1.50E-09
ΔL20transition distance for state 2 unbinding meters (m) 4.52E-10 4.52E-10
ΔL21transition distance from strong to weak meters (m) -1.20E-09 -3.80E-10
Ku kb,a (uncoiling) times per second (s-1) 4.3e6 4.3e6
Kc ka,b (uncoiling) times per second (s-1) 0.0688 0.0688
Naming what number to start appending to file name
torsional_stiffness allows scaling of torsional stiffness 0 or 1
usecoil Allowing uncoiling 1 = yes, 0 = no
maxstate2 sets the max number of bonds allowed in state 2
2.2.4 Simulation Core Code
The core simulation codes contain the main algorithms for the simulation, implementing
the aforementioned equations from section 2.1. Some important functions are
FimbrialLocations() which reads in the Fim Mesh files to construct the fimbrial surface,
Globals() and setVariables() to read in the global parameters and set the variables, and Coil() to
set the coiling parameters and adjust each time step with the added effect of coiling. The code
details can be found within the appendix.

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2.2.5 Outputs & MATLAB
The simulation generates total of six output files, called trajectory, state change, coil,
lifetime and forvmd respectively. MATLAB scripts are used to generate analysis and figures of
the data, such as the trajectory plot of force versus position as well as the bond counts versus
time. The forvmd file is used exclusively with the molecular visualization program VMD for
more interactive and visualization purposes.
2.2.6 VMD Visualization of the AFM experiment
A Python script is used to convert the forvmd output file into a large series of PDB files
for the molecular visualization program VMD. The PDB files are loaded into VMD and linked
together into a movie, which can be played at various speeds. Bond formation and breakage can
be observed through the color changes of the interacting molecules.
2.2.7 AFM pull experiments: trajectory analysis
The results from an AFM pulling experiment can be interpreted graphically by plotting
the tension force of the interaction as either a function of position or time. Two metrics, the max
Figure 5: VMD Visualization of a typical
AFM experiment pull. The blue bead
signifies a mannose bead while the white
strands represent randomly stacked
fimbriae created from the Fim_Fall
MATLAB script.
Figure 6: Simulation of AFM cantilever
being pulled in the y direction (after initial
contact with fimbriated surface). The
fimbriae are bound to mannose via FimH
while the FimA uncoils, resulting in the
elongation of the fimbrial structure

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rupture force and the work of adhesion, are used to evaluate overall adhesiveness of the
interaction. Max rupture force gives the maximum magnitude of the force that the entire
adhesion process withstood, while the work of adhesion gives a more holistic view of how many
bonds are at work and for how long. Graphically, he max rupture force is simply the highest peak
observed in the plot, while the work of adhesion is the area under the curve (with x-axis being
position).
Figure 6: A typical AFM pull experiment trajectory plot, with force being either as a function of
time or position of the cantilever.

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3. RESULTS & DISCUSSIONS
3.1 Research Data & Design Process
The most important process in designing the best parameter value sets for fimbrial
adhesion to consider the problem in the context of the following three questions:
1) What state transition rates results in the highest adhesive force?
2) What coiling parameters result in the highest adhesive force?
3) How does the pull speed affect the adhesion?
These three questions relate directly back to the two mechanisms that drive the adhesion
behavior of fimbria: catch bonds and uncoiling. The on and off rates are crucial for catch-bond
formation while the coiling/uncoiling “dissociation rates” ka,b and kb,a decide the degree of
coiling that occurs in the process. Both processes are velocity and force dependent, therefore the
interaction can also be interpreted in the context of energy barriers, as illustrated in figure 7.
The velocity at which the adhesive is pulled is the third factor that affects the uncoiling
process immensely. Recall equation 8 from Forero et al relates the extension velocity to the force
on a bond, and the entire expression is an evaluation of the relative contributions by the
uncoiling components versus the coiling components. It has been found that fimbria uncoils at
above 60 pN of force while the fimbria coils back when the force drops down to 25 pN at wild
type k1,2 and k2,1 conditions [5]. Therefore it is appropriate to consider a pull speed that will result
in the minimum of 60 pN needed to activate uncoiling.
In addition, aside from the intrinsic parameters, one should consider the physiological
condition the simulation should focus on, and in the case of designing an adhesive through
simulation AFM experiments, the pulling speed is crucial in determining the behavior of the
adhesion process (See Figure 8). The sigmoidal shape of the data suggests that max rupture force
is less dependent on pulling speed once the pull velocity reaches a threshold of about 100
microns/s, but the max rupture force is highly sensitive to velocity at much lower velocities,
where the max rupture force rises rapidly with a small increase in pulling velocity. Similar trend
is observed for the work of adhesion as velocity increases.
!
Figure 7: 3 Energy Barriers involved in fimbrial adhesion
JUST!RIGHT:!activates!state!2!catch!bonds!as!
well!as!uncoiling!of!fimbria!
TOO!LOW:!not!enough!to!activate!state!2!bonds!
TOO!HIGH:!bonds!break!prematurely!
F
o
r
c
e!

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In designing an adhesive and picking parameter values conducive to creating maximum
adhesive strength, it seems that the 100 microns/s range is ideal as the default pull speed to
further optimize the parameters on. However, a particular roadblock to this approach is that AFM
experiments in real life are not reliable beyond the 10 microns/s range as viscous forces of the
experiment (as a result of the high pull velocity) overtakes the actual forces from bond formation
and breakage, therefore creating too much noise in the force data gathered. Therefore, a choice
was made to continue explore the adhesive properties at the speed of 10 microns/s, for at 10
microns/s the maximum rupture force and work of adhesion are still statistically insignificantly
different from those at 100 microns, but it is within the threshold where simulation data could be
realistically evaluated with real life AFM experiments.
For the transition state rates, only k1,2 and k2,1 are easily modified via mutants while the
k1,0 and k2,0 mutant variants are not as easy to achieve, therefore it is prudent to start exploring
the range of values for the state 1 and 2 transition rates first. As for the coiling parameters, Ka,b
and Kb,a are of much importance as well, however the effective distances for both the transitional
bond rates and the coiling rates are not variables suited for varying in a simulation as they are
usually functions of the other variables and is determined by the force exerted.
Lastly, of the simulation parameters, none should be modified with the exception of the
radius of the bead, but at the moment the simulation is limited by computational power and is
prone to errors beyond a binding surface of greater than 16 microns. Therefore, most of the work
done will be on the 1.5 μm range for the mannose bead.

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3.1.1 Figure 8: Effect of Pulling Speed on Maximum Rupture Force and Work of Adhesion.
Using default k12 type I fimbria parameters
Next, the adhesion properties are observed for varying the transition rate k1,2 at a pull
speed of 10 microns/s. The wild type has a k1,2 value of 3*10-5, with a maximum rupture force of
about ~2000 pN. The highest force however is reached at around 1.5*105 with a max rupture
force of ~3300 pN. It makes sense that the max rupture force and work of adhesion maxes out at
k1,2 = 1010, as the exponent portion of the equation becomes constant (x1,2 is
bound to 15*10-10). Similarly, in the case of k2,1, because x2,1 is a negative number, no matter
how high the force gets the exponent portion of the rate equation will go toward a limit of exp(-
1) which is fixed around 0, which explains why changing the k2,1 has little effect on the max
rupture force.
0
5000
10000
15000
20000
25000
30000
35000
0
500
1000
1500
2000
2500
3000
3500
0.1 1 10 100 1000
WorkofAdhesion
pN
microns/s
adhesion behavior as a function of pull
speed
Max Rupture Force
Work of Adhesion







Tk
Fx
kk
B
bijo
ijij exp

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3.1.2 Figure 9: Effect of varying k1,2 on-rate and k2,1 off-rate
0.00E+00
5.00E+03
1.00E+04
1.50E+04
2.00E+04
2.50E+04
3.00E+04
3.50E+04
4.00E+04
0.00E+00
5.00E+02
1.00E+03
1.50E+03
2.00E+03
2.50E+03
3.00E+03
3.50E+03
4.00E+03
10000 1000000 100000000 1E+10 1E+12 1E+14 1E+16
WOA
pN
Microns/s
Varying k1,2: max rupture force and
work of adhesion.
n = 10 for each point
Max Rupture
Force
Work of
Adhesion
0.00E+00
5.00E+02
1.00E+03
1.50E+03
2.00E+03
2.50E+03
3.00E+03
1.00E-10 2.00E-09 4.00E-08 8.00E-07 1.60E-05 3.20E-04 6.40E-03 1.28E-01
Max Rupture Force, k1,2=3e-5, varying
k2,1

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Similarly, the effects of altering ka,b, the coiling on-rate, is explored. The resulting data has
extreme peaks in maximum rupture force and work of adhesion near the wild type default values,
which strongly suggests that uncoiling behavior of E. coli has been strongly selected by
evolution and optimized naturally to have the maximum adhesive strength (See Figure 10).
However, no such salient peaks were observed for varying kb,a as the max rupture force and work
of adhesion remained steady at wild type values regardless of the change in kb,a (see red, Figure
11).
0.00E+00
5.00E+02
1.00E+03
1.50E+03
2.00E+03
2.50E+03
3.00E+03
3.50E+03
4.00E+03
1.00E-200 1.00E-150 1.00E-100 1.00E-50 1.00E+00
pN
k2,1
Max Rupture Force: k1,2 = 3e5, varying k2,1

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3.1.3. Figure 10: Effects of Varying Coiling Parameter Ka,b
0
500
1000
1500
2000
2500
3000
1.00E-10 3.00E-07 9.00E-04 2.70E+00 8.10E+03 2.43E+07 7.29E+10
pN
ka,b
Max Rupture Force as a Function
of Ka,b
0.00E+00
5.00E+03
1.00E+04
1.50E+04
2.00E+04
2.50E+04
1.00E-03 3.00E+00
Work of Adhesion as a Function of
Ka,b (subset of interest)

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3.1.4 Figure 11: Effects of Varying Coiling Parameter kb,a
Finally, drawing conclusion from the adhesive behavior according to aforementioned
parameters, a stronger E. coli adhesive could be obtained by pulling at around 10 microns per
second from a fimbriated surface of density 240 FimH/square microns, and changing the k1,2
transition rate from weak to strong binding from 3*10-5 to 3*105, while maintain all other
parameters to their default wild type values (k2,1 = 0.11, ka,b= 0.688, kb,a = 4.3*106), in particular
the coiling parameters as they are already optimized, the resulting comparison of adhesion
improvement can be seen in Figure 3.1.5. The mean maximum rupture force for the design was
3390 pN +/- standard deviation of 246.3 (n=10), above the mean maximum rupture force for the
wild type (n=10) of 1989 pN +/- standard deviation of 425.52 pN. Similar differences can be
observed for work of adhesion as well.
0
500
1000
1500
2000
2500
3000
1.00E-100 1.00E-80 1.00E-60 1.00E-40 1.00E-20 1.00E+00 1.00E+20 1.00E+40
pN
Max Rupture Force vs Kb,a (k1,2=3e-5)
0.00E+00
5.00E+03
1.00E+04
1.50E+04
2.00E+04
2.50E+04
1.00E-100 1.00E-80 1.00E-60 1.00E-40 1.00E-20 1.00E+00 1.00E+20 1.00E+40
WOA vs Kb,a (k1,2=3e-5)

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3.1.5: Final Design & Improvements in Adhesiveness
Figure 12: Comparison of Designparameters and wild type parameters over different pull
speeds
3.1.6 Statistical Analysis
The error bars for standard deviation do not overlap, which indicates that the difference in mean
between the two groups is possibly significant. Therefore we proceed to a t-test for each of the
pull speeds.
Null hypothesis: no difference between design and wild type in adhesive strength
*α=0.5 *unpaired t-test
n= 15
Speed
(microns/s)
0.1 0.5 2.5 10 100
P values <0.0001
significant
< 0.0001
significant
<0.0001
significant
< 0.0001
significant
< 0.0001
significant
0
500
1000
1500
2000
2500
3000
3500
4000
4500
0.1 1 10 100
MRF(PN)
pull speed
MRF: design vs wild type
design MRF
wild type MRF
0.00E+00
5.00E+03
1.00E+04
1.50E+04
2.00E+04
2.50E+04
3.00E+04
3.50E+04
4.00E+04
4.50E+04
0.1 1 10 100
Worofadhesion
pull speed
WOA: design vs wild type
design
wild type

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The design choice of increasing the k1,2 parameter and maintaining the wild type value for the
uncoiling parameters increased the max rupture force and work of adhesion across a large range
of pull speeds.
We follow up the statistical analysis by examining the number of bonds forming at any given
time as well as the uncoiling forces with the variations in ka,b values.
Figure 13: Bond Activation Diagrams for the improved parameters versus wild type at
three pull speeds: medium (2.5 microns/s), fast (10 microns/s) and extremely fast (100
microns/s). All activities before 1 seconds represent the cantilever moving towards the

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fimbrial surface, blue line indicates the formation and breakage of state 1 (low affinity)
bonds, red line indicates the formation and breakage of state 2 (high affinity) bonds. Green
is the total number of bonds showing binding activity (purely as reference, as blue and red
totals should always add up to the green value at any time point).
Several conclusions can be implicated from the bond state diagrams in Figure 12. First, is
that the design model has a higher number of state 2 bonds than the wild type counterpart of the
same speed group. This is consistent given the boost from the increased transition rate from state
1 to 2, making the catch bond formation happen more frequently. In addition, it is curious to see
that state 2 bonds start forming much earlier for the design parameters( even while the cantilever
is still approaching surface), while the wild type generally must all transition into state 1 bonds
first before transitioning into state 2 bonds, as seen by the fact that almost no state 2 bonds ever
form until after the 1 second point where the cantilever starts to move away from the fimbrial
surface.
Using Equation #8, the calculated uncoiling forces for the three speeds are 82 pN, 94 pN
and 115 pN respectively. As all three of these force values are greater than 60 pN, we can
assume that uncoiling dominates coiling in these experiments and that uncoiling occurs at these
pull speeds. In addition, it is also worth noting that the calculated theoretical uncoiling force for
wild type fimbria is 91 pN, very close to the 85 ± 4 pN uncoiling force observed previously in
the Thomas lab [4]. However, when the coiling parameters are changed so that it is about ten
magnitudes higher than its wild type value, the coiling force dropped to near zero and the
ultimate result was that state 2 bond formation was severely hindered (Figure 14, 15a). In turn,
the lack of state 2 bonds meant the maximum rupture force was capped rather short as well as
seen from the blue line in the trajectory plot in Figure 14b. On the other hand, lowering the
uncoiling parameter well below the wild type value results in extremely high uncoiling forces
that would certainly break the catch bonds prematurely, as a single catch bond can only
withstand about 130 pN of force before breaking [4] (Figure 14). This is indeed the case as seen
in the trajectory plot where the green line spikes but falls in a very swift pattern (Figure 15b).
The logarithmic nature of the uncoiling force in relation to the uncoiling parameter reaffirms the
notion that the uncoiling is highly sensitive to change, and even a magnitude of change results in
an uncoiling force that either breaks the catch bond prematurely or does not activate catch bonds
at all. Perhaps the sensitivity is due to the process of evolution finely selecting for an uncoiling
behavior that best complements catch bond formation.

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Figure 14: Uncoiling force drops with increase in ka,b (note the logarithmic scale), pull
speed = 10 microns/s. Red data dot signifies uncoiling force for wild type ka,b value of
0.0688.
Figure 15 a) the lack of state 2 bond formation due to the increased ka,b value (which resulted
in an extremely low uncoiling force calculated with equation 8 mentioned previously). The
formation of catch bonds occur in the reverse case when ka,b is low and uncoiling force is
high, but the catch bonds all tend to break prematurely which also hindered adhesiveness
0
11.3
91
194
296
0
50
100
150
200
250
300
350
1.00E-12 1.00E-09 1.00E-06 1.00E-03 1.00E+00 1.00E+03 1.00E+06 1.00E+09
UncoilingForce(pN)
Ka,b values
Uncoiling Force vs. Ka,b

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Figure 15 b) corresponding results in maximum rupture force plots. Low uncoiling
parameter leads to high uncoiling force and therefore immediate bond ruptures (green)
whereas high uncoiling parameter leads to low uncoiling force that meant low catch bond
bond formation. *Red line is wild type reference
CONCLUSION & FUTURE DIRECTIONS
With the current design modifications, the adhesion capacity of the E. coli has been
increased by about 1400 pN (~75%) by increasing k1,2 to 3*105 and maintaining the coiling
parameter ka,bat the wild type value, as data suggests that evolution has already optimized
coiling to achieve the maximum adhesive strength. Although varying the off-rates k2,1 and kb,a
had little effects on increasing the maximum adhesive force, there are still many areas to be
explored for potential to increase the adhesive strength.
For example, the behavior of the other coiling parameter, kb,c and kc,b (for the worm-like
chain segments of FimA) has not been explored in depth. It is worth exploring all of the
aforementioned parameters and velocity at different bead radiuses, as scaling of the adhesive
force still remains an obstacle to creating a wet reversible adhesive at the micro-scale. Lastly, all
of the simulation experiments were performed on a single variation of the fimbrial mesh file,
while it makes sense to perform all experiments in a pair-wise manner on different fimbriated
surfaces to observe the effect of the random variations of fimbriae surface on the adhesive
behavior.

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KEY PERSONNEL & RESOURCES
Key Personnel
This project would not be possible without the generous mentoring of Professor Wendy Thomas,
graduate student Matt Whitfield, and my immediate graduate student mentor, Keith Johnson.
Assistance from Gianluca Interlandi in the technical use of all the lab resources is also crucial to
the completion of this capstone thesis. During the duration of the project, help and morale
support is also greatly appreciated from fellow teammate on the bio-adhesive project, junior
Nichole Tyler.
Resources
This project is supported by a NSF Career Award funding. The workspace where this project
takes place is within the Bioengineering department at the University of Washington, with the
utilization of on-site computers and servers as well as personal laptops. The project is able to
proceed with the availability of publications for reference as well as software such as MATLAB
and C++ and a laboratory capable of performing AFM experiments, parallel-plate flow chamber
experiments and E. Coli culturing. The simulation aspects of this project do not involve any
contact with bio-hazardous materials, but the real life implementation of the design involves
contact with biological material and requires short courses on dealing with blood-borne
pathogens.

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Works Cited
[1] Kellar Autumn, Yiching A. Liang, Tonia Hsieh, and Robert J Full, "Adhesive forceof a single
gecko foot-hair,"Nature,vol. 405, pp. 681-685, June 2000.
[2] C. Menon, M. Murphy, and M. Sitti, "Waalbot:An Agile Small-Scale Wall Climbing Robot
Utilizing Dry Elastomer Adhesives," IEEE/ASME TransactionsonMechatronics,vol.12,no. 3,
June 2007.
[3] Wendy Thomas et al., "Catch-Bond Model Derived From Allostery Explains ForceActivated
Bacterial Adhesion," BiophysicalJournal,vol.90, no. 3, pp. 753-764, February 2006.
[4] Matthew Whitfield and Wendy Thomas, "A Nanoadhesive Composed Of Receptor-Ligand
Bonds," The Journalof Adhesion,vol.87, no. 5, May 2011.
[5] Manu Forero, Olga Yakovenko,EvgeniSokurenko, Wendy Thomas, and Viola Vogel, "Uncoiling
Mechanics of Escherichiacoli TypeI Fimbriae Are Optimized For Catch Bonds," vol.4, no. 9,
September 2006.
[6] Haeshin Lee, Norbert Scherer, and Phillip Messersmith, "Single-molecule mechanics of mussel
adhesion," Proc Natl AcadSci, vol. 103, no. 35, pp. 12999-13003, August 2006.
[7] G Murat, D Brass, Messersmith, and K Shull, "Adhesion of DOPA-FunctionalizedModel
Membranes to Hard and Soft Surfaces," TheJournalof Adhesion,vol.85, no. 9, pp. 631-645,
2009.
[8] Haeshin Lee, Bruce Lee, and P.Messensmith, "A Reversible Wet/dry Adhesive Inspired By
Mussels and Geckos," Nature, vol.448, pp. 328-341.
[9] Tia Ghose, Wendy Thomas, and Matthew Whitfield, "Shear-stabilized rolling behavior of E. coli
examined with simulations," BiophysicalJournal,vol.99, pp. 2470-2478, 2010.
[10] O Bjornham and O Axner, "Multipili attachment of bacteria withhelixlike pili exposed to
stress," Journal of ChemicalPhysics, vol.130, no. 23, 2009.
[11] D Hammer and S Apte, "Simulations of cellrolling and adhesion on surfaces in shear flow:
general results and analysis of selectin-mediated neutrophil adhesion ," BiophysicalJournal,
vol.63, 1992.
[12] NSF. National Science Foundation. [Online]. NSF.gov
[13] Noel M Lee and Glenn M Eisen, "10 Years of Capsule Endoscopy:An Update," ExpertRev
GastroenterolHepatol,vol.4,no. 4, pp. 503-512, 2010.

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[14] C Hassan, A. Zullo, S. Winn, and S. Morini, "Cost-Effectivenessof Capsule Endoscopy in
Screening for ColorectalCancer," Endoscopy,vol.40,pp. 414-421, January 2008.

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APPENDIX
Code Used: version AllSims_N.cpp
#include "commonEcoli3D.h"
#include <time.h>
#include <math.h>
#include <string>
#include <iostream>
#include <iomanip>
#include <complex>
#include <cmath>
#include <unistd.h>
using namespace std;
#ifdef _MSC_VER
#include "gsl/gsl_sf_erf.h"
#endif
//This program was changed heavily during December 2011 to improve the AFM type cimulations KCJ 12/2011
//This program can now only be used for AFM simulations. For FLOW simulations, use allsims_v1 KCJ 1/2012
double randomn() {
/* The random function generates a random number between 0 and 1. The distribution is UNIFORM.
The resolution for the random function is 1/RAND_MAX most of the time; if
the generator comes up with either 0 or 1 as the random number, then the function
is implemented recursively to get very small random numbers, it is implemented recursively. */
double randomTemp= 0;
randomTemp = (double) rand()/RAND_MAX;
if (randomTemp == 0) {
randomTemp = randomn()/RAND_MAX;
}
if (randomTemp == 1) {
randomTemp = 1 - randomn()/RAND_MAX;
}
return randomTemp;
}
double simpleRand() {
double randTemp = (double) rand()/RAND_MAX;
return randTemp;
}
double ExponentialRandom(double lambda) {
double randpick = randomn();
double Exponential = -log(randpick)/lambda;
return Exponential;
}
#ifdef _MSC_VER
double erfc(double Z) {
return gsl_sf_erfc(Z);
}
#endif
/* Declares all variables that are globally accessible within all functions*/
// OUTPUT FILE HANDLES
FILE *fid1;
FILE *fid2;
FILE *fid4;
FILE *fid7;
FILE *fid9;
FILE *fid39;
FILE *fid40;
FILE *fid5;
string filename1;
string filename4;
string filename5;
string filename7;
string filename9;
string filename39;
string filename40;
string filename2;
string x_start_name;
string y_start_name;
string z_start_name;
string x_end_name;
string y_end_name;
string z_end_name;
//////////////////////////////////////////////
// Parameters read in from input file ///
//////////////////////////////////////////////
string TYPE;
double TMAX;
int fast;
double TIMESTEP;
long long PRINTSPERSECOND;

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int numfim;
double fluid_shear;
double cell_density;
double cell_radius;
double cellE;
double viscosity;
double equilib_fim_length;
double fim_stiffness;
double fimbrialE;
double kbT;
double K01;
double K02;
double K10;
double K12;
double K20;
double K21;
double deltaL10;
double deltaL12;
double deltaL20;
double deltaL21;
int naming;
double torsional_stiffness;
int usecoil;
double ku;
double kc;
int maxstate2;
double holdforce;
int interp;
double Retract;
//////////////////////////////////////////////
// GENERIC VARIABLES ///
//////////////////////////////////////////////
double time1 = 0;
double t = 0;
int recordyn=0;
int print=1;
int stopped=0;
int bentsum=0;
double bucklesum=0;
double contactsum=0;
double scn;
double mominertia;
double Inertia;
double eff_drag_radius;
double closestTip;
double Km;
double fluid_velocity;
double xrmsbound;
double dCrit;
long long PRINTFACTOR;
double cell_mass;
double drag_coeff;
double length;
std::valarray<double> Fbuckling;
std::valarray<double> fim_length;
std::valarray<double> equilib_fimL;
std::valarray<int> buckle;
std::valarray<double> k;
std::valarray<int> bent;
std::valarray<double> bound_factor;
//////////////////////////////////////////////
// BOND TRANSITION VARIABLES ///
//////////////////////////////////////////////
double timeToTransition=0;
double nextTransitionTime=0;
double stateSum=0;
double state2Sum=0;
double sqrt2=sqrt(2.0);
int nextState=0;
int transitionIndex=1;
std::valarray<double> bondrate;
std::valarray<double> bondrate01;
std::valarray<int> state;
std::valarray<int> prev_state;
//////////////////////////////////////////////
// FORCE CALCULATION VARIABLES ///
//////////////////////////////////////////////
double axial_torqueM, side3, att,atf,rotateforceM;
double force;
double thetazx;
Point tottorque;
Point atM;
Point tensionSum;

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BIOEN402
Point change_in_ten;
std::valarray<double> totten;
std::valarray<Point> tension;
std::valarray<Point> torque;
std::valarray<Point> rotateforce;
std::valarray<double> anglediff;
std::valarray<Point> tip;
std::valarray<Point> normaltip;
std::valarray<Point> attach;
std::valarray<double> tether_anglet2;
std::valarray<double> tether_anglef2;
std::valarray<Point> current_attach_angle;
//////////////////////////////////////////////
// ERMAK & ROTATION VARIABLES ///
//////////////////////////////////////////////
double position_gap=0;
int posjump;
double Tys,ah;
Point accel, vel, velo;
Point aaccel, avel, avelo;
Point center, ocenter;
Point change_in_pos;
Point position;
double c0,c1,c2,beta;
double c0r,c1r,c2r,betar;
//////////////////////////////////////////////
// COILING PARAMETERS ///
//////////////////////////////////////////////
double dxc=-1.9e-9;
double dxu=4.61e-10;
double kbc=2.63e5;
double kcb=2.19e5;
double dxbc=6.4e-10;
double dxcb=3e-10;
double xLc=.7e-9;
double xLu=8.2e-9;
double xL2=6.1e-9;
double xL3=8.2e-9;
double Lp2=3.35e-9;
double Lp3=.516e-9;
//////////////////////////////////////////////
// COILING VARIABLES ///
//////////////////////////////////////////////
double Ku,Kc,Kx,Ky,Nc,Nu,N 2,N3;
double a,a2,a3,b,b2,b3,e,e2,e3,d,d2,d3,Km2,Km3,xLc2,xLc3;
double EQLc,EQLu;
double C1,Lp;
long double Lc;
double force2;
double sq3=sqrt(3.0);
double statep;
complex <double> halfpw=.5;
complex <double> thirdpw=1.0/3;
complex <double> L1;
complex <double> L2;
complex <double> Lcc;
std::valarray<double> ratio;
std::valarray<double> NUMc;
std::valarray<double> NUMu;
std::valarray<double> NUM2;
std::valarray<double> NUM3;
std::valarray<double> NUM2p;
std::valarray<double> NUM3p;
std::valarray<int> subunits;
std::valarray<int> coiled;
//////////////////////////////////////////////
// AFM VARIABLES ///
//////////////////////////////////////////////
double xposition=0, yposition=0, zposition=0;
double xsurface,ysurface, zsurface;
double stopit=0;
std::valarray<double> xmin;
std::valarray<double> ymin;
std::valarray<double> zmin;
std::valarray<double> Dmin;
std::valarray<double> u_vect_x;
std::valarray<double> u_vect_y;
std::valarray<double> u_vect_z;
std::valarray<double> part_xmin;
std::valarray<double> part_ymin;
std::valarray<double> part_zmin;
std::valarray<double> new_xmin;
std::valarray<double> new_ymin;
std::valarray<double> new_zmin;
std::valarray<double> current_length;
std::valarray<double> c_univect_x;
std::valarray<double> c_univect_y;
std::valarray<double> c_univect_z;

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BIOEN402
std::valarray<double> new_tipx;
std::valarray<double> new_tipy;
std::valarray<double> new_tipz;
std::valarray<double> tip_angle;
std::valarray<double> Dmin_angle;
std::valarray<Point> uni_fully;
std::valarray<Point> tip_full;
std::valarray<Point> uni_tips;
std::valarray<double> normal_full;
std::valarray<double> length_min;
std::valarray<Point> dummy_tip;
std::valarray<Point> uni_force;
std::valarray<Point> uni_dummy_full;
std::valarray<Point> dummy_tip_full;
double orig_center;
double floor_k;
double xmin_g;
double tip_diff;
double attach_diff;
double testing;
double tottensum;
double rotatesum;
int increment;
int partial=0;
std::valarray<double> xdw(numfim);
std::valarray<double> ydw(numfim);
std::valarray<double> zdw(numfim);
std::valarray<double> fxs(numfim);
std::valarray<double> fys(numfim);
std::valarray<double> fzs(numfim);
std::valarray<double> fxe(numfim);
std::valarray<double> fye(numfim);
std::valarray<double> fze(numfim);
ifstream::pos_type size;
char * memblock;
void globals(){
/* The globals function reads in the parameters that are input in a separate file. This allows for changing
input parameters without directly changing the code. See globals.h for a full description*/
cin>>TYPE; // AFM or FLOW
cin>>TMAX; // Runtime of the simulation, (s)
cin>>fast; // Flag that changes the amount of outputs of simulation 0=all outputs, 1=subset of outputs
cin>>TIMESTEP; // Time step of the Simulation, (s)
cin>>PRINTSPERSECOND; // How many times per sec outputs are written to output files
cin>>numfim; // The total number of fimbriae
cin>>fluid_shear; // The fluid shear rate, (1/s)
cin>>cell_density; // Density of cell, (kg/m^3)
cin>>cell_radius; // Radius of cell, (m)
cin>>cellE; // Young's Modulus of cell, (Pa)
cin>>viscosity; // Viscosity of fluid, (Pa*s)
cin>>equilib_fim_length; // Mean fimbrial length, (m) was 0.572e-6 m
cin>>fim_stiffness; // Effective stiffness of fimbriae and linkage to cell, (N/m) 2.06 pN/nm according to Matts dissertation
cin>>fimbrialE; // Young's Modulus of fimbriae, (Pa)
cin>>kbT; // Thermal Energy: Boltzmann's constant x Temperature, (J)
cin>>K01; // State 1 on-rate, (1/s) KB91: 3
cin>>K10; // State 1 off-rate, (1/s) KB91: 6
cin>>K12; // Transition rate from state 1 to 2, (1/s) KB91: .00125
cin>>K20; // State 2 off-rate, (1/s) KB91: 5e-6
cin>>K21; // Transition rate from state 2 to 1, (1/s) KB91: .025
cin>>deltaL10; // Transition distance for state 1 unbinding, (m) KB91: 4.52e-10
cin>>deltaL12; // Transition distance for state 1-2 transition, (m) KB91: 12e-10
cin>>deltaL20; // Transition distance for state 2 unbinding, (m) KB91: 4.52e-10
cin>>deltaL21; // Transition distance for state 2-1 transition, (m) KB91: -12e-10
cin>>naming; // What number to start appending to file name
cin>>torsional_stiffness; // Allows scaling of torsional stiffness from assumed value
cin>>usecoil; // Flag for allowing uncoiling (1 yes, 0 no)
cin>>ku; //kb,a (uncoiling)
cin>>kc; //ka,b (uncoiling)
cin>>maxstate2; // Sets the maximum number of bonds that are allowed to form in state 2
if (TYPE.compare("AFM")==0) {
cin>>Retract; //was the velocity of retraction, now defined by driver wave.
cin>>interp; // if (1/Timestep) = pps (from driver file), then interp = 1.
cin>>holdforce;
}
} //End globals
void setvariables() {
// Sets the initial values of all of the variables that could not be set before input variables collected
// Inital Calculations
PRINTFACTOR = (long long) (1/(PRINTSPERSECOND *TIMESTEP)); // Calculates how many iterations between outputs
K02 = K12*K20/(K21*K10 + K12*K20)*K01; // State 2 on-rate
(essentially 0)
cell_mass = PI * 4/3 * pow(cell_radius,3) * cell_density; // Mass of cell based on radius and density and neglecting fimbraie
Inertia = PI/4*(pow(3.45e-9,4)-pow(1.25e-9,4))*torsional_stiffness; // Howard book equation hahn paper for the inner and outer diameters
eff_drag_radius=cell_radius+equilib_fim_length; // Effective drag radius taking average
fimbrial length into account

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mominertia=0.4*cell_mass*eff_drag_radius*eff_drag_radi us; // moment of inertia for rotation of cell
drag_coeff = PI*6 * viscosity * eff_drag_radius; // Drag coefficient of cell
scn = fimbrialE*Inertia*3;
// Bending stiffness spring constant numerator from Howard book
beta = drag_coeff/cell_mass; // Ermak
equations - translation component
betar = PI*8*viscosity*pow(cell_radi us,3)/mominertia; // Ermak equations - rotational component
xrmsbound = sqrt(kbT/fim_stiffness); // Describes expected
deviation of axial distance the tip of the fimbriae given its stiffness
dCrit = xrmsbound*5;
// Defines a cutoff where fimbrial tip is considered to far from surface to bind that time step
Km=fim_stiffness;
Km2=Km*Km;
Km3=Km*Km*Km;
xLc2=xLc*xLc;
xLc3=xLc*xLc*xLc;
// Sets the size of the valarrays based on the number of fimbriae
fim_length.resize(numfim);
tip.resize(numfim);
normaltip.resize(numfim);
attach.resize(numfim);
tether_anglet2.resize(numfi m);
tether_anglef2.resize(numfi m);
state.resize(numfim);
prev_state.resize(numfi m);
current_attach_angle.resize(numfi m);
tension.resize(numfim);
bondrate.resize(2*numfi m);
equilib_fimL.resize(numfim);
Fbuckling.resize(numfim);
totten.resize(numfim);
buckle.resize(numfim);
k.resize(numfim);
torque.resize(numfim);
rotateforce.resize(numfim);
anglediff.resize(numfim);
bondrate01.resize(numfim);
bent.resize(numfim);
bound_factor.resize(numfim);
NUMc.resize(numfim);
NUMu.resize(numfim);
NUM2.resize(numfim);
NUM3.resize(numfim);
NUM2p.resize(numfim);
NUM3p.resize(numfim);
ratio.resize(numfim);
subunits.resize(numfim);
coiled.resize(numfim);
xmin.resize(numfim);
ymin.resize(numfim);
zmin.resize(numfim);
Dmin.resize(numfim);
u_vect_x.resize(numfi m);
u_vect_y.resize(numfi m);
u_vect_z.resize(numfi m);
part_xmin.resize(numfi m);
part_ymin.resize(numfi m);
part_zmin.resize(numfim);
new_xmin.resize(numfi m);
new_ymin.resize(numfi m);
new_zmin.resize(numfi m);
c_univect_x.resize(numfi m);
c_univect_y.resize(numfi m);
c_univect_z.resize(numfi m);
current_length.resize(numfim);
new_tipx.resize(numfim);
new_tipy.resize(numfim);
new_tipz.resize(numfim);
uni_fully.resize(numfim);
tip_full.resize(numfim);
uni_tips.resize(numfim);
dummy_tip.resize(numfi m);
uni_force.resize(numfim);
uni_dummy_full.resize(numfi m);
dummy_tip_full.resize( numfim);
normal_full.resize(numfim);
tip_angle.resize(numfim);
Dmin_angle.resize(numfim);
length_min.resize(numfim);
//shouldn't need now that fimbriae start and end points are imported.
for (int n = 0; n < numfim; n++) {
//USED FOR TESTING
//equilib_fimL[n] = sqrt((.5e-6)*(0.5e-6)+(2e-6)*(2e-6)+(1.2e-6)*(1.2e-6));

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//Normal stuff to run
equilib_fimL[n] = sqrt((fxe[n]-fxs[n])*(fxe[n]-fxs[n])+(fye[n]-fys[n])*(fye[n]-fys[n])+(fze[n]-fzs[n])*(fze[n]-fzs[n]));
fim_length[n]=equilib_fimL[n];
k[n]=0; // Torsional spring constant (Howard book spring constant = 3*E*I/L^3 * L^2)
bound_factor[n]=(20.191- 4.6516*exp(-.048714*k[n]/( kbT))-5.6621*exp(-.30302*k[n]/(kbT)));
subunits[n]=int(equilib_fimL[n]/xLc); // Calculates the # of subunits to the nearest whole number
equilib_fimL[n]=subunits[n]*xLc; // Changes the fim length based on # of subunits
NUMc[n]=subunits[n]; // Fimbriae start fully coiled
}
// Set the position of the centroid of the cell
center.x = 0;
center.y = 3.50e-6;
center.z = 0;
ocenter.x = center.x;
ocenter.y = center.y;
ocenter.z = center.z;
//Used when bead hits the surface
orig_center = center.y;
floor_k = .020; //20 pN/nm; roughly the typical spring constant of the cantilever
// Sets the starting values of variables stored for each fimbriae
Fbuckling[n] = fimbrialE * Inertia/(equilib_fimL[n]*equilib_fimL[n])*torsional_stiffness;
attach[n].x = fxs[n];
attach[n].y = fys[n];
attach[n].z = fzs[n];
tip[n].x = fxe[n];
tip[n].y = fye[n];
tip[n].z = fze[n];
//use to set fimbria start and end points manually, USED FOR TESTING
/*attach[n].x = -.5e-6;
attach[n].y = 0;
attach[n].z = -0.7e-6;
tip[n].x = 0;
tip[n].y = 2e-6;
tip[n].z = .5e-6;
normaltip[n].x = tip[n].x;
normaltip[n].y = tip[n].y;
normaltip[n].z = tip[n].z;*/
normaltip[n].x = fxe[n]; normaltip[n].y = fye[n]; normaltip[n].z = fze[n];
tension[n].x = 0; tension[n].y = 0; tension[n].z = 0;
anglediff[n]=0;
rotateforce[n].x=0; rotateforce[n].y=0; rotateforce[n].z=0;
xmin[n]=1;
ymin[n]=1;
zmin[n]=1;
new_xmin[n]=1;
new_ymin[n]=1;
new_zmin[n]=1;
part_ymin[n]=1;
part_zmin[n]=1;
part_xmin[n]=1;
u_vect_x[n]=1;
u_vect_y[n]=1;
u_vect_z[n]=1;
c_univect_x[n]=1;
c_univect_y[n]=1;
c_univect_z[n]=1;
current_length[n]=equilib_fimL[n];
new_tipx[n]=1;
new_tipy[n]=1;
new_tipz[n]=1;
current_attach_angle[n].z = asin(tip[n].y-attach[n].y/equilib_fimL[n]);
current_attach_angle[n].y = atan2(tip[n].z-attach[n].z,tip[n].x-attach[n].x);
state[n] = 0;
prev_state[n] = 0;
bondrate[n]= 0;
totten[n]=0;
buckle[n]=0;
bent[n]=0;
bondrate01[n]=0; bondrate02[n]=0;
NUMu[n]=0;
NUM2[n]=0; NUM3[n]=0;
NUM2p[n]=0; NUM3p[n]=0;
ratio[n]=1;
coiled[n]=0;
}
tottorque.x=0; tottorque.y=0; tottorque.z=0;

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atM.x=0; atM.y=0; atM.z=0;
//Get the minimum distance to center of bead for each fimrbiae (Dmin) and x-location on fimrbiae where this minimum distance occurs (xmin)
//for each fimrbia, determine the y and z point corresponding to xmin, the unit vector between the minimum point and the center of the bead, and then if Dmin is less
//than the cell radius, pivot the fimbriae until the fimbriae contacts the surface of the bead along the extension of the unitvector.
//Update the tip of each fimbriae based upon unit vectors between the attachment points and new min points.
//solve for x-location on fimbriae which will have minimum distance to center of cell.
double xmin_g = (2*center.x + (2*(attach[n].y - tip[n].y)*(center.y - attach[n].y + (attach[n].x*(attach[n].y - tip[n].y))/(attach[n].x - tip[n].x)))/(attach[n].x - tip[n].x) +
(2*(attach[n].z - tip[n].z)*(center.z - attach[n].z + (attach[n].x*(attach[n].z - tip[n].z))/(attach[n].x - tip[n].x)))/(attach[n].x - tip[n].x))/((2*(attach[n].y - tip[n].y)*(attach[n].y -
tip[n].y))/(attach[n].x - tip[n].x)*(attach[n].x - tip[n].x) + (2*(attach[n].z - tip[n].z)*(attach[n].z - tip[n].z))/(attach[n].x - tip[n].x)*(attach[n].x - tip[n].x) + 2);
//if xmin_g is somewhere along the length of the fimbriae then the minimum distance is calculated at xmin
if (xmin_g>attach[n].x) {
if (xmin_g<tip[n].x) {
xmin[n]=xmin_g;
}
}
else if (xmin_g<attach[n].x) {
if (xmin_g>tip[n].x) {
xmin[n]=xmin_g;
}
}
//if xmin_g is not along the fimbriae, it is calculated as either the tip or attachment point.
tip_diff = abs(tip[n].x-xmin_g);
attach_diff = abs(attach[n].x-xmin_g);
if(tip_diff>attach_diff) {
xmin[n]=attach[n].x;
}
else {
xmin[n]=tip[n].x;
}
}
}
if (xmin_g<attach[n].x) {
}
else {
xmin[n]=tip[n].x;
}
}
}
//Calculate the y and z coordinates at the mininimum distance point, and then the minimum distance
ymin[n]=((tip[n].y-attach[n].y)/(tip[n].x-attach[n].x))*(xmi n[n]-attach[n].x)+attach[n].y;
zmin[n]=((tip[n].z-attach[n].z)/(tip[n].x-attach[n].x))*(xmi n[n]-attach[n].x)+attach[n].z;
Dmin[n]=sqrt((xmin[n]-center.x)*(xmin[n]-center.x)+(ymin[n]-center.y)*( ymi n[n]-center.y)+(zmin[n]-center.z)*(zmin[n]-center.z));;
//If the minimum distance point is within the bead, then take small steps outward until it is just touching the surface
if (Dmin[n]<cell_radius-dCrit) {
while (abs(Dmin[n]-cell_radius)>dCrit) {
//find direction of steepest descent (in terms of distance)
part_xmin[n]=-(2*center.x - 2*xmin[n])/(2*sqrt((center.x - xmin[n])*(center.x-xmin[n]) + (center.y - ymin[n])*(center.y - ymin[n]) + (center.z
- zmin[n])*(center.z - zmin[n])));
part_ymin[n]=-(2*center.y - 2*ymin[n])/(2*sqrt((center.x - xmin[n])*(center.x-xmin[n]) + (center.y - ymin[n])*(center.y - ymin[n]) + (center.z
part_zmin[n]=-(2*center.z - 2*zmin[n])/(2*sqrt((center.x - xmin[n])*(center.x-xmin[n]) + (center.y - ymin[n])*(center.y - ymin[n]) + (center.z
//Take small step towards edge of bead
new_xmin[n]= xmi n[n]- part_xmin[n]*(Dmi n[n]-cell_radius)*0.1;
new_ymin[n]= ymi n[n]- part_ymin[n]*(Dmi n[n]-cell_radius)*0.1;
new_zmin[n]=zmin[n]- part_zmin[n]*(Dmi n[n]-cell_radius)*0.1;
//Get unit vector between new point and attach point
u_vect_x[n]=(new_xmin[n]-attach[n].x)/sqrt((new_xmin[n]-attach[n].x)*(new_xmin[n]-attach[n].x)+(new_ymin[n]-
attach[n].y)*(new_ymin[n]-attach[n].y)+(new_zmin[n]-attach[n].z)*(new_zmin[n]-attach[n].z));
u_vect_y[n]=(new_ymin[n]-attach[n].y)/sqrt((new_xmin[n]-attach[n].x)*(new_xmin[n]-attach[n].x)+(new_ymin[n]-
u_vect_z[n]=(new_zmin[n]-attach[n].z)/sqrt((new_xmin[n]-attach[n].x)*(new_xmin[n]-attach[n].x)+(new_ymin[n]-
//Determine new tip point using unit vector
tip[n].x=attach[n].x+u_vect_x[n]*equilib_fimL[n];
tip[n].y=attach[n].y+u_vect_y[n]*equilib_fimL[n];
tip[n].z=attach[n].z+u_vect_z[n]*equilib_fimL[n];

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double xmin_g = (2*center.x + (2*(attach[n].y - tip[n].y)*(center.y - attach[n].y + (attach[n].x*(attach[n].y - tip[n].y))/(attach[n].x -
tip[n].x)))/(attach[n].x - tip[n].x) + (2*(attach[n].z - tip[n].z)*(center.z - attach[n].z + (attach[n].x*(attach[n].z - tip[n].z))/(attach[n].x - tip[n].x)))/(attach[n].x - tip[n].x))/((2*(attach[n].y -
tip[n].y)*(attach[n].y - tip[n].y))/((attach[n].x - tip[n].x)*(attach[n].x - tip[n].x)) + (2*(attach[n].z - tip[n].z)*(attach[n].z - tip[n].z))/((attach[n].x - tip[n].x)*(attach[n].x - tip[n].x)) + 2);
xmin[n]=xmin_g;
}
}
xmin[n]=xmin_g;
}
}
}
else {
xmin[n]=tip[n].x;
}
}
}
}
else {
xmin[n]=tip[n].x;
}
}
}
Dmin[n]=sqrt((xmin[n]-center.x)*(xmin[n]-center.x)+(ymin[n]-center.y)*( ymi n[n]-center.y)+(zmin[n]-center.z)*(zmin[n]-center.z));
}//end while loop
}//end if statement
}//end for loop
}// End setvariables
void catchbond(int &state, int current_fim) {
/*The catchbond function takes in the tension and current bond state of a fimbriae and uses the two-state
catch bond model to update the bond state*/
double ratesum = 0;
double force = totten[current_fim] ;
if (force>=0) { force = 0; } //Here we assume that a compressive force on the fimbriae is equivalent to 0 force on the bond
else { force = abs(force); } //Tensile force is negative here, but needs to be postive for equations
if (state ==1) {
bondrate10[current_fi m] = K10*exp(force*deltaL10/kbT);
ratesum = bondrate10[current_fi m] + bondrate12[current_fi m];
double Pbond = 1 - exp(-ratesum*TIMESTEP);
double randpick1 = randomn();
if (randpick1 < Pbond) {
// if randpick less than Pbond, system leaves the current state
if (randpick02 < bondrate10[current_fi m]/ratesum) {
state = 0;
prev_state[current_fi m] = 1;
stateSum--;
//the variable state returns 0 if the bacteria is unbound, returns 1 if
//the bacteria is weakly bound and returns 2 if
// E.coli is strongly bound,
}
else {
state = 2;
state2Sum++;

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}
nextTransitionTime=time1;
fprintf(fid1, "%dt%dt%dt%2.8ft%10.10ft%10.10ft%10.10ft%2.6ft rn",current_fim, prev_state[current_fim],state, time1,
totten[current_fim]*1e12,fim_length[current_fi m]*1e6,equilib_fimL[current_fi m]*1e6,NUMu[current_fi m]/(NUMc[current_fim]+NUMu[current_fim]));
}
else {
state = 1;
}
}
else if (state == 2) {
double ratesum = bondrate20[current_fim] + bondrate21[current_fim];
state =0;
stateSum--;
}
else {
state = 1;
}
state2Sum--;
}
else {
state = 2;
}
}
else if (state == 0) {
bondrate01[current_fi m] = 0;
bondrate02[current_fi m] = 0;
//because only half the fimbriae (or so) will have the fimh at the "tip", only allow half of the "tips" to bind
double even = 2;
double checkeven = fmod(current_fim,even);
if (checkeven>0.5) {
if (xmin[current_fim]==tip[current_fim].x){
if (abs(Dmin[current_fim]-cell_radius < dCrit)) {
double Zvalue = (Dmin[current_fim]-cell_radius)/xrmsbound;
bondrate01[current_fi m] = K01*(erfc(Zvalue/sqrt2));
bondrate02[current_fi m] = K02*(erfc(Zvalue/sqrt2));
}
}
}
ratesum = bondrate01[current_fi m] + bondrate02[current_fi m];
state = 1;
}
else {
state = 2;
state2Sum++;
}
stateSum++;
}
else {
state = 0;
}
}
bondrate01[current_fi m]=0;
} // End catchbond
void coil(int n) {
// Used to calculate percentage of fimbriae coiled and force on fimbriae
totten[n]=-totten[n];
Ku=ku*exp(totten[n]*dxu/kbT);
Kc=kc*exp(totten[n]*dxc/kbT);
Ky=kbc*exp(totten[n]*dxbc/kbT);
Kx=kcb*exp(-totten[n]*dxcb/kbT);

Ning Shu
BIOEN402
//jj=dsolve('DN1=KC-KU', 'DN2=-KC+KU','N1(0)=NUMc[n]','N2(0)=NUMu[n]')
Nc=TIMESTEP*Kc-TIMESTEP*Ku+NUMc[n];
Nu=-TIMESTEP*Kc+TIMESTEP*Ku+NUMu[n];
if (Nc>subunits[n]) { Nc=subunits[n];}
else if (Nc<0) { Nc=0;}
if (Nu>subunits[n]) { Nu=subunits[n]; }
else if (Nu<0){ Nu=0;}
NUMc[n]=Nc;
NUMu[n]=Nu;
//jj=dsolve('DN1=KC*N2-KU*N1', 'DN2=-KC*N2+KU*N1','N1(0)=NUMxp[n]','N2(0)=NUMyp[n]')
N2=(-(NUM3p[n]*Kx-NUM2p[n]*Ky)/(Kx+Ky)*exp((-Kx-Ky)*TIMESTEP)*Ky+Nu*Kx/(Kx+Ky)*Ky)/Ky;
N3=Nu/(Kx+Ky)*Ky+(NUM3p[n]*Kx-NUM2p[n]*Ky)/(Kx+Ky)*exp((-Kx-Ky)*TIMESTEP);
if (N2>Nu) { N2=Nu; } else if (N2<0) { N2=0;}
if (N3>Nu) { N3=Nu; } else if (N3<0) { N3=0;}
NUM3p[n]=N3;
NUM2p[n]=N2;
NUM2[n]=N2;
NUM3[n]=N3;
EQLc=Nc*xLc;
EQLu=N2*xL2+N3*xL3;
d=fim_length[n];
statep=N3/(N3+N2);
Lp= -0.0000000154*statep*statep*statep*statep*statep + 0.0000000487*statep*statep*statep*statep - 0.0000000607*statep*statep*statep +
0.0000000387*statep*statep - 0.0000000143*statep + 0.0000000034;
if (Lp>Lp2) { Lp=Lp2; }
if (Lp<Lp3) { Lp=Lp3; }
C1=kbT/Lp;
b=C1;
b2=C1*C1;
b3=C1*C1*C1;
if (Nu<=0) {
force2=Km*(d-equilib_fimL[n]);
}
else {
a=EQLu;
a2=EQLu*EQLu;
a3=EQLu*EQLu*EQLu;
d2=d*d;
d3=d2*d;
e=EQLc;
e2=e*e;
e3=e2*e;
L1=(b*(64.0*Km3*a3+192.0*Km3*a2*e-192.0*Km3*a2*d+252.0*b*a2*Km2-384.0*Km3*a*d*e+192.0*Km3*a*d2+192.0*Km3*a*e2+288.0*Km2*a*e*b-
288.0*Km2*a*d*b+324.0*b2*a*Km+64.0*Km3*e3+192.0*Km3*d2*e- 64.0*Km3*d3-192.0*Km3*e2*d+144.0*Km2*d2*b+144.0*Km2*e2*b-288.0*Km2*d*e*b+108.0*b2*Km*e-
108.0*b2*d*Km+135.0*b3));
L2=360.0*b*a2*Km2+192.0*Km3*a*d2-192.0*Km3*a2*d+ 192.0*Km3*a*e2-64.0*Km3*d3+64.0*Km3*e3+540.0*b2*a*Km+ 192.0*Km3*a2*e-
288.0*Km2*d*e*b+243.0*b3-108.0*b2*d*Km+ 108.0*b2*Km*e-384.0*Km3*a*d*e+192.0*Km3*d2*e-192.0*Km3*e2*d+144.0*Km2*d2*b-
288.0*Km2*a*d*b+144.0*Km2*e2*b+288.0*Km2*a*e*b+64.0*Km3*a3;
Lcc=1.0/12.0*a/(b+Km*a)*pow((L2+12.0*sq3*pow(L1,halfpw)*b+12.0*sq3*pow(L1,halfpw)*Km*a),thirdpw)+1.0/12.0*a*(-
32.0*Km2*a*d+16.0*Km2*e2+16.0*Km2*d2+24.0*b*a*Km+ 32.0*Km2*a*e- 24.0*b*d*Km-
32.0*Km2*e*d+24.0*Km*e*b+9.0*b2+16.0*Km2*a2)/(b+Km*a)/pow((L2+12.0*sq3*pow(L1,halfpw)*b+12.0*sq3*pow(L1,hal fpw)*Km*a),thirdpw)-1.0/12.0*(9.0*b*a-
12.0*b*d+8.0*Km*a2-8.0*Km*a*d-4.0*Km*a*e)/(b+Km*a);
Lc=real(Lcc);
force2=Km*(Lc-EQLc);
}
totten[n]=-force2;
if (print==1 && fast==0) {
if (state[n]==2) {
fprintf(fid2, "%2.8ft%dt%dt%2.8ft%2.8ft%2.8ft%2.8ft%2.8frn", time1, n, state[n],totten[n]*1e12,NUMc[n],NUMu[n],NUM2[n],NUM3[n]);
}
}
} // End Coil
void outputs()
{
for (int n = 0; n< numfim; n++) {
bucklesum+=buckle[n];
if (NUMu[n]>1) {
ratio[n]=NUMu[n]/(NUMc[n]+NUMu[n]);
}
else {
ratio[n]=0;
}
}
if (TYPE.compare("AFM")==0) {
fprintf(fid4, "%2.9ft%2.6ft%2.6ft%3ft%2.6ft%2.6frn", time1, center.y*1e6, tensionSum.y*1e12,contactsum,tottensum*1e12,rotatesum*1e12);
}
if (recordyn==1) {
fprintf(fid9, "%2.3ft%2.3ft%2.3ft",center.x*1e6,center.y*1e6,center.z*1e6);

Ning Shu
BIOEN402
for (int nf = 1;nf<=numfim-1; nf++)
{
fprintf(fid9, "%2.3ft%2.3ft%2.3ft%2.3ft%2.3ft%2.3ft%dt%dt%2.3ft",attach[nf-1].x*1e6, attach[nf-1].y*1e6,attach[nf-1].z*1e6,tip[nf-1].x*1e6,ti p[nf-
1].y*1e6,tip[nf-1].z*1e6,state[nf- 1],buckle[nf-1],ratio[nf-1]);
}
fprintf(fid9, "%2.3ft%2.3ft%2.3ft%2.3ft%2.3ft%2.3ft%dt%dt%2.3frn",attach[numfi m-1].x*1e6, attach[numfim-1].y*1e6,attach[numfim-1].z*1e6,tip[numfim-
1].x*1e6,tip[numfi m-1].y*1e6,tip[numfi m-1].z*1e6,state[numfi m-1],buckle[numfi m-1],r atio[numfi m-1]);
}
} // End outputs
void drive(int increment) {
//determines change in position of the bead center in x,y,z directions
xposition=(xdw[increment+1]- xdw[increment]);
yposition=(ydw[increment+1]- ydw[increment]);
zposition=(zdw[increment+1]-zdw[increment]);
} //End drive
void ErmakAFM()
{
drive(increment);
partial+=1;
if (partial==interp) {
increment+=1;
partial=0;
}
//ocenter = old center
ocenter.y = center.y;
ocenter.x = center.x;
ocenter.z = center.z;
center.y=center.y+yposition;
center.x=center.x+xposition;
center.z=center.z+zposition;
orig_center=orig_center+yposition;
} // End ErmakAFM
void bound_fimbriaAFM(int n)
{
//Calculate new fim_length. The fimbriae are bound at the attach point, so only the tips should move.
fim_length[n] = exp(.5*log(((attach[n].x -tip[n].x)*(attach[n].x - tip[n].x) + (attach[n].y-tip[n].y)*(attach[n].y-ti p[n].y) + (attach[n].z -tip[n].z)*(attach[n].z - tip[n].z))));
contactsum+=1;
buckle[n]=0;
//Angle the fimbria is currently at
tether_anglet2[n] = atan2(tip[n].y-attach[n].y,tip[n].x-attach[n].x); //phi
tether_anglef2[n] = acos((tip[n].z-attach[n].z)/fim_length[n]); //theta
if (isnan(tether_anglef2[n])){
tether_anglef2[n]=0;
}
if (isnan(tether_anglet2[n])){
tether_anglet2[n]=0;
}
//using 'coil' requires a more complex calculation for totten
if (usecoil==1) { coil(n); }
else { totten[n]= -fim_stiffness*(fim_length[n] -equilib_fimL[n]); }
//if totten is positive and above a threshold, then buckling occurs
if (totten[n]> bound_factor[n]*Fbuckling[n]) {
totten[n] = bound_factor[n]*Fbuckling[n];
buckle[n]=1;
}
//determine angle between current fimbria position and original fimbria position
anglediff[n]=acos(((tip[n].x-attach[n].x)*(nor maltip[n].x- attach[n].x)+(tip[n].y-attach[n].y)*(normalti p[n].y-attach[n].y)+(tip[n].z-attach[n].z)*(normaltip[n].z-
attach[n].z))/(fim_length[n]*equilib_fimL[n]));
if (isnan(anglediff[n])){
anglediff[n]=0;
}
if (abs(anglediff[n])>.001) {
bent[n]=1;
}
else {bent[n]=0;}
//Need to calculate forces on bead due to torional forces of fimbriae
//Determining where fimbrial tip would be if it was at its full length, calculate unit vector first
uni_fully[n].x=(tip[n].x- attach[n].x)/fi m_length[n];
uni_fully[n].y=(tip[n].y- attach[n].y)/fi m_length[n];
uni_fully[n].z=(tip[n].z-attach[n].z)/fi m_length[n];
//now calculate tip location at full-extension
tip_full[n].x=attach[n].x+uni _full y[n].x*equilib_fimL[n];

Ning Shu
BIOEN402
tip_full[n].y=attach[n].y+uni _full y[n].y*equilib_fimL[n];
tip_full[n].z=attach[n].z+uni _full y[n].z*equilib_fimL[n];
//distance between normal tip and tip_full
normal_full[n]=sqrt((normaltip[n].x-ti p_full[n].x)*(normalti p[n].x-tip_full[n].x)+(normalti p[n].y-tip_full[n].y)*(normaltip[n].y-tip_full[n].y)+(normaltip[n].z-
tip_full[n].z)*(normaltip[n].z-tip_full[n].z));
//Determine distance from attach point to xmin,ymin,zmi n
length_min[n]=fim_length[n];
//Need so that we aren't dividing by zero
if (normal_full[n]<1e-9) {
rotateforce[n].x=0;
rotateforce[n].y=0;
rotateforce[n].z=0;
}
else {
//the force vector for the rotational force will be the vector between tip_full and normal tip, get unit vectors.
uni_tips[n].x=(normaltip[n].x-tip_full[n].x)/normal_full[n];
uni_tips[n].y=(normaltip[n].y-tip_full[n].y)/normal_full[n];
uni_tips[n].z=(normaltip[n].z-tip_full[n].z)/normal_full[n];
//create dummy tip that is stepped a small amount towards normal tip
dummy_tip[n].x = tip_full[n].x+uni_tips[n].x*(1e-8);
dummy_tip[n].y = tip_full[n].y+uni_tips[n].y*(1e-8);
dummy_tip[n].z = tip_full[n].z+uni_tips[n].z*(1e-8);
//find dummy fimbria length
double dummy_tip_length= sqrt((dummy_tip[n].x-attach[n].x)*(dummy_ti p[n].x-attach[n].x)+(dummy_ti p[n].y-attach[n].y)*(dummy_tip[n].y-
attach[n].y)+(dummy_tip[n].z-attach[n].z)*(dummy_tip[n].z-attach[n].z));
//determine dummy fimbria unit vector
uni_dummy_full[n].x = (dummy_tip[n].x-attach[n].x)/dummy_tip_l ength;
uni_dummy_full[n].y = (dummy_tip[n].y-attach[n].y)/dummy_tip_l ength;
uni_dummy_full[n].z = (dummy_tip[n].z-attach[n].z)/dummy_tip_l ength;
//extend dummy tip out to equilibrium fim length
dummy_tip_full[n].x = attach[n].x+uni_dummy_full[n].x*equilib_fimL[n];
dummy_tip_full[n].y = attach[n].y+uni_dummy_full[n].y*equilib_fimL[n];
dummy_tip_full[n].z = attach[n].z+uni_dummy_full[n].z*equilib_fimL[n];
//Determine unit vector between dummy_tip_full and tip_full
double tip_full_tip_dummy_l ength = sqrt((dummy_tip_full[n].x-tip_full[n].x)*(dummy_ti p_full[n].x-tip_full[n].x)+(dummy_tip_full[n].y-tip_full[n].y)*(dummy_ti p_full[n].y-
tip_full[n].y)+(dummy_tip_full[n].z-tip_full[n].z)*(dummy_tip_full[n].z-tip_full[n].z));
//force unit vector will be vector between dummy_tip_full and tip_full
uni_force[n].x = (dummy_tip_full[n].x-tip_full[n].x)/tip_full_ti p_dummy_length;
uni_force[n].y = (dummy_tip_full[n].y-tip_full[n].y)/tip_full_ti p_dummy_length;
uni_force[n].z = (dummy_tip_full[n].z-tip_full[n].z)/tip_full_ti p_dummy_length;
//rotateforce is the force mulitplied by the force vector (uni_force)
rotateforce[n].x=-uni_force[n].x*anglediff[n]*k[n]/fim_l ength[n];
rotateforce[n].y=-uni_force[n].y*anglediff[n]*k[n]/fim_l ength[n];
rotateforce[n].z=-uni_force[n].z*anglediff[n]*k[n]/fim_l ength[n];
}
//'tension' is sum of tension and rotate forces if there is not buckling
if (buckle[n]==0) {
tension[n].x = -totten[n]*sin(tether_anglef2[n])*cos(tether _anglet2[n])+rotateforce[n].x;
tension[n].y = -totten[n]*sin(tether_anglef2[n])*sin(tether_anglet2[n])+rotateforce[n].y;
tension[n].z = -totten[n]*cos(tether_anglef2[n])+rotateforce[n].z;
}
//if buckling has occured, then there are not rotation forces
else {
tension[n].x = -totten[n]*sin(tether_anglef2[n])*cos(tether _anglet2[n]);
tension[n].y = -totten[n]*sin(tether_anglef2[n])*sin(tether_anglet2[n]);
tension[n].z = -totten[n]*cos(tether_anglef2[n]);
}
prev_state[n]=1;
} //End bound_fimbriaAFM
void unbound_fimbriaAFM(int n) {
fim_length[n] = equilib_fimL[n];
tension[n].x = 0;
tension[n].y = 0;
tension[n].z = 0;
totten[n]=0;
rotateforce[n].x=0;
rotateforce[n].y=0;
rotateforce[n].z=0;
buckle[n]=0;

Ning Shu
BIOEN402
//
//
//Reset coiling parameters
if (coiled[n]==1) {
NUMc[n]=subunits[n];
NUMu[n]=0;
NUM3p[n]=0;
NUM2p[n]=0;
}//end if statement
//Determine current length of fimbria
current_length[n]=sqrt((tip[n].x-attach[n].x)*(tip[n].x-attach[n].x)+(tip[n].y-attach[n].y)*(tip[n].y- attach[n].y)+(tip[n].z-attach[n].z)*(tip[n].z-attach[n].z));
//Extend fimbrial length out to its equilibrium fim length
c_univect_x[n]=(tip[n].x-attach[n].x)/current_length[n];
c_univect_y[n]=(tip[n].y-attach[n].y)/current_length[n];
c_univect_z[n]=(tip[n].z-attach[n].z)/current_length[n];
tip[n].x=attach[n].x+fi m_length[n]*c_uni vect_x[n];
tip[n].y=attach[n].y+fi m_length[n]*c_uni vect_y[n];
tip[n].z=attach[n].z+fim_length[n]*c_uni vect_z[n];
xmin_g = (2*center.x + (2*(attach[n].y - tip[n].y)*(center.y - attach[n].y + (attach[n].x*(attach[n].y - tip[n].y))/(attach[n].x - tip[n].x)))/(attach[n].x - tip[n].x) + (2*(attach[n].z
- tip[n].z)*(center.z - attach[n].z + (attach[n].x*(attach[n].z - tip[n].z))/(attach[n].x - tip[n].x)))/(attach[n].x - tip[n].x))/((2*(attach[n].y - tip[n].y)*(attach[n].y - tip[n].y))/((attach[n].x -
tip[n].x)*(attach[n].x - tip[n].x)) + (2*(attach[n].z - tip[n].z)*(attach[n].z - tip[n].z))/((attach[n].x - tip[n].x)*(attach[n].x - tip[n].x)) + 2);
xmin[n]=xmin_g;
}
}
xmin[n]=xmin_g;
}
}
}
else {
xmin[n]=tip[n].x;
}
}
}
}
else {
xmin[n]=tip[n].x;
}
}
}
Dmin[n]=sqrt((xmin[n]-center.x)*(xmin[n]-center.x)+(ymin[n]-center.y)*( ymi n[n]-center.y)+(zmin[n]-center.z)*(zmin[n]-center.z));;
double xmincheck=xmi n[n];
double ymincheck=ymi n[n];
double zmincheck=zmin[n];
//If the minimum distance point is within the bead, then take small steps outward until it is just touching the surface
if (Dmin[n]<cell_radius-dCrit) {
while (Dmin[n]<cell_radius-dCrit) {
part_xmin[n]=-(2*center.x - 2*xmin[n])/(2*sqrt((center.x - xmin[n])*(center.x-xmin[n]) + (center.y - ymin[n])*(center.y - ymin[n]) + (center.z -
zmin[n])*(center.z - zmin[n])));
part_ymin[n]=-(2*center.y - 2*ymin[n])/(2*sqrt((center.x - xmin[n])*(center.x-xmin[n]) + (center.y - ymin[n])*(center.y - ymin[n]) + (center.z -
part_zmin[n]=-(2*center.z - 2*zmin[n])/(2*sqrt((center.x - xmin[n])*(center.x-xmin[n]) + (center.y - ymin[n])*(center.y - ymin[n]) + (center.z -
new_xmin[n]= xmi n[n]- part_xmin[n]*(Dmi n[n]-cell_radius)*0.1;
new_ymin[n]= ymi n[n]- part_ymin[n]*(Dmi n[n]-cell_radius)*0.1;

402 CAPSTONE THESIS

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