Lewis University Dr. James Girard Summer Undergraduate Research Program 2021 Faculty Mentor - Project Application Mathematical Modeling of ...
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Lewis University
Dr. James Girard Summer Undergraduate Research
Program 2021
Faculty Mentor - Project Application
Mathematical Modeling of Self-Assembling DNA
Dr. Amanda Harsy
Department of Engineering, Computing, and Mathematical Sciences
Mathematical Modeling of Self-Assembling DNA
February 11, 2021
Abstract
Self-assembly is a term used to describe the process of a collection of components combining
to form an organized structure without external direction. The unique properties of double-
stranded DNA molecules make DNA a valuable structural material with which to form nanos-
tructures, and the field of DNA nantechnology is largely based on this premise. By modeling
nanostructures with discrete graphs, efficient DNA self-assembly becomes a mathematical puz-
zle. These nanostructures have wide-ranging applications, such as containers for the transport
and release of nano-cargos, templates for the controlled growth of nano-objects, and in drug-
delivery methods. This research project centers around the exploration of the graph theoretical
and combinatorial properties of DNA self-assembly, as well as development of computational
tools to aid in answering fundamental questions that arise.
1
1 Project Description
1.1 Introduction and Background
Motivated by the recent advancements in nanotechnology and the discovery of new laboratory tech-
niques using the Watson-Crick complementary properties of DNA strands, graph theory has be-
come useful in the study of self-assembling DNA complexes. DNA self-assembly, and self-assembly
in general, is a rapidly advancing field, with [24, 27] providing good overviews. Synthetic DNA
molecules have been designed that self assemble into given nanostructures, starting with branched
DNA molecules [14, 31], nanoscale arrays [32, 33], numerous polyhedra [3, 9, 10, 28, 39], arbitrary
graphs [13, 25, 34], a variety of DNA and RNA knots [20, 21, 30], and the first macroscopic self-
assembled 3D DNA crystals [40]. This has led to molecular sca↵oldings made of DNA which have
wide-ranging potential, such as containers for the transport and release of nano-cargos, templates
for the controlled growth of nano-objects, biomolecular computing, biosensors and in drug-delivery
methods (see [1, 6, 7, 15, 18, 22, 23, 38, 26, 36]).
Furthermore, new experiments demonstrating the feasibility of engineering synthetic DNA nanoscale
constructs o↵er great promise for emergent applications in nanoelectronics, biosensors, biomolec-
ular computing, drug delivery systems, and directed organic synthesis, all of which lead to more
e↵ective diagnosis and treatment of illness. [37], [19], [17], [35], [18] give excellent overviews of
self-assembly methods and emerging applications. In particular, DNA origami is advancing very
rapidly toward its potential as a drug delivery mechanism. For example, [35] and [41] discuss how
DNA assembly can improve the treatment of cancer and [16] outlines how self-assembled DNA can
be used to help overcome drug resistance in breast cancer cells. Already various labs have produced
closed containers self-assembled out of DNA strands which can be made to encapsulate particles,
and then their opening deliberately triggered. See [2] and [8] and the HarvarDNAno video [29]
which well illustrates both the process and promise of this new technique.
Since modeling this self-assembling process requires designing the component molecular building
blocks, which often are modeled through surface meshes, lattice subsets, and other graph-like struc-
tures, construction methods developed with concepts from graph theory have resulted in signifi-
cantly increased efficiency. For example, one recent focus in DNA nanotechnology is the formation
of nanotubes which can be modeled using a lattice graph. The rules governing the structure of
these nanotubes are not yet well understood, and this naturally o↵ers open problems in the realm
of applied graph theory.
1.2 Research Aims and Goals
In this research, we explore the underlying graph-theoretical structure of nanostructure construc-
tion and related design strategy problems. Our project focuses on a construction method for
self-assembling DNA structures which involves branched junction molecules whose flexible k-arms
are double strands of DNA. These arms have cohesive ends which can bond to any other cohesive
end with a complementary sequence of bases. We call the k-armed molecule a “tile” and represent
it as a vertex of degree k in a graph. We say a collection of tiles (a pot) realizes a graph, G, if
the collection of tiles constructs the same structure as G. Thus, we can investigate design process
questions regarding which types of final structures can be constructed from a given pot of tiles.
1
Inversely, we can also find a pot of tiles that will realize a given target graph. Our group primarily
focuses on the latter problem. Along with graph theory and combinatorics, we also use linear alge-
bra to help answer these questions through the “construction matrix” [5]. The goal in answering
these questions will be to determine the minimum number of branched junction molecules and their
combinatorial structures for self-assembling DNA nanostructures.
The central focus of our research is to find the minimum number of tile types and bond-edge
types necessary to construct a target graph G by modular assembly under a variety of laboratory
conditions, and furthermore find explicit pots realizing these minima. This includes creating general
design theory and finding accurate bounds for the number of tile types and bond-edge types for a
variety of new graphs. Thus, the fundamental questions for this research include:
• Given a target graph, G, what is the minimum number of tile types which need to be designed
to create the graph? We denote this minimum number by T (G).
• Given a target graph, what is B(G), the minimum number of bond types needed?
• Given a target graph, what is the combinatorial structure of the molecules in a minimum set
which realizes the target graph?
Questions regarding design strategies for realizing a target graph are considered under three di↵er-
ent scenarios:
Scenario 1: A graph with a fewer number of vertices than the target graph may be realized from
a pot of tiles.
Scenario 2: A graph on the same number of vertices, but not isomorphic to the target graph may
be realized from a given pot of tiles. However, no graph on fewer vertices may be realized.
Scenario 3: No graph on fewer vertices nor non-isomorphic graphs on the same number of vertices
as the target graph may be realized from a given pot of tiles.
Our goal is to provide at least tight bounds, and ideally provably optimal closed form solutions,
under each scenario. My faculty research team has recently submitted a paper discussing partial
results for grid graphs, nanotubes, and platonic solids. We ultimately intend to build on our results
to obtain provably optimal closed form solutions for Scenarios 1 and 2 for each of the graph types
mentioned. For Scenario 3, we aim to first find and then prove tight bounds on numbers of tile and
bond-edge types. We also plan to develop obstruction theorems which can help provide support
for laboratories creating these nanostructures. Specifically, we hope to find design rules to help the
laboratories avoid creating smaller graphs in some of the more restrictive scenarios. The problem
of finding optimal pots of tiles leads to one of our other goals for our research which is to create
tools to help find and check pots of target graphs for di↵erent scenarios. We have already made
progress towards this goal. My faculty team has already written two Maple programs to help us
produce and check pots for Scenario 2. One program is designed to check the minimum size graphs
a given pot of tiles will construct in Scenario 2. Specifically this program deals with the case in
which a free variable appears in the solution to the system represented by the construction matrix.
2Our second program produces all possible pots for k-regular graphs on n vertices using b bond-edge
types. The program confirms that no smaller graphs can be realized from the pots produced.
1.3 Methodology
The introduction of a graph-theoretical formalization for exploring the combinatorial properties
within the self-assembly of branched junction molecules was first introduced by Jonoska et al. in
[12]. The centers of these branched molecules form the vertices of the constructed complex. One
arm of the molecule extends longer than the other and its end forms a sticky (or cohesive) end-type
which can bond with other arms with the corresponding complementary Watson-Crick Bases. See
Figures 2 and 1.
To represent these k -armed molecules and sticky end types, we will use a vertex of degree k like
the ones in Figure 1. We call these k-armed molecules “tiles.” We represent the complementary
sticky-ends or bond-edges of these tiles using letter labels as in Figures 2 and 1. The letters a, b,
c, â, b̂, ĉ, etc. represent the un-adjoined arms sticking o↵ of the k -armed molecules. For example,
bond end types a and â represent complementary sequences of bases as in Figure 2. Thus a and â
could self assemble in order to form a bond-edge. Thus we can combinatorially represent a k -armed
branched-junction molecule with bond-end types a1 , ..., ak using a tile t = (a1 , ..., ak ).
Figure 2: Representing the
complementary cohesive end
types
Figure 1: 3 -armed branched junction molecule (left) with
example tile representation (right)
We assume these tiles have arms which can freely move and are long enough and flexible enough
to reach around and connect with complementary cohesive end-types without restriction. These
“flexible tiles” were first introduced in [11]. Thus we define a tile type t to be a flexible-armed
branched junction molecule described by a set of cohesive-end types. We call a set of tiles, a pot.
Figure 3 gives an example of a pot with 4 di↵erent tile types.
Figure 3: A pot of 4 tile types.
3Theoretically we assume that once a tile type is made, a pot with that tile type can have “infinitely
many” of those tile types. Once we have a pot, those tiles are able to self-assemble into complexes
either complete or incomplete as seen in Figure 4. Labs generally want to create complete complexes.
Figure 4: Complete vs Incomplete Complex
Linear algebra and programming can also be used to help answer these questions. In particular, the
construction matrix can be used to help determine whether a pot might generate graphs smaller
than the target graph. [5, 12]. Given a pot P of tiles, in order for a complete complex to be
formed and no smaller graph to be formed, the proportions of di↵erent tile types must satisfy a
system of equations which capture the requirements outlined in Item 2 of Proposition 2 in [5].
We often represent this system of equations with an augmented matrix called the construction
matrix, denoted M (P ). If the solution to the construction matrix provides a unique solution, then
no smaller complexes can be made from the pot that determines M (P ). Thus, the construction
matrix is a useful way to determine whether pots can generate graphs smaller than the target
graph.
1.4 Mentorship Plan
To help keep the students engaged in the research, the students will meet at least three times a
week with Dr. Harsy in addition to attending the SURE seminars and the weekly Math Research
Seminars. During the Math Research Seminars, students will present their current progress on their
research to the other students and faculty working on math research. This allows them to practice
presenting and gives them a chance to get feedback from other mathematicians. Most students
present bi-weekly. Students will also be expected to keep a research journal and complete a final
write-up and research report/paper by the end of the summer. The schedules below provide details
and timelines for the students and will help keep them accountable and organized with completing
their work. Many of the students who have worked on this research have presented their results at
local, regional, or national mathematics conferences and I would expect the students in this project
to present at similar conferences over the following year.
Students working on this project will design optimal pots for di↵erent families of graphs. 16 Lewis
students have worked on this project so far and the majority of these students have been determining
and proving optimal designs for families of bipartite graphs and lattice-based graphs in Scenarios
1 and 2. Two students have been working on computer programs to help with the design process.
Depending on the SURE student’s interests, the student may continue research already done by
past students, explore new graphs, or work to develop additional mathematical software to be used
directly by the laboratories creating the nanostructures. Ideally, we would like students to explore
lattice-based graphs or graphs that grow in more than one way since these will be most applicable
4for designs of drug delivery containers. The possible projects for the research (depending on the
student’s interest and background) are outlined below:
• Project 1: Compute T (G), B(G), and find explicit pots of tiles for families of graphs with
a lattice substructure such as Mongolian Tent Graphs, Sierpinski Carpet Graphs, Ladder
graphs, Web, Helm, Cone, Sunlet, etc.
• Project 2: Continue the progress from past student research by computing T (G), B(G), and
finding explicit pots of tiles for families of graphs which grow in multiple ways like fan graphs,
stacked book graphs, Mongolean Tent Graphs, Ladder Graphs, etc.
• Project 3: Compute T (G), B(G), and find explicit pots of tiles for families of k-regular graphs
such as Crown Graphs, Cage Graphs, and n-circulant graphs. Additionally, students can
explore relationships between pots of tiles for k-regular graphs for a fixed value of k.
• Project 4: Expand and improve the efficiency of current software used to produce and check
pots for regular graphs, in addition to computing pots of tiles for non-regular graphs.
Regardless of the project the student chooses, I have a particular way which I prepare students for
this research. Below is a detailed plan for the first two weeks (6 meetings) of the research which
provides background and allows students to learn the typical techniques, notation, and logic used
in this research.
Day 1: Introduction to research topic and notation: Students will read Using DNA self-
assembly design strategies to motivate graph theory concepts pages 96-101 along with the first three
chapters of Alan Tucker’s Applied Combinatorics.
Day 2: Scenario 1 Techniques: Students will begin reading Minimal tile and bond-edge types
for self-assembling DNA graphs pages 239-245, and the Scenario 1 section (page 247-249). Create
and justify optimal tiling for known graphs like Cn , Kn , Sn , Wn , and Pn in Scenario 1.
Day 3: Scenario 2 Techniques and Introduction to Using the Construction Matrix:
Students will continue reading Minimal tile and bond-edge types for self-assembling DNA graphs,
specially pages 245-247 and the Scenario 2 section on pages 250-253. Students will start creating
and justifying optimal tiling for known graphs like Cn , Kn , Sn , Wn , and Pn in Scenario 2. The
student will be introduced to the main linear algebra construct used in this research.
Day 4: Scenario 2 Techniques Continued: Students will finish reading Minimal tile and bond-
edge types for self-assembling DNA graphs pages 254-256, 260-264. Students should have finished
creating and justifying optimal tiling for known graphs like Cn , Kn , Sn , Wn , and Pn in Scenario 2.
Day 5: Scenario 3 Techniques: Students should have finished reading Minimal tile and bond-
edge types for self-assembling DNA graphs and should start creating and justifying optimal tiling
for known graphs like Cn , Kn , Sn , Wn , and Pn in Scenario 3.
5Day 6: Scenario 3 Techniques Continued: Students should be done creating and justifying
optimal tiling for known graphs like Cn , Kn , Sn , Wn , and Pn in all 3 scenarios and will decide on
which project/graphs they would like to explore on their own.
1.5 Proposed General Timeline with Project Goals
Weeks 1-2: Read the first three chapters of Alan Tucker’s Applied Combinatorics along with some basic
papers including [4] and [5] which introduce the project. Students should work on exercises and confirm
results from these papers.
Weeks 3-7: Develop design strategies to create minimum pots for di↵erent bipartite, tripartite, or lattice-
based graphs in Scenarios 1,2, and 3. Formalize the results by constructing proofs or counter examples.
Weeks 8-9: Finalize and summarize results, attend and present at MAA MathFest.
Week 10: Finish main draft of paper and prepare for SURE presentation
1.5.1 Timeline for Student Write-Up/Paper Task List:
Week 1: Do a literary search for the current research in Modeling DNA Self-Assembly. Read [4] and [5]
which introduce the project. Complete a typed 1-3 paragraph synopsis of what might be useful in the articles.
Week 2: Work with Dr. Harsy to decide on the focus of their research and the point of their paper. At
this point they should have a title page and the TeX template set up. This is just a working title and can
be changed.
Week 3: Make a concept map of the ideas for the paper and how they interconnect. Decide how to thread
your storyline through this map.
Week 4: First draft of definitions and background terminology and notation, three examples, or other
foundation materials should be added to the report.
Week 5: Complete further work on definitions and outline of content. Start including examples or data or
initial results.
Week 6: Write a draft of the introduction. This will incorporate material from the paragraphs you wrote
during the literary search. Continue adding supporting materials. Start working on any figures you want to
add to the paper.
Week 7: Draft a conclusion and typeset. Insert figures, review expository structure. If possible, write a
15 min talk with slides, and use what you learn from the talk to improve the organization, figures, and
exposition of the paper.
Week 8: Do an overall editing pass to complete the first draft. Continue to add further results if needed.
Week 9: Have someone give you some feedback like Dr. Dr. Harsy, Dr. Meyer, or Dr. Stephenson. Having
someone outside of your project is often helpful at this stage. Finalize SURE presentation.
Week 10: Edit the second draft. Put it away for at least a week, edit again (have someone give feedback
again). Do final editing and then possibly submit to a journal.
6References
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2 Budget
Students may need to have access to Maple, but this hasn’t been a problem in the past few years of doing
this research since Lewis has university licenses. Thus, I am requesting no funds for supplies. If possible, I
would be happy to have two SURE students for this project.
3 Description of any additional Funding you will be using for your
proposed research and how they will be used in this project.
I have no additional funding for this research.
4 Criteria for Student Applicants
This research can be approached with little mathematical background. It involves basic graph theory and
depending on the student’s background and interests may also involve programming, computer graphics, art,
biology, and geometry.
5 SURE Seminars
I am interested in Presentation Skills, Preparing for Graduate School, Literature Search and Library Re-
sources, Resume Writing and Marketing YOU, and Interview Skills. I would also be willing to do a LATEX
seminar as well.
6 SURE Agreement
The James Girard Summer Undergraduate Research Program is designed to support the execution of this
proposed project by the faculty mentor and a single undergraduate student. After review of faculty propos-
als, selected projects will be advertised to Lewis University students, and all interested undergraduates will
then be required to apply into the program, denoting the project for which they would like to be considered.
Student applications will be reviewed for completeness by the program director, and then forwarded to the
appropriate faculty mentor for final selection of a candidate. Faculty may submit up to 2 projects for funding
through the program. Although faculty mentors may also mentor additional students in the summer not
funded through the program, the weekly program events and presentations will be exclusive for students in
the program.
By submitting this application, you are agreeing to the following responsibilities of a S.U.R.E. Faculty
Mentor:
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