831 research outputs found
Asymptotic formulas for solitary waves in the high-energy limit of FPU-type chains
It is well established that the solitary waves of FPU-type chains converge in
the high-energy limit to traveling waves of the hard-sphere model. In this
paper we establish improved asymptotic expressions for the wave profiles as
well as an explicit formula for the wave speed. The key step in our approach is
the derivation of an asymptotic ODE for the appropriately rescaled strain
profile.Comment: revised version with corrected typos; 25 pages, several figure
Design, Synthesis and Analysis of Self-Assembling Triangulated Wireframe DNA Structures
The field of DNA nanotechnology offers a wide range of design strategies with which nanometer-sized structures with a desired shape, size and aspect ratio can be built. The most established techniques in the field rely on close-packed 'solid' DNA nanostructures produced with either the DNA origami or the single-stranded tile techniques. These structures depend on high-salt buffer solutions and require more material than comparable size hollow wireframe structures.
This dissertation explores the construction of hollow wireframe DNA nanostructures composed of equilateral triangles. To achieve maximal material efficiency the design is restricted to use a single DNA double helix per triangle edge. As a proof of principle, the DNA origami technique is extended to produce a series of truss structures including the flat, tetrahedral, octahedral, or irregular dodecahedral truss designs. In contrast to close packed DNA origami designs these structures fold at low-salt buffer conditions. These structures have defined cavities that may in the future be used to precisely position functional elements such as metallic nanoparticles or enzymes. The design process of these structures is simplified by a custom design software.
Next, the triangulated construction motif is extended to the single-stranded DNA tile technique. A collection of finite structures, as well as one-dimensional crystalline assemblies is explored. The ideal assembly conditions are determined experimentally and using molecular dynamics simulations. A custom design software is presented to simplify the design and handling of these structures.
At last, the cost-effective prototyping of triangulated wireframe DNA origami structures is explored. This is achieved through the introduction of single-stranded “gap” regions along the triangle edges. These gap regions are then filled using a DNA polymerase rather than by synthetic oligonucleotides. This technique also allows the mechanical transformation of these structures, which is exemplified by the transition of a bent into a straight structure upon completion of the gap filling.:Abstract v
Publications vii
Acknowledgements ix
Contents xi
Chapter 1 A short introduction into DNA nanotechnology 1
1.1 Nanotechnology 1
1.1.1 Top down 1
1.1.2 Bottom up 3
1.2 Deoxyribonucleic acid (DNA) 4
1.3 DNA Nanotechnology 6
1.3.1 Tile based assembly 9
1.3.2 DNA origami and single-stranded tiles 10
1.3.3 Some applications of DNA nanotechnology 12
1.3.4 Wireframe structures 15
1.3.5 Computational tools and DNA nanotechnology. 17
Chapter 2 Motivation and objectives 19
Chapter 3 Design and Synthesis of Triangulated DNA Origami Trusses 20
3.1 Introduction 20
3.2 Results and Discussion 21
3.2.1 Design 21
3.2.2 Nomenclature and parameters of the tube structures 23
3.2.3 Gel electrophoreses analysis 25
3.2.4 Imaging of the purified structures 26
3.2.5 Optimizing the folding conditions 28
3.2.6 Comparison to vHelix 29
3.3 Conclusions 29
3.4 Methods 30
3.4.1 Standard DNA origami assembly reaction. 30
3.4.2 Gel purification. 30
3.4.3 AFM sample preparation. 31
3.4.4 TEM sample preparation. 31
3.4.5 Instructions for mixing the staple sets. 31
Chapter 4 Triangulated wireframe structures assembled using single-stranded DNA tiles 33
4.1 Introduction 33
4.2 Results and Discussion 35
4.2.1 Designing the structures 35
4.2.2 Synthesis of test structures 37
4.2.3 Molecular dynamics simulations of 6-arm junctions 38
4.2.4 Assembly of the finite structures 40
4.2.5 Influence of salt concentration and folding times 42
4.2.6 Molecular dynamics simulations of the rhombus structure 43
4.2.7 1D SST crystals 44
4.2.8 Controlling the crystal growth 46
4.3 Conclusions 48
4.4 Methods 49
4.4.1 SST Folding 49
4.4.2 Agarose Gel Electrophoresis 49
4.4.3 tSEM Characterization 49
4.4.4 AFM Imaging 49
4.4.5 AGE-Based Folding-Yield Estimation 49
4.4.6 Molecular Dynamics Simulations 50
Chapter 5 Structural transformation of wireframe DNA origami via DNA polymerase assisted gap-filling 52
5.1 Introduction 52
5.2 Results and Discussion 54
5.2.1 Design of the Structures 54
5.2.2 Folding of Gap-Structures 56
5.2.3 Inactivation of Polymerase. 57
5.2.4 Secondary Structures. 58
5.2.5 Folding Kinetics of Gap Origami. 60
5.3 Conclusions 61
5.4 Methods 62
5.4.1 DNA origami folding 62
5.4.2 Gap filling of the wireframe DNA origami structures 63
5.4.3 Agarose gel electrophoresis 63
5.4.4 PAGE gel analysis 63
5.4.5 tSEM characterization 64
5.4.6 AFM imaging 64
5.4.7 AGE based folding-yield estimation 64
5.4.8 Gibbs free energy simulation using mfold 65
5.4.9 List of sequence for folding the DNA origami triangulated structures 65
Chapter 6 Summary and outlook 67
Appendix 69
A.1 Additional figures from chapter 369
A.2 Additional figures from chapter 4 77
A.3 Additional figures from chapter 5 111
Bibliography 127
Erklärung 13
Nonlinear and Nonlocal Eigenvalue Problems: variational existence, decay properties, approximation, and universal scaling limits
We study a class of nonlinear eigenvalue problems which involves a convolution operator as well as a superlinear nonlinearity. Our variational existence proof is based on constrained optimization and provides a one-parameter family of solutions with positive eigenvalues and unimodal eigenfunctions. We also discuss the decay properties and the numerical computations of those eigenfunctions, and conclude with two asymptotic results concerning small and large eigenvalues
A uniqueness result for a simple superlinear eigenvalue problem
We study the eigenvalue problem for a superlinear convolution operator in the
special case of bilinear constitutive laws and establish the existence and
uniqueness of a one-parameter family of nonlinear eigenfunctions under a
topological shape constraint. Our proof uses a nonlinear change of scalar
parameters and applies Krein-Rutmann arguments to a linear substitute problem.
We also present numerical simulations and discuss the asymptotics of two
limiting cases.Comment: revised version with enhanced introduction; 21 pages, several figure
Subsonic phase transition waves in bistable lattice models with small spinodal region
Phase transitions waves in atomic chains with double-well potential play a
fundamental role in materials science, but very little is known about their
mathematical properties. In particular, the only available results about waves
with large amplitudes concern chains with piecewise-quadratic pair potential.
In this paper we consider perturbations of a bi-quadratic potential and prove
that the corresponding three-parameter family of waves persists as long as the
perturbation is small and localised with respect to the strain variable. As a
standard Lyapunov-Schmidt reduction cannot be used due to the presence of an
essential spectrum, we characterise the perturbation of the wave as a fixed
point of a nonlinear and nonlocal operator and show that this operator is
contractive in a small ball in a suitable function space. Moreover, we derive a
uniqueness result for phase transition waves with certain properties and
discuss the kinetic relation.Comment: revised version with extended introduction, improved perturbation
method, and novel uniqueness result; 20 pages, 5 figure
Uniqueness of solitary waves in the high-energy limit of FPU-type chains
Recent asymptotic results by the authors provided detailed information on the shape of solitary high-energy travelling waves in FPU atomic chains. In this note we use and extend the methods to understand the linearisation of the travelling wave equation. We show that there are not any other zero eigenvalues than those created by the translation symmetry and this implies a local uniqueness result. The key argument in our asymptotic analysis is to replace the linear advance-delay-differential equation for the eigenfunctions by an approximate ODE
Nonlinear and Nonlocal Eigenvalue Problems: variational existence, decay properties, approximation, and universal scaling limits
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