Exploring the HP Model for Protein Folding

We explore the HP model not only on the square lattice as originally proposed by Ken Dill, but we also use the triangular lattice. We find upper and lower bounds on the number of self-avoiding walks. In the square lattice, we get O(b^n) for some b in [2.414, 3]. We count the number of all self-avoiding walks of length up to 16 in the square and triangular lattices by exhaustively listing them. We use these lists of self-avoiding walks to study two HP sequences, one of length 11, and the other of length 16. We show that the diameter of the convex hull of a conformation can be used as an estimate of the energy of the conformation. Our examples demonstrate that the same holds true for the area of the convex hull. Both of these measures can be easily computed for a given conformation

Structural investigations of biomolecules under extreme conditions

Scattering methods like small-angle X-ray scattering and X-ray reflectivity enable to perform in situ studies on biomolecules under various conditions of temperature and pressure. These methods are employed in this work to shed light on the changes of size and shape as well as on the interactions between biomolecules under extreme conditions. Most of these aim to simulate conditions of temperature and pressures encountered in hydrothermal vents in the deep sea, where life might have evolved. This work includes many examples from all classes of biomolecules, lipids, nucleic acids as well as peptides and proteins, ranging from fairly simple to very complex systems

Structural investigations of biomolecules under extreme conditions

Scattering methods like small-angle X-ray scattering and X-ray reflectivity enable to perform in situ studies on biomolecules under various conditions of temperature and pressure. These methods are employed in this work to shed light on the changes of size and shape as well as on the interactions between biomolecules under extreme conditions. Most of these aim to simulate conditions of temperature and pressures encountered in hydrothermal vents in the deep sea, where life might have evolved. This work includes many examples from all classes of biomolecules, lipids, nucleic acids as well as peptides and proteins, ranging from fairly simple to very complex systems

Towards Understanding the Self-assembly of Complicated Particles via Computation.

We develop advanced Monte Carlo sampling schemes and new methods of calculating thermodynamic partition functions that are used to study the self-assembly of complicated ``patchy '' particles. Patchy particles are characterized by their strong anisotropic interactions, which can cause critical slowing down in Monte Carlo simulations of their self-assembly. We prove that detailed balance is maintained for our implementation of Monte Carlo cluster moves that ameliorate critical slowing down and use these simulations to predict the structures self-assembled by patchy tetrominoes. We compare structures predicted from our simulations with those generated by an alternative learning-augmented Monte Carlo approach and show that the learning-augmented approach fails to sample thermodynamic ensembles. We prove one way to maintain detailed balance when parallelizing Monte Carlo using the checkerboard domain decomposition scheme by enumerating the state-to-state transitions for a simple model with general applicability. Our implementation of checkerboard Monte Carlo on graphics processing units enables accelerated sampling of thermodynamic properties and we use it to confirm the fluid-hexatic transition observed at high packing fractions of hard disks. We develop a new method, bottom-up building block assembly, which generates partition functions hierarchically. Bottom-up building block assembly provides a means to answer the question of which structures are favored at a given temperature and allows accelerated prediction of potential energy minimizing structures, which are difficult to determine with Monte Carlo methods. We show how the sequences of clusters generated by bottom-up building block assembly can be used to inform ``assembly pathway engineering'', the design of patchy particles whose assembly propensity is optimized for a target structure. The utility of bottom-up building block assembly is demonstrated for systems of CdTe/CdS tetrahedra, DNA-tethered nanospheres, colloidal analogues of patchy tetrominoes and shape-shifting particles.Ph.D.Chemical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/91509/1/erjank_1.pd

The development of solid-state NMR methodology to study the dynamics of proteins and ice

Solid-state nuclear magnetic resonance (SSNMR) is an excellent tool for determining the molecular motions within a dynamic system. SSNMR relaxation measurements can access a vast range of timescales (ps - ms) and are able to simultaneously determine the frequency and amplitude of the motion that a particular nucleus is undergoing. Recent developments in SSNMR instrumentation now allow for >100 kHz magic angle spinning (MAS) using 0.7 and 0.8 mm rotors. State-of-the-art MAS is especially beneficial for those wishing to investigate proteins in the solid state: only sub-milligram amounts of sample are required and the fast spinning yields incredible spectral resolution. This also enables proton detection and the associated improvements in sensitivity (for protonated samples). Unfortunately, these small rotors are extremely challenging to pack with the semi-solid protein samples. Furthermore, the proteins can become dehydrated in the slow packing process, making them unsuitable for NMR. To address this point, in Chapter 3, we present the design and application of an ultracentrifuge tool for the packing of proteins into 0.7 - 1.3 mm diameter SSNMR rotors. The tool helps to reduce the waste of expensive isotopically labelled proteins and decreases the packing time from several hours to minutes. The work in Chapter 4 takes advantage of the mentioned fast MAS developments and demonstrates the accurate measurement of site-specific, spin-lattice relaxation rates (R1) on 13Ca nuclei in a fully protonated, uniformly 13C-labelled protein at 100 kHz MAS. Our approach overcomes the averaging effect of proton-driven spin diffusion that obscures site-specific information for the relaxation rates measured at slower spinning frequencies. One area where measurements of relaxation in the solid state can yield significant insights is the understanding of the complex energy landscape describing conformational changes of proteins, which are often closely linked to their functions. In Chapter 5 we present some of the first extensive site-specific variable temperature measurements of 13C' and 15N R1 and spin-lattice relaxation rates in the rotating frame (R1r) in a crystalline protein as a way to explore its conformational energy landscape. We observe R1r more than doubling over a narrow range of temperatures and minimal variation in R1 over the same range. We model the relaxation data using an extended model free approach and Arrhenius relationship to extract activation energies for the motions dominating the dynamics, however _nd that further measurements are required for an accurate determination of the activation energies. In Chapter 6 we show that relaxation measurements in the solid state are not only useful for characterising protein motions. In this chapter, we employ variable temperature measurements, including relaxation measurements, to investigate the effects of non-colligative antifreezes on ice dynamics. Antifreeze (glyco)proteins facilitate the survival of a diverse range of organisms at low temperatures by altering the freezing point, structure and growth of ice by modifying the dynamics of water molecules. These proteins and their synthetic mimics have many vital applications throughout science and engineering, but their mechanism of action is still not completely understood. In this PhD project, a combination of variable temperature relaxation measurements and 2D exchange spectra revealed that the antifreeze glycoproteins, type I antifreeze proteins, safranin and polyvinyl alcohols were exploiting a similar antifreeze mechanism involving reversible binding to ice, whereas the type III antifreeze protein was irreversibly binding to ice

Nucleation of Minerals: Precursors, Intermediates and Their Use in Materials Chemistry

Nucleation is the key event in mineralisation, but a general molecular understanding of phase separation mechanisms is still missing, despite more than 100 years of research in this field. In recent years, many studies have highlighted the occurrence of precursors and intermediates, which seem to challenge the assumptions underlying classical theories of nucleation and growth. This is especially true for the field of biomineralisation, where bio-inspired strategies take advantage of the special properties of the precursors and intermediates for the generation of advanced materials. All of this has led to the development of "non-classical" frameworks, which, however, often lack quantitative expressions for the evaluation and prediction of phase separation, growth and ripening processes, and are under considerable debate. It is thus evident that there is a crucial need for research into the early stages of mineral nucleation and growth, designed for the testing, refinement, and expansion of the different existing notions. This Special Issue of Minerals aims to bring together corresponding studies from all these areas, dealing with precursors and intermediates in mineralisation with the hope that it may contribute to the achievement of a better understanding of nucleation precursors and intermediates, and their target-oriented use in materials chemistry

Phases of Polymers and Biopolymers

In this thesis we develop coarse grained models aiming at understanding physical problems arising from phase transitions which occur at the single molecule level. The thesis will consist of two parts, grossly related to and motivated by the two subjects dealt with above. In the first half, we will focus on critical phenomena in stretching experiments, namely in DNA unzipping and polymer stretching in a bad solvent. In the second part, we will develop a model of thick polymers, with the goal of understanding the origin of the protein folds and the physics underlying the folding \u2018transition\u2019, as well as with the hope of shedding some light on some of the fundamental questions highlighted in this Introduction. In the first part of the thesis we will introduce a simple model of self-avoiding walks for DNA unzipping. In this way we can map out the phase diagram in the force vs. temperature plane. This reveals the present of an interesting cold unzipping transition. We then go on to study the dynamics of this coarse grained model. The main result which we will discuss is that the unzipping dynamics below the melting temperature obeys different scaling laws with respect to the opening above thermal denaturation, which is governed by temperature induced fluctuating bubbles. Motivated by this and by recent results from other theoretical groups, we move on to study the relation to DNA unzipping of the stretching of a homopolymer below the theta point. Though also in this case a cold unzipping is present in the phase diagram, this situation is richer from the theoretical point of view because the physics depends crucially on dimension: the underlying phase transition indeed is second order in two dimensions and first order in three. This is shown to be intimately linked to the failure of mean field in this phenomena, unlike for DNA unzipping. In particular, the globule unfolds via a series (hierarchy) of minima. In two dimensions they survive in the thermodynamic limit whereas if the dimension, d, is greater than 2, there is a crossover and for very long polymers the intermediate minima disappear. We deem it intriguing that an intermediate step in this minima hierarchy for polymers of finite length in the three-dimensional case is a regular mathematical helix, followed by a zig-zag structure. This is found to be general and almost independent of the interaction potential details. It suggests that a helix, one of the well-known protein secondary structure, is a natural choice for the ground state of a hydrophobic protein which has to withstand an effective pulling force. In the second part, we will follow the inverse route and ask for a minimal model which is able to account for the basic aspects of folding. By this, we mean a model which contains a suitable potential which has as its ground state a protein-like structure and which can account for the known thermodynamical properties of the folding transition. The existing potential which are able to do that[32] are usually constructed \u2018ad hoc\u2019 from knowledge of the native state. We stress that our procedure here is completely different and the model which we propose should be built up starting from minimal assumptions. Our main result is the following. If we throw away the usual view of a polymer as a sequence of hard spheres tethered together by a chain (see also Chapter 1) and substitute it with the notion of a flexible tube with a given thickness, then upon compaction our \u2019thick polymer\u2019 or \u2019tube\u2019 will display a rich secondary structure with protein-like helices and sheets, in sharp contrast with the degenerate and messy crumpled collapsed phase which is found with a conventional bead-and-link or bead-and-spring homopolymer model. Sheets and helices show up as the polymer gets thinner and passes from the swollen to the compact phase. In this sense the most interesting regime is a \u2018twilight\u2019 zone which consists of tubes which are at the edge of the compact phase, and we thus identify them as \u2018marginally compact strucures\u2019. Note the analogy with the result on stretching, in which the helices were in the same way the \u2018last compact\u2019 structures or the \u2018first extended\u2019 ones when the polymer is being unwinded by a force. After this property of ground states is discussed, we proceed to characterize the thermodynamics of a flexible thick polymer with attraction. The resulting phase diagram is shown to have many of the properties which are usually required from protein effective models, namely for thin polymers there is a second order collapse transition (O collapse) followed, as the temperature is lowered, by a first order transition to a semicrystalline phase where the compact phase orders forming long strands all aligned preferentially along some direction. For thicker polymers the transition to this latter phase occurs directly from the swollen phase, upon lowering T, through a first order transition resembling the folding transition of short proteins

Maßgeschneiderte Protein Inkapsulierung in einen DNA Käfig mittels geometrisch angeordneter supramolekularer Wechselwirkung

Im Rahmen dieser Dissertation wurden verschiedene Aspekte der DNA-Nanotechnologie, Biologie und der supramolekularen Chemie miteinander verknüpft und zur Anwendung gebracht. Die Arbeit kann in drei Teilbereiche unterteilt werden: (i) Das Designen eines geeigneten hexagonalen turbularen DNA-Käfigs mittels der Software CaDNAno2, dessen räumliche Dimensionen zur Immobilisierung des Proteins DegP in seinen unterschiedlichen Oligomerisierungs-Zuständen entsprechen musste. (ii) (ii) Die Synthese eines Oligonukleotid-Heptapeptid-Konjugates, welches supramolekular an die PDZ1 Domäne des Zielproteins bindet. (iii) (iii) Das Laden des Proteins in den Käfig mit anschließendem Versuch der gerichteten Freisetzung des Proteins. Der DNA-Origami-Käfig wurde mittels der Software CaDNAno2 als einlagiges, turbuläres und hexagonales DNA-Röhrchen mit einem inneren Radius von 20 nm und einem äußeren Radius von 23 nm entworfen und erfolgreich realisiert. Die einzelnen Seitenwände des Käfigs wurden durch jeweils speziell entworfene Oligonukleotide miteinander verknüpft, so dass zum einen die Orientierung der Seitenflächen zum Zentrum der Kavität (6p120 und 6p240) determiniert werden konnte, (vgl. Kapitel 3.2.1 und 3.2.2) und zum anderen durch eine flexible Verbindung der Seitenflächen mittels drei Thyminen eine zufällige Orientierung (6p180) der Seitenflächen zum Zentrum der Kavität vorlag. Zusätzlich wurde jede Seitenfläche mit der Option entworfen, mit null, ein, zwei oder drei (insgesamt 0cA1, 6cA1 oder 18cA1) orthogonal herausragenden Oligonukleotiden (Protruding-Arme), die als Verlängerung von zentral liegenden Oligonukleotiden zu sehen sind, ausgestattet zu werden. Die Protruding-Arme dienen als Träger für die mit Peptiden modifizierten komplementären Oligonukleotid-Liganden (Kapitel 3,3), welche wahlweise mit Fluorescein (Flc-A1-DPMFKLV) oder TAMRA (TAMRA-A1-DPMFKLV) ausgestattet waren. Mittels eines Biotin-modifizierten Liganden konnte durch Zugabe von Streptavidin die korrekte Formation der Konstrukte 6p120, 6p180 und 6p240 durch Gel-Elektrophorese und Raster-Kraft-Mikroskopie (AFM) bestätigt werden. Messungen durch das Transmissions-Elektronen-Mikroskop (durchgeführt von Pascal Lill am MPI in Dortmund im Rahmen seiner Masterarbeit) und der dynamischer Lichtstreuung bestätigten ebenfalls die korrekte Formation in Lösung. Unter Berücksichtigung des antiproportionalen Verhältnisses der Diffusionsrate durch den Käfig zur Größe des Proteins wurde der DNA-Origami-Komplex so geplant, dass das 12-mer des Proteins DegP bevorzugt binden sollte. Die Synthese des Peptidfragmentes der Liganden erfolgte mittels Festphasenpeptidsynthese nach Standardbedingungen. Die N-terminale Aminogruppe der Peptidsequenz konnte direkt in eine Maleimide Funktion überführt werden (Kapitel 6.5.2.8.2). Ohne die Verwendung eines üblichen Crosslinkers konnten Thiol-modifizierte Oligonukleotide über eine kovalente Bindung an das Maleimid gebunden werden. Dieser erfolgreich etablierte Weg stellt somit eine universelle Methode dar, N-terminale Peptide direkt an Thiol-modifizierte Oligonukleotide zu binden. Eine Charakterisierung der Konjugate erfolgte anschließend mittels MALDI-TOF. Das Laden diverser DegP Protein-Varianten (vgl. Kapitel 3.1, Tabelle 1) erfolgte nur in Anwesenheit des Peptid-Liganden (vgl. Kapitel 3.4.3), was die Selektivität der Methode erfolgreich demonstrierte und unspezifische Wechselwirkungen ausschloß. Das spezifische und erfolgreiche Binden an die DNA-Nano-Käfige konnte mittels interner Totalreflexionsfluoreszenzmikroskopie durch Überlagerung der Fluoreszenzsignale in Einzelmolekül-Experimenten bestätigt werden (durchgeführt durch die AG Birkedal, Aarhus Universität). Nach statistischer Auswertung der AFM-Bilder konnte gezeigt werden, dass eine Präferenz zur Immobilisierung von 12-meren im Verhältnis 1 : 2.2 : 1.3 für DegP6, DegP12 und DegP24 vorlag, was der Zielsetzung bezüglich der Selektivtät entsprach. Durch das Assemblieren eines 6p120 Käfigs mit einer unterschiedlichen Anzahl an Protruding-Armen (vgl. Kapitel 3.4.3.5) und dementsprechend mit einer unterschiedlichen Anzahl von Liganden konnte gezeigt werden, dass zum einen ein Ligand ausreicht, um ein Protein (DegP6) erfolgreich zu immobilisieren und zum anderen ein proportionales Verhältnis zwischen Ladungseffiziens und der Anzahl der Protruding-Arme besteht. Um den positiven Effekt einer hohen lokalen Konzentration der Peptid-Liganden auf die Bindeeffiziens der Proteine zu zeigen, wurden die DNA-Käfige mit den jeweiligen Ausrichtungen der Protruding-Arme (6p120, 6p180 und 6p240, vgl. Kapitel 3.4.3.4) und gleicher Anzahl der Liganden mit DegP12/24A488SA versetzt und mittels eines Vergleiches der Intensität des Fluorescein-Signals die höchste Bindeeffiziens im Verhältnis 8 : 1.4 : 1 (6p120: 6p180 : 6p240) für die Käfige mit nach innen-orientierten Liganden bestimmt. Nach dem erfolgreichen Immobilisieren des Proteins innerhalb der Kavität des Käfigs wurde anschließend eine Freisetzung des Proteins durch einen Austausch des zum Protruding-Arm nicht vollständig komplementären, mit Fluorophoren markierten Peptid-Liganden durch ein vollständig komplementäres Oligonukleotid versucht. Der erfolgreiche Austausch der markierten Liganden konnte mittels Gel-Elektrophorese gezeigt werden, jedoch nicht unter vollständiger Freisetzung des Proteins. Eine verbesserte Freisetzung konnte trotz Änderung der Nettoladung des Proteins ebenfalls nicht erreicht werden (vgl. Kapitel 3.4.3.8). Zusammenfassend kann festgehalten werden, dass zum ersten Mal ein Protein durch schwache supramolekulare Interaktionen innerhalb einer DNA-Origamistruktur immobilisiert werden konnte. Diese Proteine konnten ohne chemische Veränderung nur aufgrund der räumlichen Nähe der Peptid-Liganden, die an die PDZ1-Domänen der Proteine bindeten, und der hohen lokalen Konzentration innerhalb der Kavität durch multivalente Wechselwirkungen mit geringer Reichweite im Käfig gehalten werden. Die Bedeutung der Oberflächenladung der Proteine für die Immobilisierung innerhalb der Kavität bedarf noch weiterer Untersuchungen. Ein Indiz für diese Bedeutung liefern die Versuche mit dem molekularen Tweezer, welche an die Lysine an der Oberfläche der Proteine bindet und deren positive Ladung abschirmt. Folglich führt dies zur Abschwächung der unspezifischen Wechselwirkungen zwischen den Proteinen und dem Käfig, so dass diese erfolgreich geladen werden konnten. Die analogen Ergebnisse der Experimente mit einer weiteren Struktur eines offenen Prismas im HoneyComb-Design (vgl. Kapitel 1.2.3) werden in Kapitel 8 gezeigt, da diese keine neuen, bzw. nur die bisherigen Ergebnisse bestätigende Resultate erbracht haben.During this PhD project, aspects of DNA nanotechnology, biology and supramolecular chemistry have been merged. This work can be divided into three main parts: (i) the design of a suitable tubular DNA cage, which is able to encapsulate all different oligomers of DegP; (ii) the synthesis of a DNA-hepta-peptide conjugate, which binds non-covalently to the PDZ1 domain of the target protein and (iii) the loading of the protein. The DNA origami cage was successfully designed and realized as a single-layer hexagonal DNA prism with an inner radius of 20 nm and an outer radius of 23 nm. Using special spatial staples strands for the face-to-face connections, the orientation of the faces towards the cavity could be programmed (6p120 and 6p240). Alternatively, stochastically oriented faces (6p180) were obtained using three thymines as flexible hinges (see chapter 3.2.1 and 3.2.2). Additionally, each face was equipped with zero, one, two or three orthogonal protruding arms (for a total of 0cA1, 6cA1 or 18cA1 arms, respectively). These arms were used for hybridization with DNA-peptide conjugates equipped with optional fluorophores (chapter 3.3). Correct formation of the different designs (6p120, 6p180 and 6p240) was proven by AFM and gel electrophoresis, after adding streptavidin to the biotinylated ligands. TEM characterization (performed by Pascal Lill at the MPI in Dortmund) and dynamic light scattering confirmed the correct dimensions of the DNA structures in solution. The peptide sequence DPMFKLV was synthesized via solid phase peptide synthesis under standard coupling conditions. Transforming the N-terminal amino group of the peptide directly into a maleimide function allowed reaction with the thiol group of an oligonucleotide without the use of any crosslinking agents. This successful method represents a general method to link the terminal amino-group of a peptide to a thiol bearing oligonucleotide. Purification via HPLC allowed characterization of the DNA-DPMFKLV ligands per MALDI-TOF. Loading of diverse DegP proteins (compare chapter 3.1, table 1) only took place in the presence of the A1-DPMFKLV ligands, demonstrating the validity of the encapsulation strategy (compare chapter 3.4.3). Successful and specific binding was shown at the single-molecule level using total internal reflection fluorescence (TIRF) microscopy, (performed by AG Birkedal, Aarhus University). Statistical evaluation of AFM images revealed preferential encapsulation of the DegP12 protein, with a ratio of 1.3 : 2.2 : 1 for the DegP6, DegP12 and DegP24, respectively. Loading experiments performed with DegP6A633SA and cages with a different number of PAs (see chapter 3.4.3.5) and a correspondingly different number of A1-peptide ligands, showed that one A1-DPMFKLV ligand is sufficient for encapsulation of the protein, with a loading efficiency proportional to the number of ligands. DNA origami cages with identical number of ligands but different orientations of the PAs (6p120, 6p180 and 6p240, see chapter 3.4.3.4) were loaded with DegP12/24A488SA. Gel electrophoresis analysis showed a highest binding efficiency for the 6p120 design in a ratio of 8 : 1.4 : 1 for the 6p120-, 6p180- and 6p240-designs, respectively, thus indicating the importance of a high local concentration of peptide ligands. After successful encapsulation of the protein, experiments to release the protein from the cage were performed, using single strand displacement reactions. Disappearance of the fluorophore signals from the cage sample showed successful displacement of the ligands; however, without releasing the protein. Variation of the pH of the solution and its ionic strength did not result in any beneficial effects (see chapter 3.4.3.8). To sum up, it could be shown that a protein can be encapsulated within a DNA origami cage by weak non-covalent supramolecular interactions without any previous chemical treatment of the protein. The arrangement of a distinct number of peptide ligands in the vicinity of the corresponding binding sites on the protein surface allowed modulation of local concentration effects and multivalent short-range interactions in a single system. The importance of the net charge of the protein for encapsulation within the cavity of the DNA nanochamber and the different binding efficiencies observed for distinct fluorophores, require further investigation. Evidence for the important role of the net charge is the successful loading of the DegP12 protein in presence of molecular tweezers, targeting the lysine residues on the surface of the protein. Results of analogue experiments performed with a second DNA cage, (HoneyComb design, see chapter 1.2.3) are shown in chapter 8