This report examines the mathematics of paper folding. One can solve cubic polynomials by folding a common tangent to two distinct parabolas. This then leads to constructions that cannot be done with a straightedge and compass such as angle trisection and doubling a cube. / text

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The HPr proteins were characterized as folding by a two-state folding mechanism. Here, we present a comparison of the equilibrium and kinetic folding for the HPr protein from Bacillus subtilis, E coli and a key variant from these proteins. For the wild-type protein we find that GHX is greater than GUDC, suggesting that the HPr does not fold by a simple two-state mechanism. This discrepancy is revealed by testing the two-state nature of the folding reaction of HPr with mutation. We show that removing a single charge side chain (Asp 69) converts the HPr protein back to a simple two-state mechanism. Ribonuclease Sa and two charge-reversal variants can be converted into amyloidin vitro by the addition of 2,2,2-triflouroethanol (TFE). We report here amyloid fibril formation for these proteins as a function of pH. The pH at maximal fibril formation correlates with the pH dependence of protein solubility, but not with stability, for these variants. Additionally, we show that the pH at maximal fibril formation for a number of ivwell-characterized proteins is near the pI, where the protein is expected to be the least soluble. This suggests that protein solubility is an important determinant of fibril formation.

Protein folding is considered to be one of the grand challenge problems in biology. Protein folding refers to how a protein's amino acid sequence, under certain physiological conditions, folds into a stable close-packed three-dimensional structure known as the native state. There are two major problems in protein folding. One, usually called protein structure prediction, is to predict the structure of the protein's native state given only the amino acid sequence. Another important and strongly related problem, often called protein folding, is to study how the amino acid sequence dynamically transitions from an unstructured state to the native state. In this dissertation, we concentrate on the second problem. There are several approaches that have been applied to the protein folding problem, including molecular dynamics, Monte Carlo methods, statistical mechanical models, and lattice models. However, most of these approaches suffer from either overly-detailed simulations, requiring impractical computation times, or overly-simplified models, resulting in unrealistic solutions. In this work, we present a novel motion planning based framework for studying protein folding. We describe how it can be used to approximately map a protein's energy landscape, and then discuss how to find approximate folding pathways and kinetics on this approximate energy landscape. In particular, our technique can produce potential energy landscapes, free energy landscapes, and many folding pathways all from a single roadmap. The roadmap can be computed in a few hours on a desktop PC using a coarse potential energy function. In addition, our motion planning based approach is the first simulation method that enables the study of protein folding kinetics at a level of detail that is appropriate (i.e., not too detailed or too coarse) for capturing possible 2-state and 3-state folding kinetics that may coexist in one protein. Indeed, the unique ability of our method to produce large sets of unrelated folding pathways may potentially provide crucial insight into some aspects of folding kinetics that are not available to other theoretical techniques.

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Members of the Sco protein family have been proposed to function in the assembly of cytochrome c oxidase in the respiratory chain of all aerobic life forms. The Sco protein in Bacillus subtilis, BsSco, is characterized for its folding/unfolding behavior in the presence or absence of Cu(II) in this study. The folding/unfolding of apo-BsSco is investigated by CD and fluorescence spectroscopies. BsSco follows an apparent two-state mechanism in both folding and unfolding processes. The two apo forms of BsSco, reduced and oxidized, exhibit similar equilibrium stabilities suggesting that the formation of an intramolecular disulfide in oxidized apo-BsSco does not add to BsSco’s overall stability. In contrast, Cu(II) binding to reduced apo-BsSco results in extreme stabilization and resistance to unfolding in urea. However, when Cu(II) is present with unfolded, reduced apo-BsSco, the protein is rapidly oxidized. Another widely used denaturant, GdnHCl, is able to unfold Cu(II)-BsSco by allowing the loss of Cu(II) from the metal/protein complex. When the presence of Cu(II)-BsSco complex and the protein’s folded state are monitored simultaneously, the unfolding of Cu(II)-bound BsSco occurs coincidently with Cu(II) dissociation. We suggest that the loss of Cu(II) binding and the loss of BsSco’s native conformation are coincident, which leads to the conclusion that Cu(II)-BsSco does not unfold until it forfeits Cu(II). The kinetics of folding/unfolding of reduced, oxidized and Cu(II) bound BsSco are explored by stopped-flow fluorescence spectroscopy. The rate constants at which the two apo forms of BsSco fold and unfold are measured and plotted versus denaturant concentration. Reduced and oxidized forms of apo-BsSco are similar in folding and unfolding kinetics. Cu(II)-involved refolding kinetics of BsSco show that Cu(II) is able to accelerate the rate of refolding. However, the involvement of Cu(II) in the refolding process results in two competing processes: oxidation and Cu(II) binding. Which process predominates depends on the refolding rate which further depends on the denaturant concentration. This study has provided direct evidence for metal-involved stabilization of BsSco which is beneficial to efficiently fulfill its copper trafficking duty in a cellular environment. / Thesis (Master, Biochemistry) -- Queen's University, 2010-12-17 17:24:09.598

As the expression of the genetic blueprint, proteins are at the heart of all biological systems. The ever increasing set of available protein structures has taught us that diversity is the hallmark of their architecture, a fundamental characteristic that enables them to perform the vast array of functionality upon which all of life depends. This diversity, however, is central to one of the most challenging problems in molecular biology: how does a folding polypeptide chain navigate its way through all of the myriad of possible conformations to find its own particular biologically active form? With few overarching structural principles to draw upon that can be applied to all protein architecture, the search for a solution to the protein folding problem has yet to produce an algorithm that can explain and duplicate this fundamental biological process. In this thesis, we take a two-pronged approach for investigating the protein folding process. Our initial statistical studies of the distributions of hydrophobic and hydrophilic residues within α-helices and β-sheets suggest (i) that hydrophobicity plays a critical role in helix and sheet formation; and (ii) that the nucleation of these motifs may result in largely unidirectional growth. Most tellingly, from an examination of the amino acids found in the smallest β-sheets, we do not find any evidence of a β-nucleating code in the primary protein sequence. Complementing these statistical analyses, we have analyzed the structural environments of several ever-widening aspects of protein topology. Our examination of the gaps between strands in the smallest β-sheets reveals a common organizational principle underlying β-formation involving strands separated by large sequential gaps: with very few exceptions, these large gaps fold into single, compact structural modules, bringing the β-strands that are otherwise far apart in the sequence close together in space. We conclude, therefore, that β-nucleation in the smallest sheets results from the co-location of two strands that are either local in sequence, or local in space following prior folding events. A second study of larger β-sheets both corroborates and extends these findings: virtually all large sequential gaps between pairs of β-strands organize themselves into an hierarchical arrangement, creating a bread-crumb model of go-and-come-back structural organization that ultimately juxtaposes two strands of a parental β-structure that are far apart in the sequence in close spatial proximity. In a final study, we have formalized this go-and-come-back notion into the concept of anti-parallel double-strandedness (DS), and measure this property across protein architecture in general. With over 90% of all residues in a large, non-redundant set of protein structures classified as DS, we conclude that DS is a unifying structural principle that underpins all globular proteins. We postulate, moreover, that this one simple principle, anti-parallel double-strandedness, unites protein structure, protein folding and protein evolution. / Thesis (Ph.D, Biochemistry) -- Queen's University, 2011-09-30 12:32:41.379

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The Major Intrinsic Proteins are represented in all forms of life; plants, animals, bacteria and recently archaebacteria have all been shown to express at least one member of this superfamily of integral membrane proteins. We have overexpressed the E. coli aquaglyceroporin, glycerol facilitator (GlpF), to use as a model for studying membrane protein structure, folding and stability. Understanding membrane protein folding, stability, and dynamics is required for a molecular explanation of membrane protein function and for the development of interventions for the hundreds of membrane protein folding diseases. X-ray analysis of GlpF crystals shows that the protein exits as a tetramer in the crystallized state [1]. However, preparations of stable aqueous detergent solutions of GlpF in its native oligomeric state have been difficult to make; the protein readily unfolds and forms non-specific aggregates in many detergents. Here, I report the study of the structure and stability of the glycerol facilitator in several detergent solutions by blue native and sodium dodecyl sulphate polyacrylamide gel electrophoresis, circular dichroism, and fluorescence. For the first time, stable protein tetramers were prepared in two different detergent solutions (dodecyl maltoside (DDM) and lyso-myristoyl phosphatidylcholine (LMPC)) at neutral pH. Thermal unfolding experiments show that the protein is slightly more stable in LMPC than in DDM and that the thermal stability of the helical core at 95oC is slightly greater in the former detergent. In addition, tertiary structure unfolds before quaternary and secondary structures in LMPC whereas unfolding is more cooperative in DDM. The high stability of the protein is also evident from the unfolding half-life of 8 days in 8 M urea suggesting that hydrophobic interactions contribute to the stability. The GlpF tetramers are less resistant to acidic conditions; LMPC-solubilized GlpF shows loss of tertiary and quaternary structure by pH 6, while in DDM the tertiary structure is lost by pH 5, however the tetramer remains mostly intact at pH 4. The implications of thermal and chemical stress on the stability of the detergent-solubilized protein and its in vivo folding are discussed.