Narrow Results

We have studied the chain length dependence of folding time for proteins by implementing a novel Monte Carlo (MC) method. The physical parameters in our model are derived from the statistics for bending and torsion angles and distances between the centers of the monomers up to the fourth neighborhood. By assigning potential wells to each of the physical parameters, we are able to use a modified Metropolis algorithm to efficiently trace the later conformations of the proteins as time evolves. Our prescription for microscopic dynamics of the protein "Crambin" results in an increase of folding times with increasing chain length. The folding times are determined via Debye relaxation process.

We are interested in methods which compute the inverse of a triangular matrix A of order n by solving the n linear systems Ax=e_i, i=1,…, n, where e_i is the i-th element of the canonical basis of Rⁿ. More precisely, we consider the dependence graph associated with algorithms where the entries of matrix A are read only once and used in pipeline for the solution of these systems. We exhibit a new scheduling which induces an algorithm with time complexity T*=2n−1. The number n²/8+O(n) of processors required by this scheduling improves the best previously known bound n²/6+O(n), and is quite close to the lower bound n²/8.5+O(n).

The nascent peptide folding in vivo is different from the denatured peptide refolding in vitro and can be divided into two stages. In the first stage, the peptide is folding as it is being synthesized until the whole peptide chain is synthesized. The final conformation formed in this stage is called as nascent state. In the second stage, the protein folds beginning with the nascent state formed in the first stage into the native state. We use a lattice model to simulate these two stages and investigate the folding time of the nascent peptide comparing with that of the denatured peptide refolding. Our results show that the synthesis process may affect the folding time of the nascent peptide. This may be helpful to understand why the former folds faster than the latter.

In this work filter structures that decrease the required number of multipliers and adders for implementation of linear-phase FIR filters using frequency-response masking techniques are introduced. The basic idea of the proposed structures is that identical subfilters are used. This leads to the same arithmetic structure can be multiplexed in the implementation, reducing the number of required multipliers and adders. The subfilters are mapped using the folding transformation to obtain an area-efficient time-multiplexed (or pipeline/interleaved) implementation. Both narrow-band and wide-band frequency-response masking as well as arbitrary bandwidth frequency-response masking techniques are considered. The filter design is discussed and for each filter structure the limits on the specifications are derived. Designed examples show the usefulness of the proposed structures.

We consider the problems of straightening polygonal trees and convexifying polygons by continuous motions such that rigid edges can rotate around vertex joints and no edge crossings are allowed. A tree can be straightened if all its edges can be aligned along a common straight line such that each edge points "away" from a designated leaf node. A polygon can be convexified if it can be reconfigured to a convex polygon. A lattice tree (resp. polygon) is a tree (resp. polygon) containing only edges from a square or cubic lattice. We first show that a 2D lattice chain or a 3D lattice tree can be straightened efficiently in O(n) moves and time, where n is the number of tree edges. We then show that a 2D lattice tree can be straightened efficiently in O(n²) moves and time. Furthermore, we prove that a 2D lattice polygon or a 3D lattice polygon with simple shadow can be convexified efficiently in O(n) moves and in O(n log n) time. Finally, we show that two special classes of diameter-4 trees in two dimensions can always be straightened.

Biedl et al.¹ presented an algorithm for unfolding orthostacks into one piece without overlap by using arbitrary cuts along the surface. They conjectured that orthostacks could be unfolded using cuts that lie in a plane orthogonal to a coordinate axis and containing a vertex of the orthostack. We prove the existence of a vertex unfolding using only such cuts.

We consider two types of folding applied to equilateral plane graph linkages. First, under continuous folding motions, we show how to reconfigure any linear equilateral tree (lying on a line) into a canonical configuration. By contrast, it is known that such reconfiguration is not always possible for linear (nonequilateral) trees and for (nonlinear) equilateral trees. Second, under instantaneous folding motions, we show that an equilateral plane graph has a noncrossing linear folded state if and only if it is bipartite. Furthermore, we show that the equilateral constraint is necessary for this result, by proving that it is strongly NP-complete to decide whether a (nonequilateral) plane graph has a linear folded state. Equivalently, we show strong NP-completeness of deciding whether an abstract metric polyhedral complex with one central vertex has a noncrossing flat folded state. By contrast, the analogous problem for a polyhedral manifold with one central vertex (single-vertex origami) is only weakly NP-complete.

We study the problem of folding a polyomino $P$ into a polycube $Q$, allowing faces of $Q$ to be covered multiple times. First, we define a variety of folding models according to whether the folds (a) must be along grid lines of $P$ or can divide squares in half (diagonally and/or orthogonally), (b) must be mountain or can be both mountain and valley, (c) can remain flat (forming an angle of $180^{\circ }$), and (d) must lie on just the polycube surface or can have interior faces as well. Second, we give all the inclusion relations among all models that fold on the grid lines of $P$. Third, we characterize all polyominoes that can fold into a unit cube, in some models. Fourth, we give a linear-time dynamic programming algorithm to fold a tree-shaped polyomino into a constant-size polycube, in some models. Finally, we consider the triangular version of the problem, characterizing which polyiamonds fold into a regular tetrahedron.

In 1985 Hopcroft, Joseph and Whitesides showed it is NP-complete to decide whether a carpenter’s ruler with segments of given positive lengths can be folded into an interval of at most a given length, such that the folded hinges alternate between 180 degrees clockwise and 180 degrees counter-clockwise. At the open-problem session of 33rd Canadian Conference on Computational Geometry (CCCG ’21), O’Rourke proposed a natural variation of this problem called ruler wrapping, in which all folded hinges must be folded the same way. In this paper we show O’Rourke’s variation has a linear-time solution.

A total of 120 ns molecular dynamics simulations was used to study the folding and conformational aspects of six peptides with different lengths (Pep^19–25, Pep^15–25, Pep^1–25, Pep^15–39, Pep^1–40, and Pep^1–50) truncated from the αβ-tubulin dimer. These truncated peptides were found to undergo distinct structural transitions, with Pep^1–25 and Pep^1–50 folding into their respective stable conformations whereas on the contrary for the others. All the six truncated peptides are more or less compact than the corresponding segments in the αβ-tubulin dimer. The most striking contraction was observed in Pep^1–25, which folds in a similar manner of β-hairpin. Pep^1–50 has the least contraction and its folded conformation is the closest to that in the αβ-tubulin dimer. Moreover, the same conversions of β12–β23 from helices to hydrogen-bonded turns were witnessed in both Pep^1–50 and the αβ-tubulin dimer. The structural instabilities of Pep^19–25, Pep^15–25, Pep^15–39, and Pep^1–40 were caused by the lack of long-distance interactions or/and the absence of key residues, with the details given in the discussions. The folding and conformational divergences of six truncated peptides were also observed in their active peptide segments (Ap^15–25). Ap^15–25 in Pep^1–50 achieves the best agreements with the αβ-tubulin dimer, implying that the local structure of Ap^15–25 in the αβ-tubulin dimer can be well reserved in Pep^1–50 rather than in the other truncated peptides. The long-distance interactions, especially the key residues (e.g. β48-Arg), play a crucial role in the correct folding of Ap^15–25. The correct folding into the stable conformations is a prerequisite for the peptides to implement their catalytic actions, and therefore the present results are helpful to the future designs of active peptides.

Recent NMR structural and dynamical data on partially folded forms of mono-heme cytochrome c provide a unifying picture of the behavior of the protein far from the native conditions and suggest useful hints to explain the redox dependent stability of the protein. A fragile hinge in the structure of mitochondrial cytochrome c is identified, which may not have correspondents in smaller type-1 cytochromes. Former spectroscopic and kinetic data are here discussed in terms of this new view.

We conducted a meta-analysis of (bacterio)chlorophyll [(B)Chl] molecules in photosynthetic pigment-protein complexes from the viewpoint of coordination chemistry. We surveyed the ligand species and site in the axial coordination of 146 Chl and 21 BChl molecules in 42 reported crystal structures of 12-type proteins. The imidazolyl moiety of histidine (His) is the most abundant ligand, and the second is water, a much weaker ligand. We focused on the positions, the circumstances, and the macrocycle sides for the coordination of the 31 hydrated (B)Chl molecules found in these proteins. A ligand water molecule of a hydrated (B)Chl is not necessarily hydrogen-bonded to the surrounding protein residues. A hydrated (B)Chl seems to occupy the redundant space where more strongly coupled His-Chl complexes cannot be formed. It is noted that 28 of 31 hydrated (B)Chl molecules (90) were coordinated from the α-side of the (bacterio)chlorin macrocycle, the opposite side from which the C17-propionic ester protrudes. Among them, all five hydrated Chl molecules at the edges of the proteins were coordinated from the α-side, suggesting that (B)Chl molecules prefer this side for the coordination bondings to the β-side. The analysis also revealed that each (B)Chl binding site was composed of both the protein residues and the neighboring pigment molecules contributing roughly equally. It can be safely said that the cofactor pigments aggregated even in the proteins. Penta-coordination is advantageous to flexible adjustment of intermolecular orientations of (B)Chl molecules in the aggregates.

In this report, we have designed and synthesized a novel switching molecule whose fluorescence can be switched via dynamic conformational change between expanded and shrunk states induced by metal complexation and decomplexation. The switching molecule is composed of three kinds of components, namely, a Zn²⁺-porphyrin fluorophore, two quinone quenchers, and their linkers containing a 4,4′-bipyridine moiety. UV-vis and fluorescence titration studies revealed that metal complexation of the bipyridine units with Zn²⁺ ions induced the dynamic structural change of the molecular shape and simultaneous enhancement of fluorescence of the Zn²⁺-porphyrin fluorophore.

p21^{Cip1, Waf1, Sdi1} (p21) is a member of the cyclin-dependent kinase (CDK) family of inhibitors in eukaryotes. We report the refolding of an inclusion body of a recombinant p21 (rp21) to its native form, under an alkaline to neutral environment, via an over-critical process describable by a first-order state transition model. The secondary structure of the refolded rp21 possesses a helical-major structure as determined by circular dichroism (CD) analysis, and its diameter is around 3 nm, as measured by dynamic light scattering studies (DLS) and atomic force microscopy (AFM). The differential scanning calorimeter (DSC) measurement indicates that the folded rp21 possesses unique but weak tertiary interactions. That the function of rp21 is reinstated upon refolding under our experimental conditions is evidenced by its binding to proliferating cell nuclear antigen (PCNA) in an immuno-co-precitptation analysis. The conformational changes of the folding intermediates of rp21 are consistent with the framework of a sequential model proposed earlier. The lack of a definitive structure of p21 in acidic condition will be discussed.

Folding in graphene sheet has been extensively observed experimentally. While it is generally recognized that such a conformational state can influence the electronic, magnetic and mechanical properties of graphene nanostructures, the mechanism driving the nonlinear mechanical deformation remains an interesting subject of study. Here we present an investigation on the folding in bi-layer graphene sheet due to in-plane compression. To describe the lattice registry effect of interlay cohesion in layered graphitic structures, a registry-dependent potential model was implemented. We have determined the critical length to stabilize the graphene folding to be 5.4~10.7 nm through both theoretical and simulation analysis. The mechanism for such a stabilized fold is attributed to the variations in the inter-layer interaction energy that produces a friction-like effect. The climbing image nudged elastic band (CINEB) calculations predicted an identical activation energy barrier associated with the transition between flat and folded configurations, 0.47 eV/Å, for graphene sheets with length of 7~10 nm. When the mechanical stimulation is high enough to overcome the energy barrier, the supported graphene sheet can be folded to form a nanotube.

The goal of this interactive online workshop was to introduce paper strips as manipulatives to foster understanding of fractions. Attendees gained appreciation for the versatility of the paper strip to visualize concepts, link fractions to the whole numbers, and build arithmetic algorithms. The operations became alive in the participants’ hands.