A Purely Functional Computer Algebra System Embedded in Haskell

We demonstrate how methods in Functional Programming can be used to implement a computer algebra system. As a proof-of-concept, we present the computational-algebra package. It is a computer algebra system implemented as an embedded domain-specific language in Haskell, a purely functional programming language. Utilising methods in functional programming and prominent features of Haskell, this library achieves safety, composability, and correctness at the same time. To demonstrate the advantages of our approach, we have implemented advanced Gr\"{o}bner basis algorithms, such as Faug\`{e}re's $F_4$ and $F_5$, in a composable way.Comment: 16 pages, Accepted to CASC 201

Involution and Constrained Dynamics I: The Dirac Approach

We study the theory of systems with constraints from the point of view of the formal theory of partial differential equations. For finite-dimensional systems we show that the Dirac algorithm completes the equations of motion to an involutive system. We discuss the implications of this identification for field theories and argue that the involution analysis is more general and flexible than the Dirac approach. We also derive intrinsic expressions for the number of degrees of freedom.Comment: 28 pages, latex, no figure

Advanced optical imaging in living embryos

Developmental biology investigations have evolved from static studies of embryo anatomy and into dynamic studies of the genetic and cellular mechanisms responsible for shaping the embryo anatomy. With the advancement of fluorescent protein fusions, the ability to visualize and comprehend how thousands to millions of cells interact with one another to form tissues and organs in three dimensions (xyz) over time (t) is just beginning to be realized and exploited. In this review, we explore recent advances utilizing confocal and multi-photon time-lapse microscopy to capture gene expression, cell behavior, and embryo development. From choosing the appropriate fluorophore, to labeling strategy, to experimental set-up, and data pipeline handling, this review covers the various aspects related to acquiring and analyzing multi-dimensional data sets. These innovative techniques in multi-dimensional imaging and analysis can be applied across a number of fields in time and space including protein dynamics to cell biology to morphogenesis

How to obtain lattices from (f,σ,δ)-codes via a generalization of Construction A

We show how cyclic (f,σ,δ)-codes over finite rings canonically induce a Z-lattice in RN by using certain quotients of orders in nonassociative division algebras defined using the skew polynomial f. This construction generalizes the one using certain σ-constacyclic codes by Ducoat and Oggier, which used quotients of orders in non-commutative associative division algebras defined by f, and can be viewed as a generalization of the classical Construction A for lattices from linear codes. It has the potential to be applied to coset coding, in particular to wire-tap coding. Previous results by Ducoat and Oggier are obtained as special cases

Interaction of microtubules and actin during the post-fusion phase of exocytosis

Exocytosis is the intracellular trafficking step where a secretory vesicle fuses with the plasma membrane to release vesicle content. Actin and microtubules both play a role in exocytosis; however, their interplay is not understood. Here we study the interaction of actin and microtubules during exocytosis in lung alveolar type II (ATII) cells that secrete surfactant from large secretory vesicles. Surfactant extrusion is facilitated by an actin coat that forms on the vesicle shortly after fusion pore opening. Actin coat compression allows hydrophobic surfactant to be released from the vesicle. We show that microtubules are localized close to actin coats and stay close to the coats during their compression. Inhibition of microtubule polymerization by colchicine and nocodazole affected the kinetics of actin coat formation and the extent of actin polymerisation on fused vesicles. In addition, microtubule and actin cross-linking protein IQGAP1 localized to fused secretory vesicles and IQGAP1 silencing influenced actin polymerisation after vesicle fusion. This study demonstrates that microtubules can influence actin coat formation and actin polymerization on secretory vesicles during exocytosis

Using enhanced number and brightness to measure protein oligomerization dynamics in live cells

Protein dimerization and oligomerization are essential to most cellular functions, yet measurement of the size of these oligomers in live cells, especially when their size changes over time and space, remains a challenge. A commonly used approach for studying protein aggregates in cells is number and brightness (N&B), a fluorescence microscopy method that is capable of measuring the apparent average number of molecules and their oligomerization (brightness) in each pixel from a series of fluorescence microscopy images. We have recently expanded this approach in order to allow resampling of the raw data to resolve the statistical weighting of coexisting species within each pixel. This feature makes enhanced N&B (eN&B) optimal for capturing the temporal aspects of protein oligomerization when a distribution of oligomers shifts toward a larger central size over time. In this protocol, we demonstrate the application of eN&B by quantifying receptor clustering dynamics using electron-multiplying charge-coupled device (EMCCD)-based total internal reflection microscopy (TIRF) imaging. TIRF provides a superior signal-to-noise ratio, but we also provide guidelines for implementing eN&B in confocal microscopes. For each time point, eN&B requires the acquisition of 200 frames, and it takes a few seconds up to 2 min to complete a single time point. We provide an eN&B (and standard N&B) MATLAB software package amenable to any standard confocal or TIRF microscope. The software requires a high-RAM computer (64 Gb) to run and includes a photobleaching detrending algorithm, which allows extension of the live imaging for more than an hour