The actin cytoskeleton forms a membrane-associated network whose proper regulation is essential for numerous processes, including cell differentiation, proliferation, adhesion, chemotaxis, endocytosis, exocytosis, and multicellular development. In this report, we show that in Dictyostelium discoideum, paxillin (PaxB) and phospholipase D (PldB) colocalize and coimmunoprecipitate, suggesting that they interact physically. Additionally, the phenotypes observed during development, cell sorting, and several actin-required processes, including cyclic AMP (cAMP) chemotaxis, cell-substrate adhesion, actin polymerization, phagocytosis, and exocytosis, reveal a genetic interaction between paxB and pldB, suggesting a functional interaction between their gene products. Taken together, our data point to PldB being a required binding partner of PaxB during processes involving actin reorganization.
A major area in neuroscience research is the study of how the brain processes spatial information. Neurons in the brain represent external stimuli via neural codes. These codes often arise from stereotyped stimulus-response maps, associating to each neuron a convex receptive field. An important problem consists in determining what stimulus space features can be extracted directly from a neural code. The neural ideal is an algebraic object that encodes the full combinatorial data of a neural code. This ideal can be expressed in a canonical form that directly translates to a minimal description of the receptive field structure intrinsic to the code. In here, we describe a SageMath package that contains several algorithms related to the canonical form of a neural ideal.
The brain processes information about the environment via neural codes. The neural ideal was introduced recently as an algebraic object that can be used to better understand the combinatorial structure of neural codes. Every neural ideal has a particular generating set, called the canonical form, that directly encodes a minimal description of the receptive field structure intrinsic to the neural code. On the other hand, for a given monomial order, any polynomial ideal is also generated by its unique (reduced) Gröbner basis with respect to that monomial order. How are these two types of generating sets -canonical forms and Gröbner bases -related? Our main result states that if the canonical form of a neural ideal is a Gröbner basis, then it is the universal Gröbner basis (that is, the union of all reduced Gröbner bases). Furthermore, we prove that this situation -when the canonical form is a Gröbner basis -occurs precisely when the universal Gröbner basis contains only pseudo-monomials (certain generalizations of monomials). Our results motivate two questions: (1) When is the canonical form a Gröbner basis? (2) When the universal Gröbner basis of a neural ideal is not a canonical form, what can the non-pseudo-monomial elements in the basis tell us about the receptive fields of the code? We give partial answers to both questions. Along the way, we develop a representation of pseudo-monomials as hypercubes in a Boolean lattice.
In this paper, we provide a natural bijection between a special family of block circulant graphs and the graphs of critical pairs of the posets known as generalized crowns. In particular, every graph in this family of block circulant graphs we investigate has a generating block row that follows a symmetric growth pattern of the all ones matrix. The natural bijection provides an upper bound on the chromatic number for this infinite family of graphs.
The sandpile group of a graph is a well-studied object that combines ideas from algebraic graph theory, group theory, dynamical systems, and statistical physics. A graph's sandpile group is part of a larger algebraic structure on the graph, known as its sandpile monoid. Most of the work on sandpiles so far has focused on the sandpile group rather than the sandpile monoid of a graph, and has also assumed the underlying graph to be undirected. A notable exception is the recent work of Babai and Toumpakari, which builds up the theory of sandpile monoids on directed graphs from scratch and provides many connections between the combinatorics of a graph and the algebraic aspects of its sandpile monoid. In this paper we primarily consider sandpile monoids on directed graphs, and we extend the existing theory in four main ways. First, we give a combinatorial classification of the maximal subgroups of a sandpile monoid on a directed graph in terms of the sandpile groups of certain easily-identifiable subgraphs. Second, we point out certain sandpile results for undirected graphs that are really results for sandpile monoids on directed graphs that contain exactly two idempotents. Third, we give a new algebraic constraint that sandpile monoids must satisfy and exhibit two infinite families of monoids that cannot be realized as sandpile monoids on any graph. Finally, we give an explicit combinatorial description of the sandpile group identity for every graph in a family of directed graphs which generalizes the family of (undirected) distanceregular graphs. This family includes many other graphs of interest, including iterated wheels, regular trees, and regular tournaments.
Tight control of actin cytoskeletal dynamics is essential for proper cell function and survival. ARNO, a mammalian guanine nucleotide exchange factor for Arf, has been implicated in actin cytoskeletal regulation but its exact role is still unknown. To explore the role of ARNO in this regulation as well as in actin-mediated processes, the Dictyostelium discoideum homolog, SecG, was examined. SecG peaks during aggregation and mound formation. The overexpression of SecG arrests development at the mound stage. SecG overexpressing (SecG OE) cells fail to stream during aggregation. Although carA is expressed, SecG OE cells do not chemotax toward cAMP, indicating SecG is involved in the cellular response to cAMP. This chemotactic defect is specific to cAMP-directed chemotaxis, as SecG OE cells chemotax to folate without impairment and exhibit normal cell motility. The chemotactic defects of the SecG mutants may be due to an impaired cAMP response as evidenced by altered cell polarity and F-actin polymerization after cAMP stimulation. Cells overexpressing SecG have increased filopodia compared to wild type cells, implying that excess SecG causes abnormal organization of F-actin. The general function of the cytoskeleton, however, is not disrupted as the SecG OE cells exhibit proper cell-substrate adhesion. Taken together, the results suggest proper SecG levels are needed for appropriate response to cAMP signaling in order to coordinate F-actin organization during development.
The q-analog of Kostant's weight multiplicity formula is an alternating sum over a finite group, known as the Weyl group, whose terms involve the q-analog of Kostant's partition function. This formula, when evaluated at q = 1, gives the multiplicity of a weight in a highest weight representation of a simple Lie algebra. In this paper, we consider the Lie algebra sl4(C) and give closed formulas for the q-analog of Kostant's weight multiplicity. This formula depends on the following two sets of results. First, we present closed formulas for the q-analog of Kostant's partition function by counting restricted colored integer partitions. These formulas, when evaluated at q = 1, recover results of De Loera and Sturmfels. Second, we describe and enumerate the Weyl alternation sets, which consist of the elements of the Weyl group that contribute nontrivially to Kostant's weight multiplicity formula. From this, we introduce Weyl alternation diagrams on the root lattice of sl4(C), which are associated to the Weyl alternation sets. This work answers a question posed in 2019
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