The Lefschetz-Hopf theorem and axioms for the Lefschetz number

The reduced Lefschetz number, that is, the Lefschetz number minus 1, is proved to be the unique integer-valued function L on selfmaps of compact polyhedra which is constant on homotopy classes such that (1) L(fg) = L(gf), for f:X -->Y and g:Y -->X; (2) if (f_1, f_2, f_3) is a map of a cofiber sequence into itself, then L(f_2) = L(f_1) + L(f_3); (3) L(f) = - (degree(p_1 f e_1) + ... + degree(p_k f e_k)), where f is a map of a wedge of k circles, e_r is the inclusion of a circle into the rth summand and p_r is the projection onto the rth summand. If f:X -->X is a selfmap of a polyhedron and I(f) is the fixed point index of f on all of X, then we show that I minus 1 satisfies the above axioms. This gives a new proof of the Normalization Theorem: If f:X -->X is a selfmap of a polyhedron, then I(f) equals the Lefschetz number of f. This result is equivalent to the Lefschetz-Hopf Theorem: If f: X -->X is a selfmap of a finite simplicial complex with a finite number of fixed points, each lying in a maximal simplex, then the Lefschetz number of f is the sum of the indices of all the fixed points of f.Comment: 9 page

Cell polarization in budding and fission yeasts

Polarization is a fundamental cellular property, which is essential for the function of numerous cell types. Over the past three to four decades, research using the best-established yeast systems in cell biological research, Saccharomyces cerevisiae (or budding yeast) and Schizosaccharomyces pombe (or fission yeast), has brought to light fundamental principles governing the establishment and maintenance of a polarized, asymmetric state. These two organisms, though both ascomycetes, are evolutionarily very distant and exhibit distinct shapes and modes of growth. In this review, we compare and contrast the two systems. We first highlight common cell polarization pathways, detailing the contribution of Rho GTPases, the cytoskeleton, membrane trafficking, lipids, and protein scaffolds. We then contrast the major differences between the two organisms, describing their distinct strategies in growth site selection and growth zone dimensions and compartmentalization, which may be the basis for their distinct shape

Recent advances in understanding Candida albicans hyphal growth

International audienceMorphological changes are critical for the virulence of a range of plant and human fungal pathogens. is a major human fungal Candida albicans pathogen whose ability to switch between different morphological states is associated with its adaptability and pathogenicity. In particular, C. albicans can switch from an oval yeast form to a filamentous hyphal form, which is characteristic of filamentous fungi. What mechanisms underlie hyphal growth and how are they affected by environmental stimuli from the host or resident microbiota? These questions are the focus of intensive research, as understanding hyphal growth has broad implications for cell C. albicans biological and medical research

A Septin from the Filamentous Fungus A. nidulans Induces Atypical Pseudohyphae in the Budding Yeast S. cerevisiae

BACKGROUND: Septins, novel cytoskeletal proteins, form rings at the bases of emerging round buds in yeasts and at the bases of emerging elongated hyphal initials in filamentous fungi. METHODOLOGY/PRINCIPAL FINDINGS: When introduced into the yeast Saccharomyces cerevisiae, the septin AspC from the filamentous fungus Aspergillus nidulans induced highly elongated atypical pseudohyphae and spore-producing structures similar to those of hyphal fungi. AspC induced atypical pseudohyphae when S. cerevisiae pseudohyphal or haploid invasive genes were deleted, but not when the CDC10 septin gene was deleted. AspC also induced atypical pseudohyphae when S. cerevisiae genes encoding Cdc12-interacting proteins Bem4, Cla4, Gic1 and Gic2 were deleted, but not when BNI1, a Cdc12-interacting formin gene, was deleted. AspC localized to bud and pseudohypha necks, while its S. cerevisiae ortholog, Cdc12, localized only to bud necks. CONCLUSIONS/SIGNIFICANCE: Our results suggest that AspC competes with Cdc12 for incorporation into the yeast septin scaffold and once there alters cell shape by altering interactions with the formin Bni1. That introduction of the A. nidulans septin AspC into S. cerevisiae induces a shift from formation of buds to formation of atypical pseudohyphae suggests that septins play an important role in the morphological plasticity of fungi

Orientation of Mitotic Spindles during the 8- to 16-Cell Stage Transition in Mouse Embryos

Background: Asymmetric cell divisions are involved in the divergence of the first two lineages of the pre-implantation mouse embryo. They first take place after cell polarization (during compaction) at the 8-cell stage. It is thought that, in contrast to many species, spindle orientation is random, although there is no direct evidence for this. Methodology/Principal Findings: Tubulin-GFP and live imaging with a spinning disk confocal microscope were used to directly study spindle orientation in whole embryos undergoing the 8- to 16-cell stage transition. This approach allowed us to determine that there is no predetermined cleavage pattern in 8-cell compacted mouse embryos and that mitotic spindle orientation in live embryo is only modulated by the extent of cell rounding up during mitosis. Conclusions: These results clearly demonstrate that spindle orientation is not controlled at the 8- to 16-cell transition, but influenced by cell bulging during mitosis, thus reinforcing the idea that pre-implantation development is highly regulative and not pre-patterned

Impaired Function of HDAC6 Slows Down Axonal Growth and Interferes with Axon Initial Segment Development

The development of morphological neuronal polarity starts by the formation and elongation of an axon. At the same time the axon initial segment (AIS) is generated and creates a diffusion barrier which differentiate axon and somatodendritic compartment. Different structural and functional proteins that contribute to the generation of neuronal action potential are concentrated at the axon initial segment. While axonal elongation is controlled by signalling pathways that regulate cytoskeleton through microtubule associated proteins and tubulin modifications, the microtubule cytoskeleton under the AIS is mostly unknown. Thus, understanding which proteins modify tubulin, where in the neuron and at which developmental stage is crucial to understanding how morphological and functional neuronal polarity is achieved. In this study performed in mice and using a well established model of murine cultured hippocampal neurons, we report that the tubulin deacetylase HDAC6 is localized at the distal region of the axon, and its inhibition with TSA or tubacin slows down axonal growth. Suppression of HDAC6 expression with HDAC6 shRNAs or expression of a non-active mutant of HDAC6 also reduces axonal length. Furthermore, HDAC6 inhibition or suppression avoids the concentration of ankyrinG and sodium channels at the axon initial segment (AIS). Moreover, treatment of mouse cultured hippocampal neurons with detergents to eliminate the soluble pool of microtubules identified a pool of detergent resistant acetylated microtubules at the AIS, not present at the rest of the axon. Inhibition or suppression of HDAC6 increases acetylation all along the axon and disrupts the specificity of AIS cytoskeleton, modifying the axonal distal gradient localization of KIF5C to a somatodendritic and axonal localization. In conclusion, our results reveal a new role of HDAC6 tubulin deacetylase as a regulator of microtubule characteristics in the axon distal region where axonal elongation takes place, and allowing the development of acetylated microtubules microdomains where HDAC6 is not concentrated, such as the axon initial segment