K1,3-covering red and blue points in the plane

We say that a finite set of red and blue points in the plane in general position can be K1, 3-covered if the set can be partitioned into subsets of size 4, with 3 points of one color and 1 point of the other color, in such a way that, if at each subset the fourth point is connected by straight-line segments to the same-colored points, then the resulting set of all segments has no crossings. We consider the following problem: Given a set R of r red points and a set B of b blue points in the plane in general position, how many points of R ¿ B can be K1, 3-covered? and we prove the following results: (1) If r = 3g + h and b = 3h + g, for some non-negative integers g and h, then there are point sets R ¿ B, like {1, 3}-equitable sets (i.e., r = 3b or b = 3r) and linearly separable sets, that can be K1, 3-covered. (2) If r = 3g + h, b = 3h + g and the points in R ¿ B are in convex position, then at least r + b - 4 points can be K1, 3-covered, and this bound is tight. (3) There are arbitrarily large point sets R ¿ B in general position, with r = b + 1, such that at most r + b - 5 points can be K1, 3-covered. (4) If b = r = 3b, then at least 9 8 (r + b- 8) points of R ¿ B can be K1, 3-covered. For r > 3b, there are too many red points and at least r - 3b of them will remain uncovered in any K1, 3-covering. Furthermore, in all the cases we provide efficient algorithms to compute the corresponding coverings

The \u3cem\u3eChlamydomonas\u3c/em\u3e Genome Reveals the Evolution of Key Animal and Plant Functions

Chlamydomonas reinhardtii is a unicellular green alga whose lineage diverged from land plants over 1 billion years ago. It is a model system for studying chloroplast-based photosynthesis, as well as the structure, assembly, and function of eukaryotic flagella (cilia), which were inherited from the common ancestor of plants and animals, but lost in land plants. We sequenced the ∼120-megabase nuclear genome of Chlamydomonas and performed comparative phylogenomic analyses, identifying genes encoding uncharacterized proteins that are likely associated with the function and biogenesis of chloroplasts or eukaryotic flagella. Analyses of the Chlamydomonas genome advance our understanding of the ancestral eukaryotic cell, reveal previously unknown genes associated with photosynthetic and flagellar functions, and establish links between ciliopathy and the composition and function of flagella

Phylogenomic analysis of the Chlamydomonas genome unmasks proteins potentially involved in photosynthetic function and regulation

Chlamydomonas reinhardtii, a unicellular green alga, has been exploited as a reference organism for identifying proteins and activities associated with the photosynthetic apparatus and the functioning of chloroplasts. Recently, the full genome sequence of Chlamydomonas was generated and a set of gene models, representing all genes on the genome, was developed. Using these gene models, and gene models developed for the genomes of other organisms, a phylogenomic, comparative analysis was performed to identify proteins encoded on the Chlamydomonas genome which were likely involved in chloroplast functions (or specifically associated with the green algal lineage); this set of proteins has been designated the GreenCut. Further analyses of those GreenCut proteins with uncharacterized functions and the generation of mutant strains aberrant for these proteins are beginning to unmask new layers of functionality/regulation that are integrated into the workings of the photosynthetic apparatus

Evolution of reproductive development in the volvocine algae

The evolution of multicellularity, the separation of germline cells from sterile somatic cells, and the generation of a male–female dichotomy are certainly among the greatest innovations of eukaryotes. Remarkably, phylogenetic analysis suggests that the shift from simple to complex, differentiated multicellularity was not a unique progression in the evolution of life, but in fact a quite frequent event. The spheroidal green alga Volvox and its close relatives, the volvocine algae, span the full range of organizational complexity, from unicellular and colonial genera to multicellular genera with a full germ–soma division of labor and male–female dichotomy; thus, these algae are ideal model organisms for addressing fundamental issues related to the transition to multicellularity and for discovering universal rules that characterize this transition. Of all living species, Volvox carteri represents the simplest version of an immortal germline producing specialized somatic cells. This cellular specialization involved the emergence of mortality and the production of the first dead ancestors in the evolution of this lineage. Volvocine algae therefore exemplify the evolution of cellular cooperation from cellular autonomy. They also serve as a prime example of the evolution of complex traits by a few successive, small steps. Thus, we learn from volvocine algae that the evolutionary transition to complex, multicellular life is probably much easier to achieve than is commonly believed

Ion homeostasis in the Chloroplast

peer reviewedThe chloroplast is an organelle of high demand for macro- and micro-nutrient ions, which are required for the maintenance of the photosynthetic process. To avoid deficiency while preventing excess, homeostasis mechanisms must be tightly regulated. Here, we describe the needs for nutrient ions in the chloroplast and briefly highlight their functions in the chloroplastidial metabolism. We further discuss the impact of nutrient deficiency on chloroplasts and the acclimation mechanisms that evolved to preserve the photosynthetic apparatus. We finally present what is known about import and export mechanisms for these ions. Whenever possible, a comparison between cyanobacteria, algae and plants is provided to add an evolutionary perspective to the description of ion homeostasis mechanisms in photosynthesis

3-symmetric and 3-decomposable geometric drawings of Kn

Even the most super cial glance at the vast majority of crossing-minimal geometric drawings of Kn reveals two hard-to-miss features. First, all such drawings appear to be 3-fold symmetric (or simply 3-symmetric). And second, they all are 3-decomposable, that is, there is a triangle T enclosing the drawing, and a balanced partition A, B, C of the underlying set of points P, such that the orthogonal projections of P onto the sides of T show A between B and C on one side, B between A and C on another side, and C between A and B on the third side. In fact, we conjecture that all optimal drawings are 3-decomposable, and that there are 3-symmetric optimal constructions for all n multiple of 3. In this paper, we show that any 3-decomposable geometric drawing of Kn has at least 0.380029 () n 3 4 + Θ(n) crossings. On the other hand, we produce 3-symmetric and 3-decomposable drawings that improve the general upper bound for the rectilinear crossing number of Kn to 0.380488 () n 3 4 +Θ(n). We also give explicit 3-symmetric and 3-decomposable constructions for n < 100 that are at least as good as those previously known.