Graphs are represented visually by drawing a dot or circle for every
vertex, and drawing an arc between two vertices if they are connected by
an edge. Graph drawing also can be said to encompass problems that deal
with the crossing number and its various generalizations. If the graph
is directed, the direction is indicated by drawing an arrow.
Platonic graph corresponding to the skeleton of a Platonic solid. The
five platonic graphs, the tetrahedral graph, cubical graph, octahedral
graph, dodecahedral graph, and icosahedral graph. They are special cases
of Schlegel graphs. Graph-theoretic methods, in various forms, have
proven particularly useful in discrete structures.
with a planar graph drawn on a flat piece of rubber, then we can draw a
large circle around the graph, cut it out, and then stretch the disk of
rubber around a large sphere (without crumpling - though we will have to
do some stretching and compressing). There will be a small isolated
part of the sphere not covered. The book describes the process in
reverse, starting by drawing a graph on a balloon and then punching a
hole in the balloon and stretching it flat (so these processes are
reversible). So any graph that can be drawn in the plane with no
crossing of edges, can also be drawn on a sphere with no crossings (and
vice-a-versa). Notice also that the number of regions is preserved. (Gifs: Scanftree-Graph-Theory)
No, not BK as in Burger King. This virus was isolated in 1971 from the urine of a patient with the same initials as the fast food restaurant. Typically, infection with this virus is nothing to worry about: it is thought that up to 80% of the population carries a latent form.
BK is a polyomavirus, which means it has a circular double-stranded DNA genome, an icosahedral capsid, and no lipid envelope:
Those histones shown are cellular, by the way. BK virus and its relatives replicate in the nucleus and steal them from the host cell. A neat trick.
Most infections with BK virus are asymptomatic, although a small percentage of people present with respiratory infections or fevers. This scenario is referred to as a ‘primary’ BK infection. Following primary infection, BK proceeds to the kidneys and urinary tract where it will persist for the life of the individual. If he or she becomes immunosuppressed, as in the case of a kidney transplant, the virus can cause serious complications. As many as 10% of renal transplant patients progress to the stage of BK virus neuropathy and may lose their graft.
To prevent BK virus-induced complications, many transplant strategies attempt to reduce the overall load of immunosuppressive drugs. Certain pharmaceuticals may also be employed to reduce the viral load. The key, of course, is to prevent rejection of the transplant while maintaining an appropriate level of immunity to pathogens.
Oddly enough, it is not known how BK virus is transmitted. It passes directly from person to person, with no animal source. Infected individuals do periodically shed virus in their urine, which is the most likely explanation for its transmission. Respiratory fluids are also a possible vehicle.
Given their importance in science, I really haven’t written enough about bacteriophage. It’s hard, though: their names aren’t sexy, their life cycles are usually short, and their stories don’t have an obvious hook. But let’s give it a try today, shall we?
Enterobacteria phage P2 is temperate (i.e. it displays both lytic and lysogenic stages) in its infection of E. coli. It has a double-stranded DNA genome in an icosahedral capsid connected to a tail. A prototypical P2-like virus is shown below:
The species belongs to the Myoviridae, which includes other familiar members like the T-even and SPO1-like phage.
Phage P2 was isolated in 1951 by Giuseppi Bertani, he of LB broth fame. It is best characterized in E. coli, but can also infect a number of other Enterobacteria including strains of Serratia, Salmonella, Klebsiella, and Yersinia. It accomplishes this by adsorbing to the core region of the bacterium’s LPS and injecting its DNA into the cytoplasm. Two mutually exclusive promoters determine whether the phage will enter the lytic or lysogenic cycle.
P2 can serve as a helper phage for P4, a species that cannot lyse a host cell on its own. P4 has evolved several ways to modulate P2 expression, including forcing P2 to form smaller capsids in a co-infection.
Here’s to phage, which are already among the most useful viral tools, and the hope that they become more useful in the future!
Starting from the top, Heterocapsa circularisquama is a dinoflagellate, one of those little guys that cause the massive red tides. The algal blooms it creates are toxic to certain oyster, mussel, and clam species. The dinoflagellate can be infected by a couple of species of virus, one of which is today’s microbe.
Heterocapsa circularisquama RNA virus (HcRNAV) is a single-stranded RNA virus with an icosahedral shell:
The genus to which this virus belongs? Dinornavirus. Dino(flagellate)RNA!
HcRNAV can lyse certain strains of H. circularisquama, and by doing so can control the population dynamics of algal blooms. It’s unlikely that HcRNAV will rid the oceans of harmful dinoflagellates, though: some host strains are only lysed by a second kind of virus, while some are resistant to both.
The virus is (+)RNA, so it has a simple genome and life cycle. It just needs to encode a replicase to copy its genetic information and a capsid in which to package it. Shown here is a reconstruction of the virion, colored by density and cross-sectioned at various points: