ag.algebraic geometry - Is the complex moduli of Quintic Calabi-Yau toric?

The complex moduli space does not admit a toric strucutre, since the orbifold fundamental group of a toric orbifold must be abelian. Indeed, $pi_1(mathbb C^*)^n$ surjects on the orbifold fundamental group. Also, the orbifold stabisier of each point on a toric orbifold
is a finite abelian group. At the same time the stabiliser of the quintic $sum_i z^5=0$
is a non-comutative group. Also I am sure that the orbifold fundamental group of the moduli space of quintics contains free (non-abelian) subgroups, but I don't know how to prove it.

Also it should be true that the Tiechmuller space is not algebraic. It least this happen in lower dimensions for cubics in $mathbb CP^2$ and for quartics in $mathbb CP^3$.
In the first case the Theichmuiller space is a disk, and in the second it is
a hermitian domain of type IV. Moduli spaces of polarised K3 are discussed here
for example, here:

http://people.bath.ac.uk/masgks/Papers/k3moduli.pdf

structural biology - How would one describe the R-factor in crystallography?

Crystallography requires the collection of many measurements (could be a few thousand to even millions depending on the size of the molecule and complexity of the crystal (technically speaking, the size of the crystal's unit cell is a major determining factor for the size of the data set). I'm not going to assume this is a small molecule crystal like a salt or a large molecule like a protein. its not necessary here.

I'll call this a set of intensity measurements, even though what is actually used is the square root of the intensities measured, called the structure factor.

The only way we really know that the molecular structure model is correct is that it generates as accurately as possible the intensities that were experimentally measured.

R, which I believe stands for residual, is a fractional difference between the measured intensities and what any proposed molecular model gives.

The residual is calculated as the absolute value of F(model)-F(measured). This means that 0.0 is a perfect match while 1.0 is a perfectly awful fit that shows the model is perfectly awful.

In practice 0.60 is usually as bad as random model will give you (there's a statistical argument as to why it doesn't go higher). Also the measurements are usually not perfect - they contain errors in measurements or artifacts from imperfect crystals or the detector, so an R value of < 0.20 (20%) is typically what you see in a reasonable paper. I think its commonly less than 0.15 for most structures now in fact.

molecular biology - How does translational coupling work in prokaryotes?

Translational coupling describes in how some cases an mRNA will code from more than one protein (i.e. will be polycistronic). Translational coupling is thought to be mostly used as a way to make a set of genes are translated at roughly the same amount in the cell.

Translational coupling is very common in prokaryotes and nearly half of e coli genes are found in a polycistronic operon. What we know about them is revealing. There are some fancy mechanisms to adjust the ratios of these adjacent genes, which are still coupled, but with ratios that are not just 1:1. Its shown that the later genes are often translated at somewhat lower frequency because the first genes are available more quickly before the mRNA degrades.

Eric Alm @ MIT wrote this great paper on how operons evolve.

I've only been able to find this reference to a eukaryotic case of "translational coupling" which is very rare, but does exist. The most common cases of translationally coupled genes in eukaryotes are RNA viruses which usually contain only a single full length mRNA which codes for all the genes in the virus. The selective pressures to keep the viral genome small and constrain the ratios of these genes, will cause these genes to even overlap, starting the next one before the current gene finishes.

molecular biology - How do I prepare and clone from E. coli DNA?

Yes, you must fragment the genome in order to insert it into a vector for cloning; you can't "insert" the whole 5 Mbp genome of E. coli into a vector. It's difficult to transform cells with huge plasmids, 2-20 kbp is an optimal range. In any case, if you want a clone of the whole genome, wait 30 minutes and the cell will happily oblige you.

Most procedures that isolate genomic DNA will fragment it in the first place, as it is much too large and fragile to stay together, and if it did it, the majority would be caught up in other cell debris and discarded. Vortexing with glass beads is a typical first step to randomly fragment it. After this, you can digest the DNA and your vector to place matching ends on it, ligate it, and transform host cells.

If you want a targeted approach (to extract a specific gene, a technique with which I am unfamiliar), you may be able to use PCR on your extracted, fragmented DNA, as there would hopefully be one intact segment spanning what you want.

ho.history overview - A mathematical idea "abstract enough to be useless for physics"

Dear Jérôme, I doubt that Grothendieck ever said that.

However, in an analogous vein, Jean Leray, a brilliant French mathematician, was taken prisoner by the Germans in 1940 and sent to Oflag XVIIA ("Offizierslager", officers' prison camp) in Edelsbach (Austria), where he remained for five years till the end of WW2.

He managed to hide from his captors that he was an expert in fluid dynamics and mechanics, lest they would force him to contribute to their war effort (submarines, planes).
Instead, he organized a course, attended by his fellow prisoners, on the foundations of Algebraic Topology, a harmless subject for applications in his eyes. It is in these courses that he introduced sheaves, cohomology of sheaves and spectral sequences.

His strategy worked out fine since these discoveries didn't play any role in the construction of weapons by the German enemy, who never cared about Leray's courses and findings. On the other hand, these theoretical tools have had a non entirely negligible role in pure mathematics since.

biochemistry - Protein construct design

I am trying to create some constructs of a certain protein deleting well defined domains (at either terminus) to determine interaction regions with other proteins etc., 3 constructs with varying start/end sites have ended up being aggregated in E coli (while the full length protein expresses reasonably well). My question is what considerations one should use to determine start/end sites to maximize chances of getting soluble, purifiable protein. The criterion I used were:

1. Preferably loop region in known x-ray structure


2. Use hydrophobicity plots to minimize hydrophobic residues at both termini


3. 3-4 residue linker between protein and affinity tag (GST/6xHis)

From personal experience with an earlier construct, this can turn out to be an idiosyncratic exercise but I was wondering if I was missing any crucial parameters.

EDIT: I also played around with the PDB file of the non-truncated protein to check whether huge hydrophobic patches were exposed on deletion of the domain(s), and I didn't find any such patches that could potentially be exposed and lead to aggregation

Hochschild/Cyclic Homology of von Neumann Algebras: Useless?

Some further thoughts: the most striking results I know of on "purely algebraic cyclic/Hochschild homology" are due to Wodzicki, see e.g.

Homological properties of rings of functional-analytic type, Proceedings of the National Academy of Sciences USA 87 (1990), 4910-4911

which states that stable C*-algebras have trivial cyclic homology. Obviously this doesn't answer your II_1 factor question...

Also: your remark that in some cases, we can ignore the analysis and make the situation a bit simpler confuses me a little. To get anywhere with cyclic or Hochschild homology, we need to do some kind of comparison of resolutions, or construction of contracting homotopies, or something like that. My intuition - but I don't work much on operator algebras, so I could well be wrong here - is that a von Neumann algebra is such a big object we usually can only get a handle on it by looking at suitable subsets which generate its unit ball in the WOT/SOT. So for group von Neumann algebras, one tries to see what's going on for translations, and thence to deduce more general results by exploiting w*-w* continuity; or else use projections and approximation arguments. If we go to a purely algebraic category, then it is no longer sufficient to define things on dense subsets - one really needs a global definition, one really needs to verify that certain putative identities are satisfied by each element of the von Neumann algebra.

Sorry if that's a bit waffly. I think my point is that imposing continuity restrictions actually makes things easier, because - intuitively - more things are going to be projective/injective/flat relative to one's restricted class of short exact sequences. This is why, for instance, we know that $H^n_{cb}(M,M)=0$ for any von Neumann algebra M, but why the analogous claim without the 'cb' is open and back-breaking. In a similar vein, if you work in a restricted category then one does indeed get some known instances of homological non-triviality (though at the level of modules, not at the level of cyclic homology):

M. E. Polyakov, An Example of a Spatially Nonflat von Neumann Algebra

I should also say that the Hilbert module stuff you mention doesn't really connect to your original question about cyclic (co)homology. It's interesting, and I think more has been done, but it's just different - so if that's what interests you, cyclic and Hochschild homology may be something of a distraction.