Tag Archives: Jordan Curve Theorem

Are you sure you know everything your protein is up to?

Just because you know one function of a protein doesn’t mean you know them all. A recent excellent review of the (drumroll) executioner caspases [ Neuron vol. 88 pp. 461 – 474 ’15 ] brings this to mind. Caspases control a form of cell death called apoptosis, in which a cell goes gently into the good night without causing a fuss (particularly inflammation and alerting the immune system that something bad killed it). They are enzymes which chop up other proteins and cause the activation of other proteins which chop up DNA. They cause the inner leaflet of the plasma membrane to expose itself (particularly phosphatidyl serine which tells nearby scavenger cells to ‘eat me’).

The answer to the mathematical puzzle in the previous post will be found at the end of this one.

In addition to containing an excellent review of the various steps turning caspases on and off, the review talks about all the things activated caspases do in the nervous system without killing the neuron containing them. Among them are neurite outgrowth and regeneration of peripheral nerve axons after transection. Well that’s pathology, but one executioner caspase (caspase3) is involved in the millisecond to millisecond functioning of the nervous system — e.g. long term depression of neurons (LTD), something quite important to learning.

Of course, such potentially lethal activity must be under tight control, and there are 8 inhibitors of apoptosis (IAPs) of which 3 bind the executioners. We also have inhibitors of IAPs (SMAC, HTRA2) — wheels within wheels.

Are there any other examples where a protein discovered by one of its functions turns out to have others. Absolutely. One example is cytochrome c, which was found as it shuttles electrons to complex IVin the electron transport chain of mitochondria.Certainly a crucial function. However, when the mitochondria stops functioning either because it is told to or something bad happens, cytochrome c is released from mitochondria into the cytoplasm where it then activates caspase3, one of the executioner caspases.

Here’s another. Enzymes which hook amino acids onto tRNA are called tRNA synthases (aaRs for some reason). However one of the (called EPRS) when phosphorylated due to interferon gamma activity, became part of a complex of proteins which silences specific genes (translation — stops the gene from being transcribed) involved in the inflammatory response.

Yet another tRNA synthase, when released from the cell triggers an inflammatory response.

Naturally molecular biologists have invented a fancy word for the process of evolving a completely different function for a molecule — exaptation (to contrast it with adaptation).

Note the word molecule — exaptation isn’t confined to proteins. [ Cell vol. 160 pp. 554 – 566 ’15 ] Discusses exaptation as something which happens to promoters and enhancers. This work looked at the promoters and enhancers active in the liver in 20 mammalian species — all the enhancers were rapidly evolving.

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Answer to the mathematical puzzle of the previous post. R is the set of 4 straight lines bounding a square centered at (0,0)

Here’s why proving it has an inside and an outside isn’t enough to prove the Jordan Curve Theorem

No. The argument for R uses its geometry (the boundary is made of straight
line segments). The problem is that an embedding f: S^1 -> R^2 may be
convoluted, say something of the the Hilbert curve sort.

An incorrect proof of the Jordan Curve Theorem – can you find what’s wrong with it?

Every closed curve in an infinite flat plane divides it into a bounded part and an unbounded part (inside and and outside if you’re not particular). This is so screamingly obvious, that for a long time no one thought it needed proof. Bolzano changed all that about 200 years ago, but a proof was not forthcoming until Jordan gave a proof (thought by most to be defective) in 1887.

The proof is long and subtle. The one I’ve read uses the Brouwer fixed point theorem, which itself uses the fact that fundamental group of a circle is infinite cyclic (and that’s just for openers). You begin to get the idea.

Imagine the 4 points (1,1),(1,-1),(-1,1) and (-1,1) the vertices of a square centered at ( 0, 0 ). Now connect the vertices by straight lines (no diagonals) and you have the border of the square (call it R).

We’re already several pages into the proof, when the author makes the statement that R “splits R^2 (the plane) into two components.”

It seemed to me that this is exactly what the Jordan Curve theorem is trying to prove. I wrote the author saying ‘why not claim victory and go home?.

I got the following back

“It is obvious that the ‘interior’ of a rectangle R is path connected. It is
only a bit less obvious – but still very easy – to show that the ‘exterior’
of R is also connected. The rest of the claim is to show that every path
alpha from a point alpha(O)=P inside the rectangle R to a point alpha(1)=Q
out of it must cross the boundary of R. The set of numbers S={alpha(i) :
alpha(k) is in interior(R) for every k≤i} is not empty (0 is there), and it
is bounded from above by 1. So j=supS exists. Then, since the exterior and
the interior of R are open, j must be on the boundary of R. So, the interior
and the exterior are separate components of R^2 \ R. So, there are two of
them.”

Well the rectangle is topologically equivalent (homeomorphic) to a circle.

So why isn’t this enough?  It isn’t ! !

Answer to follow in the next post. Here’s the link — go to the end of the post — https://luysii.wordpress.com/2015/11/10/are-you-sure-you-know-everything-your-protein-is-up-to/