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That just sounds like a second-degree differential equations with extra steps.

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Be careful saying that beneficial traits will necessarily become more prevalent. It's all probability. If getting the long-term better trait necessitates going through a worse trait in the short-term, it may never happen. As an example with arbitrary numbers, say staying in fever another 1 to 5 hours is actually disadvantageous, even if staying 5-10 hours is advantageous.

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Short answer: no.

but some mutations are more likely in smokers, others in drinkers, drug users etc etc etc. A whole bunch of statistics can show these patterns.

Back from patterns to the individual; Definite proof that your tumor came from smoking is, again, not possible. But it's a likely factor that had an influence.

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It's how you "make" ice cream at home. You use salt + ice to drop the temperature really cold, really fast.

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so if you spent winter in the southern hemisphere and then winter in the northern hemisphere would a doctor recommend getting both vaccines?

That's another good question, which again deals with unambiguous signaling. Remember that molecules have not only size in space, but also a shape and a pattern of electron density. So, yes, you could make a protein as big or bigger than T3, but will it fit into the necessary active site the same way? And will it electronically interact with the active site the same way?

For peptides, the bond angles of the main chain are given by something called a Ramachandran Plot, which works out the most likely shapes a peptide bond can form in space. In this case, maybe you could have a protein with a pair of tyrosines that could meet up and maybe look kind of like T3? Well, the necessary bond angles make that unlikely, so such a shape would not persist in space for enough time to matter. Even if it did, you'd have the rest of the protein hanging out at weird angles, preventing a good fit with the intended receptor.

Finally, even if all that worked, there's still some intentional interaction with the iodine substituents necessary to induce a tight fit. You could get a couple tyrosines in there to fake the rough shape, but how are you going to get protein parts in there to fake the iodine interactions? There's no way. (A) nothing else is even kind of shaped like those iodine substituents, and (B) even if there were, you couldn't stick them into that binding site without popping the whole protein out. Thus, the iodine groups help ensure that only T3 will reliably fit into the intended receptor site.

And it really slows down the rate that viruses multiply which allows your immune system to reduce the number of viruses that are able to multiply.

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It's random in the short term. In the long term, beneficial traits will improve survivability and be selected for. If staying in fever benefitted surviving it's reasonable to assume it would have arisen by chance and then been selected for by now.

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I can't say that was ever discussed during patophysiology.

One case which I can speak about is malaria. There, it is due to Plasmodium's life cycle. It takes 3 days for them to mature in a cell. So day 1 they enter the cells, they 4 they burst the cells open, which releases all its crap simultaneously from many cells. Day 7 it's hatching time again etcetc.

I reckon there's also 1 with 4 day life cycle.

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There also isn't a reason to think the body would evolve a way to stay in fever after vital load drops. Remember, evolution is random and if this never randomly occurred we wouldn't evolve it.

I hate to sound silly, but where does the extra cold come from? Is this assuming the ice/salt mixture is still in a sub-freezing environment? How does putting salt on ice and then onto your skin make the ice colder? Colder enough to cause such damage?

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