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Defining "species" (Are species an emergent property or an ensemble of quantitative differences?)
Interbreeding
Defining species by the criteria of not being able to breed is problematic:

The concept loses its meaning in phylogenetic prospective, since closely related species were able to breed between themselves in past, during the speciation event (an "event" that often lasted hundreds or thousands of generations)
The definition is problematic for simpler life forms, such as bacteria exchanging genetic material, or viruses where different species from the same family may differ only quantitatively (e.g., HIV vs. SIV).

(Excellent exposition of this view on the difficulty/impossibility of defining species is presented in this answer, as well as in the answer by @SPr below.)
Emergence
On the other hand, species are not reducible to merely accumulation of genetic variation. When looked from more general prospective (e.g., here), species are an example of an emergent phenomenon (the same inability to interbreed is a qalitative rather than quantitative difference.) We thus should be able to define a critical boundary between what is the same species and not, although this boundary does not necessarily lie in time domain. Inability to interbreed is a vague definition, since it may be due to very different reasons: phisiological, genetic, assortative mating, etc.
Operational concepts
In some domains we could choose to define species based on a criteria of convenience. E.g., considering an ecological community one may define as species the organisms that cannot interbreed, but also those that do not share the same habitat. While such an operational definition may work well for practical studies, it is not unique and subject to change from one study to another.
Are species an emergent property or an ensemble of quantitative differences?
Can we give a robust definition of species/speciation? Can we define a domain of applicability of such a definition? (e.g., defining the forms of life for which species can be defined)
Background
When speaking of emergent properties, I tried to skip some techncialities that led me to this question, which is why this part of my question may appears somewhat vague. So I would like to add more details here.
Phil Anderson in his article Broken symmetry, emergent properties, dissipative structures, life: are they related? formulates emergence in mathematically rigorous terms as a kind of phase transition in a non-equilibrium system. Phase transitions are characterized by summetry breaking, i.e., appearance of a completely new property in a system. What I termed previously robustness is, in Anderson's language, called phase rigidity.
The question is thus: whether speciation is a kind of such phase transition (as claimed, e.g., in the Melamede's article, already cited earlier) OR whether it is merely incremental accumulation of changes. (However, in such a formulation question is unlikely to be answered either in physics or biology communities.)
For more context: in his article Anderson criticizes Ilya Prigogine's school on a few technical points, but it is Prigogine and his co-workers, who are at the origin of such thinking. In particular, they have multiple publications on the subject of such disspative structures, and non-equilibrum thermodynamics more generally. Many of their articles are available by googling.
Update
After some discussions in the comments: the crux of the question is whether (or tow hat extent) speciation occurs via abrupt or incremental changes.

Organisms differing by abrupt acquisitions, such as presence of a nuclear membrane or an extra chromosome, are unambiguously assigned to different species (and cannot breed or reproduce viable offspring).
On the other hand, most ambiguous cases seem to be those where the "species" differ only by a incremental changes (such as the number of nucleotide substitutions/insertions/deletions).

Ring species could be a possible test case: if the species at the beginning and the end of the ring cannot breed, do they always differ only by incremental changes? If one species differed by a number of chromosomes, it likely wouldn't be able to breed with any of the others.
Update 2
Here is a useful answer: species can be defined from many different viewpoints.

Ew "closely related species were able to breed between themselves in past", it's also the case in the present.  See for instance the development of the "coywolf" in eastern North America: https://en.wikipedia.org/wiki/Coywolf

@jamesqf sure, horse and donkey can breed, but two breeds of dogs sometimes can't. This underscores the insufficiency of this definition. But the point I was making was dufferent - if you have suggestion of more precise formulation, I am listening.

@Vadim often the difference with dogs is physical rather than "can't". Imagine a female chihuahua trying to breed with a male great dane, or vice versa. I don't think the genitalia would fit, nor would the pups in the first instance.

@bob1 physically can't :) the point is that any two nearest relatives could cross at some "point". It is the chimpansee-girilla-human problem - which of the three are closest relatives?

@Vadim my point was that they can if you take the sperm from one and use it to fertilize the other, it's just they they don't (not for lack of trying I am told) because things don't fit, or can't reach... They are definitely not separate species.
Horse and donkey can interbreed but offspring are usually sterile - mule and hinny respectively, so definite separate species. But the boundaries between other organisms aren't so clear.

@bob1: Yes, just about everyone regards dog breeds as varieties of a single species, yet the "don't fit" problems ensure that some breed pairings don't produce offspring that are viable without human assistance.  (Indeed, I've read that some show breeds like bulldogs need human assistance.)  OTOH, dogs, wolves, and coyotes are regarded as different species, yet can interbreed and produce viable offspring.

@bob1 my understanding is that horse and donkey have different number of chromosomes, whereas dogs differ by a number of point mutation (nucleotide substitutions/insertions/deletions). To retake the main theme: this is an example of abrupt vs. incremental changes. Are any known ring species more than dogs in this respect?

Another example of different species that freely interbreed are trout.  Particularly within the US Great Basin, where the drier climate after the Pleistocene created a lot of isolated stream/lake systems, which often evolved their own trout species.  Then eastern brook & rainbow trout were introduced, and freely bred with the native species, and with each other, producing viable hybrids such as the tiger trout: http://www.ndow.org/Species/Fish/Tiger_Trout/

@jamesqf by what criteria are these "species" are then judged to be different species?

@Vadim: 1) Taxonomists and people who discover (or want to have discovered) new species give them names.  In that context, see the arguments between "lumpers" (who think the minor differences between some of those species should better be called varieties) and "splitters" (who think minor differences justify calling them new species).  2) In the US, the EPA, Forest Service, various environmental groups &c who use the "splitter" side of the argument as justification for policies protecting the supposed species.

Can we give a robust definition of species?

No.

Species constantly evolve, diverge, converge, interbreed, and mix and shuffle and trade and spread genes. To draw a box at any particular point in time around a population or lineage, and to say, "all contained within are members of such-and-such species" fails to appreciate that this would only work for that particular case, and arbitrarily so; it is obviously futile to have a sound microtaxonomy that would be generalizable to all animals, or microbes, or plants, or viruses for that matter. What consensus there might be for (e.g.) beetles is a poor yardstick for judging whether two alike bacteria ought to be lumped together as a species. Also, life, genes, individuals and populations are dynamic over time, so whatever box we draw will by definition require adjustment, and you can see that this adjustment will have to be arbitrary in many respects. This is an old problem, and it is more philosophical and ever-debatable, rather than something to be settled by empirical science triumphantly. From Nicholson's 1872 A manual of zoology (two centuries ago now):

"No term is more difficult to define than "species," and on no point
are zoologists more divided than as to what should be understood by
this word."

And to echo all that has been said from Piglucci (2003) Bioessays:

"First, the species problem is not primarily an empirical one, but it
is rather fraught with philosophical questions that require — but
cannot be settled by — empirical evidence."

Which also summarizes a few major biological definitions, some of which do not align:

And also a few philosophical issues that are worth considering when discussing the species problem:

This is not to say that conventions don't exist, or that there is a lack of an operational and specific definition of a species in each respective field; however, there is no overarching definition that applies robustly, hence the answer at the top!

Also, search around on this SE for similar Q&As. I find no duplicates but some similar ones.

Thank you for the answer. It provides many useful insights. However, what I really would like to understand is when quantitative differences between two organisms transform into qualitative ones... or whether such a transformation never takes place and the differences are always quantitative.

@Vadim Can you not think of species that have qualitative differences between them? Even individuals of the same species can have qualitative differences. Maybe you have a different idea of what that means?

@BryanKrause Good point. Perhaps, it is more precise to ask what quality makes organisms belong to the same species or not. I think not being able to breed is a wrong quality, but the idea is correct: the quality is either there or not. If we can show that such a quality cannot exist, it is in itself a strong claim.

@Vadim I think SPr's answer and some other Q&A here make it pretty clear there is no single definition, and can't be one that is agreed upon.

Remi's answer here is excellent: https://biology.stackexchange.com/a/39669/27148

@BryanKrause the linked answer is indeed very complete. However, it addresses only one line of thinking - the one that I outlined in the first part of my question. And this line of thinking, it seems, is in contradiction with how we think about the emergence of life, which impliues a series of abrupt qualitative changes.

@Vadim What does "emergence of life" have to do with "species"? Who is the "we" that implies abrupt qualitative change? I think Remi's answer there makes clear that "species" is just a classification we draw for convenience when what we are really dealing with is "lineage" which can't be separated except by (relatively) arbitrary means.

@BryanKrause I cited Phil Anderson and Ilya Prigogine in my question. Although neither might have mentioned species - this specific question is based on the quoted Melamede's article, where speciation is listed as one of the abrupt transitions between the emergence of cell and the evolution of body systems.

@Vadim I can't access that article and don't see any quote from it. I'm guessing something is getting lost in the details. I would not describe speciation as "abrupt" except in special cases or using special definitions of species (or a geological meaning of "abrupt", perhaps).

@BryanKrause I suppose that it can be found for download by googling - (but I wouldn't post link for downloading copyrighted material). Anyhow, I see your point of view, and I tend to agree with it... but it still seems as a grey area to me.

@Vadim Dug a bit further and did find the article.. so, okay, this seems to me like a lot of word salad and not a lot of substance, at least with a casual glance. The author also seems to be mixing the words "species" referring to biological species and "species" in the context of "chemical species", the more generic "kind/sort", which has little relationship to the concept in biology. I wouldn't recommend taking it too seriously. In any event, I don't see any reference to anything "abrupt", if there's a particular quote that is relevant you should include it in your question.

@Vadim - check out ring species as an example of the difficulties in determining what is a species and where. The problem is that you are approaching it in a computational manner, where things have exact definitions. Species do not have exact definitions precisely for the reasons S Pr outlined. Biology is messy! An example of the difficulty is cancer research, where cell lines evolve in the lab so that results from one lab are not reproducible in another, even though both are working on the same cell line.

@bob1 I am not looking for exact universal definition of species. The point of the question is whether species a result of small incremental changes or an abrupt jump - like appearance of a cellular membrane. Depending on which corner you look from (taxonomy or abiogenesis) the answer is different (in my view).

@Vadim - both is the answer. Abrupt jumps are seen in some circumstances, incremental in others.

@Vadim You say in my view, species definition depends whether one considers it from a taxonomical vs. abiogenetic point of view. I think we would all disagree with you. Defining species has exactly the same problems regardless of whether you believe that at some point (i) life was designed to evolve, (ii) created and subsequently evolved, or (iii) evolved out of inorganic material. Evolution exists and that makes defining species boundaries arbitrary, since all life, populations and lineages are in constant flux over time. That's it. See, whether it's abrupt or not is irrelevant.

@SPr Perhaps abiogenic is too general a word for the veiw that I expressed, but I think I have described clearly enough the two contradictory viewpoints in my question. We have a good reason to belive that there is abrupt transition between being alive or not, between being cellular organism or not, between having a nucleus or not, etc. Any division in species that is not grounded in evolution is indeed arbitrary. And I don't see any problem with populations and lineages being in flux - so the species change, some disappear, some split into more species.


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Why do mitochondria have a phospholipid bilayer?
So, a thought came up and I couldn't find all that much info online, so I thought I'd ask some professionals here!
The endosymbiont theory states that: mitochondria came to be ingested by bigger prokaryotic cells about 1.8 bYa, and by chance of luck came to a mutualistic relationship.
Now, mitochondria are said to have been archaea, right? But the mitochondria in our cells have phospholipid bilayers, with ester bonds in them, like all eukaryotes, but when you look at archaea, you see that they have monolayers because of their ether bonds, with rings and all sorts of branching, which is what gives them that extremophile-acclaimed resistance.
Question is if the endosymbiont theory is so widely accepted and mitochondria are meant to be archeae, why do they not present a monolayer with ether bonds?
Thanks for your time!

Only some archaea have a monolayer others have a bilayer, and the monolayer is believed to be a derived adaptation for extreme environments.

@xusr — I was going to make the same point, suggesting the poster change his question to why don't eukaryotes have an archae-like cell membrane. But rather as a joke. If you think about it, eukaryotes are assumed to have evolved from prokaryotes but in either case they don't have a cell wall. So the enclosing structure of eukaryotes evolved from that of their prokaryotic ancestor at some time. Anyway, that's a separate question.

I think there is a typo or a misconstruction in the sentence before the one I take issue with. "The endosymbiont theory states that: mitochondria came to be ingested by bigger prokaryotic cells ". Whatever was ingested was not a mitochondrion, but a prokaryote that became a mitochondrion. You should tidy this up.

I think the question is based on a false premise:

Poster: Now, mitochondria are said to have been archaea, right?
Me: Wrong, I’m afraid.

The closest bacterial relation of mitochondria is  Rickettsia, an alpha-Proteobacterium (see Lang et al. for a review). Rickettsia is a eubacterium, not an archaebacterium.
The confusion is probably due to misreading one of the two alternative theories of the origin of mitochondria. This is the theory that the host for the original mitochondrion was an archaebacterium (rather than a primitive nucleated eukaryote). In both theories this host aquired a eubacterium related to Rickettsia, which gave rise to the mitochondrion.

Ah, great reply! The actual misunderstanding had come from one of my teachers actually (actually just a PhD who got to give us a few lectures), you cleared it up!

@FedericoParravicini — Don't be too hard on your teachers. It's easy to make a mistake, especially if the main focus of your lecture is elsewhere, or even to trip over your tongue (a verbal typo). What I would recommend is to use web resources to check these things. I prefer text books to random websites or Wikipedia, as they are integrated and subject to editorial review and often well illustrated. If you don't have access to a good library try NCBI Bookshelf, which has old editions of some reputable texts such as Berg, Alberts and Lodish et al.

During the process of phagocytosis, the host membrane surrounds the food. So in this case, these bacteria were surrounded and encased in the host membrane, which pinched off internally, and became the outside membrane of the mitochondria.

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Beta-diversity for data with zeros
Background
I am looking into enterotyping the gut microbiome data, obtained by shotgun sequencing. This essentially means performing the Principal Coordinates Analysis (PCoA/NMDS) of the beta-diversity matrix (distance matrix), based on the abundance table of species/gene-counts for all possible species/genes in all samples: for sample $i$ the counts of species/genes are $\mathbf{x}_i=(x_{i1}, x_{i2},...,x_{im})$. Such a table necessarily contains many zeros, whenever a species/gene is not present in a sample. Some of the possible beta-diversity measures are listed in table 1 of this article.
Problem
Statistically speaking, such data are considered to be an example of compositional data, i.e., the meaningful information is contained only in the ratios of the counts, which makes log-based distances the most suitable ones. These are used in many statistical methods (see, e.g., here). However, the log-transofmation is meaningful only when all the entries in the count table are strictly positive $x_{ik}>0$. One thus faces a problem of dealing with zero entries.
Possible solutions

Using metrics not sensitive to zero entries (excluding alr, ilr and other similar log-transformations, highly recommended by statisticians - this limits the choice of statistical methods)
Imputation of zeros: this requires a principled/biologically-sound way of imputing the zeros, since the results may be potentially very sensitive to this choice.

Question
What are the commonly used imputation methods? What is their advantages/disadvantages and justification for their use?
Some references

Here is a statisticians' view on imputation in compositional data: the methodology may get rather involved, but does not have any specific biological basis.

An appropriate way to handle this is with a so-called zero-inflated model: https://paul-buerkner.github.io/brms/articles/brms_families.html#zero-inflated-and-hurdle-models-1.

I'll often use the pseudocount method for exploratory analyses. Basically just apply a pseudocount of 1 to every count in the table (since Log(1) = 0), so the action you actually end up performing is Log(x+1).

@MikeyC Pseudocount is what I meant by "imputation of zeros": it is certainly a good method when one is interested only in the most abundant species (those that have counts), but I see how it may create problems when detecting low abandance species is important.

@vkehayas if I understand correctly, zero inflation corrects for undersampled species - one will still have actual zeros in the count table?

Correct, you still keep them in the sample. The model estimates parameter values for a separate process leading to ratios at zero and parameter values for a separate process leading to values $\in (0, 1)$. A great introduction to similar models can be found in McElreath's Statistical Rethinking book (slides, lecture 13).

To be honest I feel like this question would be better on CrossValidated ...

I'll also point out that most of the suggestions here focus on what to do about zeros, but don't consider compositionality. You'd ideally want a solution that considered (1) presence of zeros (via inflation or just low means), (2) discrete nature of data (i.e., potentially higher sampling variance at low densities) and (3) compositionality. This example uses Dirichlet-multinomial mixtures, but then reverts to Bray-Curtis-based PCoA for ordination.

I would recommend using a Bray-Curtis dissimilarity matrix since it accounts for abundances and can handle the so called double zero problem. I'm on my commute now and so must return to this for a complete answer later

I am looking forward to more details!

@Roger I'm not trying to ignore you. Started teaching summer classes + summer research students, so I've gotten a bit tied up. I'll get back to this soon.

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Apportionment models, niche theory, neutral theory
I am trying to understand the relation between apportionment models, niche theory, and neutral theory of biodiversity. Wikipedia seems to suggest that the first two are the same thing, whereas the last one is controversial

The theory has sparked controversy, and some authors consider it a more complex version
of other null models that fit the data better.

This impression is further enforced by the fact that the Hubbell's original books seems to be almost the only source of information about this model.
On the other hand, this article suggests that modern niche theory is a synthesis of the apportionment models and the neutral theory

Niche-based theories have emphasized the role of competition for resources, while neutral theory has
highlighted the role of stochastic effects, and several works have investigated the interplay
between stochasticity and competition.Many of these theoretical insights have been synthesized
in what is commonly referred to as contemporary niche theory. Contemporary niche theory highlights
the role played by equalizing mechanisms, processes that decrease fitness differences between
organisms, and stabilizing mechanisms, processes that decrease competition for resources. These basic
organizational schema have been successfully applied to understand community structure in a wide range
of settings.

I am looking for guidance in this theory landscape, comments on what is mainstream and what is not, and suggestions for deeper reading (math is welcome).
Remark:  It is understood here that neutral theory of biodiversity is not the same as the neutral theory in population genetics (although the latter might have inspired the former).
Additional references:

The unified neutral theory of biodiversity and biogeography at age 10 - Article by Hubbell, outlining successes of the neutral theory.
Conceptual synthesis in community ecology - this articles draws extensive parallels between the community ecology and population genetics, but light on math.
The transition between the niche and neutral regimes in ecology - a physics-minded model, demonstrating a transition between niche and neutral behavior depending on the population size and environmental factors.

I think I would start with Letten et al.  https://doi.org/10.1002/ecm.1242  . BTW I wouldn't say "modern" niche theory, "contemporary niche theory" appears to be the standard terminology ...  I wouldn't say Hubbell's book is the only source - there are lots of research papers ...

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Long-term efficacy and absence of side effects of anti-covid vaccines
The question is about the methodology/biostatistics of clinical trials (I state this beforehand to avoid accusations of being an anti-vaxxer).
As multiple anti-COVID vaccines are offered on the market, the question naturally arises about their long-term efficacy and possible side effects. As SARS-CoV-2 virus has been known for a bit more than a year, neither of these could be tested directly, so one probably has to rely on indirect knowledge due to the experience in developing other vaccines. Given that multiple vaccines have been approved for a wide use, there should exist well-established procedures to assure their efficacy and safety.
The question is: how does one prove the long-term efficacy and absence of side effects during a short period of time (short in comparison to the expected duration of immunity and absence of side effects).
Update
As @MattDMo have correctly pointed in their answer, the absence of long-term effects is established in the last phase of the clinical trials, while observing cohorts of vaccinated individuals during a long period of time. @MaximilianPress, in their comment, has also correctly brought up the fact that this is done within the framework of survival analysis. In particular, this means that it is sufficient to follow the vaccinated cohort during several years to evaluate the side effects that occur during their lifetime, as the survival analysis keeps track of the sensored data.
Here we are dealing with a situation where such a study could not be done - the best we can guarantee is the absence of the side effects in the next few months. We thus need to rely on a different method, which would account for the similarity between the established and the new vaccines.
I am less concerned about the vaccine efficacy, since the inefficacy of vaccines is usually against different strains of virus, as is in the case of influenza and Dengue.
Update 2
The accepted answer points out two important things:

The long-term side effects of a vaccine/medication can be truly understood only in the Phase IV of the clinical trials, i.e., after the vaccine is in widespread use.
The threshold for approving this specific vaccine is lower due to the emergency of the current health crisis.

This however still leaves open the question of the safeguards undertaken in the Phase III of the clinical trials, before the vaccine approval. I.e., the question is about ranking the candidate vaccines against the above-mentioned threshold.
Remarks:

I understand why this question irritates, but I believe that it is not the asking, but rather refusing to seek the answer that plays in the hands of anti-vaxxers.
I did follow the suggestions in the comments to pose this question in the medical and the statistics communities, but it seems to get there even less traction.

Update 3
Recent suspensions of Astrazeneca vaccine in 9 European countries demonstrate the importance of this issue.
Update 4 Pfizer vaccine has been fully approved - that is from now on it is considered a fully tested vaccine (till now it had been used under the emergency authorization).

What is your question?

I’m voting to close this question because this is a statement of opinion, not a question.

@David The question is: how does one prove the long-term efficacy and absence of side effects in such a short time.

@Vadim: One doesn't.  It is simply a question of whether the small risk of such long-term effects is less than the immediate risk of serious illness or death from the virus.

@jamesqf to make such a statement one has to estimate and compare the risks

Your question is then one of medical policy and public health, not a specific question of biology and is therefore off-topic here. It may be on topic on SE Medical Sciences, but you would be better to read one of the many detailed accounts available via reputable news sources.

Everyone, lets please be civil to one another.

I’m voting to close this question because it's probably more appropriate to SE Medical Sciences or a statistical forum. I believe that it is a question of how to handle censored data, but I'm not an expert in those spaces: https://en.wikipedia.org/wiki/Censoring_(statistics)

@MaximilianPress one cannot speak of "censored data" here, since there's no data to speak of. The risk-assesment methods are probably indirect, based on comparison with the existing vaccines, modelung, etc. But I am engaging in a guesswork here - which is why I am looking for an educated answer.

If what you want to ask is whether vaccines have long-term effects, then that is definitely a question for SE Medical Sciences. I guess that you can try to interpolate COVID vaccines somehow into the space of other vaccines, though it is starting to flirt with anti-vaxxers a little bit. It's basically asking "what is the long-term effect of an arbitrary strong immunologic challenge", except imputing that to an emergent property of vaccination. Maybe such emergent properties exist, but the place to get answers on that is from practicing vaccinologists.

@MaximilianPress as I explained in the update to my question, long-term effect of vaccines is my secondary concern. I think that knowledge is the best response to anti-vaxxers, although my mentioning of them has probably given a wrong direction to the whole discussion.

What do you mean with "This however still leaves open the question of the safeguards undertaken in the Phase III of the clinical trials"? General safety is tested in Phase 1, overall safety is also a part of Phase 3, here all observations are collected. So far, no serious side effects have been found. For long-term, Phase 4 studies are up and running.

@Chris I suppose that one aporove a vaccine, being confident that in a few years people won't start dying from it at a higher rate than they now die from coronavirus. If I seem stubborn, it is because I come from a quantitative field: I want to see the protocols and the numbers.

@Chris a friend of mine brought up a possibility that this is not possible simply because it is impossible, like saying that the sun wouldn't rise tomorrow. Perhaps, for people familiar with immunology/farmavology my question simply seems naive?

What protocols and numbers are you talking about? The protocol for the phase 3 trial? Can be found here. The published results from the phase 1/2/3 trials? All are linked at the bottom of this page.

You are correct that we have been creating and approving vaccines for a very long time, so the procedures for looking at efficacy and side effects of vaccines in general long-term are quite well developed. They occur via so-called "Phase IV" trials, also known as post-marketing surveillance.
However, one complicating factor with at least the Pfizer/BioNTech and Moderna vaccines is that they're the first mRNA vaccines ever approved. While I hope the companies have been following some of the volunteers from previous clinical studies of other mRNA vaccines, we're in somewhat uncharted territory right now. The vaccines appear to be safe and effective, at least on the scale of tens of thousands of Phase III test subjects, but we just don't know exactly how they'll behave long-term. They were approved in the US as "Emergency Use" products, meaning the upside of having them available outweighs any known or potential downsides at this point.
So, the technical answer to your question "how does one prove the long-term efficacy and absence of side effects in such a short time?" is that they can't prove it. Based on the available data to date, there's no reason to suspect that some previously-unobserved deleterious effect will become common, but we can't be 100% certain until the products have been in widespread use for years and possibly decades. However, given the seriousness of COVID-19 and its currently unchecked spread across much of the world, the benefits of having "highly, strongly, very most likely safe" vaccines (that are also very effective) are huge.

I will try here to sketch an answer starting from where @MattDMo left off.
Question:
How do we evaluate possible long-term side effects of a vaccine? (With specific application to the recently authorized vaccines for SARS-CoV-2).
Phase IV clinical trials
As @MattDMo has pointed out, the definitive answer can be given only in the Phase IV of the clinical trials, which is the follow-up of the vaccine/medication after its commercialization, which typically lasts throughout the lifetime of the medication. Obviously, such data are not available for the vaccines against SARS-CoV-2 (or any new vaccines). Thus, we have to rely on the information obtained in the earlier stages of the vaccine development. This leaves us with the following options:
Phase III of the clinical trials
Vaccine development process typically lasts 5-10 years, and the phase III of the clinical trials typically lasts about 6 months. Thus, typically one has the data for at least 6 months. This is however not the case for the vaccines against SARS-CoV-2, which were given emergency approvals after only 8 weeks of Phase III trials.
Animal models
Several years of vaccine development also allow for additional information: e.g., the vaccine could have been extensively tested on relevant animal models for extended periods of time. How one quantitatively estimates human risks from those for the animal model deserves a separate question, but the established animal model likely means decades of data avalable for comparing it with human. Again, this is not the case of SARS-CoV-2, which has been known for a bit more than a year, and for which no reliable animal model has been tested yet.
See this, this, and this articles for the general review of the use of animal models for vaccine development, and this one for the specifics of the animal models for COVID-19.
Similar viruses
In our case there are at least two very similar viruses that have been studied for years: SARS-CoV (first outbreak in 2003) and MERS-CoV (first outbreak in 2012). Although no vaccine has been available for either virus, much research effort has been put into this direction, and the data (at least on animal models) should exist. There might be even data for mRNA vaccines, which have been under development for some time already.
Nature of vaccines
Finally, the very nature of vaccination may explain why vaccines are approved faster than other types of medications, which can be for years in clinical trials. Adaptive immune system develops response to a new pathogen within about a week time. Thus, any adverse side effects are likely to appear during a week after vaccination, whereas the later developments have to do with the functioning of the immune system (other medications can have cumulative effects, if taken over a long period of time, or stay in the body for longer periods - such as medical implants). How one models quantitatively the long-term immune response is another question that deserves to be asked separately.
On this respect the new SARS-CoV-2 vaccines are taking a conservative approach, with the vaccines being counter-inducated to those with possibly different immune system responses: allergics, pregnant women, children.
Disclaimer
This answer could benefit from more reliable references, which I hope to pinpoint later. Also, it doesn't contain quantitative methods of risk estimation that I hoped for - rather it opens way to several more precise questions.


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What is neutral theory of evolution?
The question is not about the very basics, but more about where the line separating the neutral theory from the rest of population genetics lies.
One often reads/hears claims that "the findings of the neutral theory are still disputed" or that "there's no experimental proof for its validity". It is hard to accept these claims wholesale, since:.

It is hardly in doubt that some mutations are neutral
It is hardly in doubt that neutral or even slightly deleterious mutations may fix in the population due to genetic drift, or that beneficial mutations may fail to fix for the same reason.

I therefore interpret such statements as attacks on specific points of the neutral theory, or its most sweeping claims (e.g., that most mutations are neutral), or perhaps as attacks on specific meaning of the term. Moreover, such wholesale rejection seems to ignore that the theory may have very different degrees of validity for different organisms (say, more likely to be true for RNA viruses than for mammals).
I will appreciate clarifications of these points.
Update
Here is an article that raises several issues regarding the applicability of the neutral theory framework to bacteria: Neutral theory, macrobial practice: challenges in bacterial population genetics

I don't often read or hear these claims, but maybe I don't read the correct references.  Are these claims being made by population geneticists? Can you provide a few links?

@BenBolker It recently surfaced quite forcefully in the comments to my answer to another question in this community: https://biology.stackexchange.com/a/97296/59521 There's a more cautious statement in the Gellespie's little book (that the neutral theory is not generally accepted), and I remember reading in more recent reviews that the debate between the proponents of the neutral theory and the neo-darwinists is not over.

@BenBolker this is a very good way to think about it! Thanks.

@BenBolker I think that does deserve to be an answer

A short answer (converted from a comment): I would call neutral theory a useful null model: in particular, it may be less important that it be a factually correct description of natural systems (i.e. "most alleles can  be treated as being neutral (or weakly deleterious") than that it provides a useful standard for distinguishing which alleles are clearly behaving otherwise.
Satta et al (2018) say:

A decade ago, Crow (2008) viewed the then present status of the neutral theory of Kimura (1968) as a standard null model in evolutionary genetics: “Vast regions of the genome are near enough to being neutral for neutrality to be assumed. In fact virtually every study of molecular variability, evolutionary rates, and coalescence is based on this assumption.” At the same time, since he knew Kimura’s devotion to the theory, he felt irony when seeing that reduced neutral variation around a focal site often provided the strongest evidence for selection acting on the site.

In contrast, Gillespie (1994) says (see here):

... we can conclude that both [silent and replacement mutations] are undergoing the same brand of neutral evolution. There cannot be one neutral theory to accept or reject.
The case for neutrality of replacement subsitutions seems particularly weak ... In my view, the neutral theory must be abandoned for replacement substitutions.
Silent substitutions, on the other hand, may well be neutral, although the evidence at the time of this writing is incomplete ... patterns of codon usage and GC% evolution imply that at least some of the silent variation is moderately selected ... we cannot rely on statistical procedures that assume that silent variation is weakly selected and at equilibrium ...

Gillespie, John H. The Causes of Molecular Evolution. Oxford University Press, 1994.
Satta, Yoko, Naoko T Fujito, and Naoyuki Takahata. “Nonequilibrium Neutral Theory for Hitchhikers.” Molecular Biology and Evolution 35, no. 6 (June 1, 2018): 1362–65. https://doi.org/10.1093/molbev/msy093.

Your question would deserve a long essay. R.C. Lewontin wrote a book ("The genetic basis of evolutionary change" - Columbia U.P) on this subject in 1974, right after the neutral theory had been explicitly formulated in the previous few years. The reasons for the controversy over this theory are still alive some 50 years later, despite the enormous advancements in DNA technology. For example the not so hidden polemical tone of this recent paper would not be understandable if some profound ideological issues were not involved; these issues involve something more than "attacks on specific points of neutral theory, (...) or perhaps as attacks on the specific meaning of the term".
In essence, Lewontin argues that the neutral theory has its philosophical roots in the Platonic notion of "ideal" or "type", for which real objects (the genes, in the present case) are only imperfect approximations. This view, applied to the genetic variation in animal species, implies that the role of natural selection is mainly “purifying”, that is, it acts by removing the slightly deleterious mutations that happens continuously and degrade all genomes (when mutations are perfectly neutral, selection does not act at all). As Kimura and Ohta have said, quoting Muller, “the gene through the long course of evolution has finally found itself in man” (quoted in Lewontin, above, p. 30).
The opposite view (can we refer to it as Democritean?) sees all natural populations, man included, as being continuously modeled by unstable and ever-changing environments, so that any attempt to define an “optimum” is intrinsically meaningless. Genomes are highly integrated pieces of firmware, but they are much more fluid, over the generations, than once thought. This means that all genetic variation may have or may acquire meaning, as their molecular or ecological context changes.
Consider an applied consequence of the two views. One of the claims of most anti-racist geneticists is that the variance component among populations of gene frequencies is much higher than the component within populations; this allegedly falsifies the racist argument according to which the differences observed among human population (races) are mostly genetically determined. But if you claim that most of that genetic variation is irrelevant (“neutral”), what remains are the genes responsible for the differences in phenotypic traits, and it becomes obvious that the variance component among populations in the frequency of these genes is lower with respect to the component within populations; and this may be interpreted as supporting a racist view.
It is not automatic that scientists who prefer one theory to the other also support the philosophical implications of that theory. However, the dispute about the neutral theory of evolution cannot be understood without reference to the above conceptual basis.

This is an extremely interesting prospective! This is not what I had in mind when asking the question, but it does give food for thought. Thanks!

Also, thank you for the references to Lewontin and the Jensen et al.


text:
Viruses in human history
How much do we know about ancient viruses and the viral evolution throughout the human history?
To my knowledge the HIV history has been rather well documented for about hundred years back (e.g., see here and here) - via the phylogenetic reconstruction, massive study of this virus since it was discovered, and the analysis of the records and blood samples from the countries where the virus first spread (before arriving to the West and being discovered).
Ebola has been well documented as well over the last half a century or so, although it hasn't changed much, since the repeated outbreaks are spills of very similar strains from the nature.
Finally, there have been obviously many viruses well documented since the advent of massive sequencing.
The question is thus about the viruses with longest history:

What are such well-documented viruses?
How much do we know about them?
How do we know this?
How much have they changed ? ( In terms of their danger to humans, e.g.)

Disclosure: the question is motivated by this one. Here is a somewhat similar question (but rather limited in scope).
Update
The article on early HIV spreading along rail- and waterways, mentioned in the comments.

I'd reckon that the evolutionary history of retroviruses can be traced back pretty far given their persistence and co-option in the genomes of their hosts.  I'll link sources if I can find any.

https://www.nature.com/articles/d41586-020-02083-0

Could you maybe be a little bit more specific about the sources of evidence you're interested in/would accept (e.g. archaeological/ancient DNA vs phylogenetic)?

@BenBolker thank you for this nature article - it is great! I am mostly interested in epidemiological angle: how the virus spread and how it evolved in human population. I think one cannot go far back without ancient DNA... but I may be wrong. (I recall that for HIV one could even trace its spread along the rail- and waterways in colonial Africa.)

Smallpox - not origins, but history, evolution are reasonably well known. Not really human, but also Orf virus (sheep, goats, also PoxV). No time now for full answer, will try to come back and write a more complete one later.

Influenza of course

@reuns could you give a reference? I mean genetic evidence.

@RogerVadim https://nextstrain.org/community/neherlab/allflu/h1n1_ha?m=div https://www.ncbi.nlm.nih.gov/genomes/FLU/Database/nph-select.cgi?go=database https://www.pnas.org/content/pnas/suppl/2009/07/13/0904991106.DCSupplemental/0904991106SI.pdf mostly for H1N1 but there are also strains such as A/FPV/Dutch/1927(H7N7) A/chicken/Brescia/1902(H7N7) A/chicken/Rostock/45/1934(H7N1)

We don't know a whole lot about the direct evolution of the genomes of many viruses yet, though as methods for recovery of ancient nucleic-acid sequences improve, I would expect to see increases in this sort of information over the next few years.
We do know the history and evolution of a limited number of viruses. Included in this are some of the common-cold coronaviruses, such as the OC43 type, which as been dated by molecular clock methods to have evolved all the current genotypes since the 1950's(1) (see fig below from paper).

One example that we do know about is Polio (Poliovirus, Picornaviridae), where we know for instance that it affected a number of people in ancient history. The most famous of these is probably the stele below from ancient Egypt, which shows the classic withered leg and crutch used by the subject of the stele. We don't have any genetic information about these cases though.

Compared with:

It is fairly likely that localized (e.g. city wide) epidemics of polio occurred prior to recent history, but there are no credible reports of mass epidemics beyond the early 20th century in Europe and northern America. Which may be an indication that, while it was a problem, until the advent of rapid travel and high density populations, polio was a localized event and possibly only limited numbers were exposed to it.
However, one virus that we do know a fair bit about genetically is Smallpox (Variola virus Poxviridae). This, as you probably know, was the basis for the very first true vaccine (cowpox for cross-protection), and was also the subject of variolation, a primitive form of vaccination using dried smallpox scabs, for some centuries before the advent of Jenner and Jesty(2).
In 2016 smallpox was found in a Lithuanian mummy dating back to the 1600's(3), at the time this was the oldest DNA evidence for smallpox, but a recent report in Science(4) provides complete genomes for smallpox dating back as early as AD 600 in Viking individuals, meaning that we can compare at least these viruses with those present at the time of elimination.
In addition for smallpox, we know that there were two forms, Variola Major and Variola Minor (alastrum). These differed in pathogenicity and survival rate, having ~1% for the minor form vs 30% for the major. It was thought by some that the minor form was the virus adapting to its host, but more recent analysis indicates(5) that it was likely merely phenotypic variation.

Lau et al. J Virol. 2011 Nov; 85(21): 11325–11337.
Hamarstein et al. Trans Am Clin Climatol Assoc. 1979; 90: 44–55.
Duggan et al. Curr Biol. 2016 Dec 19;26(24):3407-3412.
Muhlemann et al. Science. 2020; Vol 369, Issue 6502
Li et al. PNAS October 2, 2007 104 (40) 15787-15792


text:
What breaks hydrogen bonds while producing sticky ends using restriction endonucleases?
I am a high school student and I am little confused about the uses of restriction endonucleases. Why do hydrogen bonds(base pairing)
break when restriction endonucleases produce sticky ends? If they only break phosphodiester bond, should hydrogen bonds not break on their own? I searched it on few sites but didn't find any discussion.

Which hydrogen bonds are you referring to? If you're referring to base-pairing in the sticky end section, how many adjacent paired bases do you need to keep the sticky ends "stuck" at physiological temperature or at room temperature?

If you could just provide a certain example/illustration for the bonds which you're referring to, then it may help others to understand what issue you're facing.

I am referring to H bonds in base pairing? you should understand it because there are no other H bonds

The short answer is heat. Compared to the covalent phosphodiester bonds, hydrogen bonds (H-bonds) are very weak, and they're much easier to break and re-form. A long enough piece of double-stranded DNA will stick together at physiological temperature because it contains thousands of highly coordinated hydrogen bonds. But even long pieces of double-stranded DNA can easily be denature into single-stranded DNA by heating them up to ~95°C. The handful of H-bonds on a sticky end will spontaneously disassociate quite easily at the restriction enzyme's optimal temperature (usually ~37°C).
A good way to think about this is to imagine the sticky ends as PCR primers. When I'm planning a PCR reaction, I first need to calculate the melting temperature (Tm) of my primers (the single-stranded oligonucleotides that form duplex DNA on a reverse-complimentary sequence of a strand of template DNA). Tm is the temperature at which half of the duplexed DNA primers will disassociate and become single strands. As the temperature increases above the Tm, an even greater proportion of those primers will disassociate into single-stranded DNA.
If I take EcoR1 as an example, the sticky end sequence is AATT. Just four bases with 2 H-bonds per base, for a total of 8 H-bonds (A-T pairs have two H-bonds, G-C pairs have three). When I plug this sequence into a primer Tm calculator like OligoCalc, I get a Tm of 8°C. Meaning, if you kept this reaction at 8°C, half of your sticky ends would become unstuck. This is why the ligation of sticky ends is often performed at a much cooler temperature than the restriction digestion (either room-temperature, or 4-8°C).

but the temperature/heat will be distributed to all nucleotides i.e kinetic energy of all nucleotides will increase not just those cutted by restriction endonuclease so how can only those get seperated?


text:
Osmotic pressure and turgor pressure details
I am a high school student and I am very confused in how we define osmotic pressure?
Osmotic pressure is defined as the pressure required to stop the diffusion of solvent into a solution by applying external pressure over solution, which means that when the pure solvent is diffusing into the solution then it will apply pressure on the semi permeable membrane and when we have applied the same external pressure over the solution then there is no net movement of solvent occurs across the membrane and mathematically it is equal to P=cRT, where c is the concentration of solution , T is temperature in kelvin and R is the universal gas constant, if we go by this equation and calculate the osmotic pressure of 1 molar concentrated solution at room temperature it comes out to be almost 2495Pa which means 2495Newton/m^2 which is very high pressure and it doesn't seem to be true as osmosis is very slow process it surely cannot generate that much of pressure( just to imagine how large is that pressure assume an area of 1m^2 and place Mass of 249.5kg on it), so I think that it should not be that higher, so ,I am unable to understand what is actually osmotic pressure ,on which does it actually act and how do we actually calculate it?
we say that when a cell is placed in hypotonic solution then osmotic pressure is greater than Turgor pressure and water will flow inside the cell, but I have a confusion that like turgor pressure acts on the wall of cell ,osmotic pressure should also act on the wall and in opposite direction of turgor pressure, so by this logic wall should compress more as osmotic pressure on it will cause it to compress and turgor pressure will cause it to expand but Osmotic pressure is greater than turgor pressure it should compress, but it doesn't happen? we only talk of turgor pressure to explain the expansion but not osmotic pressure why? I know I am going wrong somewhere if Osmotic pressure would really leads the cell to compress then an animal cell would have never burst in hypotonic solution. and if osmotic pressure is also acting on the wall or membrane then net pressure on the wall is O.P-T.P (in inward direction) , and not only T.P (outward) so the wall pressure here also should not be same as turgor pressure?
If anyone will explain me this by giving as many examples as possible than it would be very helpful for me to understand because its my first time I came to know about these things.

when we talk about external force applied in this case we talk about a setup that is made by us. like a U tube experiment that is filled with water and separates both the hands of the tube via semipermiable membrane and we add solute one side and due to this water starts to rise in that hand of the tube to avoid this rise in water level an external pressure is applied tht prevents this water rise and this pressure is termed as osmotic pressure. THIS IS THE CASE OF A ENVIROMENT WHERE EXTERNALFORCES CAN BE APPLIED BY US AND IT IS VERY MUCH DIFFERENT FROM THAT OF THE CELL WORLD.
in case of cells osmotic pressure is directly proportional to the concentration of the solution i.e more the solute more the OP and visa versa. in cells osmotic pressure are internal force that is caused due to  difference in concentration of the solutions.
and as we know water move from low osmotic pressure to high osmotic pressure
{ from less concentrated to more concentrated solution} thus cell swells up because of entering of water when cell is placed in pure water or hypotonic solution.
CONCLUSION : CELL DO NOT SHRINK BECAUSE NO EXTERNAL FORCE IS APPLIED BY US THE DEFINATION U GAVE IS CORRECT BUT IT IS NOT APPLIED IN CELL WORLD. IN CELL WORLD OP IS A PRESSURE DEVELOPED DUE TO DIFFERENCE IN CONCENTRATION OF SOLUTIONS AND THIS PRESSURE IS CREATED INORDER TO EQUAL THE CONCENTRATION OF SOLVENT (WATER) IN BOTH THE SOLUTIONS.

Welcome to the Biology Stack Exchange. You might want to consider your answer formatting and provide a bit more clarity to assist the question poster's question. Good luck!


text:
What is this arthropod? (found in Lapland - Finland)
I found this arthropod climbing the wall of my room in Lapland, Finland.

(the max width is about 0.5 cm)
What is it? Is it dangerous?

The picture is not great but I'd assume a pseudoscorpion of some sort. Here is one common species found where humans live but I can't be sure about Lapland specifically and doubt ID is possible at the species level from your image: https://en.m.wikipedia.org/wiki/Chelifer_cancroides
If so, I'm not aware of any harm from these directly, but they can be predators of insects that eat rotting wood, which in turn may indicate a problem with moisture in your structure. A single sighting is not something I would worry about unless you have other signs from those sorts of prey themselves.

thank you the image in the wikipedia link you shared matches

According to this observations map, pseudoscorpions do live in Lapland, as does C. cancroides specifically.


text:
Bats observed hunting insect prey - any evidence of "handedness" (laterality) in circular flight pattern?
Whilst watching a bat hunting on the wing at dusk (most likely was a species common to my urban UK location, e.g. pipistrelle) its flight pattern around the garden comprised circles, several metres in diameter, but I'm reasonably sure going in the same direction - in this case it seemed always to be turning to its left. (The circular flight pattern is something I've observed many times.) Is there any evidence in bats of a preferred direction of circular "sweep" whilst hunting? More generally, are bats known to exhibit left/right- handedness (or laterality in technical-speak) and does/could this manifest itself in flight patterns?
What I've already tried to answer my question:
Searched this site for related questions/answers + tried Internet searches with variations on circular flight for bats- results included a 2013 Nature paper, Bandyopadhyay, P., Leinhos, H. & Hellum, A. Handedness helps homing in swimming and flying animals. Sci Rep 3, 1128 (2013). https://doi.org/10.1038/srep01128 - fascinating stuff re. handedness and advantages for homing, but the brown bat featured was trained and observed under controlled lab conditions. Hoped that someone here with knowledge of the subject could provide a research paper/book ref., etc. I posted this question out of simple curiosity - I'm not a biologist and so don't have institutional access to many papers (just the Abstracts).

Welcome to Biology.SE! Please take the [tour] and then go through the help pages starting with [Ask] questions effectively on this site and [edit] your question accordingly. In particular, we expect you to demonstrate that you have done some research on your own and then, informed by what you have learned, ask any questions you still have (ideally with references to reliable sources). ——— In this case, editing your question to include a summary of and reference (plus a functional link) to the paper you mentioned would make this much better. Thanks!

@Jimbo, welcome to Bio.SE! I've edited your post to include the info from your comment -- thanks for the update! Could you please update the link to your Nature paper? It's broken. A citation would also be good to include to avoid broken links in the future. Interesting question -- thanks for asking.

This article shows a lefthanded bias for climbing and eating, with one bat species:
Paolo Zucca, Alessandra Palladini, Luigi Baciadonna, Dino Scaravelli.
"Handedness in the echolocating Schreiber's Long-Fingered Bat (Miniopterus schreibersii)"
Behavioural Processes,
Volume 84, Issue 3,
2010,
Pages 693-695,
ISSN 0376-6357,
https://doi.org/10.1016/j.beproc.2010.04.006.

Paywalled - only the abstract is visible.

Found using Google Scholar with search terms: handedness, microchiroptera

You could mention sci-hub as a way to bypass paywall.

I hadn't seen this site before. I will have to seek advice as to where I stand legally.


text:
Why does touch the face make more likely to be infected?
Why is it risky to touch the face although there is no direct contact with into the nose, eyes and mouth?
Is there any possibility that pathogens infect us from our facial skin?
Suppose we have viruses or bacteria on our hands and we touch the side of our nose with our hands.
Can these bacteria reach our noses by their own?

Welcome to Biology.SE! Please take the [tour] and then go through the [help] pages starting with [Ask] questions effectively on this site and [edit] your question accordingly. In particular, each question should be posted separately — this improves the chances that you will get answers for each question and makes the answers more accessible for future users. ——— We also expect you to do some research on your own and then, informed by what you have learned, ask any questions you still have (ideally with references to reliable sources). Thanks!

Do you have evidence or a citation to prove your claim that it is risky to touch the face without direct contact with nose, eyes, or mouth?


text:
Skin (epithelial) stem cells: unipotent or multipotent?
In a video on the Khan platform on stem cells, epithelial stem cells are described as unipotent stem cells, i.e. only producing one kind of specialised cell: epithelialor skin cells.
However, on a website describing the hierarchy of [stem] cell potency, skin cells are shown to be the product of multipotent stem cells:

Which category is correct for skin stem cells - unipotent or multipotent?
UPDATE

In order for the epidermal stem cells to become multipotent or pluripotent, they have to be induced. This involves some precisely timed chemical exposures that don't occur in nature. I think when some people refer to unipotent stem cells, they are referring to them by their natural function.

In this article on sciencedirect.com epithelial skin cells are described as the products of epithelial multipotent stem cells and/or unipotent "progenitor" cells (which I presume is a way of describing a stem cell that only produces one type of cell - in this case an epithelialor skin cell). The epithelial multipotent stem cells can produce a variety of cells.

Most epithelial tissues self-renew throughout adult life due to the
presence of multipotent stem cells and/or unipotent progenitor cells [emphases mine].
Epithelial Stem Cells and Their Lineages Epithelial stem cells can generate tissues that display a fascinating array of cellular
architectures, each of which are specifically tailored for distinct
functions. In this review, we will focus on four well-characterized
epithelial stem cells whose tissues possess diverse architectural
designs and physiology: intestine, epidermis, mammary gland, and
cornea.

So it seems the answer to my question is: both.

I think the main cause of the discrepancy is a lack of formal definition for "unipotent". We don't really talk about "unipotent" cells, there are "multipotent" ones like in the figure above, and then "progenitor cells". We don't really think of progenitor cells as multipotent even if they technically produce more than one final cell type; they're just the last level of stem cell before you have the final differentiated lines. There's a lot of "lumper vs splitter" issues when you get down to that level. It's not really even that clear how many "cell types" a human has.

@BryanKrause OK, thanks I get it.


text:
To a population with no immunity, why is smallpox or measles more deadly compared to COVID-19?
Specifically, this is not a question asking how easily a virus spread in a
population (airborne, asymptomatic spread, etc), but regarding the
mechanism or the "havoc" it wreaks once inside a person's body.
Edit: The question is based on the assumption that, once a human body can readily recognize the virus signature and starts flooding the blood with anti bodies, the virus won't be a problem.

Today, We have no prior immunity to COVID; 400 years ago, Americans
had no prior immunity to smallpox either.
Both virus can be effectively combatted once the body has immunity
(recognization and anitbody production).
Yet, smallpox is so much more lethal and its damage horrifying.

I would suggest not asking multiple questions at once as it runs a risk of getting flagged as too broad. In each point you have made some assertions - please add references for the same. e.g. for point 2 add references for the paper/book/article which details which cell types are affected by covid/measles etc. For the third point you have your own hunch - have you done any research for the same? If yes, add references; if no, mention that. Finally please complete the tour from your profile to understand what qualifies as a good question/answer.

I think this is an interesting question, and I've been struggling to put together a good answer to it for a while. Unfortunately, everything I came up with just boils down to "some viruses are more deadly because they kill more cells faster" which is pretty circular. A detailed understanding at the subcellular level of what happens during virus infections doesn't exist.
It's instructive here to look at the evolutionary pressures on human-infecting viruses. SARS-1 was about 10x as deadly as SARS-2, but it was contained quickly, while SARS-2 is on track to infect a substantial portion of all humans on the planet. SARS-2 is more successful because of, not in spite of, its lower lethality. Because many infections are asymptomatic, the infection can't easily be contained. If we look at things this way, the asymptomatic infections are the ones that go "according to plan" for the virus, and the severe cases and deaths are unwanted collateral damage. Severe cases of COVID are associated with the presence of viral RNA in the blood. But SARS-2 primarily enters new hosts through droplets expelled from infected hosts' respiratory systems. Any viral activity outside of the respiratory system cannot produce these droplets, and is therefore "wasted effort". This is similar to polio, which causes no symptoms when it transmits successfully by replicating in intestinal cells, but causes paralysis when it gets stuck in the dead end of its host's nervous system.
So, if viruses are most successful when they cause no disease, shouldn't they all quickly evolve into harmless versions? They don't because the "sweet spot" of asymptomatic transmission is the unstable result of an immune response that is just effective enough. If a virus fights the immune system either too hard or not hard enough, the infection will end too fast to spread effectively; due to the death of the host or the virus, respectively. Similarly, if a virus is too "aggressive" in entering cells it will damage tissues that don't lead to transmission or new infections, but if it is not aggressive enough it will not be enter those tissues it needs to make, e.g. respiratory droplets carrying new virus in the case of SARS. Since a lethal disease still results in some transmission but complete suppression by the immune system doesn't, it's more evolutionarily favorable for viruses to err on the size of causing worse disease.
Edits for changes in the question:
I think the differences between covid-19 and smallpox are a bit misleading. The big differences between the two diseases is that, while the human populations most affected by smallpox were naive to the virus, the virus was not naive to them. Smallpox is believed to have been infecting humans since at least the 3rd century BCE. It had a lot of time to evolve tricks to defeat the human immune system. And as a dsDNA virus with a 186,000 base pair genome, it had plenty of space to store those tricks. For comparison, SARS-CoV-2 has a 30 kb RNA genome, and poliovirus has a 7.5 kb RNA genome. However, Herpesviruses have been infecting humans since before humans were human, have large genomes, but only cause disease when the immune system is suppressed or nonfunctional. In those cases, however, it is quite effective at spreading throughout the host and can kill. So we see that herpesviruses also rely on the immune system of the host to stay in their sweet spot, they just have a different strategy than smallpox. Since smallpox has been exterminated and herpesvirus is in nearly every human on earth, we can say the mild strategy is the unequivocal winner, at least for now.

This is true. There were mysterious burst of horrible disease burst into the scene and just killed droves in short period of time from the well-known Athens, Justinian to less well known ones as late as the (1800s?). These disease don't last long. And modern version of the disease we are dealing with today like Cholera are less lethal as well. There is this tendency to converge to the sweet spot. This "meta" is indeed very interesting, thank you for mentioning it. However, my question is really about the details  inside one individual, not the meta.

Yes, unfortunately, how viruses interact with cells and the immune system is mostly only known in broad strokes.

actually, if you want to focus on a comparison between covid and smallpox, there are a few things we can say. I'll edit them in.

I have some somewhat-informed hunches. My original question before edit focused on "attacking of the adaptive immune system", and the depth the attack goes, do you just infect front line workers and presenters? or do you attack the antibody factory?. Also I don't believe in "killing body cells" is the cause of death, more like "causing disorder". Septic shock for example, it pretty much immune system overreacting and killing the host instead of what baterias are doing.

It might have something to do with what particular cells a virus can attack.  E.g. just abou everyone gets cold viruses, yet they seem to only attack the respiratory system, and mostly cause a week or so of mild discomfort.  HIV attacks the immune system, and can proliferate there for years before causing accute symptoms.  A lot seems to depend on the specific method a virus uses to enter cells.  The COVID virus latches on to specific receptor molecules, the much more deadly Ebola virus uses a "non-specific engulfing process called macropinocytosis".  (Not enough room for links :-()

@jamesqf that kind of information is exactly what I was hoping for. thank you for listing terms I can google with. That engulfing thing is absolutely fascinating by the way! by some odd chance you think we can have our own pet virus in our cells like mitochondria! with enough time we may be able to have our pet virus clone anti bacteria phages and release them into blood stream


text:
why secondary/booster vaccine shots may sometimes induce worse side effects?
Background:
I had my first shot about 1.5 months ago, I just had my shot yesterday. Unmistakable fever like symptom started within 4 hours. within 10 hours the whole body started aching, accompanied with chill, tiredness and headache.
2 details in this chain of event I can't explain:
(I believe those 2 details are related, I just couldn't connect the dots)
1
base on the 2 conclusions I just accepted without truely understanding them:

fever is an innate immune system function
adaptive immune system kicks in after days

it is hard to explain the lack of fever after the first dose (maybe the body is just treating the first dose as a local wound with some oppotunist bacteria getting in). It is even harder to explain the extreme quick onset of fever after the second dose (maybe I have left over CD4 CD8 plasmaB cells in response to first shot still wondering around or not commit suicide yet in the body?)
2
muscle and joint pain should be due to white blood cells releasing cytokine. The vaccine's lipid particles are injected into my arm muscle, not major blood vessels. Those lipid particles couldn't travel far into my whole body, lipids alone do not last long. They should only merge with or "infect" mostly local cells around the injection site. local pain, local ache makes sense, why did the whole body's immune cells ALL decide to release cytokine? shouldn't the aching be a gradient though out the body? starting from the site of injection, it should hurt the most, gradually the pain will be lowered the further away from the site of injection.
P.S.
the whole question is based on the knowledge that adaptive immune response takes days to take effect.  I just assumed the 2nd shot's side effect which takes merely hours to manifest couldn't be the adaptive immune response. However, I am thinking now, maybe the side effect is the continuation of the adaptive immune response from the first shot?
PS2 NEW INFORMATION
apparently nanolipids don't just stay locally
body ache is immensely way more complicated

This is a personal medical question, so it may get rightfully closed. But I think the underlying biological question, and how you pose it, is excellent. I also think that the understanding we have on the mechanisms of immunogenicity of the COVID-19 vaccines is still limited; it takes time to study them to a level that would satisfy your answer. I hope you understand that. I think a question about why secondary/booster vaccine shots may sometimes induce worse side effects is a good one, perhaps you can try to do your own research on that, and ask here once more, if this question gets closed.

To summarize the question, you are asking why:

the first and second vaccine shot did not induce the same immune response in terms of fever.
you experienced muscle and joint pain throughout your body even though the vaccine was injected into your arm muscle.

Regarding #1, your guess is mostly correct. According to Dr Melanie Swift, co-chair of the COVID-19 Vaccine Allocation and Distribution Work Group at Mayo Clinic:

When you take two doses of a COVID-19 vaccine, the first dose is the first time for your body to see the spike protein that the COVID-19 vaccines produce, and your body begins to develop an immune response. But that happens slowly. Then when you come back with a second dose, your body is ready to attack it. Your body is primed by that first dose of vaccine. The second vaccine dose goes into your body, starts to make that spike protein, and your antibodies jump on it and rev up your immune system response. [Source]

As for #2, I believe that the pain and soreness near the injection site is a separate symptom from the muscle and joint aches throughout the body.
The paper COVID-19 vaccine-related myositis has an explanation for the localized symptoms near the injection site:

With intramuscular vaccination, muscle is exposed to modified contaminant agents through direct inoculation, which elicit an immune response to the injected antigen. [...] Although similar to the COVID-19 infection itself the exact mechanism causing damage to the injected muscle is not known, it is worth reporting our observation of myositis suggesting a causal relationship to the novel, modified mRNA COVID-19 vaccine-adverse event, myalgia.

So, the arm symptoms seem to mostly be a localized reaction to modified contaminant agents which are injected.
As for the muscle and joint aches throughout the body, these are part of the general immune response which the body exhibits against the spike proteins which are eventually created. You are right that these symptoms are caused by cytokines, which are created by macrophages and neutrophils.
While it is true that the spike proteins will be manufactured by cells near the injection site, and will be detected there, this will trigger an immune response all throughout the body. This is similar to how a respiratory infection triggers an immune system throughout the body, and not just near the respiratory system.
As explained in an article on Breakthroughs.com, cytokines are chemical messengers which tell the body to produce more immune cells to help combat the infection. Their effects are intended to impact the whole body, which is why they cause symptoms on the level of the whole body, such as fever and muscle aches.

To summarize, "adaptive immune response takes days to take effect" is true, but that's the time it takes to adapt, not to react. Once the adaptation is done, it can respond immediately to the known antigen. The purpose of the vaccines is to eliminate that delay.


text:
After removing the stigma from a flower, specifically the Crocus flower, does it eventually grow back or does the plant die?
After removing the stigma from a flower, specifically the Crocus flower, does it eventually grow back or does the plant die?  The reason why I am asking is I am trying to find a way to mass produce saffron.

Welcome to Bio.SE! Please limit each post to a single question and demonstrate your own research effort. See our [tour] and [help] for more information. We will be glad to help if 1) you [edit] your post to focus on a single question and 2) you explicitly demonstrate some of your own research effort (i.e., where have you already looked for an answer and what did you find out?). Failure to follow-through with these edits will likely result in downvotes and post closure, so please take a moment to [edit] accordingly. thanks!

Someone will find the pigment genes and engineer them into another crop like wheat or corn. that's the only solution. So if you want to do some crispr then that project would be worth a lot, but it's legally a minefield. saffron is a multi million dollar business and people with degrees in botany are trying to do the same as you, they have lots of tricks like molecular engineering and polyploidy by colchicine. https://www.actahort.org/books/650/650_31.htm

Neither.  (From personal experience growing a few saffron crocus in my garden.)  Like other bulbs (or in the case of crocus, corms, to be technical), the saffron crocus has an annual life cycle.  The bulb sends up leaves & flowers according to seasonal* cues.
The flowers will die whether or not you pick the stigmas. Indeed, the stigmas are usually the largest and most pickable just at the point when the flower is about to start senescing anyway.  At that point the plant is about to go into winter dormancy anyway.
The plant does not die, but stays in a dormant state over the winter, then grows leaves in the spring.  My experience, though, is that they will eventually deterioriate over several years unless the corms are lifted and divided.
The problem with mass production is not growing the plant: it grows readily in many areas of the world - but the labor-intensive harvesting needed.  The stigmas need to be plucked individually, and carefully.  Growing in a greenhouse would be possible, but uneconomical, since you need lots of plants to provide commercial quantities.  (70,000 plants for 1 lb/0.45 kg of saffron, per Wikipedia: https://en.wikipedia.org/wiki/Saffron ) It's unlikely that you could speed up the annual growth/flowering cycle enough to matter.
*Saffron crocus (C. sativus) is a bit unusual in that it flowers in the fall rather than spring.

Thank you very much for your answer!  But couldn't we build a massive greenhouse in Arizona, provided we have enough water, plant the corms in trays of soil, and divide them along a very slow and long assembly line.  The workers can sit comfortably along the narrow bottom-neck, listen to music and talk to each other while separating the saffron by hand?

@Michael Lee: I don't think (though I'm by no means an expert) that would be technically possible, since you couldn't speed up the annual cycle that much.  It sure wouldn't be cost effective.  I think you might have better luck genetically engineering bacteria to produce the specific chemicals you want.

You are correct in saying we cannot speed up the annual cycle, that I understand is six weeks, but I am frequently off on my numbers.  Build a massive saffron green house in Arizona, if we can provide it with enough water, grown in trays, and passed along parallel conveyor belts, pick the stigma by hand with workers seated comfortably, and treated humanely, I know this will work, and those chemists and biologists, who think bacteria is going to get us out of this mess we are currently in, are currently out of their minds!

@Michael Lee: No, the annual cycle I meant is not six weeks (that's approximately the fall blooming time), it's a year.  The corm is dormant in winter, sends up leaves in spring which nourish the plant.  The leaves may die back in summer, as it enters a second dormant period.  Environmental cues of some sort then trigger the fall bloom.  You might be able to speed that up a bit, but IMHO not much.  WRT workers, what's "humane" about sitting in a greenhouse all day, picking those thread-like saffron stigmas?  I'd rather be out in the fresh air, at least.


text:
Do the nasal cilia move the particles trapped in the mucus out of the nose or down the throat?
Do the nasal cilia move the particles trapped in the mucus out of the nose or down the throat?
What is the direction of the pulse of the cilia? Is this changeable?

Mucus generally moves down the esophagus. If you notice , you will make many small swallow actions during the day, generally not enough volume to notice . It includes some saliva. As a result of cancer ,I cannot swallow so I am very aware of the mucus flow.

The movement is towards the pharynx (to be then swallowed).
It's useful, in the case of respiratory cilia, not to think of it as 'up' and 'down,' but rather as 'toward the pharynx.' In the lungs and the trachea, the movement is 'up' to move the mucus to the throat, and in the nasal cavity, the movement is 'down,' but again toward the throat. The aim is the movement of the mucus out of the respiratory tract and into the esophagus.

I knew that the returning is needed. The direction of the movement of particles that is caused by cilia is my question.

Where did this information come from? I'm wondering because there's not much detail and I don't know enough about the subject to know if it's reliable or not.

The nasal cilia beat toward the rear of the nasal cavity, driving nasal mucus into the throat (pharynx). From there, it is mostly swallowed. Cilia never drive mucus out the front of the nose (nostrils).

Where did this information come from? I'm wondering because there's not much detail and I don't know enough about the subject to know if it's reliable or not.


text:
Did Mendel believe that proteins are hereditary substances?
Scientists first thought that proteins, which are found in chromosomes along with DNA, would turn out to be the sought-after genetic material. Proteins were known to have diverse amino acid sequences, while DNA was thought to be a boring, repetitive polymer, due in part to an incorrect (but popular) model of its structure and composition.
Was Mendel (1822-1884) one of these scientists?
Note that in 1869, Friedrich Miescher isolated "nuclein," DNA with associated proteins, from cell nuclei. He was the first to identify DNA as a distinct molecule.

Did Mendel really even know about proteins?  AFAIK they aren't really relevant to his work - just as Darwin didn't need to know the mechanisms of heredity to know, from observation, that it exists.

@jamesqf You ignored the note that I added to this question. He probably heard about proteins and nucleic acids in his lifetime.

I would say "possibly" rather than probably, given his apparently rather isolated life.  But the point I was trying to make is that (at least AFAIK) nothing in Mendel's work requires theorizing about the actual underlying mechanism of heredity.

Your sense of scientific history is a little disoriented. Few geneticists were interested in the chemistry of the phenomenon until after the second world war. Genetics textbooks in the 1960s hardly concerned themselves with this. Mendel was a monastic plant breeder with acute observation and deductive powers. He was not a scientist in the  modern sense, and cerainly had no interest in (and probably knowledge of) chemistry.

The first person to directly indicate a linkage between the hereditary material and enzymes was Garrod in 1902, based on the observation of the hereditary enzymatic disorder alkaptonuria. This is after Mendel's death.
The tetranucleotide hypothesis (i.e. DNA is not informative) that you refer to seems to have been formulated around 1910. So there is no way that Mendel knew about that.
It was not even proposed that proteins were composed of amino acids before 1902. No protein was sequenced before 1950. So Mendel had no idea what sequences or amino acids were, for any biopolymer.
It is formally possible that Mendel knew at some point about Miescher's result, but I'm not aware that he was at all interested in the chemical nature of Mendelian factors (what we now call "genes"). I agree with commenter @jamesqf on this point.
It was not immediately obvious to anyone that there was a discrete mapping between phenotypes and specific agents such as proteins. The major theories of inheritance at the time came from biometricians such as Galton, who believed that inheritance was driven by a blending-like mechanism (more or less), based on phenomena such as regression to the mean. They weren't really interested in the chemical basis either.
Mendelian genetics was only rediscovered around 1900 (Mendel was basically ignored during his lifetime), after which the debate started to consider the question of discrete genetic factors.
In short, I don't think that there is any evidence that Mendel had any interest in chemistry or the chemical relations of the organismal characters he studied.


text:
Can LSD in very small amounts increase mental ability?
Can one take a small amount of LSD(acid) and not trip but still benefit from the intended purpose? It was said the military experimented with it for extra sensory abilities. It has also been said it could increase your hearing or sight?

What is the source of this rumor? Could you link it here?

LSD was developed by the chemist Albert Hoffmann. Please have at least a look into the relevant Wikipedia article, which gives you a lot of information. It can be found here.

@Chris I have seen a documentary on it. I stated the reason it was developed. All I can site that it was om the discovery channel.

The acute mind enhancing effect of an LSD microdose is less than running off 1000 calories. LSDm acute effects are mood related, it's not racily imaginative. Einstein didn't microdose, nor do most great scientists. Physical fitness, mental effort and cerebral tasks and challenges increase IQ. Microdosing enhances serotonin and is appreciated for depression and positivity. it doesn't score for creativity. If the person has a very healthy, interesting and energetic research lifestyle, they could try a drug for imaginative dizzyness. High drug lifestyles often correlate with demotivated people.

There is no evidence so far that this is the case. Try reading https://en.wikipedia.org/wiki/Psychedelics_in_problem-solving_experiment and especially  https://peerj.com/preprints/1202.pdf

I saw a video of a woman who microdoses on TV, it was a documentary interview. Her mental clarity was... "Fucked"... she looked like she had low energy and was fairly confused and full of ticks. I understand that people dying of cancer should definitely take mushrooms one time to give them a new perspective of their mental map, but microdosing LSD is probably only good for artistic persuits and people who can't achieve high physical health through natural means.. artistic, and not logical or rigorously intelligent ones. if you want to do some surrealism and forget your keys in your car, ++lsd

Did you read about project MK-Ultra, that is when the US military experimented with reprogramming brains. https://en.wikipedia.org/wiki/Project_MKUltra

The phenomenon you are referring to is called microdosing. It was brand-new in terms of research focus when you initially asked your question, and it is still a fairly novel and little-studied subject.
A lot of interest in this phenomenon stems from anecdotal evidence by microdosing users, who report beneficial effects. This is however not necessarily an indicator of neuroenhancing potential, since psychopharmacological agents are known to modulate subjective evaluations of performance more than constrained objective measures of it (compare tables 3 and 4 in this article, which does not discuss LSD or microdosing per se, but illustrates this psychopharmacological phenomenon).
Anecdotal findings regarding LSD are substantiated in an article, which found that microdosing LSD improved the self-reported mood of participants.
Additionally, this article showed an improvement of vigour (as per the POMS scale) at the highest dose.
This is, however, still not a behavioural effect, as POMS is a psychometric assessment method.
Still, this further substantiates the mood-enhancing effects beyond direct self-reporting.
Another article reported increases in creativity, focus, and well being, though again, it should be noted that all of these metrics were assessed using participant reporting, and not evaluated based on behaviour.
So to conclude, there is to date no convincing evidence that microdosing LSD improves “mental ability”.
There is however a growing amount of evidence that LSD microdosing improves mood, which in and of itself can be a desirable effect, and can further be argued to contribute to improved “ability” in more complex behaviours such as sustained deductive reasoning or social interaction.
Conjecturing from the first cited article on methylphenidate, modafinil, and caffeine, it is conceivable that a similar effect:

In conjunction with results from neuropsychological testing we conclude that modifying effects of stimulants on complex cognitive tasks may in particular result from more reflective decision making processes. When not under time pressure, such effects may result in enhanced performance.

as the one seen with modafinil and methylphenidate could be present with LSD microdosing.
Appropriate data to address such complex evaluations is however very scant in general, and to date wholly absent for the emerging phenomenon of LSD microdosing.
TL;DR
No but maybe.

There was UK program interviewing people that microdose and they appeared slightly/considerably insane.

@LifeInTheTrees that hardly sounds rigorous. People willing to experiment with currently-unproven cognitive supplements might not appear to be the most "sane" people, even if their actual experimentation is minimal. The perceived sanity of self-experimenters in a documentary is no indicator of the pros or cons a substance might offer when used in a scientifically-optimized manner.

Aldous huxley was a famous psychadelics experimentor who took quantities that most scientists would consider high, since he was 40-50. He did appear cognitively able and a bit disctracted, fairly on the ball in his later days. Carlos Castenada, on the other hand, told one of his disciples, a little blonde girl, to drive to the middle of Mojave desert and walk to meet him in the center, after he had died, and they would both dissapear together in a ball of light everywhere in the univese, and so she go there. That is not sane advice.

all_licenses: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/3.0/


text:
Get "DNA" of one parent based on its children
I'm no biology expert so bear with me, but I've been given a task to estimate alleles of an animal based on the animal's children and the children's other parents.
I've looked up Mendel's laws but I'm not sure I can possibly determine the correct result - better show an example:
Let us say that we want to get alleles of a mother. We know the following:
Child 1 has alleles: AG, the father of the child has alleles: AG
Child 2 has alleles: GG, the father of the child has alleles: GG
Child 3 has alleles: GG, the father of the child has alleles: GG
According to Mendel's laws and the example above it would seem that
For the first child x father pair - the mother would have alleles AG
For the second and third child x father pair - the mother would have alleles either GG or it could have AG, CG or TG.
How can I determine which anwser is correct?
If needed I can provide more examples.

EDIT: More examples / clarification

I have a database of animals (cows/bulls). Each animal has ~50 000 SNPs, each SNP has 2 alleles.
However, some of the animals data are missing, for instance I do not have SNPs for a certain bull, but I do have SNPs of the bull's children and their mothers.
Now I need to figure out an algorithm that based on the SNPs of said children and the SNPs of their mothers tries to compute the SNPs of the bull.
Additional example: (each row represents a child - mother pair)
SNP                                 | Child alleles | Mother alleles | Possible father alleles
ARS-BFGL-BAC-19454  | GG               | AG                   | AG, GG
ARS-BFGL-BAC-19454  | AA                | AG                   | AG, AA

From the table above it is clear that the bull can only have AG.
However I do not know how to handle the following case:
SNP                                 | Child alleles | Mother alleles | Possible father alleles
ARS-BFGL-BAC-14220  | AG               | AG                   | AA, AG, GG
ARS-BFGL-BAC-14220  | GG               | GG                   | GG, AG, CG, TG
From the table above it looks like the bull can have either AG or GG - how do I figure out whether the father's value is AG or GG?

It requires a bit of Bayesian statistics.

It would help to get the actual prompt or question that you are trying to answer. For example, do the alleles {A, G} cover the total set of alleles at this locus? In that case some genotypes could be ruled out. There are cases in which you can infer parental genotypes, but some more information is required. Is this a homework question?

You can't be certain. Is this a real data set you are working on or something used in, say, a course?

@MaximilianPress I've tried to make it more clear in the original post. I do not know if the alleles cover the total set of alleles at this locus, frankly I do not even know what a locus is (as I said, I'm not a biologist).

@BryanKrause It is a real data set.

If you have real data you can't always find the answer definitively. Vadim gives good advice for getting the probabilities.

@BryanKrause I'm not sure how would I get the probability for the SNP ARS-BFGL-BAC-14220. Three of the children - mother pairs are AG - AG, while nine of them are GG - GG. Howver all twelve pairs can get me either a AG or GG combination. To me it seems like it is always 50:50 between AG and GG, am I missing something?
(In this case the GG is the correct answer as I am testing the algorithm on data that I do have).

@toucheqt You aren't being very clear about the data you have, and I don't really want to go around and around trying to find out and solve your problem for you. I think you've got some information to run with - see where that gets you after you give it a couple days thought. You'll learn a lot more trying to figure it out.

Ok, I think I can answer somewhat usefully with the update.
It's helpful in this case to think in terms of sampling alleles. It's true that the cases that you give are not 100% unambiguous, but it is not true that we can't give a somewhat confident estimate. We can still make a statistical prediction of the most probable genotype.
The first thing to note is that you are working with pedigrees. That is, you have parents and multiple children. Pedigrees are great because they allow you to pool information across individuals.
The next thing to note is that your alleles are almost certainly not of uniform frequency in the population. In other words, If I sample a random cow from the herd, the probability of allele $A$ at some arbitrary locus is almost certainly not the same as the probability of allele $G$. You can use that information in inferring the most likely allele. Moreover, you can put a precise number on the probability of any given allele.
The simplest method I can think of is to use genotype likelihoods, to account for pedigree information, and to incorporate the background allele frequency as a prior probability (h/t Vadim for the Bayesian approach).
A likelihood is the probability of some dataset given a specific hypothesis. In other words, your data is your offspring genotypes and your hypothesis would be the different offspring.
In your first example, we can logically rule out alleles other than $AG$. (That's actually not true; you should account for the possibility of genotyping error, but we'll ignore that for now.)
In your second example, we can't logically rule out all but one genotype. But we can say some things nonetheless:

you have said that there are only 2 alleles per locus (i.e. $A$ and $G$ are the only possible alleles).
$A$ and $G$ have distinct frequencies in the population.
we now at least one paternal allele ($G$).

Let's take a simple case of the likelihood application, in which we say that frequency of $Pr(A) = 0.9$ and $Pr(G) = 0.1$ (I am making these numbers up). Now in the ambiguous case:
$$Pr(paternal=GG | offspring\_genotypes=GG,AG) = \\
Pr(offspring\_genotypes = GG,AG|paternal=GG) * Pr(paternal=GG) = \\
Pr(offspring=GG | paternal=GG, maternal=GG) \\
* Pr(offspring=AG | paternal=GG, maternal=AG) * Pr(paternal=GG)
$$
and
$$Pr(paternal=AG | offspring\_genotypes=GG,AG) = \\
Pr(offspring\_genotypes = GG,AG | paternal=AG) * Pr(paternal=AG) = \\
Pr(offspring=GG | paternal=AG, maternal=GG) \\
* Pr(offspring AG | paternal=AG, maternal=AG) * Pr(paternal=AG)
$$
Which evaluate to:
$Pr(paternal GG) = 1.0 * (0.5 * 0.5 + 0.5 * 0.5) * (0.1 * 0.1) = 0.005$
and
$Pr(paternal AG) = (1.0 * 0.5) * (0.5 * 0.5 + 0.5 * 0.5) * (0.5 * 0.1) = 0.0125$
You can see that there are multiple ways for the offspring to get genotypes in the paternal $AG$ case, which is what leads to the terms with addition. Note also that the $Pr(paternal genotype)$ at the end handles the genotype frequency, as all the other terms are conditioned on the parental genotypes.
So in this highly artificial example, we prefer genotype of $AG$ somewhat (note that these likelihoods do not sum to 1.0). But, if we were to alter the genotype frequencies to 50% frequency of each allele, we would get:
$Pr(paternal GG) = 1.0 * (0.5 * 0.5 + 0.5 * 0.5) * (0.5 * 0.5) = 0.125$
and
$Pr(paternal AG) = (1.0 * 0.5) * (0.5 * 0.5 + 0.5 * 0.5) * (0.5 * 0.5) = 0.0625$
and you would come to the opposite conclusion!
So it really matters what the background frequency is, if you choose to consider that term.
You could also think about genotyping error or whatever as part of these calculations.
I am certain that there is a more sophisticated way to do this, that takes better advantage of available information, etc. Maybe I've even made a mistake in the computation- but this is an approach that you might use to infer paternal genotypes.

Thank you for your detailed answer, it helped a lot.

Good! I should note that this is a Bayesian approach to compute a posterior probability, and therefore probably similar to what Vadim first suggested.

I think you deserve the full credit for working through the calculations.