text:
Can neurotransmitter depletion cause short-term drug tolerance?
I'm curious to know if neurotransmitter degradation factors into drug tolerance and neural conduction and, if so, to what extent. As I recall from textbooks I read as an undergrad, one mechanism of drug tolerance works by increasing the necessary thresh-hold at a synapse such that larger concentrations of neurotransmitter or more-frequent action potentials in the upstream axon were required to depolarize the membrane and propagate the signal. It seems to me that the same effect would occur if there were fewer available molecules of the required neurotransmitter.
Does over-stimulation of a neural pathway (through the use of a drug or other stimuli that up-regulates a given neurotransmitter) shorten the half-life of the neurotransmitters it uses?
Could a diet which is poor for necessary precursor nutrients cause a person's neurotransmitter concentrations to drop and reduce signaling along specific neural pathways?
Could lack of sleep or other physiological stressors have a similar effect?
I am not a neuroscientist so please correct (and forgive) me if I'm using any terms incorrectly or am flat-out wrong in any of my assumptions.

Have a look at this article. Is this the direction you are thinking about?

As I recall from textbooks I read as an undergrad, one mechanism of drug tolerance works by increasing the necessary thresh-hold at a synapse such that larger concentrations of neurotransmitter or more-frequent action potentials in the upstream axon were required to depolarize the membrane and propagate the signal.

Yes - usually through a reduced expression of receptors on the surface. Increasing the drug dosage merely increases the probability that one of the molecules will find the receptor. Chronic exposure of increasingly large doses can dramatically reduce receptor expression. In time, sans the drug, the receptor count will climb back up.
This is why a common cause of chronic drug addicts who have just gone through rehab is overdose: After months in rehab or free of a drug they immediately go back to the amount of drug they were doing before rehab - with some severe side-effects at the least, death at the worst.

Does over-stimulation of a neural pathway (through the use of a drug or other stimuli that up-regulates a given neurotransmitter) shorten the half-life of the neurotransmitters it uses?

No. The half-life of a neurotransmitters is not a major concern; your body will construct them from stored resources if necessary, and healthy neurons have a significant store of neurotransmitters to prepare for multiple firings.
A neuron can be artificially over-stimulated and the neurotransmitter pool emptied (or all the neurotransmitters is bound to its receptor) - but only for Ach receptors at neuromuscular junctions. It hasn't been observed naturally, and not for other types of synapses. This does not result in the muscle no longer responding, but rather it will not respond any faster.

Could a diet which is poor for necessary precursor nutrients cause a person's neurotransmitter concentrations to drop and reduce signaling along specific neural pathways?

Yes, but by that point you're not in good shape, if not courting death. Neurotransmitters are either very common in the diet - like ions - or are constructed from very common constituents of a normal diet - like Glutamine, the most common amino acid.
It would not be a therapy to treat drug addiction.

Could lack of sleep or other physiological stressors have a similar effect?

You would die of lack of sleep first. Your body generally keeps a 30 day supply of absolutely essential vitamins and minerals, and will easily digest itself for precursors to construct neurotransmitters out of.
I have not heard of any physical or mental stress which has reduced the amount of neurotransmitters released into the synapse beyond absolute starvation or artificial scarcity.


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Enzyme kinetics at the chemical level
I am stumped by two questions:

Why do we take only the initial 10%(or may be 9.99999....%) of S conversion as the rate of the enzyme reaction. why not more than 10%?
Why doesn't the velocity keep on increasing linearly until the Vmax is reached (as all the S molecules get exhausted)

Research and attempts to answer:
Why v0's 10%?

Steady state kinetics apply when the [ES] is constant, on reaching 10% [P] (for a uni uni reaction) the backward reaction increases, increasing the [ES]?
On reaching 10% conversion the rate decreases, so we measure the optimum activity.

Why doesn't the velocity increase linearly until all the molecules are used up?

Since an enzyme's turnover number (kcat) is constant...
...

Disclaimer :

'have put two Qs together.
Didn't find relevant Q on BioSE.

related

For 2. Think A. about the importance of MM kinetics, which was to establish that the reactions were catalysed by a single active site on the enzyme. Simple non-enzymic chemical kinetics would, instead, predict an order proportional to the product of concs of reactants. B. About the interaction of a substrate with the active site and why this has a hyperbolic shape. Consider collisions between entities (here a substrate and active site) to understand the hyperbola. 1 is trivial.

@AlanBoyd — But the answer to that question, despite its votes and acceptance is unsatisfactory. It does not explain why the curve is hyperbolic rather than linear.

@David 'trivial' is a subjective comment, which in itself is trivial according to what BioSE is for ('... biology researchers, academics, and students.') When there's a Q there's either an answer or it is yet to be unraveled.

Hint. Read sections 8.3.1 and 8.4 (start) of Berg online. Oh, and I don't know why you have a fixation on 10%. It's the initial velocity while the rate is linear, whatever that % may be.

You write: "Why doesn't the velocity increase linearly until all the molecules are used up?" Which molecules? The substrate? What other molecules are involved? Think!

I have decided to answer your second question. I now notice that your question says "…Vmax is reached (as all the S molecules get exhausted)". That's your basic misconception. It is the enzyme molecules that are saturated. You are adding more and more substrate to less and less effect — it can't be the substrate that is exhausted!

For a user with a high reputation you have been rather quiet in reponse to the answers suggested. Did either of them solve your problem? If so you might consider accepting the one that did. If not, perhaps you could explain the problem you still have.

The answer to the first part of your question is that we don't take the initial 10% of a progress curve (velocity vs time) as a measure of activity, but we measure the initial rate of the reaction.  We do this by drawing a tangent at the origin.
It is merely a 'rule of thumb' that progress curves are practically linear provided that not more than 10% of substrate has been used up. If there is significant slowing down before 10% is used up - and there may be if the substrate concentration is way below Km value - then it is not valid to draw a line through the points: we must draw the tangent to the curve. That is, it is not acceptable to take the average rate during the first 10% of the progress curve.
For this reason it is desirable to use as sensitive a method as possible when measuring an enzymic rate, and to never rely on a single time point (as is common in radioactive assays) without sufficient controls to justify that it it is the initial rate, and not the average rate (or something else) that is being measured.
The reason we measure initial rates is that is makes things simpler on two fronts.
Firstly, the decrease in rate with time of an enzymic reaction may be due to many factors: product may be inhibiting the reaction, the reverse reaction may become important, the enzyme or substrate may be unstable. It is only during the initial rate period that conditions are accurately known, and such  effects may be legitimately ignored. It was Michaelis and Menten that first realized the importance of measuring initial rates, and of the great simplification principle that ensued, in enzyme kinetics. (see Cornish-Bowden, 2004).
Secondly, the rate laws themselves are greatly simplified.  Provided that eo << So (the initial enzyme concentration is very much less than the initial substrate concentration), we may take the So as being equal to the substrate concentration and 'plug in' this value into our rate law (such as the Michaelis-Menten equation), and ignore the effect of time on the value of S.  In addtion, we can set all product terms to zero. This can greatly simplify things.
For example, the rate law for a reversible single-substrate mechanism (see here) is the following:
$$ v = {
  {{{V_{max}^f}\over{K_{m}^s}}\ S\ -{{V_{max}^r}\over{K_{m}^p}}\ P }\over{1 + {{S}\over{K_{m}^s}} + {{P}\over{K_{m}^p}}}}\ \ \ \ \ (1)$$
This one is not as complicated as it looks: we have just defined a Km and Vmax for the forward and reverse directions. But let's set product to zero:
$$ v_i = {
  {{{V_{max}^f}\over{K_{m}^s}}\ S_o\ }\over{1 + {{S_o}\over{K_{m}^s}} }}\ \ \ \ \ = \ \ \ {{V_{max}^f S_o}\over{{K_{m}^s} + S_o}}\ \ \ \ (2)$$
We now obtain the familiar Michaelis-Menten Equation (where $v_i$ is the intial velocity).
The second part of your question is more fundamental. I'll take it to mean the following. Why does an enzymic reaction not follow second-order kinetics at all substrate concentrations? Why doesn't doubling the substrate always double the rate (as is often the case in chemical kinetics)? A. J. Brown in 1902 suggested that the reason for this is the formation of an enzyme-substrate complex and that at sufficiently high substrate concentration all the enzyme would exist in such a complex and the enzyme would become 'saturated'. The formation of an enzyme-substrate complex, of course,  is now a cornerstone of enzyme kinetics. One of the early key pieces of evidence for an enzyme-substrate complex was the observation that an enzyme is much more heat-stable in the presence of its substrate than in its absence.
Finally, it is probably worth pointing out that some enzyme operate way below Km values and at 'physiological' concentration of substrates do obey (to all intents and purposes) second-order kinetics. Catalase, for example, has a Km for hydrogen peroxide of about 1 Molar (Ogura, 1955), and one assumes that physiological concentrations never reach this level. (We need to be a little careful about catalase, however: it is one of the few enzymes that is inactivated by substrate).
A great reference on enzyme assays is the following:

Principles of enzyme assay and kinetic studies by K. F. Tipton in Enzyme Assays: A practical approach

For an introduction to enzyme kinetics, and the early history of enzymology, I like the following:

Cornish-Bowden (2004) Fundamentals of Enzyme Kinetics 3rd Edn. Portland Press, London.

This answer is far too complex to be comprehensible to a student who is obviously doing a first course in enzymology as part of a general biological sciences degree and is asking very basic of questions. Recommending she read Tipton or Cornish-Bowden is absurd. The answer to her question at the level she requires can be found in the Berg reference in my comment. It would have been better to let her read that (it is on line) and learn for herself.

No, she's just starting enzyme kinetics and her question, though understandable, is a poor addition to SE Biology. I've seen the subjects she has been asking questions on over the last year and have a shrewd guess as to her background. I think it's wise to try to establish first what the poster is asking before writing over 700 words with two equations in which the terms are not even defined. I surmise that you have never taught university students if you imagine that they will even buy a biochemistry text such as Berg when doing a broad biology course, never mind advanced texts.

Her answer is correct but shows only that she has learned the reactions of glycolysis and gluconeogenesis, not that she knows anything of kinetics or even the thermodynamics involved in the irreversible reactions. What Berg is trying to do is provide a visual way for the beginner to think about simple kinetic constants and, most important, that the saturation curve indicates the existance of an ES complex, the kinetics of which validate this. Many students, lacking chemical kinetics, do not understand the significance of the hyperbola, which is the point that needs explaining to them. I may.

The kinetics of enzyme catalysed reactions is a specialized area of biochemistry. It was historically important when much less was known about enzymes and the tools for studying them were limited. Why, then does it retain a place in contemporary text books? I would suggest that it illustrates how many of our ideas of enzymes and their reactions were developed, and that on a more mundane level it provides a cheap basis for laboratory experiments where students can encounter some of the pitfalls of experimental science and also learn to test ideas numerically.
For the average innumerate contemporary student of biology struggling with the Michaelis–Menten equation it can be easy to miss what I regard as the most important concept that this illustrates. This is that the kinetic analysis of simple enzymes supported…

…a model of the enzyme in which the substrate bound and the reaction
occurred at a specific site — the active site.

How does it do this? To understand that one must first contrast these enzyme kinetics with those of uncatalysed chemical reactions. It is assumed that chemical reactions occur due to the collisions of the reacting molecule. Depending on the reaction, the rate (r) of a chemical reaction and will be proportional to some power (n) of the concentration of the  reactant(s) (S):
r = k[S]n
and the ‘order of the reaction’ is n — often 1 or 2 (first and second order). (k is the rate constant, which reflects how this proportionality is manifested in different cases.)
What this means is that if one plots r (product per time) against the concentration of the reactant one gets a line of a particular shape (linear or curvilinear — see below) in which the rate continues to increase as the concentration of reactants increase (assuming the formation of product is small in comparison).

[Taken from Askiitians]
In contrast, the rate of enzymic reactions (velocity v) reach a maximum (Vmax), approaching this asymptotically as shown. (This is for a fixed amount of enzyme, as shown for an enzyme obeying simple Michaelis–Menten kinetics.)

The existence of a Vmax is consistent with the idea of a single (or small finite number of) active sites on the enzyme — when all enzyme molecules are saturated (all functioning) increasing the concentration of reactant (substrate) can provoke no further increase in rate.
Finally to the poster’s question (or at least the part that is most interesting conceptually): Why is this maximum velocity approached asymptotically? To appreciate this one should be aware of the disparity in numbers of molecules of reactant and enzyme — for glucose at 5mM and hexokinase at 20µM this is about 250 to 1. As the concentration of reactant is increased the number of ‘free’ enzyme molecules will decrease and a corresponding greater number of reactant molecules will be needed to produce a successful collision with, and binding to, the active site. This effect is small initially, but becomes successively more severe, so that in practice Vmax is never  reached and has to be calculated by extrapolation.
And however much the purists may object, the two kinetic constants that emerge from numerical analysis — KM and kcat — illuminate for the student two key aspects of the catalytic site. These are that it has a particular binding affinity (which may vary for different substrates) and that it has an intrinsic catalytic capability. So for those who wish to learn to walk before they can run, I recommend section 8.4 of Berg et al., which can be read online.
Addendum
And the answer to the first ‘boring’ part of the question is shown in the insert added to the second figure.The kinetic analysis is only valid if you consider the initial velocity (for which you have presumably been given the arbitrary figure of 10%) while the rate is linear with time. (There are various practical reasons linearity is lost with time — removal of substrate, back reaction as products build up, denaturation of the enzyme etc.)

Having given the poster time to think about it, I decided to post an answer to the conceptually interesting part of the question because the second question recently that does not seem to understand it, and it is generally taken for granted in texts. I suppose it needs a numerical explanation, but I didn't want to get bogged down in whether it was a hyperbola or not. I just wanted to get over the idea of why this kind of phenomenon has this sort of curve.

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Why do mutations in Drosophila dsx (double sex) affect both males and females?
Reason: Loss-of-function mutation of dsx gene in female embryo leads to production of a nonfunctional protein that fails to repress male specific gene expression. So somatic characters of both the sexs persist forming an intersex.
...This is all that I have figured out.
But what is the reason of appearance of intersex characters in male drosophila? Is it due to gain-of-funnction mutation.

Alternate splicing  of the dsx gene leads to sexual differentiation.

My answer may be too late for your homework deadline, but I'd appreciate your seeing whether you understand it and whether it answers your question.

Expression of the double-sex gene is rather unusual, so it is not surprising that you are having difficulty understand mutations in it. The simplified diagram below (adapted from Gilbert’s Developmental Biolgy — available on-line) should help clarify the matter.

The key concept is that although the double-sex gene (dsx) is expressed in both male and female flies, the gene products (DsxF add DsxM) are different in the two cases. This is achieved by alternative splicing of the RNA transcript, controlled ultimately by X and Y chromosome dosage.
The second important concept (the details of which, I admit, I am less familiar with) is that sex is determined by both activating genes specific for the particular sex and inactivating genes for the opposite sex. Thus, the diagram shows that DsxF both activates expression of genes for female sex differentiation and represses expresses of genes for male sex differentiation. Likewise DsxM activates expression of genes for male sex differentiation and represses expresses of genes for female sex differentiation.
Thus, according to this schema, if dsx is completely knocked out there will be consequences in both male and female. In the female, in the absence of DsxF, the genes for female sex differentiation will not be activated, but those for male sex differentiation will not be repressed. This presumably results in the intersex phenotype. Likewise, in the male, the absence of DsxM prevents the expression of the genes for male sex differentiation, but does not repress those for female sex differentiation. Again, I presume this results in the intersex phenotype.

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Which strand of a gene specifies a protein?
A gene is a segment of DNA (I hope double stranded) and each strand when transcribed form a mRNA and after being translated a protein. As from each gene there are two proteins that are being formed. Now how do these two proteins determine the expression of the gene?
I may be wrong. Please explain if so.

Usually (I don't know about any counter examples), only one of the DNA strands encodes a mRNA.

@Maljam What happens to the other strand ? Why is it not transcribed? How does a cell know which strand to transcribe and which strand not ?

Genes have promoters and enhancers upstream of the start site, all on the same strand. It is not unusual to have two (sometimes completely different) genes overlapping in a certain segment of DNA.

All enzymes related to DNA require a 3'-> 5' strand as they can synthesize product only in 5'-> 3' direction. Same happens with RNA polymerase too. It synthesizes mRNA in 5'-> 3' direction and thus requires only 3'-> 5' strand of DNA.

If a piece of DNA were to be transcribed on both strands, at the same time, the transcription complexes would collide and inhibit each other, like steric hindrance.
If you could overcome this interference and synthesize primary transcripts from both strands, it follows that the two RNAs would hybridize to each other and form complementary RNA-RNA hybrids.  These dsRNAs are very poor substrates for translation by the ribosomes. This was the initial mechanism proposed for anti-sense RNA to down regulate gene expression.
In many, if not most, EUKARYOTIC cells, a dsRNA will trigger the siRNA/shRNA/miRNA response mediated by Dicer et al. leading to degradation of both RNA strands.
So in virtually all instances, only one strand will be transcribed, and encode a mRNA with a significant ORF.

But this only applies to eukaryotes. Prokaryotes do have overlapping ORFs and don't have the dicer system.

Normally only one strand of DNA is transcribed. This is possible because for the RNA polymerase to bind and start transcribing the DNA into mRNA (or pre-mRNA) a special sequences on the DNA called a promoter is needed. Promoters will normally only be positioned to allow transcription in one direction, i.e. of one DNA strand.
Bacteria and their viruses sometimes have slightly overlapping genes so that, depending on their direction, the end of one mRNA may be from the same stretch of DNA as the start of another. This may not matter if one or both is not translated in this region.
There are even some cases where the end of one protein is from the same stretch of DNA as the start of another, but the amino acid sequences are different because they come from opposite strands. In such cases one assumes that the sequence of the protein is unimportant in that particular region. One does generally find most variation in the N- and C-terminal regions of homologous proteins in different species.


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Is it possible for human beings to feed through the nose?
There is a prediction on a futurology site that food can be eaten by inhalation
https://2050.earth/predictions/smell-can-feed-you
Although aerosol foods do exist they are still eaten by the mouth.
Is there any biological evidence that human beings can feed through their noses?

I'm voting to close this question as off-topic because it belongs to SE Worldbuilding.

@David I've come to Worldbuilding to say that this is not a worldbuilding question, as it deals with real human biology, and not fiction.

@AndyD273 — You could have fooled me!

Do you actually mean inhalation, ie that you'd absorb nutrients through your lungs? Because otherwise, the nose is connected to the stomach via the esophagus and you can be fed through that: https://en.wikipedia.org/wiki/Nasogastric_intubation

Certain vitamins, for example, vitamin B12, can be absorbed through the nasal mucosa by simple diffusion.
Many nutrients can be absorbed through the intestinal mucosa only by the help of specific carriers. These carriers are not likely present in nasal mucosa, so the nutrients in question may not be absorbed there.
The other problem, especially with macronutrients (carbs, proteins, fats), is if the surface area of nasal mucosa is big enough to absorb large amounts of macronutrients (carbs, proteins, fats) needed to meet nutritional demands. In one experiment, only 0.1 mg of glucose from a given dose was absorbed through the oral mucosa.
In another experiment, some amino acids were absorbed from oral mucosa.

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What is the evolutionary or biological origin of apathy and indifference. Is it uniquely a human thing?
I believe that social apathy is one of the main reasons why changes cannot easily be brought about in contemporary society due to the general lack of interest of the public, and this has ties with many mental diseases
But does the act of indifference and apathy actually has an evolutionary advantage and more curiously, do animals known how to demonstrate that in some manner?
A Google search on "evolutionary origins on apathy" returns no relevant hits other than the links to schzophrenia and other related neuropathies

Are there known studies on apathy and indifference in animals and their evolutionary origins?

I have edited your post, partly to correct the English, and in particular the misuse of the word disinterest, which means lack of bias from having an interest (financial etc.) in something. I have also changed "We knew" (presumably meant as We know) to "I believe". I believe it is bad practice to make unsupported assertions of this sort, attributing them to others, rather than taking responsibility for them yourself. I personally neither know what you assert, nor do I believe it.

You need to ask the reverse question why would a species evolve to care about individuals not related to it, the only examples are reciprocal altruism which has so strict requirements.

one has to try to define "apathy" and thats tricky, but maybe you really are looking for "low altruism" where altruism is well explored in evolutionary theory. on other hand maybe you are overthinking this, what you are really describing is "survival of the fittest" which involves/ or is at core an individual-centric drive to compete/ survive ie "zero sum game". there seems to be some interspecies altruism where for example in africa, sometimes different animal species will occasionally assist each other evading predators. so there seems some tradeoff between altruism and survival-of-fittest.

While I'm really unsure as to what you mean by "social apathy". even a cursory observation of animals will show that they're apathetic about many things.  (For instance, of my two dogs, one gets excited when I throw a tennis ball, the other could care less unless I throw it into water.)  The parts about changes to contemporary society might be better addressed on the Politics site, though I must say that it's quite possible that large numbers of people like society the way it is better than they do your proposed changes :-)


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How to recognize a conserved motifs of the protein
I would like ensure that my reasoning is correct. Assuming that I know the aminoacids sequence of the protein of interest. I can't say anything about the structure looking only at the aminoacids sequence of this protein. But if I know this protein from another organism and the structure of this protein is known, then I can compare both of the sequences and conclude something, right? what I mean, is that there is no specific sequence corresponding to, for example, helix-two- turn-helix motif, and that I can take this sequence, check if my protein has it and say that there is helix-two- turn-helix motif or not. I can do this only by comparison to the protein which structure is already known, right?

Yes. You have to know a sequence motif (another example for that would be transcription factor binding motifs) and then you can do sequence comparisons. They will give you a clue, what structures might be present, but you will still have to do some crystallography to make a definite proof. But it will give you a direction. The other possibility would be to look at homologous proteins from other species where more is known about the structure if possible. Sequence comparisons will give you here a more accurate picture.

Thank you. 'You have to know a sequence motif' -- do you mean that, for example helix-two- turn-helix motif has the specific  the same for every protein sequence? or that I need to identified that motif in homologous protein and then check if I have the same sequence in my protein?

I have done this for helix-loop-helix motifs. Here we aligned one known protein to a number of different species. And we were able to identify some motifs that were conserved between species while other regions were not conserved. I think homologous proteins would be better, but you can also try with other helix-turn-helix proteins.

It seems to me that you're asking about homology modelling. In that case, yes you need to compare your protein of interest to a protein (or proteins) of known structure. Homology modelling in a nutshell includes three (four?) steps: template identification/template alignment, modelling, quality assessment.
You start with finding a template for your modelling. This is usually done by sequence alignment, for instance BLASTing. Preferably you use multiple sequence alignment which more sensitively aligns conserved regions. You then want a template with as high sequence identity as possible (above 50 % usually produces models with about 1 Å RMSE [1] in main chain atoms. Avoid less than 30 % where modelling errors rapidly increase.)
There are then a number of different modelling strategies (wikipedia). But basically, they all aim to predict the structure of the conserved protein core as well as possible (which usually is what you're really interested in). Peripheral amino acids are more dynamic and more prone to evolution and are therefore more difficult to predict. Then, most importantly, you assess the quality of your model. This can be done by calculating violations of statistical potentials or physics based conformational energies (or using more advanced methods like multivariate regression methods). As in all modelling, this really is a most crucial step because prediction with a poor model is misleading and utterly useless.
If you don't find any template you could resort to the exciting field of De novo protein structure prediction, where the aim is to predict the structure from the amino acid sequence alone. I am not very familiar with their methods, but de novo prediction is hard (!). I don't remember any exact numbers but the number of conformations in a normal sized protein is astronomically large, which leads to great algorithmic and computational challenges. Additionally, without any reference sequence the model assumptions are greater than those of homology modelling. Although, I have heard that the field has been making great progress the last few years.

Edit:
It struck me that you might be asking about protein fold recognition as well. There exists a large number of different tools and methods for recognizing and locating protein domains using the amino acid sequence as input. Many of them are available as web servers. For instance phyre which uses the amino acid profile and predicted secondary structures to search structure libraries. Threading based methods like MUSTER. A number based on Hidden Markov Models (HMMs) also exist. For instance FISH which uses structure anchored HMMs.


text:
Index of ecological niche overlap
What is the Pianka's Index?

How does it work?
How can one calculate this index?
How often is it used?
In what context do people usually use it?
Can one infer any statistical significance with this index?

Do you plan to apply it to a particular dataset or problem, or are you asking out of general curiosity? Have you looked at Pianka's paper?

No I haven't had a look to Pianka's paper! I might not have looked thoroughly but I haven't found a paper that only presents such equation. I found some equations without really knowing what was called the Pianka's Index. I might aime to apply it on a dataset but for the moment my question is not specific. I just want to know how it works and why do poeple usually use this index.

Pianka's index of niche overlap is defined in his papers from 1973 and 1974, as:
$O_{kl}=\dfrac{\sum_i^n{p_{il} p_{ik}}}{\sqrt{\sum_i^n{p_{il}^2} \sum_i^n{p_{ik}^2}}}$
where $O_{kl}$ is the resource overlap between species $k$ and $l$, and since the index is symmetric $O_{kj} = O_{lk}$. $p_{ib}$ represents the proportion of resource $i$ that is used by species $b$. The index is a modification of one presented in MacArthur & Levins (1967).
As apparent from the formula, the only thing you need to calculate the index are estimates of resource use from the resources that both species depend on. The tricky part can be to determine how you calculate proportianal use of all resources.
I haven't used the index myself, and cannot really say anything about how common it is. I have seen references to it though. Pianka (1973) use it to study overlap between lizard species, and how the distribution of niche overlaps differ between habitat types. I haven't seen any statistical tests on it, but you should be able to use resampling or bootstrap to determine the uncertainty in the index.

Thank you! Shouldn't $j$ be $l$ in your formula? Or where is $l$ in the right-hand-side and what does $j$ mean?

@Remi.b Yes it should be. Fixed now.

Just to make I understood it correctly. $p_{ij} + p_{ik}$ does not necessarily equal to 1 even if there are no other species (than $j$ and $k$) using the resource $i$ because one should take into account that not all resources are used at 100%. Is it correct? Thank you for your help!

And the index vary between 0 and n/4?

If yes, do you think it would make sense to divide the index by n in order to have comparable indices?


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Why is the Pentose Phosphate Pathway so active in erythrocytes?
Is it because glyceraldehyde -3-phosphate (a molecule which when enter glycolysis help produce ATP through substrate level phosphorylation) can be prepared without losing an ATP through this process?

I've replaced the abbreviation in your title. Avoid all but the most familiar abbreviations (such as DNA) unless you define them (and this cannot be done in titles).

I've modified your title again. The use of "predominate" could have been taken to imply it processed more glucose than glycolysis. However in resting erythrocytes the figure seems to be 10% PPP. Still a lot compared with many other cells, but less than glycolysis.

Thanks for the correction though I actually meant 'Why is the Pentose Phosphate Pathway so active in erythrocytes?' so its ok.

Can you please cite a reference which says that erythrocytes use PPP relatively more than other cells? I was actually not aware of this; I don't think this some common textbook fact that I missed.

No. Your supposition is incorrect — the phosphate in glyceraldehyde 3-phosphate has to come from somewhere, and it comes from glucose 6-phosphate. The reason a second ATP is required before you get to the triose phosphate stage in glycolysis is that you are generating two molecules of triose phosphate. In the pentose phosphate pathway (energy-producing non-oxidative branch) you are not generating two molecules of triose-P from one hexose-P, you are generating two hexose-P and one trisose-P from three hexose-P, as shown in my Diagrams 2 and 3 below. (The other three carbon atoms are lost as CO2.)
A good way of approaching this question is to ask “What can be produced by the pentose phosphate pathway that cannot be produced by glycolysis?”.
Glycolysis may be summarized in this context as:

Glucose + NAD+ ➝ Pyruvate + NADH + 2ATP

But to allow glycolysis to continue the NADH is reoxidized to NAD+:

Pyruvate + NADH ➝ Lactate + NAD+

So the sole product of glycolysis for the erythrocyte is ATP (primarily for active cation transport to maintain cell shape) lactate passes into the blood for recycling.

The Pentose Phosphate Pathway (the oxidative stage) is shown in Diagram 1, above, and may be summarized in the way done for glycolysis as:

Glucose + ATP + 2NADP+ ➝ Ribulose 5-P + CO2 + 2NADPH

Ribulose 5-P has two possible fates, but only one differs from glycolysis, so the two distinguishing products of the pathway are ribose and NADPH.
Ribose is important for nucleic acid synthesis, particularly in dividing cells, explaining the increased activity of the pentose phosphate pathway in those cells. This cannot be the reason for the high activity of the pathway in erythrocytes as they have no nuclei and do not divide. In fact the ribulose 5-P is fed back into glycolysis for generation of ATP as shown in diagrams 2 and 3.
NADPH, then, is the answer in this case. It is the reducing agent used in the cytoplasm for synthetic processes (unlike NADH used particularly for ATP generation in the mitochondria), and so the pentose phosphate pathway is found in cells such as adipose, mammary gland and liver, that synthesize fatty acids, and cells that synthesize steroids. However that is not there is no fatty acid or steroid synthesis in the erythrocyte.
NADPH is important in erythrocytes as it is the specific source of reducing power required to keep the molecule glutathione in a reduced state. This has an important protective role in reducing cellular molecules that have become oxidized by molecular oxygen, a problem that is more acute in erythrocytes than in perhaps any other cells as they are the carriers of molecular oxygen. The cell membrane is particularly prone to oxidative damage. You can read about this online in this chapter in Berg et al.

I came across this line 'This stress is most acute in red blood cells because, lacking mitochondria, they have no alternative means of generating reducing power.'.
Does it mean that normal cells with mitochondria are better equipped for combating oxidative stress? Is it due to NADH and FADH2 formed in the mitochondrial matrix during Beta oxidation and Krebs cycle? Can NADH and FADH2 act as reducing agents for glutathione just as NADPH?

I quoted the line from that chapter you linked.
However my question still remains the same Can NADH and FADH2 act as reducing agents since they say that in absence of Mitochondria RBCs have no alternative means of generating reducing power?

I'll reply in two comments because of length restriction. I don’t find the line you quote at all convincing or helpful in any way. First, if what was required were NADH, that could be generated in glycolysis as far as pyruvate, with the pyruvate going into the blood and returning to the liver. (Pyruvate can pass in and out of cells.) Second, I imagine protection against reactive oxidative species in the mitochondrion is required because the mitochondrion is the site of the electorn transport chain at which oxygen acts. Thus protection in the mitochondrion is only designed to be used there.

The mammalian glutathione reductase only uses NADPH. It does not use NADH or FADH2. This is entirely consistent with the differential use of these two reducing nucleotide coenzymes in mitochondrion and cytoplasm.

I have now updated my answer, including my own illustrations, to make it slightly more comprehensive and, I hope, of more general use. It is meant to be an introductory general answer to the question, but if anyone thinks there are defects or points that should be added, please comment.

An anonymous user keeps submitting edits which in my opinion diminish rather than improve the answer. His explanation is the cryptic "decryption". If he explains himself in a comment I will be happy to discuss any points he raises. Otherwise I shall continue to reject his proposals.


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Where do transamination and deamination take place?
The only information I know is about deamination is that it occurs in the liver and kidney. But in which part of the cell does deamination occur? To which tissues is transamination specific, and in which part of the cell does it occur?
I have searched Google and Google books and have come across some information like L-Glutamate dehydrogenase occurs in both cytoplasm and mitochondria and transamination occurs in the cytoplasm of all cells. But they were not from authentic sources, so I am asking it here.

http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed, https://scholar.google.com

Those processes happen in just about  every single cell in your body

According to David H. Vroon and Zafar Israili aminotransferases (transaminases) are widely distributed among tissues, and are found in both cytoplasm and mitochondria, although this may vary between different aminotransferases:

Aminotransferases catalyze the redistribution of nitrogen between
  amino acids and corresponding oxoacids participating in both protein
  metabolism and gluconeogenesis. They are ubiquitous in their cellular
  distribution.
Tissue activity for AST (aspartate aminotransferase) is as follows in decreasing concentration:
  heart, liver, skeletal muscle, kidney, pancreas, spleen, lung, and
  erythrocyte. Two distinct forms have been identified: a cytoplasmic,
  or soluble isoenzyme, and a mitochondrial isoform...
The distribution and relative tissue concentration of ALT (alanine aminotransferase) is similar
  but importantly different. Highest activity is found in the liver,
  followed by kidney, myocardium, skeletal muscle, pancreas, spleen,
  lung, and erythrocyte. ALT activity is found in the cytosol; organ- or
  organelle-specific isoenzymes have not been demonstrated. The
  concentration of ALT in hepatic cell cytoplasm is comparable to AST;
  however, a mitochondrial ALT isoform is not found. In all other
  tissues, ALT activity is significantly less than AST.

As regards deamination, there is a Wikipedia page for glutaminase, which indicates that it is found in the mitochondria of a variety of tissues. There are two isoforms, one especially active in the kidney, and the other in liver (although both are found in other tissues).

The oxidative deamination takes place inside the mitochondria, because the
glutamate dehydrogenase is localized in the mitochondria.
About the transamination, I'm not sure if it occur inside the mitochondria or in the cytoplasm.
The information about the oxidative damination and the enzyme could be found at this link:
http://www.bioinfo.org.cn/book/biochemistry/chapt17/sim2.htm

Please add citations to support your answers. See the accepted answer to this question for a good example. Scientists (at least good ones) don't accept statements just on their face - they must be supported with evidence. If you are making a claim, back it up.


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How does our body produce new antibodies?
My professor said that in human body nearly 10$^9$-10$^1$$^2$  antibodies are produced by VDJ recombination and all the antigens that a person encounters in his lifetime are dealt by antibodies already present in his body. But he didn't explain what if the body in infected by a new mutated microbe(antigen) whose corresponding antibody is not present in the body.
My question is does our body have the mechanism to judge the epitope of the new antigen and synthesise an antibody with completely new paratope? If so what is it?
After reading the following I was forced to wonder whether the specificity of the antibodies improve(improve by synthesising new paratopes) after encountering antigens in the course of response.

The human immune system is capable of producing far more than 1e9 antibodies - it's more on the order of 1e12.

"Holes" in peoples' antibody repertoire are quite common -- that is, it isn't unusual for individuals to fail to make an antibody for a particular epitope.  However, pathogens generally have dozens to thousands of epitopes (many per protein) and it's rare that a person fails to make antibodies to enough epitopes to protect against the pathogen.

Sanjukta, your English is usually better than this. I don't understand what you are trying to say in the phrase "does our body have (not has) the mechanism to JUDGED the epitope". "Judged" is wrong. Can you clarify what you are actually asking. And when you say "child" and then "lifetime", do you mean "individual" (baby, child, adult) or are you talking about early life?

The simple answer is that the variant library is independent of antigen exposure, and is created before. Your body makes as many random variants as it can, specifically for cases when new antigens (e.g. new mutated microbe) are present. It's basically a very extensive and, barring immunodeficiency, constant library. Because there are so many variants present, chances are you have one to bind any foreign antigen. So once again, all these random antibodies are made without knowledge of antigens. They're always there to bind most foreign antigens.

:p @David I've corrected the mistakes and have improved the question. Go through it,if you please. I hope it makes more sense.

That's clearer. There's nothing against the rules in your question so I've cut that bit out. Accepted answer seems OK. But be clear that the immune system is not "judging" the antigen or using it as a template in any way. Those pre-existing cells the antibodies of which can bind to some extent will expand and mutate, makingit  likely that some of those produced will have higher affinity.

@MattDMo You were right. I read in Kuby Immunology that the antibodies produced by B-cells are within the range- 1e12 to 1e14.

@SanjuktaGhosh I just read your updated question, and the idea that the innate response is not highly specific is getting out of date, as is the strong dichotomy between innate and adaptive responses. Surface and intracellular receptors for PAMPs are incredibly specific for their targets, and have a broad range of effects that varies greatly from receptor to receptor, depending on the type of microbe associated with each PAMP. More understanding about NK, NKT, and γ-δ T cells (among others, such as the interaction of immature B cells with subcapsular sinus macrophages in the spleen to (...)

ferry immune complexes in a non-specific manner across the mantle zone to B cell follicles in germinal centers) is really blurring the lines between innate and adaptive responses.

In case of T-cells and Abs our organism first generates a great diversity of T- and B-cell receptor variants so that any new foreign antigen is likely to be recognized by some of them. When a B-cell encounters its antigen it begins proliferating and changes it phenotype. Additionally, B-cell (but not T-cell) receptor starts to mutate (so-called somatic hypermutation) generating new B-cell receptor variants that differ a bit from the original one. This process acts like evolution, with B-cell carrying a receptor with higher affinity being more likely to proliferate. So the specificity of antibodies improves indeed, this is called affinity maturation.

Could you please add some references?

I've added the wikipedia link to Somatic hypermutation. Hope you don't mind. Besides welcome to Bio stackexchange and thank you very much for the life-saving answer. :) Hope you have a nice experience here.

@L.B. This topic is nicely covered in Janeway Immunobiology http://www.garlandscience.com/product/isbn/9780815342434 . This book really helped me when I've transferred to the immunology field having a pure physics background.

@51102 that's not the point. We require answers on this site to include relevant citations to the scientific literature to back up non-obvious claims. Of course there's always some disagreement about what "non-obvious" means, but it's often interpreted to mean things that a person who's had a basic biology course wouldn't understand. So, you don't need to provide evidence that DNA is the carrier of genetic information, but a link to the Wikipedia pages for T cells, B cells, and antigens (maybe) would be appreciated. Somatic hypermutation would definitely need a citation.

Also, you can find an older version (the 5th edition from 2001, unfortunately) of Janeway on the NCBI Bookshelf here if you have a particular passage or figure that would illustrate your point well.

Maybe it is worth adding that B cells in the germinal centers, after having migrated their, will be nurtured in its journey between present T cells and reticular cells, where the B cells receptor's ability to bind to antigens will judge how well it will be nurtured (with various cytokines). Hence, one is able to get rid of receptors which are not sought. (With reference to Abbas' Basic Immunology)


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Muller's ratchet and viruses
Do viruses suffer from Muller's ratchet?  Could it have significantly influenced the 2020 coronavirus pandemic?

As with most things in the natural world, there isn't really a single straightforward answer to this question! It depends largely on the ecological contect and the population dynamics.
Basically, in organisms with high mutations rates (such as RNA viruses), Muller's Ratchet suggests that the mean fitness in a population will always decrease. In small populations, mutation-free individuals are rare, and they will be lost through genetic drift. In 1990, Chao showed that deleterious mutations are generated at a sufficiently high rate to advance Muller's ratchet in an RNA virus.
However, in general, Muller's ratchet can be argued to be ineffective in viral populations because populations are never 'small', and drift is usually very weak (or selection is strong, in other words). Chao also notes that the effective population size in nature, 8x10^9 is often irrelevant to whether or not Muller's Ratchet is active, because Viruses often pass through the bottleneck of a single individual. Additionally, frequently recombining viruses such as HIV can also be subject to Muller's Ratchet Yuste et al (1999). The results of Chao have been generalised to many other viruses.
However, these studies are lab-based and don't pertain to your question about what might happen to SARS-CoV-2 'in the wild'. RNA viruses show the highest mutations rates in nature (Sanjuan et al 2010). This can lead to Muller's Ratchet in small populations where selection is weak and drift is prominent, but can lead to high levels of adaptation in large populations. It is not just the census population size which is important, but the effective population size, which can be thought of as a proxy for the amount of genetic diversity within a population. If a population rapidly expands from a single origin, as is expected to have happened with the SARS-CoV-2 outbreak, then the census population size would be larger than the expected diversity. Novella (1995/1996) showed that there is an interplay with the size of the bottleneck and the fitness of the surviving clones; when a more fit viruses passes through the bottleneck, they are more likely to avoid Muller's Ratchet. So in the case of SARS-CoV-2, whether or not the virus would be subject to Muller's Ratchet would depend on both the variation and fitness of mutation of the viruses who jumped from the intermediate animal to humans.
At the risk of overcomplicating things, population/deme structure also plays a role. If the virus evolves in different demes within a structure population, viruses within a deme can evolve population specific mutations. If there is then gene-flow between the demes, there can exist selection between the different mutations, driving up the average fitness across demes (Mirales et al 2009) and avoiding the Ratchet.

Muller's ratchet is more relevant/more important in organisms with small effective population sizes. While the effective population size of the worldwide SARS-COV-2 population is smaller than (billions of copies per person [a typical detection threshold is 10^4 copies/milliliter of blood]) x (millions of infected people), it's very likely to be so large that Muller's ratchet is irrelevant.

Other answers cover well Muller's ratchet of the worldwide virus population.
However, drug-induced Muller's ratchet (aka mutational meltdown) might be considered on the level of the viral population that infected one patient. A recent publication Considering mutational meltdown as a potential SARS-CoV-2 treatment strategy talks about potentially inducing mutational meltdown on the within-patient viral population.
Note it's not research but a comment paper! Authors review the theory and empirical evidence that makes them think that mutational meltdown has the potential to become a coronavirus treatment.

It has been implemented in the drug marketed under the name Molnupiravir, with questionable success. Viable mutant lines showing traces of Mulnupiravir induced mutations emerged and spread somewhat in the wild.

Time has told us that Muller's ratchet is not relevant to the Corona virus. It is compensated by the frequent occurrence of recombinant virus lines that combine the best parts of their ancestors.

Recombination would be an important rescue mechanism if Muller's ratchet were an important phenomenon in the first place, but you haven't presented any evidence that it is ...


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Structure of mitochondrial ribosomes
Is the ribosome found in mitochondrion and choloroplast formed of 30S and 50S subunits as in prokaryotes. I couldn't find any information on internet concerning it.
The general wikipedia page has no information about it though it says

The ribosomes found in chloroplasts and mitochondria of eukaryotes also consist of large and small subunits bound together with proteins into one 70S particle.[9] These organelles are believed to be descendants of bacteria (see Endosymbiotic theory) and, as such, their ribosomes are similar to those of bacteria.

The wording of the question suggests that it may be useful to clarify the meaning of the numbers in the designations 70S, 50S, 30S etc. These are sedimentation coefficients in Svedberg units as determined in the analytical ultracentrifuge, and are an indication of size.
All ribosomes have two subunits. The reference eubacterial ribosomes from E.coli are 70S with a large subunit of 50S and a small one of 30S. The reference eukaryotic cytoplasmic ribosomes (rat liver, say) are 80S, with large and small subunits of 60S and 40S respectively. The actually sizes (S-values) of the subunits in other eubacteria (say mycoplasma)  and different eukaryotes (say yeast) may differ, but the eubacterial ribosomes share an antibiotic sensitivity (e.g. to streptomycin, erythromycin) which differs from that of eukaryotic cytoplasmic ribosomes (e.g. to cycloheximide) and so it is often convenient to refer to 70S-type ribosomes or 80S-type ribosomes, despite the size difference.
Mitochondrial and plastid ribosomes share some of the antibiotic susceptibility of eubacterial ribosomes (e.g. sensitive to streptomycin, but not to erythromycin) and their rRNA is more closely related to the latter. So they have 70S-type ribosomes with 50S-type and 30S-type subunits. However they vary greatly in size. For example, pig mitochondrial ribosomes are smaller at 55S, with large and small subunits of 39S and 28S, respectively. In contrast, yeast mitochondrial ribosomes are larger at 74S, with large and small subunits of 54S and 37S, respectively. More important, they differ from eubacteria in their ratio of RNA to protein, unlike chloroplast ribosomes, which workers in the field have classified as bacteria-like. This is shown in the table:
 [Adapted from Supplementary Table S1 in Science (2014) vol 343, 1485–1489]
The striking thing about most mitochondrial ribosomes is, whether or not they have undergone a decrease in their rRNA, they have additional proteins with no analogues in bacterial ribosomes. Structural work on both mammalian and yeast mitochondrial large ribosomal subunits show that these additional proteins are associated with an altered exit tunnel for the growing polypeptides, which appears to be tailored to conduct the latter directly to the mitochondrial membrane.
Answers (depending on intent of question)

Yes, mitochondrial and plastid ribosomes have large and small subunits.
No, their sizes are not 50S and 30S, but vary greatly.
Yes, mitochondria and plastids have 70S-like ribosomes, rather than 80S-like ribosomes, but these have evolved considerably from their eubacterial precursors.

Sources
Some of my assertions are based on recent papers and earlier work referenced in them. Also access to them may not be free outside university libraries. Apologies for that — if I find more accessible sources I will add them later.

Mammalian mitochondrial ribosomes: Nature (2014) vol 505, pp. 515–521.
Yeast mitochondrial ribosomes: Science (2014) vol 343, 1485–1489.
Spinach Chloroplast ribosomes: Proc.Nat.Acad.Sci. (2007) vol 104, pp. 19315–19320.
Antibiotics and ribosomes (review): EMBO Reports (2001) vol 2, pp. 318–323.


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Why are the genomes of some trees so large?
For example, the current largest known genome belongs to a tree: http://motherboard.vice.com/read/the-largest-genome-ever-sequenced-belongs-to-a-tree
I have heard that this could potentially be because trees require many duplicates of certain genes because they live their entire life exposed to direct sunlight.
Does anyone know of some specific scientific hypotheses for why this might be the case?

Old question, and obviously nobody else knows about the hypothesis quoted, but I changed longest to largest as its a question of polyploidy not length.

This study of the recently sequenced pine species states that 82% of genome is repetitive. This is characteristic of any complex genome, including humans. Such sequences have often been considered "junk DNA", though any scientist will tell you that just because we don't know its purpose doesn't mean it doesn't have one. That said, a good portion of repeats are due to transposable elements, effectively intracellular parasites, whose only purpose is presumably self-amplification and are more often than not deleterious. These interspersed repeats make up ~45% of the human genome. That paper mentions that, based on sequence homology, ~65% of the pine's genome is made of these repeats (that's 15 billion base pairs).
It's interesting that, even though the pine's genome is ~7 times larger than a human's, it has only twice as many predicted genes. The pine tree has ~50 000 hypothetical genes but only ~16 000 are what the authors call high confidence. When the human genome was first sequenced, there were ~35 000 hypothetical genes but now, some 10 odd years later, there are only ~21 000 predicted genes. It seems too early to compare the two. This also brings up the important point that the number of genes is not necessarily related to organismal complexity: the worm C. elegans has ~20 000 genes.
From the paper:

The large genome size has primarily been attributed to an extensive contribution of interspersed repetitive content (Morse et al. 2009; Kovach et al. 2010; Wegrzyn et al. 2013). Assembly of the Norway spruce genome has shown that LTR retrotransposons in particular are frequently nested within the long introns of some gene families (Nystedt et al. 2013). In addition, there is evidence for gene duplication, pseudogenes, and paralogs, although the extent of these is not clear (Kovach et al. 2010; Pavy et al. 2012).

I have never heard of the hypothesis you mention (not that that means anything), but it seems to me that it's far to early to tell.

There is another interesting thing to be noted — Plants (and yeasts too) tolerate polyploidy. I don't know the reason for it. There is one interesting fact that plants and yeasts don't have nuclear lamins; but I don't know how that connects to polyploidy. This article relates loss of lamins to polyploidy. I speculate (more of a wild guess) that these structural constraints could also limit the genome size.

+1. Another interesting point is that there is some evidence that the retrotranspositions of repetitive sequences such as Alu (which represents 10% of the human genome) are activated under conditions of cellular stress and can help shuffle the genome, providing new variety.

@WYSIWYG Yes that is interesting. Feel free to edit the answer if you see fit (I might add it too given the time).

@terdon That's also very interesting. There was a 2003 study of 3 Bordatella species that found that B. perrtussis (the cause of whooping cough) evolved from B. bronchispetica due to massive genome shuffling mediated by transposable elements.

@Jagoe you may be right. It was just a speculation. If you wish you can expand the points you raised in the comment and post an answer. Remember to include the links to suitable references.


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What is the energy required for storing and performing computations in synthetic computational biological cells?
I want to know the energy costs of various computations performed through synthetic biological circuits in terms of ATP. I am primarily interested in storing of bits, addition, and multiplication operation.
Though I am able to get various papers on half-adders and logarithmic analog addition, I find there is a lack of precise energy costs in these papers. Especially I am not able to find the energy cost of multiplication in any of the papers, though I believe that multiplication can be done by adding two logarithmic terms, but is this the only way available?
There is one paper that answers some of my queries: Analog synthetic biology by R. Sarpeshkar.
Here the author gives at least an idea about the number of ATP required for doing additions for a certain biological environment for both digital and analog scenarios and also mentions some facts about how increasing the precision of bits lead to increase in protein copy number.
But I need more such papers that deal with multiplication, storing of bits in cells and their energy requirements in terms of ATP.
Note that I am merely interested in having some literature that deals with computation and its energy cost, as most of the literature that I encountered doesn’t mention how much energy in ATP is consumed in performing certain computation in a given biological environment.

Biochemical systems use ATP to perform chemical reactions or other energy-requiring processes. Addition and multiplication are abstract mathematical concepts that can be mimicked in computers by specific circuitry. If you wish to do likewise in biology (synthetic or otherwise) you need to set up some system that can perform some chemical transformations. The energy cost of such transformations will depend on their specific physical nature. It is therefore nonsensical to ask in abstract the energy cost of this. You will have to get to grips with the biochemistry of specific systems. Good luck.

The reason you aren't finding this information in the literature is that this metric doesn't match the way that most genetic circuits work.
You appear to be thinking about operations in genetic circuits as being like instructions on an electronic processor. In that world, an instruction starts, executes, and stops, and you can measure the amount of energy consumed between the start and the stop.
Most genetic circuits, however, are more like wiring up electronic circuits like NOT and NOR gates. The computation is thus not discrete but continuous in time. One can certainly still consider the operating burden of each device on the cell, but (1) the load is strongly dependent on the particular state of the device (e.g., high output vs. low output), and (2) ATP availability is typically not the limiting factor.
Thus, your question is simply not one that most practitioners would find meaningful to answer. Even in the Sarpeshkar paper that you cite, by the time he gets to discussing actual experimental results, the numbers are expressed in terms of fluorescent protein concentration rather than ATP.
If you want to understand the burden of genetic circuits on cells, a much more relevant starting point is the recent paper "The Effect of Loads in Molecular Communications" from McBride, Shah and Del Vecchio

This addresses, rather than answers, the question; but is, I think, preferable to continuing to discuss the problem in comments.
As I understand it, the question deals with the use of genetic circuitry in synthetic biology to perform mathematical calculations in an analogous way to that done in computer systems. By genetic circuitry is meant the induction and feed-back regulation of transcription of certain engineered genes in the DNA of living (presumably bacterial) cells by protein molecules (together perhaps with small-molecule inducers). Such systems are modelled on natural bacterial operons.
While some of the users of this site may be familiar with synthetic biology of this type to produce bacteria with practical applications based on sensing environmental molecules, I imagine that very few will be familiar with their abstract use for mathematical operations. Certainly, although I subscribe to Nature, I had missed the 2013 article on this topic by Daniel et al. (Nature 497, 619–624), which it would take me some effort to master. Therefore the poster should not be surprised if his question has few takers.
Those who are interested or involved in synthetic biology are unlikely to have been concerned with the energetics of these circuits. Certainly RNA transcription involves the hydrolysis of phosphodiester bonds of nucleoside triphosphates, but one takes this as read, and assumes the bacterium will be in an environment where energy sources are available for growth.
Out of interest I glanced at the introduction to another paper involving Sarpeshkar — Philosophical Transaction of The Royal Society A (2014) 372 1–22 — and read the following:

Every living cell within us is a hybrid analog–digital supercomputer that implements highly computationally intensive nonlinear, stochastic, differential equations with 30000 gene–protein state variables that interact via complex feedback loops. The average 10μm human cell performs these amazing computations with 0.34 nm self-aligned nanoscale DNA–protein devices, with 20 kT per molecular operation (1 ATP molecule hydrolysed), approximately 0.8 pW of power consumption (10 M ATP s−1) and with noisy, unreliable devices that collectively interact to perform reliable hybrid analog–digital computation.

So it would appear that Sarpeshkar or others are citing the energy used per molecular operation taken apparently as 1 ATP equivalent needed for each NTP addition. (This seems to me incorrect, as the cleavage of NTP to NMP + PPi means 2 phosphodiester bond equivalents of energy are used.)
As far as I can see — and I may well be wrong — the answer to the poster’s question would seem to lie in learning how many molecular operations are involved in performing the particular mathematical operations in the systems described in these papers. I suggest he has more chance of discovering that by reading the papers and calculating how much RNA is synthesized in each operation — or even contacting their authors — than by asking on a general biology forum, like this.


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Could removing non-functional parts of coenzymes improve enzyme function?
As pointed out here, many important co-enzymes (essential enzyme cofactors) such such acetyl-CoA and vitamin B12 contain nucleotide portions that do not function in the enzyme catalysis. They have been suggested to be relics of the RNA world, so it would appear to me that they are redundant. Suppose we were to remove these nucleotide portions of the co-enzymes, would that improve the working of the proteins (and perhaps our health) or wouldn’t it make any difference?

Those aren't enzymes but rather coenzymes, and the nucleotide moiety is necessary for function.

@canadianer — I agree that this question is based on a fallacy. Surely that point should be made in an answer or the question should be voted to be closed. The only point of a comment to the poster would seem to me to be to ask the poster to withdraw the question, which he is unlikely to do if he doesn't understand that the nucleotide part of coenzymes interact with proteins rather than provide the functionality.

@David Surely the point should be made in an answer. Actually I think this question is a duplicate but I cannot find the other post.

@canadianer —  Don't know about the duplicate. But, yes, probably best as an answer. Are you going to answer or me. In any case I think I'll edit it so it's more accurate and explains the problem in more detail. (I've a little time to kill while waiting for a program to run.)

@David I don't have the time; feel free.

The video cited in the question is to be commended and presents a number of points relevant to questions about DNA and RNA posted here previously.
The problem with the question sems to be an incomplete understanding of the role of these coenzymes/cofactors in the function of the enzyme. (The original question implied that acetyl-CoA and vitamin B12 are the enzymes.) In contemporary protein enzymes, these non-protein cofactors are bound to the protein part of the enzyme and complement it in its function. Typically, the protein will have binding sites for the substrates (the reactants) and the coenzyme will provide the sophisticated catalysis or other essential function.
The coenzyme must obviously interact with the protein, and in a way that leaves its catalytic or functional part free to operate. This is what the nucleotide portion of the coenzyme does — in fact it is performing the same role as has been suggested it played in the RNA world, but instead of interacting with an RNA enzyme (ribozyme) by base pairing, it interacts with amino acid side-chains by a variety of weak interactions. An example of this is shown for coenzyme-B12 in the enzyme, methylmalonyl-CoA mutase (my annotations in red and brown).

To quote from the figure legend in the paper: “The lower axial ligand to the cobalt is histidine HisA610, which is in turn bound through AspA608 to LysA604. The pseudo-nucleotide tail is bound in a pocket between the sheet and the C-terminal helix.”
The situation for coenzyme-A (CoA) is slightly different, in that it is involved in the transfer of acetyl and other groups, rather than catalyis per se. However the principle is the same — there is a chemically functional portion and a nucleotide portion that binds it to the protein.
Thus, the nucleotide portion of coenzymes is not redundant and without it they could not bind to the enzyme and enable it to perform its function.


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Meaning of “cognate genes encode the basic biological functions of proteins”
In the introduction to a review article entitled “Interpreting the protein language using proteomics” the author uses the term ‘cognate genes’ in a manner which I do not understand:

Whereas cognate genes encode the basic biological functions of
  proteins, the real-time dynamics and regulation of protein structure
  and function are generally carried out by specific post-translational
  modifications (PTMs) of proteins such as phosphorylation,
  glycosylation and acylation.

What does it mean?

Could you provide a reference. The word "cognate" implies relatedness, so that "cognate gene" on its own is meaningless. It has to be cognate to something else. Unless the preceeding sentence inidicates this, I suspect you have a poor text using terms that you should certainly not use yourself. I personally have never heard cognate used in this way and it would seem to me to be incorrect.

Perhaps the reason you ignored my request is that you are on a different time zone. An internet search found the article you (mis)quote. I have rewritten your question citing this article, which is what you should have done in the first place if you wanted people to help you. My brief opinion is the author (not a native English speaker) is misusing the word for effect. I will compose a detailed answer when I have time.

The sentence quoted accidentally misuses the word ‘cognate’ because in attempting to appear profound it has become overweight and off-balance. Notice the sentence also contains the meaningless nonsense of “real-time dynamics”, and in the rest of the paper the word ‘cognate’ is (fortunately) nowhere to be found.
What does the author mean? His review is on proteomics, and he wishes to emphasize the importance of post-translation modification of proteins. He needs to refer to the unmodified form of proteins, and so he associates them with the information in their genes, which is limited to the unmodified amino acids. The word ‘cognate’ in plain English has the meaning of ‘related’ or ‘corresponding’, so the author would appear to be using it to imply the relationship between particular genes and the individual proteins that they encode. What he may have meant to say is:

“Whereas genes encode the basic biological functions of their
  cognate proteins…”

This would have been linguistically correct, although the word ‘cognate’, in my opinion is unnecessary as it adds nothing to the sentence.
You can find a biological definition of cognate on GenScript:

cognate
A term borrowed from linguistics, signifying a correspondence;
  e.g. a receptor and its cognate ligand, a tRNA and its cognate amino
  acid.

These two examples are of the most common uses of ‘cognate’ in molecular biology — that for tRNA can be found on the Wikipedia page for aminoacyl tRNA synthetase. A quick search brought up few examples of ‘cognate proteins’ or ‘cognate genes’, and in these cases I found the word was used to refer to similar genes or proteins in different organisms or strains.

I think it may be referring to constitutive expression of those "cognate genes". So, cognate genes are those who codify for essential proteins which are constantly produced in the cell, and their function is regulated by said PTMs (Post Translational Modifications). Sorry if I had grammar mistakes in this text.

Please add some references to your answer.

yeah this is confusing - the writer just means 'the primary structure of proteins' probably.  the primary structure directs the 'job' of the protein, but the small modifications to the protein, such as those PTMs s/he mentions, are responsible for directing the protein to/from its 'workplace' and determining its level of activity and its specific shape.


text:
Why can blood group O be given to all blood groups?
The blood of persons of blood group O contains antibodies against antigens A and B. The red blood cells of persons of blood group A have A antigen on their surface. If someone with blood group A receives blood from a donor of group O, then anti-A antibodies in the donor blood should attack the A antigens on the recipient's red cells. Shouldn’t this be a problem?

Good question! Let's start with an overview and explanation of the ABO blood grouping system (Dean, 2005):

Blood groups, antigens and antibodies: Subjects with blood group A carry the A antigen on their red blood cells (RBCs) and have antibodies to antigen B; subjects with blood group B have the B antigen and anti-A antibodies, blood group AB carries both the A and B antigens and have no antibodies, and finally folks with group O have no antigens but both antibodies.
Blood groups and transfusion: Circulating antibodies in a person's blood will attack transfused RBCs from unmatching blood groups. These antibodies fix complement and cause rapid intravascular hemolysis, triggering an acute hemolytic transfusion reaction that can cause disseminated intravascular coagulation, shock, acute renal failure, and death (Dean, 2005). As people with blood group AB have no antibodies, they can receive all blood groups without issues. People with blood group O can only receive RBCs of blood group O, because they carry both antibodies. Blood group A can receive A and O, and blood group B can recive B and O. Hence, blood group O can be received by all other groups, and is therefore the universal donor.
Typically, literature on transfusion deals with the haplotyping the antigens of the donor RBCs. The question whether the antibodies in the donor's blood with blood group O would start attacking cells in recipient with blood group A, B and AB is, therefore, a very good question. And the answer is: yes, they will. Why then is O still the universal donor as they have both anti-A and anti-B antibodies?
First, whole blood is not used for "routine" blood transfusions (Torrens University Website). Instead, blood component therapy has largely replaced the use of whole blood (Patient.co.uk website). For routine transfusions, only the fraction containing packed red blood cells is transfused (Mayo Clinic website). Hence, in these cases, the blood plasma containing the antibodies is (virtually) removed after centrifugation. Since O-type red blood cells contain no antigens (Dean, 2005), and the plasma with the antibodies are not transfused, it can be donated to everyone, barred that the rhesus factor matches.
Secondly, even if plasma is transfused, antibodies in the donor's plasma are a minor problem, because of the small amount of antibody present in the donated plasma, which is further diluted on transfusion into the recipient's circulation (Dean, 2005).
Reference
- Dean (2005) Blood Groups and Red Cell Antigens. Chapter 5.The ABO Blood Group

Excellent overview of this concept. Also relevant, typically before transfusion (or a procedure that carries a risk of transfusion) we actually test a sample to make sure there won't be any hemolysis. So a recipient's blood is taken, typed (e.g., ABO group identified), and then mixed, or crossed with a sample from the donor pool before packed cells are given.


text:
Do all prokaryotes have ATP synthases and an electron transport chain?
I was looking this up and I'm wondering if so far I'm correct about the following:

For the most part eukaryotes do aerobic respiration, which involves glycolysis, the Krebs cycle and oxidative phosphorylation involving ATP synthase and an ETC
Prokaryotes can either do aerobic respiration, anaerobic respiration or fermentation, the first two of which require an ETC and ATP synthases, which would be located in the plasma membrane of the prokaryote

If that is the case, I was wondering if there are prokaryotes that only do fermentation, and if so do they have ATP synthases and an ETC? In other words, are there any organisms without ATP synthases and an ETC (which would force them to survive on fermentation alone, unless there is some other method of energy production in prokaryotes that doesn't use ATP synthases and an ETC that I'm missing)?
Thanks

This is an interesting question (I really mean this — see below), for which a straight answer is remarkably difficult to find on the web. When I googled for it I got pages with statements that obligate anaerobic bacteria still had the electron transport chain (ETC) and ATP synthase as there were different electron acceptors other than oxygen. Yes, we know that things are different in thermal vents and hot springs, but what about gas gangrene? I am not a microbiologist, but I did play with the metabolism of some bacteria for a biochemical bioinformatics lab some years ago, so I can give two instances where anaerobic bacteria appear to lack either the ETC or both the ETC and ATP synthase.
Clostridium perfringens
This is the anaerobic fermenting bacterium that leads to gas gangrene in infected wounds and was a major cause of limb loss and mortality in the First World War. The DNA sequence of Clostridium perfringens has been determined. I quote at length from the paper as it describes the fermentation, but italicise the key statement for those who wish to skip this:

We could not find any genes coding for tricarboxylic acid (TCA) cycle- or respiratory-chain-related proteins, in contrast to C. acetobutylicum, which has incomplete TCA cycle enzymes. Similar to C. acetobutylicum, we could construct a pathway map for anaerobic fermentation resulting in the production of lactate, alcohol, acetate, and butyrate, all of which have been commonly detected in C. perfringens cultures. In the fermentation pathway, pyruvate is converted into acetyl-CoA by pyruvate-ferredoxin oxidoreductase (CPE2061), producing CO2 gas and reduced ferredoxin. Electrons from the reduced ferredoxin are transferred to protons by hydrogenase (CPE2346), resulting in the formation of hydrogen molecules (H2) that are released from the cell together with CO2.

The related bacterium, Clostridium acetobutylicum, mentioned in the extract, is also an obligate anaerobe. (It ferments a variety of plant mono- and poly-saccharides to acetone, butanol and ethanol — used to produce explosives in the First World War.) Although it has some enzymes of the TCA-cycle, it is unable to use the cycle oxidatively, and, like Clostridium perifringens, lacks the enzymes of the electron transport chain and an ATP synthase.
Ureaplasma urealyticum (Ureaplasma parvum)
This micro-organism is a micoplasma (Mollicute) rather than a bacterium — i.e. it lacks a cell wall. It infects the human urogenital tract. It lacks the components of an electron transport chain but does possess a functional ATP synthase. It generates a hydrogen ion gradient, not by increasing the hydrogen ion concentration within the inter-membrane space by the oxidation of NADH in the ETC, but by reducing the intracellular hydrogen ion concentration by generating ammonia from urea (plentiful in its habitat) in a reaction catalysed by the urease it encodes. The sequence of the organism and references to previous work on its urease activity can be found here.
Evolutionary considerations
The reason I find this question interesting is that anaerobic organisms preceded aerobic organisms, so the question arises whether there are any contemporary anaerobic bacteria that have evolved from these primaeval anaerobes and have never possessed an electron transport system — or are all contemporary anaerobic organisms lacking an ETC derived from organisms with an ETC (aerobic or using some other electron acceptor) and have just lost these functions through non-use (as is most likely in the examples above)? This question must surely have been considered by those in the field.

Very nice! These organisms are super interesting.

Yes. I find them fascinating too.

Yes there are, though I had to dig a lot to find them. They are called strictly fermentative bacteria.
Lactic acid bacteria (Lactobacillales (Firmicutes)) ferment glucose through pyruvate to lactate using the glycolytic pathway, and include the genera Streptococcus, Lactococcus, Lactobacillus, and Leuconostoc.
From Madigan et al. 2014 Brock biology of microorganisms, 14th edition (PDF):
p. 66.

"Microbial eukaryotes that contain hydrogenosomes carry out a strictly
  fermentative metabolism. Examples include the human parasite
  Trichomonas (Sections 17.3 and 32.4) and various protists that
  inhabit the rumen of ruminant animals (Sections 1.5 and 22.7) or
  anoxic muds and lake sediments."

p.93.

"Reversibility of the ATPase explains why strictly fermentative
  bacteria that lack electron transport chains and are unable to carry
  out oxidative phosphorylation still contain ATPases. Many important
  reactions in the cell, such as flagellar rotation and some forms of
  transport, are coupled to energy from the pmf rather than directly
  from ATP. Thus, the ATPase of organisms incapable of respiration, such
  as the strictly fermentative lactic acid bacteria, functions
  unidirectionally to generate this required pmf from ATP formed during
  substrate-level phosphorylation in fermentation."

Trichomonas (Excavata) is an eukaryote.

I'm no expert, but looking at the English, The statement "" strictly fermentative bacteria that lack electron transport chains"   <--  doesn't state that all strictly fermentative bacteria lack an electron transport chain, and doesn't state that strictly fermentative bacteria lack an electron transport chain.   If I say "birds that have long legs", it suggests there may be birds that don't have long legs.

That said, I can see how perhaps strictly fermentative bacteria don't have an ETC(perhaps it'd mean they had an ETC but couldn't ever use it, which would perhaps be strange). But anyhow, while it may well be that they don't have one,  that quoted text doesn't say that they don't have one.

This is a very late answer, but many Mycoplasmas lack most major metabolic pathways beyond glycolysis; so no electron transport chain, and no tricarboxylic cycle (refer Pollack et a. 2002).


text:
Do some viruses that replicate in the cytoplasm contain RNA polymerases in their virions?
As emerges from this previous Stack Exchange question and its answer, viruses in the family poxviridae have the potential to replicate in the cytoplasm because they encode their own DNA polymerases and RNA polymerases.
However, it seems to me that there's a bit of a the-chicken-and-the-egg problem here: DNA viruses needs RNA polymerase to make the mRNA encoding their proteins (including RNA polymerase), but they can't make these proteins without RNA polymerase to transcribe their DNA first!
Viruses that replicate inside the cell nucleus clearly don't run into this problem as they can use the host RNA polymerase already there. But how do viruses that replicate in the cytoplasm solve this conundrum? Do they carry sufficient RNA polymerase in their virions when the enter the cells or is there some other explanation?

I have edited your question using more precise wording to clarify the problem. This is part of the way SE Biology operates. You have the option of rolling back these changes, but I think you would be wise to alert me if you think I have changed your meaning or you have a query with the rewording. Use my @ handle so that I am alerted to your comment.

Let us consider some different classes of viruses.
DNA viruses that do not encode their own DNA-dependent RNA polymerase replicate in the nucleus and so are able to utilize the host enzyme (and the DNA-dependent DNA polymerase). So there is no problem regarding production of RNA polymerase in this case.
The (double-stranded DNA) pox viruses, which do encode both their own DNA-dependent DNA polymerases and RNA polymerases, are unusual in that they replicate in the cytoplasm of the host. As the question states, in this case there is a problem of transcribing the viral genes encoding the DNA-dependent RNA polymerase that performs this transcription. As is explained in, e.g. this review by Bernard Moss not only the RNA polymerase, but approximately 20 other proteins devoted to synthesis and modification of mRNA. I reproduce part of Fig. 1 of that review below.

In positive-stranded RNA viruses the viral genome generally acts as mRNA, which can be translated into proteins (including the RNA-depenedent RNA polymerase — replicase — required for replication) by cytoplasmic ribosomes.
For negative-strand RNA viruses, the negative strand must be first replicated to produce a positive RNA strand (acting as mRNA) before synthesis of the RNA-dependent RNA polymerase can occur. As this requires viral RNA-dependent RNA polymerase, the circular problem of the source of the enzyme also arises in this case. As explained in an article by S. Payne, such viruses also carry the enzyme within their virion.

Since negative-strand RNA viruses carry the necessary enzyme in their virion, I suppose it must be happening something similar in Poxviridae

@CapivaraCientista Yes — see the modification of my answer. However I would repeat my comment to your previous question. In future please do some research first and show us what you have found and why you still have a question. This not only complies with our model, but it helps you learn to answer your own questions and gives you the opportunity of reading the sources carefully. I just did simple Google searches using the virus name or type and the enzymes or the word "transcription".


text:
Are there multicellular isogamous species?
Are there multicellular isogamous species?
Seeking through the examples of wikipedia I would tend to think that there are no multicellular isogamous species.

"Isogamy is common among algae, lower fungi, and many protozoans (Rhizopoda, Radiolaria, and lower Gregarinida) but absent in multicellular animals. In isogamy the copulating gametes differ in their biochemical and physiological properties." - but this is not from a scientific article...

Our knowledge about the sexual transmission of mitochondria and
  plastids (hereafter organelles) in isogamous eukaryotes comes mostly
  from studies of the yeast Saccharomyces cerevisiae and the green alga
  Chlamydomonas reinhardtii which are both unicellular species. To
  investigate organelle inheritance in a multicellular organism with
  morphological isogamy, we studied the filamentous brown alga
  Ectocarpus siliculosus, in which each gamete contributes one plastid
  and at least one mitochondrion to the zygote. ...

2004 - Inheritance of organelles in artificial hybrids of the isogamous multicellular chromist alga Ectocarpus siliculosus (Phaeophyceae)

So it is possible, but it seems like very rare.


text:
Is there an association between environmental and mutational robustness?
The robustness of a genotype is the ability of this genotype to resist (always produce the same phenotype) to various parameters such as mutations and environment. The ability of a genotype to resist to mutations is called mutational robustness while the ability of a genotype to resist to varying environmental variables is called environmental robustness.
Is there a correlation between environmental and mutational robustness?
In other words the above question is: is there a among-species/populations/genotype correlation between mutational robustness and environmental robustness? i.e., Knowing that species $A$ has high mutational robustness, is species $A$ more or less likely than a randomly picked species $B$ to have a high environmental robustness? Or are the concepts of environmental and mutational robustness completely different and involve completely different genes and genetic architecture? Both theoretical and empirical studies that offer insights to this question are welcome.

Hansen (2006) says (middle of page 140):

A link between genetic and environmental robustness is plausible, because genetic en environmental disturbances may often affect the same functional pathways in the organism, and ny increase in the robustness of a pathway leads to both genetic and environmental canalization. Evidence for a correlation between environmental and genetic robustness has been fund in studies of RNA folding (Ancel & Fontana 2000), in the effects of heat-shock proteins (Stearns et al. 1995).

He also cites Rifkin et al. (2005) who also showed evidence for this correlation. Finally Hansen cites Dworkin (2005) who did not find this link.
Hansen does not cite any theoretical paper and does not offer clear arguments of why this link exist but it is the best I could find


text:
Soft and Hard selection
Seems to me that these two sources (M. Whitlock, B. Wallace) use different definitions of soft and hard selection.
M. Whitlock: Soft selection occurs when the relative fitness of an individual is calculated relative to the fittest individual of the local patch. Hard selection occurs when the relative fitness of an individual is calculated relative to the fitness of the fittest individual in the whole metapopulation.
B. Wallace: Soft selection occurs when the relative fitness of an individual is calculated relative to the fittest individual in the population. Hard selection occurs when the relative fitness of an individual is calculated relative a hypothetical fitness. Therefore, extinction only occurs under hard selection but never (except eventually when all individuals have fitness of 0) in the soft selection case.
Are there other definitions? What is the most common definition?

I think Wallace (1968) developed the ideas of hard and soft selection, as they relate to genetic load. He further explains the concept in Wallace (1975). As the idea was his, I'd go with Wallace's definition over Whitlock's.  I haven't had time to watch the Whitlock video to see if they are saying more or less the same thing overall.
Consider a hypothetical population of individuals, each with 1000 loci (I'm basing example and ideas on an example given by Ridley 2004, pgs. 162-167). Heterozygotes at each locus has the greatest fitness (heterozygote advantage). Homozygotes at any locus will have reduced fitness. Therefore, the fitness of each genotype is given as:
AA: $1-s_{AA}$
Aa: $1$
aa: $1-s_{aa}$
Even if every individual in the population at time zero was heterozygous at each locus, Mendelian inheritance ensures that no individuals will be heterozygous at all loci after the second generation.  Assume that fitness reduction at each homozygous locus is just 1%. Given that well over half of the loci will be homozygous, fitness of those individuals will be reduced to 0. They will die before reproducing.  This is hard selection.  If hard selection of this magnitude occurred in a population, then extinction would occur. This is what Wallace (in your link) is saying.  This mortality due to the genetic load is added to the normal background mortality.
Soft selection compares the fitness of an individual to the genotype with the greatest fitness in the population, even if that genotype does not the theoretical optimum fitness. Soft selection replaces some of the background mortality with selective mortality, so the population size is not decreased overall.
Literature Cited
Ridley, M. 2004. Evolution. 3rd ed. Blackwell Publishing, Malden, Massachusetts, USA.
Wallace, B. 1968. Polymorphism, population size, and genetic load. pp. 87-108 in R.C. Lewontin, ed. Population biology and evolution. Syracuse University Press, Syracuse, New York, USA.
Wallace, B. 1975. Hard and soft selection revisited. Evolution 29: 465-473.


text:
Selection on linked loci in a diploid population
Let’s consider two linked loci $A$ and $B$ that are both bi-allelic. In consequence, we have four different possible haplotypes $A_1B_1$, $A_1B_2$, $A_2B_1$, $A_2B_2$, which frequencies are $X_1$, $X_2$, $X_3$, $X_4$ respectively. Let the linkage disequilibrium $D$ be defined as:
$$D=X_1X_4 - X_2X_3$$
The change in haplotype frequencies through time is given by:
$$\delta X_1 = \frac{X_1(w_1 - \bar w) - Drw_{14}}{\bar w}$$
$$\delta X_2 = \frac{X_2(w_2 - \bar w) - Drw_{14}}{\bar w}$$
$$\delta X_3 = \frac{X_3(w_3 - \bar w) - Drw_{14}}{\bar w}$$
$$\delta X_4 = \frac{X_4(w_4 - \bar w) - Drw_{14}}{\bar w}$$
, where

$r$ is the recombination rate between the loci $A$ and $B$
$\bar w$ is the mean fitness (sum of the fitnesses of each haplotype weighted by their relative frequencies)
$w_{14}$ is the fitness of the (diploid) individual that has the 1st and the 4th haplotypes
$w_i = X_1w_{i1} + X_2w_{i2} + X_3w_{i3} + X_4w_{i4}$

The change in allele frequency of locus $A$ is $x$ and $1-x$. The change of allele frequency of locus $B$ is $y$ and $1-y$. Those changes are given by the addition of haplotypes frequencies:
$$\delta x = \delta X_1 + \delta X_2$$
and
$$\delta y = \delta X_1 + \delta X_3$$
Question
I don’t fully understand these formulae! Could you please help me making sense of them. Especially of the presence of the expression $-Drw_{14}$.

source

These equations describe how the haplotype frequencies will change over time due to a combination of recombination and natural selection.
Before I proceed, I need to change your four $\delta X_i$ formulas above. Lewontin and Kojima (1960) writes the equations as:
$$\Delta X_i = \frac{X_i(w_i - \bar w) \pm Drw_{14}}{\bar w}$$
where the minus sign is used for $i = 1,4$ and the plus sign is used for $i=2,3$. (See also this chapter by Sergey Gavrilets.) Decreasing the frequency of a haplotype through recombination will necessarily increase the frequency of a different haplotype (adjusted by selection).
$D$ is the measure of linkage disequilibrium. It ranges from $-0.25$ to $0.25$ and indicates how much the haplotype frequencies deviate from an expectation of no linkage ($D=0$, the loci are independent).
First, assume no natural selection. That is, $w_i = 1$ and therefore ${\bar w} = 1$. When fitness is equal, the equations reduce down to
$$\Delta X_i = \pm Dr$$
Some recombination will occur between the linked loci proportional to the distance between them. In a large randomly mating population with no selection, $D$ will eventually decay to zero.  The larger the value of $r$, the quicker that decay will occur. Eventually, the haplotype frequencies will appear as if they are not linked. (See this figure.).
For example, assume that $A$ and $B$ are linked, with these four haplotype frequencies:
$$A_1B_1 = 0.5$$
$$A_1B_2 = 0.38$$
$$A_2B_1 = 0.05$$
$$A_2B_2 = 0.07$$
In the absence of natural selection and assuming that $r=0.1$, then $\Delta X_i = \pm 0.0016$, again with plus or minus depending on the value of $i$. This is the rate the frequencies will change each generation.  As $r$ decreases, the rate of decay decreases.
Differential fitness among the haplotypes will also affect the rate of decay. The fitness interactions among loci on the same chromosomes and between homologs can get rather complex depending heterozygote advantage, epistasis, etc. (see Lewontin 1964 for several fitness models).
The equation $w_i = X_1w_{i1} + X_2w_{i2} + X_3w_{i3} + X_4w_{i4}$ is the average fitness of the each haplotype in combination with all the other possible haplotypes. For example, for $w_1$, you're calculating the average fitness of four genotypes: $A_1B_1/A_1B_1$, $A_1B_1/A_1B_2$, $A_1B_1/A_2B_1$, and $A_1B_1/A_12B_2$.
I am uncertain why $w_{14}$ is used in the second term of the numerator. I thought it might be because that combination is heterozygous at both loci so recombination generates all four possible haplotypes. However, both Lewontin and Kojima (1960) and Lewontin (1964) use $w_{11}$. It may also be that a relative fitness component is necessary for comparison so the fitness of any one of the genotypes could be used there. I don't know.
Regrdless, as a simple example to illustrate the effects of selection, assume $r=0.1$, $\bar w = 0.5$, and $w_{14} = 1$ (heterozygote advantage, given maximum relative fitness). For $w_1 = 0.9$ then $\Delta X_1 = 0.397$. Selection strongly favors this haplotype in all genotypic combinations so it will increase in frequency.  For $w_1 = 0.3$ then $\Delta X_1 = -0.203$. The haplotype is selected against and decreases much more rapidly, relative to no selection at all.  The same idea would apply for the other haplotypes.
References
Lewontin, R.C. 1964. The interaction of selection and linkage. I. General considerations; heterotic models. Genetics 49: 49-67.
Lewontin, R.C. and K.-I. Kojima. 1960. The evolutionary dynamics of complex polymorphisms. Evolution 14: 458-472.