@kleinman, nice meet you. It seems you might be this Alan Kleinman:

That is usually the best approach. Seeing as you are scientist, trying to make sense of what I’ve said elsewhere, I’ll answer your questions. This is your area, so feel free to teach me the nuances.

Yes. Doubling the population size doubles the probability of a beneficial mutation, all else being equal.

In combination therapy, HIV faces multiple drugs at the same time, so it takes multiple mutations to occur at the same time to survive. This dramatically reduces the probability of resistance. That is why combination therapy prevents resistance from emerging, while individual therapy causes resistance.

This is too poorly specified for me to answer without fearing a “gotcha” for just not using the same definition as you. I’d tentatively say “yes”, but this is a poorly specified question. What I would say is that there is overwhelming evidence for common descent and that the overwhelming majority of mutations are spontaneuos.

Well, biological evolution does not violate the basic rules of probability, and the multiplication rule certainly applies to mathematical models of biological evolution. However, at the same time, it is common for the multiplication rule to be misapplied to biological evolution. Once again, you’ll have to specify more clearly the examples you are thinking about. It is easy to find examples of appropriate and inappropriate application of the rule.

There, I answered your questions. Let me offer a few of my own:

Do you mind telling us about yourself?

Are you generally arguing for or against mainstream evolutionary science?

This is a common error made by people not familiar with probability theory. You are trying to use the addition rule for complimentary events (that is the probability of the beneficial mutation occurring or not occurring).
If you want to see how to compute the probability correctly, check out this paper:

That is essentially correct and Edward Tatum talked about this effect in his 1958 Nobel Laureate Lecture. But why 3 drugs and not 2? And how do you determine the number of drugs necessary to treat any particular infectious disease or cancer? In other words, how does evolution work?

It’s not intended to be a “gotcha” question, it is intended to be a question to get you to start thinking about how evolutionary steps are linked. Again, it comes back to the principles of probability theory which govern the mathematics of adaptation.

That’s correct, the multiplication rule of probabilities applies to biological evolution. Aside from the empirical example of the evolution of HIV to combination therapy, I’ll show you how it applies in the Lenski LTEE and Kishony mega-plate experiments. Once you understand how these two experiments work, you will understand the fundamental principles of how evolution works.

Do you mind telling us about yourself?

Not at all, I’ve already answered some questions for Patrick about my background. What else would you like to know?

Are you generally arguing for or against mainstream evolutionary science?

I argue that Darwin got the fundamentals of evolution qualitatively correct but biologists and population geneticists have made a fundamental error in quantitating Darwinian evolution. I’ll show you where they made their fundamental error as the discussion goes on. But first, I want you to understand how random mutation and natural selection works.

What brings you here?

Your forum was mentioned on TalkOrigins and when I read your home page and what you think, I thought it might be interesting.

And, also, welcome. Glad to have you with us.

Thank you for your kind reception and having the courage to stick your neck out on my questions.

So @kleinman, you should back off a little here. I am a computational biologist, and use probability theory through out my work. You come in questioning my competence, and this seems without merit.

It is not an error where probabilities are very low. Beneficial mutations are very rare. When all P are close to zero:

1 - \prod (1 - p) \approx \sum p

I know the more complex formula, but I wouldn’t fault someone from saying that doubling the population doubles the probability of a rare event, as your question supposed. So no, it was not an error on my part, but a misleading question. You didn’t ask for the precise mathematical formula, nor did you choose an example where the probabilities were high (say, for example, coin flips).

I’m pretty sure I understand. You certainly haven’t show any errors in my understanding. You best just make your point.

Well, we moved on from darwinian evolution a long time ago, when Kimura ended it in 1968. Why don’t you engage with modern evolutionary theory instead? I’ve no interest in engaging a dead theory, except as a pedagogical tool.

Why would we focus on the rare case? The most common microevolutionary event is a neutral mutation. That makes up the majority of microevolutionary and macroevolutionary change. There is a immense amount of work that shows how this takes place. I’ll point you to two:

Now I can start piecing together your point. You are taking us to the so called “waiting time” problem. I’m very familiar with this, and have discussed it directly with people like Sanford, @Agauger , and @pnelson. My guess, piecing together some of your statements, is that you are going to argue they must all happen sequentially, and they cannot happen in parallel.

You are right. This is a deficiency with Darwinism. Which is why we left Darwinism behind a long time ago.

Kleinman’s model of mutation is simple enough. Mutations are independent and have a particular frequency. He models the probability of at least one particular mutation happening in a population of constant size in some number of generations. He doesn’t consider the probability of more than one such mutation occurring, and he doesn’t consider changes in frequency (or absolute number) due to selection, though he thinks he does.

For the record, this is addressed in medical school. It is also addressed in biology departments. It seems you weren’t taught this in medical school? Perhaps from a different era are you. Seriously though, this is just basic material.

Let me retract that invitation. You don’t have much to teach me here. Though I am happy to hear out your argument.

To give you an idea just how basic an idea it is, I cracked open my old undergrad evolutionary biology textbook: “Evolutionary Analysis” (fifth edition) by Herron and Freeman.

Here’s the contents page (ignore the chunk of Ediacaran microbial mat in the bottom left that I used to hold the page open):

Yes, I’m the one who failed the first part. I gave the same answer you did, for a different reason. I was thinking of the expected number of mutations rather than the probability of at least one. And in fact I think the expected number is the better one to use in any computations. And of course you’re also right for small mutation rates.

It’s been a while since I’ve posted on a BBS system, please be patient while I get the formatting correct. And yes, I do use these question quite often and with good reason.

I use the first question to identify those who can recognize complementary events.

The second question to identify those who can recognize that the treatment of hiv is being done using the multiplication rule of probabilities to reduce the probability of adaptation.

I use the third question to identify those who recognize that a microevolutionary event is a random event and therefore the joint probability of multiple microevolutionary events will be computed using the multiplication rule.

And the fourth question I use to identify those who think that somehow biological evolution is not subject to the multiplication rule.

And I should say Dr. Swamidass that you are awfully quick to discard Darwinian evolution when it is this mechanism which causes drug resistance and cancer treatments to fail. In a nutshell, Darwinian evolution breaks down into two categories. The first category is what Darwin calls the struggle for existence (competition) and the second category is adaptation. They are different phenomena with different governing physics and mathematics. We can also address Kimura’s concept of drift and see how this fits in.

Well, the diagnostic failed. It is a trap question, apparently. I got the answer correct, but you scored it incorrect.

Sort of. It is not precisely multiplication. You are leaving out some key parts of the correct modeling. It is a good enough approximation for now though.

This is valid reasoning.

Once again, this was a poorly specified question. The multiplication rule is often misapplied to biological evolution, and appears that you are misapplying it.

Oh really? What is your modern evolutionary theory? Is it Kimura’s model of fixation and drift? Show us how you apply your “modern evolutionary theory” to predict the behavior of the Kishony and Lenski experiments and I’ll show you how to apply the mathematics of Darwinian evolution to very accurately predict the behavior of these experiments as well as predict when you are going to have treatment failure for a cancer treatment based on the number of drugs used.

@kleinman , I’m not paid to be your tutor in evoltuionary science. I encourage you to pick up the book that @evograd pointed to, and take a look at some of the threads on this blog, two of which I already linked:

And you easily fell into the trap. Have you ever actually plotted out this probability curve. It is a sigmoidal curve and your linear approximation is only accurate when the probability is close to 0. The interesting part of that curve occurs in the sigmoidal portion because that is where the probabilities of the particular mutation occurring become reasonable for that event occurring. You have chopped off that portion of the curve and with your approximation will not know how many replications are needed for that event to occur.

It is precisely multiplication and I have left out no key parts. My math has been examined by experts in probability theory. But if you think I’ve left out something, show us.

This is valid reasoning.

I’ll give you a small point here. Competition is not a stochastic process where the multiplication rule would be applied but adaptation is a stochastic process where the multiplication rule must be used to compute the joint probability of microevolutionary steps.

@kleinman, I’d like to take a brief pause from the debate here and ask what you are hoping to accomplish with your participation here. It’s clear that you aren’t hoping to learn anything since you already have all the answers. It is also quite unlikely that you have secret knowledge to impart that is unavailable to the PhD researchers and graduate students that regularly contribute here. Perhaps you just like to argue. Or perhaps there is some other reason I haven’t yet considered.

You don’t have to write a simulation. There are many programs out there. Take a look at SLIM 3, if you like, or ask @davecarlson and @evograd for help.

Your challenge is to identify a well specified question, answerable with a simulation, where you and the experts here disagree on the results. Several outcomes are possible from this experiment.

You will not be able to find such a case, which would demonstrate you cannot substabtiate any of your disagreements are salient.

Your prediction on the experimental results could be correct, and ours would be wrong. We would learn something.

Your prediction is wrong, and ours is right. You would learn something.

You will ignore this test of your work, missing your best opportunity in a long time to be heard.

So, let’s see how this plays out. Please produce the parameters of a well specified experiment, and a prediction we can verify with simulation. Check to be sure we are making a different prediction than you. The. Let’s put it to the test.