Understanding through Discussion


Welcome! You are not logged in. [ Login ]
EvC Forum active members: 85 (8984 total)
48 online now:
PaulK, Phat (AdminPhat) (2 members, 46 visitors)
Newest Member: Jerry Johnson
Upcoming Birthdays: Diomedes
Post Volume: Total: 877,570 Year: 9,318/23,288 Month: 333/1,544 Week: 47/561 Day: 0/47 Hour: 0/0


Thread  Details

Email This Thread
Newer Topic | Older Topic
  
Author Topic:   Do you really understand the mathematics of evolution?
Kleinman
Member
Posts: 386
From: United States
Joined: 10-06-2016
Member Rating: 1.1


Message 121 of 239 (877571)
06-18-2020 3:37 PM
Reply to: Message 120 by Taq
06-18-2020 2:56 PM


Re: Does competition accelerate DNA evolution?
Kleinman writes:

If you think that DNA evolution works differently in bacteria than any other replicator, produce your mathematical and empirical evidence to prove this.

Taq writes:

Would you agree that you mutation A and mutation B can occur in different individuals and then be combined in a single descendant? If so, that differs greatly from E. coli, does it not?


Yes, mutations A and B can occur in two different individuals and that by some form of lateral transfer they then can be combined in a single descendant. But there are some variants of e coli that can do bacterial conjugation, so no, it doesn't differ greatly. What you should try to learn is how to calculate the probability that that lateral transfer of genetic material can occur. And the probability of that occurring depends on the multiplication rule of probabilities. I posted a link to a paper where they reported that DNA evolution for eukaryotes works the same as for bacteria. Did you bother looking at that link? They looked at the DNA evolution of yeast cells subject to starvation and they saw the same evolutionary behavior that Lenski is seeing with his bacterial starvation evolutionary process.
Kleinman writes:

The only correction you need to do to your previous math is that humans are diploid so it only takes 1.5e9 member replications to get the 3e9 genome replications.

Taq writes:

That 3e9 number is for E. coli, not humans. You keep making this mistake. For humans, you only need 0.18e9 replications.


If you want to use a mutation rate of 5.56E-9, then your calculation is correct. But you still need to learn that the mutation rate is not the dominant factor in DNA evolution, its the multiplication rule which dominates DNA evolution.
Kleinman writes:

It's the second and ensuing beneficial mutations in that lineage that have a low probability of occurring.

Taq writes:

You are ignoring sexual reproduction. The first mutation can happen in one individual and then spread through the population. The second mutation can happen at the same time as the first mutation and spread through the population. Descendants of both lineages can mate and produce offspring with both mutations. With sexual reproduction you can have selection for each mutation in parallel instead of in series.


I am not ignoring sexual recombination. You make these vague claims about mutation spreading through the population and then members from both lineages mate and produce offspring with both mutations. Why doesn't this happen with hiv causing combination therapy to fail? It is clear you need more hints to do this math. You have a population size "n", in that population you have a variant with beneficial allele A and the number of members with that allele is nA, you have another variant in the population with beneficial allele B and the number of members with that allele in nB, the remainder of the population has neither allele, call them C and the number of members without either allele is nC. Then nA/n + nB/n + nC/n = 1. Write out the distribution function for this situation for the probability of an nA member recombining with an nB member to give an AB descendant. Remember, an nA member can recombine with another nA member, an nB member, or an nC member, likewise for the nB and nC members. Let's see if we can get you out of this vague understanding of DNA evolution, recombination and genetic transformation.

This message is a reply to:
 Message 120 by Taq, posted 06-18-2020 2:56 PM Taq has responded

Replies to this message:
 Message 122 by Taq, posted 06-18-2020 5:18 PM Kleinman has responded

  
Taq
Member
Posts: 8409
Joined: 03-06-2009
Member Rating: 1.9


Message 122 of 239 (877581)
06-18-2020 5:18 PM
Reply to: Message 121 by Kleinman
06-18-2020 3:37 PM


Re: Does competition accelerate DNA evolution?
Kleinman writes:

Yes, mutations A and B can occur in two different individuals and that by some form of lateral transfer they then can be combined in a single descendant.

Lateral genetic transfer occurs between species, not within species. The word you are looking for is "sexual reproduction".

But there are some variants of e coli that can do bacterial conjugation, so no, it doesn't differ greatly.

I know all about this. I have used this process in the lab to shuttle plasmids between different species of bacteria.

What differs greatly is the rate at which this occurs, and it also depends heavily on the strain of E. coli. Many lab strains of E. coli are missing the genes necessary for conjugation. I would suspect that this is the case with the strains used in the the only two experiments that exist in your universe.

I posted a link to a paper where they reported that DNA evolution for eukaryotes works the same as for bacteria.

Then discuss it.

But you still need to learn that the mutation rate is not the dominant factor in DNA evolution, its the multiplication rule which dominates DNA evolution.

Once again, you ignore sexual reproduction.

I am not ignoring sexual recombination. You make these vague claims about mutation spreading through the population and then members from both lineages mate and produce offspring with both mutations. Why doesn't this happen with hiv causing combination therapy to fail?

Because HIV is not a diploid eukaryote. That should be obvious to anyone with a tiny inkling of how genetics works.

Write out the distribution function for this situation for the probability of an nA member recombining with an nB member to give an AB descendant.

You do it. I don't see any reason to do more math problems for someone who doesn't understand why HIV is not diploid.


This message is a reply to:
 Message 121 by Kleinman, posted 06-18-2020 3:37 PM Kleinman has responded

Replies to this message:
 Message 123 by Kleinman, posted 06-18-2020 6:40 PM Taq has responded

  
Kleinman
Member
Posts: 386
From: United States
Joined: 10-06-2016
Member Rating: 1.1


Message 123 of 239 (877584)
06-18-2020 6:40 PM
Reply to: Message 122 by Taq
06-18-2020 5:18 PM


Re: Does competition accelerate DNA evolution?
Kleinman writes:

Yes, mutations A and B can occur in two different individuals and that by some form of lateral transfer they then can be combined in a single descendant.

Taq writes:

Lateral genetic transfer occurs between species, not within species. The word you are looking for is "sexual reproduction".


Too bad you don't have such precision when it comes to the mathematics of DNA evolution. And I said "some form of lateral transfer". But if you are going to nitpick on points like this, I'm going to nitpick you on the mathematics, only your mathematical errors are not nits.
Kleinman writes:

But there are some variants of e coli that can do bacterial conjugation, so no, it doesn't differ greatly.

Taq writes:

I know all about this. I have used this process in the lab to shuttle plasmids between different species of bacteria.

What differs greatly is the rate at which this occurs, and it also depends heavily on the strain of E. coli. Many lab strains of E. coli are missing the genes necessary for conjugation. I would suspect that this is the case with the strains used in the the only two experiments that exist in your universe.


Lenski specifically chose strains of e coli which don't do conjugation, I don't recall what Kishony said about his strain but I suspect he also chose a variant that doesn't do conjugation to prevent that complication. I doubt it would make a difference in either experiment anyway because before you can laterally transfer a resistance allele with a phage, the allele must evolve by the DNA evolutionary process. In your universe, you probably would introduce a resistance allele from outside the experiment on a phage and say see how quickly DNA evolution works when you take into account worldwide populations. By the way, there is plenty of other empirical data that supports the math that I've presented. It happens these two experiments are among the best-measured examples.
Kleinman writes:

I posted a link to a paper where they reported that DNA evolution for eukaryotes works the same as for bacteria.

Taq writes:

Then discuss it.


I am discussing it. In fact, I'm trying to explain to you why recombination has virtually no effect on DNA evolution. I've even gone so far to set up the mathematical problem. The only thing remaining is to take the variables I've given you and formulate the equation. It really is just an example of a random card drawing problem. Have you ever studied probability theory? You should if you want to understand how stochastic processes work.
Kleinman writes:

But you still need to learn that the mutation rate is not the dominant factor in DNA evolution, its the multiplication rule which dominates DNA evolution.

Taq writes:

Once again, you ignore sexual reproduction.


Nope!
Kleinman writes:

I am not ignoring sexual recombination. You make these vague claims about mutation spreading through the population and then members from both lineages mate and produce offspring with both mutations. Why doesn't this happen with hiv causing combination therapy to fail?

Taq writes:

Because HIV is not a diploid eukaryote. That should be obvious to anyone with a tiny inkling of how genetics works.


So, now you are claiming that hiv doesn't do recombination because it isn't a diploid eukaryote? So tell us, plants are eukaryotes. Why do combination herbicides inhibit the evolution of herbicide-resistant weeds? Some plants are even polyploid. And they do recombination. Why doesn't a variant weed with a resistance allele to one herbicide recombine with another variant with a resistance allele to a second herbicide to give a variant with resistance to both herbicides?
Kleinman writes:

Write out the distribution function for this situation for the probability of an nA member recombining with an nB member to give an AB descendant.

Taq writes:

You do it. I don't see any reason to do more math problems for someone who doesn't understand why HIV is not diploid.


Don't be silly, I know what kind of virus hiv is and it does recombination. But I can see you need help writing out the probability distribution function for random recombination. So open wide, I'm going to spoon-feed you:

f(x,y) = 2!/(x!y!(2−x−y)!) *(nA/n)^x *(nB/n)^y *(nC/n)^(2−x−y)

(x+y ⩽ 2)

And if you want to compute the probability of an A variant and B variant recombining to give an AB offspring, set x=y=1

When you do that, what you will find is that the only time you will have a reasonable probability of that happening (randomly) is if the A and B variants are at high relative frequency in the population. If the C variant represent most of the population, most recombination events will be AC, BC, or CC with the majority being CC recombintion. This is why recombination is not a factor in hiv evolution to 3-drug therapy. You just don't have a high enough frequency of the A and B variants to have a reasonable probability of that recombination event.


This message is a reply to:
 Message 122 by Taq, posted 06-18-2020 5:18 PM Taq has responded

Replies to this message:
 Message 124 by Taq, posted 06-19-2020 11:34 AM Kleinman has not yet responded
 Message 125 by Kleinman, posted 06-19-2020 1:32 PM Kleinman has not yet responded

  
Taq
Member
Posts: 8409
Joined: 03-06-2009
Member Rating: 1.9


Message 124 of 239 (877629)
06-19-2020 11:34 AM
Reply to: Message 123 by Kleinman
06-18-2020 6:40 PM


Re: Does competition accelerate DNA evolution?
Kleinman writes:

In fact, I'm trying to explain to you why recombination has virtually no effect on DNA evolution. I've even gone so far to set up the mathematical problem.

Great. Let's see it.

So, now you are claiming that hiv doesn't do recombination because it isn't a diploid eukaryote?

I am claiming that HIV is not a diploid organism that reproduces through sexual reproduction. Rare recombination events are nowhere close to mixing two haploid genomes together every generation, as well as about one recombination event per chromosome per generation. It is a question of scale.

Why do combination herbicides inhibit the evolution of herbicide-resistant weeds?

Because that would require evolution against both herbicides at once. How does this apply to human evolution? Which human adaptations required the same strict selection regime?

Don't be silly, I know what kind of virus hiv is and it does recombination.

What is the rate of recombination in HIV?


This message is a reply to:
 Message 123 by Kleinman, posted 06-18-2020 6:40 PM Kleinman has not yet responded

  
Kleinman
Member
Posts: 386
From: United States
Joined: 10-06-2016
Member Rating: 1.1


Message 125 of 239 (877646)
06-19-2020 1:32 PM
Reply to: Message 123 by Kleinman
06-18-2020 6:40 PM


Re: Does competition accelerate DNA evolution?
Kleinman writes:

In fact, I'm trying to explain to you why recombination has virtually no effect on DNA evolution. I've even gone so far to set up the mathematical problem.

Taq writes:

Great. Let's see it.


I've presented it, you just are having difficulty seeing it. Consider what will happen in the Kishony experiment if he uses a diploid eukaryote instead of a haploid prokaryote. For example, yeast, and instead of antibacterial agents, he uses antifungal agents. All else being the same with two antifungal agents being used. Assume a mutation rate of 1e-9.

The wild-type yeast has no beneficial mutations and a colony starts to grow in the drug-free region and after 1.5e9 replications, every possible mutation has occurred on different members of that population. In that set of mutants, there is a member that has a beneficial mutation for one antifungal agent (call it A) and a different member has a beneficial mutation for the other antifungal agent (call it B). Now I've shown how to compute the probability of those two members randomly meeting (Message 123) and recombining to give an offspring AB variant. If you plug x=y=1 into that equation from the earlier message, you get:

f(1,1) = P(A,B) = 2*(nA/n)*(nB/n)

The population size is n=1.5e9, nA=nB=1 then, nA/n=nB/n=1/1.5e9

The only way there is going to be a reasonable probability of the AB recombination event occurring is if nA/n and nB/n are at high frequency.


This message is a reply to:
 Message 123 by Kleinman, posted 06-18-2020 6:40 PM Kleinman has not yet responded

Replies to this message:
 Message 129 by Taq, posted 06-22-2020 12:14 PM Kleinman has responded

  
vimesey
Member
Posts: 1131
From: Birmingham, England
Joined: 09-21-2011
Member Rating: 3.5


Message 126 of 239 (877681)
06-20-2020 6:34 AM


If it helps to short circuit the run-around, you can see here what seems to me to be a re-hash of this discussion with a number of other scientists:

Kleinman: Four Questions About Evolution

Similar patterns to this thread.


Could there be any greater conceit, than for someone to believe that the universe has to be simple enough for them to be able to understand it ?

Replies to this message:
 Message 127 by PaulK, posted 06-20-2020 1:19 PM vimesey has not yet responded
 Message 128 by Kleinman, posted 06-21-2020 8:59 PM vimesey has not yet responded

  
PaulK
Member
Posts: 16316
Joined: 01-10-2003
Member Rating: 1.7


Message 127 of 239 (877705)
06-20-2020 1:19 PM
Reply to: Message 126 by vimesey
06-20-2020 6:34 AM


And quite enlightening it is, although hardly surprising, especially in light of the previous thread.

This message is a reply to:
 Message 126 by vimesey, posted 06-20-2020 6:34 AM vimesey has not yet responded

  
Kleinman
Member
Posts: 386
From: United States
Joined: 10-06-2016
Member Rating: 1.1


Message 128 of 239 (877786)
06-21-2020 8:59 PM
Reply to: Message 126 by vimesey
06-20-2020 6:34 AM


Any of you posters understand introductory probability theory?
vimesey writes:

If it helps to short circuit the run-around, you can see here what seems to me to be a re-hash of this discussion with a number of other scientists:

Kleinman: Four Questions About Evolution

Similar patterns to this thread.


Swamidass is about as familiar with the mathematics of evolution as the posters on this forum. For example, from the exchange we had on that forum:
Kleinman writes:

If you double the population size, do you double the probability of a beneficial mutation occurring?

Swamidass writes:

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


Swamidass is on the faculty of a medical school yet makes an undergraduate level blunder here. He is confusing additive events with complimentary events and it very easy to show why he is wrong. Let's say for population size N, the probability of a beneficial mutation occurring is 0.6. If you double the population size to 2N the probability of a beneficial mutation occurring goes to 1.2????? If you want to understand the mathematics of DNA evolution, you have to understand the fundamentals of probability theory. Swamidass has demonstrated he doesn't understand those fundamentals and so far, none of the posters on this forum have done much better. And by the way, I asked the same question from PhD evolutionary biologist John Harshman and he makes the same blunder. If you want to understand DNA evolution, you need to understand introductory probability theory.

This message is a reply to:
 Message 126 by vimesey, posted 06-20-2020 6:34 AM vimesey has not yet responded

  
Taq
Member
Posts: 8409
Joined: 03-06-2009
Member Rating: 1.9


Message 129 of 239 (877831)
06-22-2020 12:14 PM
Reply to: Message 125 by Kleinman
06-19-2020 1:32 PM


Re: Does competition accelerate DNA evolution?
Kleinman writes:

I've presented it, you just are having difficulty seeing it.

Then present it in a way that I can see it.

Consider what will happen in the Kishony experiment if he uses a diploid eukaryote instead of a haploid prokaryote. For example, yeast, and instead of antibacterial agents, he uses antifungal agents. All else being the same with two antifungal agents being used. Assume a mutation rate of 1e-9.

If we are talking about double resistance, then it wouldn't be long after the single resistant mutants emerge that we would have doubly resistant organisms as the descendants of the singly resistant founders start to mate with one another. Of course, yeast can reproduce asexually and sexually, so we would be talking about purely sexual reproduction.


This message is a reply to:
 Message 125 by Kleinman, posted 06-19-2020 1:32 PM Kleinman has responded

Replies to this message:
 Message 130 by Kleinman, posted 06-22-2020 12:36 PM Taq has responded

  
Kleinman
Member
Posts: 386
From: United States
Joined: 10-06-2016
Member Rating: 1.1


Message 130 of 239 (877835)
06-22-2020 12:36 PM
Reply to: Message 129 by Taq
06-22-2020 12:14 PM


Re: Does competition accelerate DNA evolution?
Kleinman writes:

Consider what will happen in the Kishony experiment if he uses a diploid eukaryote instead of a haploid prokaryote. For example, yeast, and instead of antibacterial agents, he uses antifungal agents. All else being the same with two antifungal agents being used. Assume a mutation rate of 1e-9.

Taq writes:

If we are talking about double resistance, then it wouldn't be long after the single resistant mutants emerge that we would have doubly resistant organisms as the descendants of the singly resistant founders start to mate with one another. Of course, yeast can reproduce asexually and sexually, so we would be talking about purely sexual reproduction.


Using your math, the single resistant mutants would emerge after 3e9 genome (gene) replications, it would then take another 3e9 replications of that variant for the second beneficial mutation to occur on the variant with the first beneficial mutation. If the experiment is performed with a single selection pressure as done with the present Kishony experiment, that 3e9 replications will occur in the next higher drug concentration in about 30 generations of doublings. If it requires both mutations before the variant can grow in the next higher drug-concentration region (higher concentration of drugs or two drugs) then the 3e9 replications of that variant must occur in the original founder colony which already has only slightly less than 3e9 members and it will also be doing 30 doublings. That's why the colony size must be in the trillions before the increase in fitness can be achieved. I hope you now understand why recombination does not make a difference in the DNA evolution process. Unless the members with the individual beneficial mutations are at high frequency, the probability of that recombination event will be minuscule. Even if the yeast are replicating sexually, it has a negligible effect on the DNA evolutionary process.

This message is a reply to:
 Message 129 by Taq, posted 06-22-2020 12:14 PM Taq has responded

Replies to this message:
 Message 131 by Taq, posted 06-22-2020 12:42 PM Kleinman has responded

  
Taq
Member
Posts: 8409
Joined: 03-06-2009
Member Rating: 1.9


Message 131 of 239 (877836)
06-22-2020 12:42 PM
Reply to: Message 130 by Kleinman
06-22-2020 12:36 PM


Re: Does competition accelerate DNA evolution?
Kleinman writes:

Using your math, the single resistant mutants would emerge after 3e9 genome (gene) replications, it would then take another 3e9 replications of that variant for the second beneficial mutation to occur on the variant with the first beneficial mutation.

That's not my math. The single resistant mutants would occur separately in two individuals, and soon after each one emerges, about 3e9 replications, the descendants of those two mutants would mate and have offspring with both mutations.


This message is a reply to:
 Message 130 by Kleinman, posted 06-22-2020 12:36 PM Kleinman has responded

Replies to this message:
 Message 132 by Kleinman, posted 06-22-2020 1:52 PM Taq has responded

  
Kleinman
Member
Posts: 386
From: United States
Joined: 10-06-2016
Member Rating: 1.1


Message 132 of 239 (877840)
06-22-2020 1:52 PM
Reply to: Message 131 by Taq
06-22-2020 12:42 PM


Re: Does competition accelerate DNA evolution?
Kleinman writes:

Using your math, the single resistant mutants would emerge after 3e9 genome (gene) replications, it would then take another 3e9 replications of that variant for the second beneficial mutation to occur on the variant with the first beneficial mutation.

Taq writes:

That's not my math. The single resistant mutants would occur separately in two individuals, and soon after each one emerges, about 3e9 replications, the descendants of those two mutants would mate and have offspring with both mutations.


Sure it is your math Message 55.
Taq writes:

The final equation is 1.38E7/(4.6E6*mu) as the number of replications needed.


And of your 3e9 members in that population, you are going to have one member with the A mutation and a different member with the B mutation. And the rest of the population will have neither mutation. So what do those A and B variants do? Is there an internet web site where yeast can meet so that the correct variants mate?

This message is a reply to:
 Message 131 by Taq, posted 06-22-2020 12:42 PM Taq has responded

Replies to this message:
 Message 133 by Taq, posted 06-22-2020 3:12 PM Kleinman has responded

  
Taq
Member
Posts: 8409
Joined: 03-06-2009
Member Rating: 1.9


Message 133 of 239 (877847)
06-22-2020 3:12 PM
Reply to: Message 132 by Kleinman
06-22-2020 1:52 PM


Re: Does competition accelerate DNA evolution?
Kleinman writes:

So what do those A and B variants do? Is there an internet web site where yeast can meet so that the correct variants mate?

If conditions are driving an increase in both variants then it won't take long for their numbers to increase and random mating matching two organisms that each have one of the variants.

This is basic sexual reproduction. I'm not sure what you are arguing against here. If there are two separate beneficial alleles in the population then it won't take long for those alleles to become common and for both of them to be found in a single offspring. You certainly don't have to wait for serial mutations in a single lineage.


This message is a reply to:
 Message 132 by Kleinman, posted 06-22-2020 1:52 PM Kleinman has responded

Replies to this message:
 Message 134 by Kleinman, posted 06-22-2020 4:44 PM Taq has responded

  
Kleinman
Member
Posts: 386
From: United States
Joined: 10-06-2016
Member Rating: 1.1


Message 134 of 239 (877855)
06-22-2020 4:44 PM
Reply to: Message 133 by Taq
06-22-2020 3:12 PM


Re: Does competition accelerate DNA evolution?
Kleinman writes:

So what do those A and B variants do? Is there an internet web site where yeast can meet so that the correct variants mate?

Taq writes:

If conditions are driving an increase in both variants then it won't take long for their numbers to increase and random mating matching two organisms that each have one of the variants.


Sure, that is Darwin's finches example. The environment selects out all variants that can't feed on the particular food source because of the incorrect beak shape and the remaining variants increase in frequency and the population inbreed for those phenotypes. But what that has done is reduced the diversity of the individual populations.
Taq writes:

This is basic sexual reproduction. I'm not sure what you are arguing against here. If there are two separate beneficial alleles in the population then it won't take long for those alleles to become common and for both of them to be found in a single offspring. You certainly don't have to wait for serial mutations in a single lineage.


Here's where your argument falls apart. Unless those alleles already exist in the population such as with Darwin's finches, you still need to get the appropriate mutations for the next evolutionary step. And you know that it takes 3e9 replication to get those variants. You need to learn that the correct alleles don't necessarily exist for a particular evolutionary process. If they did, you could start a dog breeding program and breed cats from those dogs The same principle applies to the evolution of reptiles to birds. The problem for you is that the correct alleles don't exist in the gene pool to make that kind of genetic transformation. You still need DNA evolution to get those alleles and as you know, it takes 3e9 replications for each one of those mutations. Recombination only works under very specific circumstances and does little to change DNA evolution.

This message is a reply to:
 Message 133 by Taq, posted 06-22-2020 3:12 PM Taq has responded

Replies to this message:
 Message 135 by Taq, posted 06-22-2020 5:20 PM Kleinman has responded

  
Taq
Member
Posts: 8409
Joined: 03-06-2009
Member Rating: 1.9


Message 135 of 239 (877862)
06-22-2020 5:20 PM
Reply to: Message 134 by Kleinman
06-22-2020 4:44 PM


Re: Does competition accelerate DNA evolution?
Kleinman writes:

The environment selects out all variants that can't feed on the particular food source because of the incorrect beak shape and the remaining variants increase in frequency and the population inbreed for those phenotypes. But what that has done is reduced the diversity of the individual populations.

If there were two traits under selection, wouldn't that increase the number of individuals for each of those traits, and increase the chances of two organisms mating and having those traits combined in their offspring?

Unless those alleles already exist in the population such as with Darwin's finches, you still need to get the appropriate mutations for the next evolutionary step. And you know that it takes 3e9 replication to get those variants.

So after 3e9 replications we have nearly every possible beneficial point mutation in eat least one individual. What happens when they start mating and mixing those beneficial mutations together?


This message is a reply to:
 Message 134 by Kleinman, posted 06-22-2020 4:44 PM Kleinman has responded

Replies to this message:
 Message 136 by Kleinman, posted 06-22-2020 5:55 PM Taq has responded

  
Newer Topic | Older Topic
Jump to:


Copyright 2001-2018 by EvC Forum, All Rights Reserved

™ Version 4.0 Beta
Innovative software from Qwixotic © 2020