RufusAtticus

EvoMath 2 – Classical (Constant) Selection Model

This is the second installment of EvoMath. The first can be found <a href="http://iidb.org/cgi-bin/ultimatebb.cgi?ubb=get_topic&f=58&t=000020" target="_blank">here</a>.

Assumptions:
Same assumptions as the Hardy-Weinberg model except selection occurs.

• autosomal, diploid locus
• infinite population size
• constant viability selection
• no mutation
• no migration
• no fertility differences
• random, panmictic mating
• discrete, non overlapping generations
• adult census

Setup

Generation cycle: zygotes -&gt; variation selection -&gt; adults -&gt; random mating -&gt; zygotes

Let A1 and A2 denote the two alleles at the locus under investigation.
Let wij be the viability (fitness) of genotype AiAj; i,j=1,2.
Let p=freq(A1) and q=freq(A2); p+q=1.

Genotype: Zygote Freq, Viability, Adult Freq
A1A1: p^2, w11, (w11*p^2)/wbar
A1A2: 2pq, w12, (w12*2p(1-p))/wbar
A2A2: (1-p)^2, w22, (w22*(1-p)^2)/wbar

wbar = w11*p^2+w12*2p(1-p)+w22*(1-p)^2
This is the average fitness of the population.

What is the frequency of A1 in the next generation?

The frequency of the A1 genotype in the next generation (p’) will be determined by the frequency of adult genotypes and the probability that they produce an A1 gamete.

p’=(1*w11*p^2+0.5*w12*2*p*(1-p))/wbar = p*w1/wbar

w1 = p*w11+(1-p)*w12
w2 = p*w12+(1-p)*w22
wbar = p*w1+(1-p)*w2

w1 and w2 are the marginal fitness of allele A1 and A2 respectively. They represent the weighted average of the fitness that an allele receives depending on the type of gamete it fuses with.

What is the change in allele frequency between generations?

Let the change in freq(A1) between subsequent generations be denoted by d.

d = p’- p = p*w1/wbar – p = p*(w1/wbar – 1) = p*(w1 – wbar)/wbar
=p*( w1-p*w1-(1-p)*w2)/wbar = p*((1-p)*w1-(1-p)*w2)/wbar
=p*(1-p)*(w1-w2)/wbar

Obviously, d &gt; 0 if w1 &gt; w2 and d &lt; 0 if w2 &gt; w1. In other words, freq(A1) will increase if its marginal fitness is greater than the marginal fitness of A2. Freq(A1) will decrease if its marginal fitness is lower than the marginal fitness of A2. If the marginal frequencies are the same, the population is at equilibrium. The population also has two trival equilibria at p=0 and p=1, i.e. the population is fixed for an alleles.

More to come.

Any Questions?

~~RvFvS~~

(This followed a treatment by my advisor, Dr. Marjorie Asmussen.)

[ February 08, 2002: Message edited by: RufusAtticus ]

[ December 23, 2002: Message edited by: RufusAtticus ]</p>

Oolon Colluphid

Wow Rufus! <img src="graemlins/notworthy.gif" border="0" alt="[Not Worthy]" /> <img src="graemlins/notworthy.gif" border="0" alt="[Not Worthy]" />

Yeah, just one... what the hell was that?

More seriously, if I, who like to think I understand a little about evolution, didn't know what it meant (I gathered, very roughly, but only from the bits of English in between), then the cretinists won't have a hope, and will likely dismiss it as scientists being deliberately obscurantist to hide the flaws in evolution (which are of course widely known... ).

Oolon the mathematically inept

Dr Rick

RufusAtticus; your mathematical prowess exceeds that of the average life-sciences major. For instance, I learned most of my math with the aid of Big Bird.

If you have the time, would you please "walk" us through those calculations; I would like to try to understand them better.

Thanks, Rick

hammegk

Good stuff (well at least the parts I actually understood).

You are just the man to demonstrate why Haldane's Dilemma is NOT applicable to homo sap sap evolving from that fabled 'common ancestor' in time available.

What time frame do you prefer to use? 6 myr maybe?

RufusAtticus

Bump for EvolSkeptic's attention.

LiveFreeOrDie

What are these mathematical models used for? The reason I ask is because the list of assumptions seems rather difficult to meet in a real-world scenario.

KC

The assumptions are used to reduce or eliminate factors (such as genetic drift) which can also affect the results, so as to more clearly illustrate the point. Rufus wanted to show Evolskeptic how even marginal differences in fitness can result in evolutionary change. Extreme selection isn't required.Of course, in reality many other factors are involved as well, which doesn't help Evolskeptic's assertion anyway.For example, Evolskeptic seems to think genetic drift only applies to small populations. Actually, genetic drift applies to any finite population.The study of population genetics revolves around studying the factors that disturb the ideal equilibrium model (such as selection and genetic drift), and applying them so as to more closely describe and predict genetic change in natural populations.

KC

RufusAtticus

Well modeling does a lot of things. They help explain the real world. They can generate predictions that can be tested. They a very useful null models that can be tested. Their significance actually comes when a population doesn't fit a model, because then you can test and see what assumptions it violates.

The assumptions listed are not unreasonable at all. They don't apply to the population as a whole, but rather to the genetic locus in question. The obvious one that is always violated is "infinite population size." My advisor prefers to change that requirement to "near infinite/ very very large" because on the short term, compared to selection drift in those populations is so negliable that it might as well not exist. I stick to "infinite population size" because that is techniquely the actual assumption of the model. Real world populations can fit that assumption if they are considerablly large.

Hopefully I can make some more Evomaths.

~~RvFvS~~