## Further explanation of the population simulation

This page gives some more detail on the thought experiment showing how a low number of founders with fused chromosomes could result in an entire population of fused chromosome individuals. In the example scenario there are two couples with one individual in each couple being a carrier of a fused chromosome. The workings below explores what might happen if these people became the founders of a new human population.

In the Punnett squares used below, a normal pair of separate chromosomes is represented by the letter N and the letter F stands for a fused version of the same chromosomes. Remember that each person has two copies of all genetic material, so NF means they have one fused chromosome and one pair of normal chromosomes.

The first generation (F1) of children from the founding couples would have the following chromosome count proportions. Each child would have a 50% chance of inheriting a fused chromosome, so therefore they have a 50% chance of having 45 chromosomes rather than 46.

```
50% NF = 45 chromosomes
50% NN = 46 chromosomes
```

Moving ahead in time to when the children are old enough to have children of their own, and using a Punnett square, we can see that 25% of the random pairings would be between pairs of 45 chromosome individuals.

First generation pairings
NF(45) NN(46)
NF(45) NF x NF NF x NN
NN(46) NN x NF NN x NN
```25% NN x NN pairings (46 x 46 chromosomes)
50% NF x NN pairings (45 x 46 chromosomes)
25% NF x NF pairings (45 x 45 chromosomes)
```

The following Punnett squares show what happens when people with the various abnormal chromosome counts mate, it shows what percentage of second children end up with fused chromosomes from each union.

46 crossed with 45
Fused Normal
Normal NF NN
Normal NF NN
50% NF (45 chromosomes)
50% NN (46 chromosomes)

45 crossed with 45
Fused Normal
Fused FF FN
Normal NF NN
25% FF (44 chromosomes)
50% FN (45 chromosomes)
25% NN (46 chromosomes)

So there are three possible chromosomes counts in the second (F2) generation. These are 44, 45 and 46 in these proportions:

```
6.25% of = FF = 44 chromosomes
37.50% of = NF = 45 chromosomes
56.25% of = NN = 46 chromosomes
```
There was a 25% chance of a pairing between two 45 chromosome individuals, and of that pairing 25% of the offspring would have 44 chromosomes. That means that 6.25% (25% of 25%) of the next generation would have 44 chromosomes. For the next generation we need another two Punnett squares because there are now 44 chromosome individuals in the population.

45 crossed with 44
Fused Fused
Fused FF FF
Normal NF NF
50% FF (44 chromosomes)
50% NF (45 chromosomes)

46 crossed with 44
Fused Fused
Normal NF NF
Normal NF NF
100% NF (45 chromosomes)

The ratio of each chromosome count stays the same as the previous generation, and if all things were equal then further generations have the same proportions forever. However those ratios would be very sensitive to both random drift and any advantages or disadvantages caused by the fused chromosomes. These advantages could be absolutely anything that changed the reproductive success (number of children) of the individuals, for example stronger muscles, better reasoning skills or simply an attractive face. Natural selection over time would ensure that the fused chromosomes either prospered or disappeared from the population.

The effect of small changes in reproductive success can be demonstrated using a simple population simulator written in perl (Ref 1) to model a population where 50% of the founders had a fused chromosome. The graph below is the result of running this program 100 times with a population of 1000 individuals each time. Each run continued until either the fused chromosomes took over completely, or vanished from the population.

For this graph the fused chromosomes were given a reproductive advantage by supplying a 1% chance of a single additional child at each mating.

Simulation of fused chromosomes having a 1% advantage

The graph is the average shape of the population change when fused chromosome individuals won and dominated the entire population. The fused chromosome won 75% of the time and did so by generation 2607 on average. The affect of reproductive advantage is even more pronounced if the percentage chance of an extra child per mating is increased to 2%. At that probability the fused chromosomes won 98% of the time, and they did so by generation 2006 on average.

The control case of no reproductive advantage showed that the fused chromosomes win in the same proportion as they exist in the original population, this is due simply to genetic drift. In our simulation of two fused chromosome carriers (45 chromosomes) and two individuals with a normal chromosome count (46 chromosomes) the fused chromosomes win 33% of the time.

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References:
• Ref 1 : chromosome_44_sim.txt is a perl program which needs a perl interpreter to run.