How do the processes of binary fission and mitosis, despite their differences, support the theory that eukaryotes evolved from prokaryotes?

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As you said, the processes of binary fission and mitosis (or meiosis-you'll see why) are different in many aspects. But it's those differences that give us evidence to support our theory!

Let's first see how the theory tries to explain how eukaryotic cells developed from a larger prokaryotic cell.

The theory we're talking about is the Endosymbiosis Theory, which states that the eukaryotic cell developed from a larger prokaryotic cell engulfing (or eating, if you prefer) a smaller prokaryotic cell (or a bacteria) WITHOUT digesting it. The larger cell gave protection to the smaller one, which in turn gave energy to the larger one. Aerobic bacteria which have been "engulfed" eventually became what today is known as the mitochondria, while the photosynthetic ones became the chloroplasts.

Now, the most straightforward way to obtain evidence to support (or not) this theory is to look at present-day eukaryotic cells. One of the methods is to look for bacterial genes in the cell's DNA - and we did find a lot of bacterial genes in eukaryotic cell's DNA!

Another way is to compare the characteristics of bacteria with the characteristics of the mitochondria and the chloroplast. This is where mitosis and binary fission come into play! It is known that eukaryotic cells multiply through mitosis or meiosis while bacteria divide by binary fission. And we know that both the mitochondria and the chloroplast replicate INDEPENDENTLY of the eukaryotic cell by BINARY FISSION, just as bacteria do! Not only that, but they also are a similar size to bacteria and share many DNA and structural similarities! Thus, we have found good evidence to support our theory by looking at how these organelles divide and how this process is the same as that of bacteria.

I hope you now understand why this fact is seen as evidence to support the theory!

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