What I wonder is this: How did the first single-celled organisms gradually change to start off with? If some managed to replicate more copies of their DNA, what mutations aided them in this journey?
There's no way to do more than speculate on this; even if we could recreate it, the evidence would only show it was a workable mechanism, not
the mechanism.
However, we do know what sort of changes could have conferred advantages. But lets start at the beginning.
The fundamentals of life are 1) a replicating unit, 2) mechanisms of energy transfer and 3) structures which protect and facilitate 1) and 2). Basically all living things are complex, imperfectly replicating chemical reactions.
However, at its very core it is conceivable to have the above occur without a contained cellular unit. Basically a form of protolife could focus on no 1) only. To replicate, energy merely in the form of temperature fluctuation could be sufficient (if slow and inefficient). Enzymes and supporting structures are definitely useful, but it pays to remember enzymes are biological catalysts which help reduce time and energy factors. They are not 100% necessary for a replicating unit to replicate, given time and energy.
So, we can safely see the first living components as something of a polymer which in a changing environment will itself create small, imperfect copies. Most polymers would immediately fall apart without replicating again. But some sequences, through virtue of variation in their sequence, would have an advantage of lasting longer and replicating more. Here is a very basic form of natural selection through competition - those that can last longer to replicate will steal the monomers of those that didn't. Hence natural selection isn't just a biological thing, but a function of any competing systems, such as chemical reactions.
Some of those stable polymers might be more unstable than others, replicating less faithfully. Some might lengthen, some might conform to more stable shapes.
Given time and enough variations through imperfect replications, some might form with physical properties which could allow it some simple enzymatic function. All of this is possible without the need of a cell or even recognisable metabolism. Those sequences which take advantage of environmental features such as other naturally occurring chemical formations (perhaps the ability to 'hide' inside micelles, which are single-layered fat bubbles) would again have the edge on other sequences.
It pays to remember that any such replicating sequence would need to have physical properties before it had coding properties. Coding for protein-based structures is useful and a great evolutionary feature, but it is not 100% necessary to have inefficient replicating nucleic-acid polymers.
Plus, I'm still no closer to understanding mutations. When Henry VIII's wife was declared a witch because of her sixth thumb, is that an example of random mutation that could in theory help her?
A mutation is any change in the sequence of our coding. It can confer a change to the polymer's physical structure (it introduces a bias in the way it physically bends) or it can confer a change in the coding sequence (protein structures it codes for will be different).
Most single examples of mutation, by virtue of an evolutionary 'redundancy', do not result directly in a change in the organism. If there is a change, it will probably be bad and the organism will not continue into any functioning maturity, usually ceasing replication after several cellular splits (depending on what no longer works due to the mutation). However, on occasion a mutation will occur which confers a slight advantage. Occasionally significant advantages occur when a mutation occurs which makes other 'neutral' mutations provide an advantage as well (you'll have to forgive me for not whipping out an example here, as I'm trying to remember the paper I read last year which went into detail on this).
Advantageous mutations are a numbers game. Small differences that give you the slightest edge, of even a few calories a year being retained instead of used for survival, could allow you to have perhaps a few more offspring than your competitors. It doesn't sound like much, but mathematically it could provide just enough of an edge so in several generations your variation has a significant representation in a population. Of course, it will all depend on the mutation, too. Sometimes a single mutation is enough to inspire a speciation event across a few generations.
I was born with two toes that point away from all my other toes, bending in a strange half moon shape. Is that an example of random mutation?
It would depend on whether it was the result of genetic variation or the environmental factors present while you were forming. It would also then depend on whether by having slightly different toes made a difference to how the genes coding for that variation were passed on (do they give you a chance to reproduce more through providing more energy? Do they allow you to help your siblings to reproduce more, increasing the chance that a future generation will have that variation either through the gene or through the mutation that gave you that gene). In itself it also depends on the environment and whether your toe variation worked well with it or not.
Athon