- [Instructor] In the last video we saw that
when a creature's replication chance is higher
that its death chance its numbers can grow exponentially.
But we were left with a mystery.
Since complex organisms can't form without replication
it doesn't matter how good they are at replicating
if there aren't any around to replicate.
So how do they get their start?
So far, each time one of our creatures
has replicated it's done a perfect job.
They make an exact copy every time.
In the real world, though, nobody's perfect.
Sometimes a mistake during replication causes a new
kind of creature to appear.
We'll call these mistakes mutations.
Whatever's different about that kind of creature,
color in this case, will likely have some effect
on the statistics of that creature.
The blue kind of creature has a spontaneous
birth chance of one which means that every time step
in the simulation one of them will appear.
And each time step each blue creature has a 10% chance
of dying and a five percent chance of replicating.
So how would a green creature be different?
Well to be totally honest we're just
making things up right now.
We're not looking at actual creatures
in an actual environment so it's totally up
to us to decide what this change might do.
And then we'll use our equations and simulations
to see how things play out.
Eventually we will tie this to the real physical world,
stay tuned, but making stuff up and looking
at the consequences helps us understand the deep
mathematical truths that apply to all replicators.
Anyway, let's pick stats that aren't
too different from the blue kind.
After all it comes from a mistake in a process
that normally works.
So let's keep the same death and replication chances
per creature but get rid of
that non-replication birth rate.
So the green and blue creatures will be essentially
the same when they're alive but for some reason
green blob skin doesn't form
unless it comes from replication.
Oh, and we should als keep track
of the chance of a mutation.
Let's say it's 10%.
This means that whenever a blue creature replicates
there's a 10% chance that a green creature
will be produced instead of a blue one.
Alright let's look at a simulation.
After watching for a while we can see that the green ones
are having a hard time compared to the blue ones.
No offense to the green creatures
but this wasn't a very good mutation.
It's easy to think of evolution as a forward march
through making better and better organisms
but it's actually a bumbling mess
that just gets lucky sometimes.
We'll have a look at some good mutations in a moment
but first let's add these mutations
to our equations from last video.
Just like in the last video the total expected change
will be equal to the non-replication birth rate
plus a term that depends on the current number of creatures.
In the last video this was the replication chance
per creature minus the death chance per creature
times the total number of creatures.
Mutation affects what happens when a creature replicates
so to add mutations into this model we should
do something to this replication piece.
But what exactly?
To think about this say 10 blue creatures
replicate at a certain time.
Without mutations all 10 of the new creatures
will contribute to the number of blue creatures
at the next time step.
But since blue has a 10% mutation chance, on average
we'd expect one of those creatures
to actually come out green.
So only nine of the 10 new creatures will be added to blue.
We can account for this loss
by multiplying R by one minus M.
With a 10% mutation chance 90% of blue replications
will convert into blue creatures.
Alright, what about the equation for green creatures?
Again, we'll start with the equation from last video.
But before we do anything else notice how we're using
the same symbols in both equations.
The equations are actually talking about different
kinds of creatures, though, so let's add some labels
to keep things straight.
Okay, because the green creatures aren't mutating
we'll leave the replication chance alone
in this equation but we'll add another term
to the end to account for the new creatures
that appear due to the blue creatures mutating.
What should this term be?
Well if we multiply out the blue equation
to get rid of all the parentheses we can see this term here
which stands for the blue creatures that would have
come from replication but were instead
born green due to mutation.
These are the same creatures being added
to green's numbers so we can use the exact
same expression except add it.
This might not seem like it but this is a big moment.
The green creatures can't form on their own
but this mutation term shows
how the green creatures get above zero.
Their existence depends on replication alone
but in a way they've hacked the system
by depending on the replication of a different
kind of creature.
From their perspective it's basically the same
as being able to form without replication.
So even though there weren't too many green creatures
this is a moment to remember.
Alright, so that's one possible mistake a blue creature
can make while replicating but as you may know
from being a human there are many, many possible
mistakes you could make.
So let's add a few more which will lead us
to new kinds of creatures.
First, let's say blue can also mutate into this red kind,
also with a 10% chance each replication.
And we have another labeling issue here.
Both of these mutation chances belong to the blue
kind of creature so they should keep the label one.
But since there are two of them we should add another
label to tell them apart somehow.
The first mutation chance leads to green creatures
so we can add a two to keep track of that fact.
And this second mutation leads to the red kind
of creature which we can label with the number three.
With that sorted out this new red kind can only
come from replication just like the green kind.
It can't form on its own.
But it has a difference.
Its death chance is lower than that
of a green creature or a blue creature.
For some reason red blob skin is stronger
than green or blue blob skin or something like that.
Again, we're imagining a situation and seeing
what our model predicts about that kind of situation
to explore the deep mathematical truths behind replicators.
Next, this orange kind.
The blue creatures aren't the only ones that make mistakes.
The orange ones come from red ones
with a chance of, say, five percent.
The label three four here keeps track of the fact
that it's about creature type three, which is red,
mutating into creature type four.
The orange ones are different from the red ones
in that they have a higher replication chance,
10% instead of five percent.
Continuing with our imaginings, orange blob skin
is also strong with a low death chance
but it's also easier to make more of it
once it exists or something.
If you saw the last video you might notice
that the replication chance being higher
than the death chance means this creature type
has a shot at growing exponentially.
Alright, let's run a simulation starting with no creatures.
As expected the blue creatures are strong
coming out of the gate.
Like before we have green here and there.
Okay, there's some red so now we have
a chance to see some orange.
And now that we have a few oranges we can really
see that high replication chance doing work.
Blue looks like it's around its equilibrium number
but orange is growing exponentially so it blows right by.
Looking back at this tree we started building
two of these kinds of creatures are extra special.
The orange kind is the first type that grows exponentially.
Now that it's around there's lots and lots
of replication happening and that means lots
more chances for mistakes.
And that means lots of new kinds of creatures
have a chance to develop, eventually leading
to some pretty complex creatures.
And this blue kind is the first replicator.
It's simple enough to form without replication
in the red environment but it's also complex enough
to make more of itself.
And even though there's never a huge number of them
it's the seed that everything else comes from.
So if we were imaging this situation why did we
bother with the green and the red creatures?
Wouldn't it be more convenient to just have a blue creature
mutate directly into an orange creature?
Well maybe but that feels a little bit
too convenient, don't you think?
The real power of this way of looking at replicators
comes from the fact that even a messy system
full of bad mutations can lead
to some exponentially growing replicators.
In fact, the reality of how life started
is probably much more complex than what we built here.
But no matter how messy things get, as long as there's
a first replicator and it makes mistakes, odds are good
that we'll eventually stumble on to a creature
whose replication chance is higher than its death chance
and then the tree of life begins to sprout.
We're still working out the details of how this played
out in the real world but we do have a leading candidate
for the first replicator.
It's a molecule called RNA.
The basic story with RNA is that it's simple enough
to form without having to come from replication
and yet it replicates, creating more of itself
just like our friends the blue blobs.
You might be looking at a portrait of one of you ancestors
which is absolutely astounding if you ask me.
Thanks for watching.
In the next video we'll continue adding
to our model and we'll break the news to this orange blob
that exponential growth can't go on forever.
See you then.