- [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?

(gentle music)

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.

(gentle music)

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.

(gentle music)