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PROFESSOR: Our goal for today is to basically analyze
this simple model to death.
So we're first going to try to understand
the deterministic behavior of this model gene
expression, where we just get transcription of mRNA,
and then translation of protein.
And after we think we understand the mean behavior,
the deterministic dynamics, then we
will try to understand just stochastic behavior
in this model.
So we're going to try to understand
what's the distribution of mRNA in a cell
in this simple situation.
What's the distribution of protein?
What's going to be the bursting behavior?
Everything you can possibly think of
to ask about this model, we will hopefully
have asked by the end of today's class.
This simple model of gene expression,
as was indicated in the review, is perhaps
a reasonable description of gene expression
in bacteria, when the gene is in some active state.
So there's no repressor, for example, bound.
Although maybe even in the presence
of a repressor, if it's binding and unbinding,
maybe you still end up getting some sort of renormalization
that looks like this.
But this is first order, a reasonable description
of gene expression in bacteria.
And it's the model that was basically
used in the Sunney Xie paper that we talked about
on Tuesday.
And hopefully this model will allow
us to think a little bit more deeply about the data
that they obtained in that paper.
As always, we want to start by understanding
the basic aspects of the model.
So what we're going to do, is we're
going to go through a series of questions
of increasing difficulty.
And in some of them, we are indeed,
the answers will end up being something divided by something.
In which case you take advantage of your cards,
and illustrate that by putting something on top, something
below.
But just first in this model, what is the unit of time?
So if I say t is equal to 1, or delta t is equal to 1,
what am I referring to?
So we're not use the cards.
But in particular, the question is, is delta t equal to 1,
is that a cell cycle necessarily?
Yes or no, ready, three, two, one.
Well I guess now maybe I've complicated things by-- well,
this was really going to be relevant for the later ones.
All right.
Now I've totally confused you.
But can somebody offer why it may or may not
be-- how do we think about the unit of time in this model?
AUDIENCE: Usually the lifetime of one
of the species [INAUDIBLE].
PROFESSOR: Right.
OK so indeed, what we often do in these non-dimensionalized
models is we set something equal to 1.
Have we set anything equal to 1 here?
No.
So in principle, we've said there's
some degradation rate of the mRNA, some degradation
rate of the protein.
And in general, those will be given in some units involving
seconds or minutes or hours.
So in general, so at this stage, we
have not yet-- we have not actually
gotten to this sort of non-dimensionalized version
of any model.
So in this case this is going to be something
like a seconds, or minutes, or hours,
whatever units we use for those degradation rates.
So we have not done anything where it's the cell generation
time, or the protein lifetime, mRNA lifetime, or anything
like that.
Here everybody happy with this statement so far?
So we'll go ahead and vote here.
So we're going to do some A, B, C, D's.
And you can always combine anything you want.
So we'll go ahead and say this is
the synthesis rate of the mRNA.
This is the degradation rate for the mRNA, the synthesis
rate for the protein, the degradation
rate for the protein.
And if you're just confused, you can just do this.
But in general, for any of the questions we're going to do,
you can do some combination of these guys
by putting things in numerator and denominator.
Yes?
AUDIENCE: Calculate the population of the cells
that were hidden?
PROFESSOR: Yes.
Question is, if you just look at the cell population,
and you find it's growing exponentially,
the question is what is going to be that rate
of exponential growth.
Have I done something wrong already?
AUDIENCE: [INAUDIBLE]?
PROFESSOR: OK.
But I am going to say that now for this we're
going to assume that the protein is stable.
So it's not actually degraded.
This is to remind you of what we read about in chapter one,
maybe of Uri's book, maybe chapter two.
I'll give you 10 seconds to think about this.
Do you need more time?
All right, Ready, three, two, one.
OK.
We got a bunch of C's and a bunch of D's and some E's.
All right.
So the E's are going to argue with me, presumably rather than
a neighbor.
OK.
I think that there are enough people that
are disagreeing on this to maybe go ahead, and turn.
You should be able to find somebody
that disagrees with you.
The distribution was a bit patchy, unfortunately.
Did you guys-- you guys are worried that you're not going
to be able to find somebody.
OK.
Fine, fine.
Yeah?
AUDIENCE: So if the protein is stable, ah,
so the mRNA may not be stable?
PROFESSOR: The mRNA may not be stable.
AUDIENCE: Ah, OK.
That makes sense.
PROFESSOR: And in general which one typically
has a longer lifetime?
AUDIENCE: Proteins.
PROFESSOR: Proteins typically have a longer lifetime.
Right.
So mRNA are actively degraded, typically.
They're also just kind of less stable intrinsically.
But what we're going to assume for now
is that we're working with stable proteins.
In which case the growth rate of the population
will just be this effective degradation
rate of the protein.
So in this model, even if we say there's
no active degradation of the protein,
still there's going to be some effective degradation that's
due to dilution.
So we can say effective, if you like.
So the rate of exponential growth of the population
will be equal to this effective degradation
rate for the protein, if it's stable.
AUDIENCE: So you're talking about the population
of the protein?
PROFESSOR: No.
The growth rate of the cell population.
So this is if we go in there, and you
go into your spectrophotometer.
And you measure population-- numbers in function of time
is growing exponentially.
It'll grow exponentially with this rate.
Because this is what's causing the dilution.
In some ways if you stop making the protein,
and you double the number of cells, and that means
the concentration of the protein in each cell
has to go down by a factor of two.
So that's the statement.
Are there any questions about why I'm making this argument?
Yes?
AUDIENCE: What was the relevance of the protein being stable?
PROFESSOR: All right.
So the relevance of the protein being
stable, because this is in general, this delta, this
is the effective rate.
This is going to be equal to the growth rate of the population.
So you might call it gamma growth
plus the actual degradation.
I don't want to use the same, but I'll just
say plus the degradation rate.
And this is a true physical degradation, true degradation
rate of the protein.
So if it's stable, then we say that this thing is zero.
So when we say stable protein, it
means there's no degradation of the protein.
So this physical degradation rate is zero.
And then the effective degradation rate of the protein
is just equal to the growth rate of the population.
AUDIENCE: OK, so no degradation means stable, basically?
PROFESSOR: Yes, sorry, yeah.
Any other questions about what I mean by this?
So now what we want to do is ask a few other quantities
about this model.
So for example, what will be the number of mRNA per cell?
And this is always going to be the mean.
I'll give you 20 seconds.
In this model what is the mean number of mRNA per cell?
All right.
Ready?
Three, two, one.
And we have let's say a majority of the group
is saying it's A over B, which corresponds to the synthesis
rate of the mRNA divided by the degradation.
Some people are this one?
Yes, so this is indeed, synthesis rate
divided by the degradation rate.
Now this is saying that what happens later
doesn't really matter, for the mean mRNA number.
Because it's just that it's going to be made at some rate.
Its lifetime is given by 1 over delta m.
Now this thing, of course, is again as always,
the effective degradation rate.
So it's the sum of the sort of physical degradation rate,
plus this dilution due to growth.
But in general, the true degradation,
the physical degradation is much faster than the cell division
rate.
So this is very close to actually just
the physical degradation rate.
But in any case, it's just delta m, regardless.
Are there any questions about why this is the way it is?
Yes?
AUDIENCE: Does it matter whether it's only physical?
Because wouldn't it be the same if it were--
PROFESSOR: It doesn't matter that it's only-- exactly.
That's what I was trying to say.
So the way that this is written, it
doesn't matter whether the-- this is the answer regardless
of whether the physical degradation
rate is much larger than the growth rate or not.
Yeah.
All right.
What is this protein molecules per mRNA?
How many protein molecules are made from each mRNA?
Protein produced-- Do you need more time?
Remember.
This is again, the mean number of proteins
produced from a single mRNA or each mRNA.
Let's go ahead and vote, so I can see where we are.
Ready?
Three, two, one.
OK.
So we have, I'd say, so at least a majority
are saying it's going to be C over B Now.
All right.
So this is interesting.
So this is saying that really what's happening
is that there's a competition once you
make an mRNA that the proteins are going to be getting
fired off at some rate.
But eventually it's going to be degraded.
It's a competition between those two rates
that determines basically how many proteins,
how many times do you fire off a protein before you
get degraded.
Any questions about that logic?
AUDIENCE: Can you please just repeat that one more time?
PROFESSOR: Sure.
Right, so what we're assuming is that OK, an mRNA is produced.
And that's already happened.
So it doesn't matter what Sm is anymore.
So now we have an mRNA.
Eventually this mRNA will be degraded.
But before that happens, we want to know, basically
how many proteins do we expect to be made.
Now if Sp and delta m are the same that means
you kind of expect one protein to be made on average,
before it's degraded.
Or if Sp we're twice delta m, then you
would get two proteins made before it was degraded.
Now this is a mean statement.
We're about to start thinking-- in 10 minutes,
we'll think about this distribution.
And so we have to be careful.
But in terms of mean behavior, this thing is true.
AUDIENCE: So is this different than it
has been for the number of proteins per mRNA in the cell?
PROFESSOR: Is this different from the number
of proteins in the cell?
AUDIENCE: Number of proteins per mRNA in the cell.
Because then you will have to do the protein concentration
over mRNA concentration.
PROFESSOR: OK.
Right.
So this is not the same thing as asking about the ratio
of the number of proteins.
And we can calculate that as well.
Yeah these are different.
This is the number of protein molecules
produced from each mRNA.
So this is just talking about production.
Because indeed, the degradation rates
are going to be different.
So then we can see what that ends up being.
Any other questions about why this one is what it is?
How about the number of mRNA produced per cell cycle?
And for now we're going to ignore factors of log two.
Do you need more time?
So another 10 seconds.
AUDIENCE: Produced but not degraded?
PROFESSOR: Produced, yes.
We're just talking about production.
Because we've already calculated a number of mRNA in the cell.
But now we want to know the mean number produced.
For example, this is the same as the mean number
of protein bursts observed in Sunney Xie's paper.
But this is just the number of mRNA produced per cell cycle.
All right.
Let's see where we are.
Ready?
Three, two, one.
All right.
So we've got lots of A's over D's.
That's sounds nice.
So this is going to be some synthesis rate.
But now the relevant thing is this delta p.
Because that's the cell division rate.
So it's barring issues of log two,
it's approximately the synthesis rate
of the mRNA divided by delta p.
Because this is the growth rate of population.
Cell generation time is log two off of that.
Are there any questions about that statement?
All right, so this is the mean number.
Now from the paper, we know how this thing is distributed.
We should probably-- we're going to use a bunch of distributions
over the next couple.
So we can-- we like exponential distributions.
We like geometric distributions.
We like Poisson.
We like Gaussian.
And we like gamma.
These are various probability distributions.
The question is, how is it that now, not the mean,
but how is the number of mRNA produced
per cell cycle distributed.
Ready?
Three, two, one.
All right.
We've got some-- this side of the rooms a little bit
slower, maybe.
But that's OK.
So maybe some people are not confident of this statement.
OK.
So this one ends up being Poisson.
So this is indeed how the number of-- this
is number mRNA per cycle.
Now this is-- so Poisson, in general,
that's what you get if there's some probability per unit time
that something's going to happen,
and you want to know how many of them happen in some finite time
period.
That's basically the definition of a Poisson.
And this is, if you recall, this is what we talked about
on Tuesday.
The probability observe n, it's given by this mean number.
So if lambda is the mean, then we
get lambda to the n, over n factorial,
e to the minus lambda.
If you go ahead and calculate the mean of this,
you indeed get lambda.
So lambda is equal to the mean, which in this case
was around, well in the case of Sunney's paper,
does anybody remember what that roughly was?
It was around one.
Now what about this other one?
So we also have another mRNA problem,
which is that we calculated the mean number of mRNA per cell.
If you look at a cell, the mean number is this.
But what's the probability distribution
of the number of mRNA per cell?
So we probably-- I'm trying think it-- you probably
don't yet know this answer.
This ends up also being Poisson.
We're going to calculate this in a bit.
But this is very confusing somehow.
That both this thing and this thing, are Poisson.
But they're not the same Poisson,
in the sense they have different lambdas.
Which one is going to be larger?
This one or this one?
The bottom one, right?
And that's because delta m is much larger than delta p,
typically.
So indeed, if you ask, in Sunney's paper, for example,
there was just over one mRNA produced per cell cycle.
But the mean number of mRNA might have been 1/30th of that.
Because the degradation rate was just 1 and 1/2 minutes.
What that's saying is that in a typical situation
you would not see an mRNA in a cell in that condition.
We're going to calculate this in a moment.
So don't worry if you don't see why it's a Poisson.
But don't get confused.
There are two different distributions
that arise from the mRNA in the cell or in the cell cycle.
And they're different Poissons.
And I think that-- I mean I'm sure that in some deep sense
there's a reason that they're the same.
But it's somehow not immediately obvious.
So there was another one that we might have wanted to do,
which is the mean number of proteins in each cell.
Now this one is a bit harder.
And this one is going to take full advantage of the cards
that you have in front of you.
So be prepared.
I'm going to give you 30 seconds.
Because this one you might-- well
you might need a little bit more time.
AUDIENCE: This is hard.
PROFESSOR: Yeah.
Although I think that it's useful to see that
it can be a bit tricky.
Because this really is the simplest possible model.
We're going to talk about some models that
get to be horribly complicated.
And so it's useful to just make sure
you can nail down the intuition on this model.
All right.
Do you need more time?
It's OK if this is escaping you at this moment.
Why don't we go and see where we are?
Ready?
Three, two, one.
All right.
So you know all the naysayers on the cards,
now that you've done this, you feel
like it's an amazing system.
So it's AC over DB.
So the two, the product of the synthesis rates
divided by the product of degradation rates.
So what we have is the synthesis rate
for the mRNA divided by the degradation rate for the mRNA.
Synthesis rate for the proteins divided by the degradation rate
for the proteins.
Can somebody give us a verbal explanation
for why this might have been, or why this is?
Yes?
AUDIENCE: It's the same reasoning
as the number of mRNA per cell.
But instead of just a basel-- like a synthesis rate
doesn't depend on the concentration.
You're just multiplying the synthesis rate
by the number of mRNA.
PROFESSOR: Yeah, that's great.
OK.
So what you're saying is that this thing here was indeed,
we calculate that was the mean number of mRNA in the cell.
If you just start with something,
and you have a production and degradation rate.
OK.
Well that means that if you had one mRNA, then indeed that's
what the concentration would be, is this Sp divided by delta p.
But now we-- well we multiply that by the number of mRNA,
and then we are set.
All right.
Now another question.
We have a distribution, or a mean protein--
wait sorry, mean number.
We have the mean number of protein produced
from each mRNA, is something.
And the question is, is this the most likely number of proteins
to observe.
Is the distribution here, now this is a mean,
but now we want to start thinking
about the probabilistic stochastic elements.
Is this the most likely, is it like the number of proteins
observed from an mRNA?
The question is, is this most likely.
By which I mean is the probability distribution
peaked here.
So we're going to do an A as a yes, and B is a no.
Does everybody understand the question I'm trying to ask?
So an mRNA is here.
There's going to be some proteins made from it.
This is the mean.
What I want to know is, is that we should somehow expect?
In a sense, is the distribution peaked, the probability
distribution peaked around this value?
And C is-- do I want to do a depends?
Well you can always argue after.
Do you need more time?
AUDIENCE: Is this, you're saying,
is this the most likely number of proteins?
PROFESSOR: Yeah.
What I'm wondering is the mode there?
AUDIENCE: Right.
But only for this quantity?
PROFESSOR: Only yeah.
So now we're not doing means anymore.
We want to know if the probability distribution
of the protein produced from each mRNA
is the mode around this.
Ready?
Three, two, one.
All right.
We got a lot of no's, but some yeses.
So this is actually going to be a no.
And this was because the probability distribution.
The question is, what is the probability distribution
for the number of proteins produced from each mRNA.
It's going to be one of these.
Ready?
Three, two, one.
All right.
So we've got some difference.
But I'd say that most of the group
is saying it's going to be A or B.
And indeed these are almost the same distributions.
What's the difference between them?
AUDIENCE: One's discrete--
PROFESSOR: Right.
So this guy's discrete.
This guy is continuous.
Right.
Indeed when we're taking about the numbers,
then we should get-- it's a geometric.
But often we're kind of a little bit loose about these things.
So it's not a disaster if you said exponential.
But the key thing is that the distribution
looks something like-- so now I've certainly
drawn it as an exponential.
This is the probability of n as a function of n.
Of course, the geometric thing it looks--
AUDIENCE: Sorry.
So why should we expect that distribution?
PROFESSOR: Why should we expect the distribution?
So one answer is that because that's
what you read on Tuesday.
But let's go ahead and-- yes, but let's go ahead
and calculate it.
That's useful.
The way to think about this, in some ways,
there's another way to write this perhaps.
Which is that imagine you have an mRNA.
Now at some rate it's going to be degraded.
And maybe we'll keep the degradation rate down,
just for-- so there's a degradation rate.
But then if you'd like, we could draw it like this.
Where this is the synthesis rate for a protein.
And out pops a protein.
And so the idea is that is here we're in some state where,
OK, here we have an mRNA.
Here's the state where we don't have an mRNA.
Now this is the competition between those two rates
that I was telling you about.
There is some degradation rate for the mRNA.
Or there's a synthesis rate where we go around this loop.
If we come around this loop, we come back
to the state with an mRNA.
There still is an mRNA intact.
Just out pops a protein.
So then what we want to do, is we
want to think about what's the number of proteins
that we expect, not just the mean,
but the actual distribution.
So it's useful in these situations
to define some probability rho, which is the probability
you actually, if you're here, it's
the probability that you produce one protein at least.
The question is, which path do you take initially.
Well that's just given by the ratios.
So there's the rate that we take this circular path divided
by the sum of these two other rates.
And then what we can do is we can ask, well
what is the probability that 0 proteins are produced,
probability that we get 0.
Well if we take this path initially,
will we have 0 proteins?
AUDIENCE: No.
PROFESSOR: No.
Right, so this probability is indeed
simply equal to the probability that we do this first, which
is 1 minus rho.
Now what's the probably that we get 1 protein?
Well, only 1?
That's equal to the probability that we first take this path,
and then we take this path.
Well we can multiply those probabilities.
Because we first take the circular path
to make a protein.
And then we take the degradation path.
Well what's the probability we get 2?
Well that's just that we come around here once,
twice, and then degrade.
Now if you're not seeing a pattern here,
then we're in trouble.
So this says the probability of n
will then just be equal to rho to the n, 1 minus rho.
And indeed, it's always useful in order
to warm up your probability muscles,
to check to make sure that this is a normalized probability
distribution.
So sum over all possible ends indeed goes to 1.
And that's just because the sum over a bunch of rho to the n's
is equal to 1 divided by 1 minus rho, which is the term there.
And that goes to 1.
AUDIENCE: So this is making a pretty strong assumption
that they're all independent?
PROFESSOR: Yep, yep.
Yep.
This is assuming that if you've gone around once, you return.
But I've come back to the original state.
AUDIENCE: But do mRNAs like actually
get caught in ribosomes--
PROFESSOR: There are a lot of things that can be true.
And I would say that in biology and in life,
what you do is you first write down
the simplest possible model.
And then you go and you make measurements.
And you ask whether the simplest possible model can adequately
explain the data.
And if the answer is no, then you're
allowed to start thinking about other things.
Because everything's is in principle true.
In that mRNA, maybe it's this or that.
The question is whether it's significant.
And at least from the data from Sunney's group
would say that in that condition, in those cells,
that those things are not significant,
in the sense that you still get a geometric distribution.
Of course it could also be that those other things actually are
true and are significant.
But then you end up with some new parameters
that describe how things look as a result of all the complexity.
That's also OK in the sense that I'd
say that you can get a quantitative description
of the process by describing it as a geometric with just
a single free parameter.
And they found that the mean was four, or four or five.
The mean number of proteins produced from each mRNA.
But they got this geometric distribution in that paper.
Yes?
And I'll just mention here that the mean of this
is rho divided by 1 minus rho.
So what you see is that as rho goes to 1,
then this thing is going to diverge.
And that makes sense.
because as rho goes to 1, it's saying that you essentially
always synthesize another protein rather than degrading.
And before I move on, I just want
to say one more thing, which is that there
are many different definitions of the geometric distribution,
depending upon whether the probability of rho
is the probability of terminating,
or the probability of going around, and also depending
on whether you're asking what is the-- here
we're talking about the probability
distribution for the number of proteins produced.
Whereas we could have talked about the probability
distribution for the number of times we go around this loop
before, no, no sorry.
That is for the number of proteins produced.
So the other way you could have defined this is the number
of times where it's-- the number of cycles that you had to go
before you went here, in the sense that if you first go
here, you can either call that a 0 or a 1.
Do you see what I'm saying?
And reasonable people can disagree.
But you end up getting distributions that are
just a little bit different.
So watch out.
If you just memorize something, you
might have memorized the equation
for a different definition of this distribution.
Does everyone understand what I tried to say there?
Maybe?
Yeah?
AUDIENCE: When there's no degradation
is it still a Poisson?
PROFESSOR: Ah, if there's no degradation then would this
be a Poisson?
I mean, this would be infinity, right?
AUDIENCE: Right, [INAUDIBLE] protein,
this is done independently, like there's an mRNA.
[INAUDIBLE] proteins independently.
PROFESSOR: OK.
So I want to be clear.
This is, p of n is, this is the probability
distribution for number of proteins n,
produced from a single mRNA.
Now if there's no degradation of the mRNA,
then this thing is not even, I think,
defined in that the number of proteins
produced from that mRNA just really goes to infinity.
If you wanted to ask about the probability distribution
for the number of proteins produced
in some unit, some period of time that would indeed
be a Poisson distribution, assuming
that there's no degradation.
Do you understand what I'm trying to say?
AUDIENCE: So Sp is like 0?
PROFESSOR: If Sp, I'm sorry.
If Sp were 0?
AUDIENCE: Yeah. [INAUDIBLE].
PROFESSOR: OK.
And you're saying that what would be Poisson distributed?
AUDIENCE: The number of proteins.
PROFESSOR: Yeah, I think that actually-- no I
think that-- I think that you're probably right.
That as Sp goes to 0-- I'm a little bit worried that--
AUDIENCE: No, no, no.
Zero-th order, sorry.
PROFESSOR: Oh.
AUDIENCE: Not 0.
[INAUDIBLE]
PROFESSOR: OK.
Right, so the mRNA distribution we're about to find
is indeed going to be a Poisson at steady state.
And so if there's some process by which the protein
distribution is really just mirroring
the mRNA distribution, then it will also be Poisson.
Although I think you have to be careful about how you actually
implement that.
Because even in the absence of this geometric bursting,
different things, I think, can happen.
Because for example, if there were exactly 10 proteins
produced from each mRNA, then that probability distribution
is a shift.
But then it's no longer actually going
to be Poisson, because the mean and variance are going to scale
differently if you do that.
Let's maybe do the Poisson distribution for the mRNA
first.
And then we can try to touch back on this.
So this is a plot of kind of geometric distribution
with a mean of 3-4-ish.
Is everybody happy with where we are now?
OK.
Now from this, what we've said so far
is it obvious what the distribution of proteins
will be in a cell?
We can say obvious, yes.
Or not obvious, no.
Ready, just verbal, yes?
Ready or no?
All right.
Ready?
Three, two, one.
AUDIENCE: No.
PROFESSOR: No.
Right.
So we've said that the distribution
of size of protein bursts from single mRNA is geometric.
But that doesn't mean that that's
going to be the distribution of proteins in the cell.
And indeed after, we're going to find
that the distribution of mRNA is going to be Poisson.
But even then it's not obvious what the distribution
of proteins is.
All right.
So what we want to do now is we want
to introduce kind of a simple version of what's
known as the Master Equation.
Now you guys are going to do more reading on this
for the lecture on Tuesday.
Where we're going to talk about the Master Equation, as well
as the Fokker-Planck approximation.
Maybe the Gillespie algorithm, and so forth.
But I want to start by thinking about the this notion
in the simplest possible context.
So what we're going to do is we're
going to think about the world.
So we want to know the steady state, or the equilibrium
distribution of mRNA numbers in the cell, given this process.
So that's great.
We can-- so mRNA distribution, question mark.
Now in this case we don't care about Sp, delta p,
because the only things that are relevant are going to be these.
Now what we're going to do is we're
going to think about the world in which we just
defined states corresponding to the different numbers
of these mRNAs.
So there's a state where there's 0.
We can't go to the left, less than 0,
but we can go to the state where there's 1, or the state where
there's 2, and so forth.
Now the description here is supposed
to be the analog of this over there.
So this is trying to understand the situation where
the deterministic equations would be described
by m dot is equal to this some synthesis rate,
minus a degradation rate that's proportional to the number,
so minus delta m times m.
So what you can see is that the deterministic equations
are very simple.
We already calculated the equilibrium.
So when this thing is equal to 0,
then we get that m equilibrium is just
going to equal to the synthesis rate divided by the degradation
rate.
If we're away from the equilibrium
in this deterministic approximation,
how long is it going to take us to kind
of approach our equilibrium?
Verbal answer, ready?
Three, two, one.
[STUDENTS RESPOND]
PROFESSOR: Right.
So it's going to be 1 over delta m.
So this tells us the characteristic timescale
to come back.
So if we're-- this is the equilibrium Sm over delta m.
This is m as a function of time.
If we're below, we come here.
If we're above, we come here.
And this time is 1 over delta m.
Are the any questions about?
Now this-- so I want to highlight
that this is like the world's simplest dynamical equation,
almost the world's simplest.
Yet what we're going to find is that once we go over
and we try to understand the full probability
distribution of the stochastic system
then it's a little bit more complicated.
In particular, we end up with an infinite set
of differential equations.
So in general the Master Equation format,
where we're going to write differential equations for how
these probabilities change over time.
Now what we've done is we've traded a single differential
equation for an infinite number of differential equations.
So that's a bummer.
But on the other hand, it will allow
us to do the full stochastic treatment.
And it's also a nice, to me, the master equation
is useful in kind of two ways.
One is that it's going to be a tool for us to do
analytic calculations.
But it's also kind of a principled way
of organizing your thoughts so that you can go and do
stochastic simulations, if that's what you want to do.
So it's also just kind of like a weigh station
to kind of help you set up your simulation.
So what we're going to do is we're
going to ask about the general way
that this thing is going to move between different states.
In particular, we are going to have
some general state in here.
Mn, which can go forward or back, Mn plus 1.
Now what we want to do is think about how those probabilities
are going to change over time.
So we typically have fn.
So this is often written as an fn and fn minus 1.
And then this is a g.
I want to make sure I get the n's and n minus ones
correct here.
Typically we write gm plus 1, gn.
So these are telling us about the rates
of being in this state, with say, n mRNAs, as
compared to going here.
We're going here.
So then what we can do is we can write
the change in the probability of mn with respect to time.
Well there are just a few different ways
that the probability can change.
So we can leave the state in two different ways.
fn, gn.
So the way that we lose the probability
is that we have fn plus gn times the probability
that we are in mn.
That's an n there.
And then there are going to be two ways
that we gain probability.
We can gain probably from the mn minus 1.
So this is fn minus 1, plus we can get probability
from the upper state.
That's a gn plus 1 mn plus 1.
So this is just saying that the change in the probability
of being in this state is going to be given by the probability
that we leave the state.
Sorry.
The probability that we enter the state,
minus the probability that we're leaving
the state, kind of the rates.
Now this is going to be true for all n,
except for n equal to zero, we don't have
the terms over on the left.
So this is kind of for all n.
So this is, in particular this is for n,
basically 0 on up to infinity.
So this is a differential equation for the probability
for having an mRNA.
But this is, we have to have a different equation for each n,
0, 1, 2, 3, 4, 5 on up.
So this is what I mean by converting single differential
equation, which is actually an exceedingly simple one, for one
that is for an infinite set.
And each one is even a little bit more complicated.
In general, these f's and n's can be pretty complicated.
In this situation they're not so bad.
But let's make sure.
Can somebody say what fn and gn are equal to?
Any volunteers?
AUDIENCE: [INAUDIBLE]
PROFESSOR: Right so fn, this is rate that we add a new mRNA.
Well that's just synthesis rate for mRNA.
And this guy is what?
AUDIENCE: Delta.
PROFESSOR: So this is degradation rate.
And we actually do have to multiply still by the number n.
And that's because as we go further out here to the right,
then it is true.
The rate at which we come back to the left is increasing.
Because there's just more mRNA that can be degraded.
Now it's worth saying that you can, for example, use
this to simulate the probability distribution
if you start from any distribution you like.
So for example, you could start M0 equal to 1.
And then just simulate how the probability recalibrates
and comes over here.
Similarly, you could do it over here.
You could start with any probability distribution
you want.
And you could use this as a framework
to calculate what the probability distribution will
be at any time later.
But you can also use this just as a way
of figuring out what the equilibrium distribution is
going to be.
Because at equilibrium, we can just ask,
for each one of these arrows, the probability
of moving to the right has to be equal to the probability
of moving to the left, otherwise we wouldn't be at equilibrium.
And that's true for every one of these kinds of pairs of arrows.
And in particular, what we can get,
and I want to make sure that-- so but it's
not that fn is equal to gn-- so it's really going to end up
being that if you see what fn and gn,
so that fn is going to have to be equal to g of n plus 1
for all n.
Yes?
AUDIENCE: Do you also need to multiply
like natural probabilities--
PROFESSOR: Ah, Yes, yes, indeed.
So that sorry, times mn times m of n plus 1.
So it's the kind of probably flux so we have to equalize.
So this is nice because this gives us a ratio of things.
In particular, this tells us that the probability
of being in the n plus 1 divided by the probability of being n,
and this is at equilibrium.
Is going to be fn divided by gn plus 1.
Which is this synthesis rate.
And then down here is going to be this degradation rate
times, in this case, n plus 1.
So this is useful.
Because for example, if we start at m,
we could say that m1 over m0-- well maybe
we'll even put the m0 over on the right.
So then m1, what is that equal to?
That's going to be synthesis rate divided
degradation rate, times m0.
But then we also know that m2, well that's going to be again,
synthesis rate divided by degradation rate.
And we're going to get a squared.
But then now we have to divide by 1/2 times m0.
Continuing on, m3 we get Sm over delta m cubed,
divided by 1 over 3 times 2 times m0.
So in general, we get the probability
of being in the nth state, is going to be this thing.
We'll call it lambda for now.
Lambda to the n, divided by n factorial, times m0.
Now what's the-- and I'll-- remember lambda here
we've defined it to be the ratio Sm over delta m.
Now if we sum over all these probabilities,
what should we get?
AUDIENCE: 1.
PROFESSOR: 1.
Right, if we sum over this thing, what does that equal to?
It's what?
AUDIENCE: Eta lambda.
PROFESSOR: Eta lambda, right?
So just remember in this world-- the sum over lambda
to the n, n factorial, from n equal to 0 to infinity,
this is indeed the definition of e to the lambda.
So what that means is that the normalization condition is
that m0 has to be equal to e to the minus lambda, which is
indeed a Poisson distribution.
I'll raise it up a little bit.
So this is saying, OK, to back up.
If we just have constant rate of creation of something,
constant rate of degradation of that thing,
on a per item basis, per unit basis,
then you end up getting a Poisson distribution,
at equilibrium for the number of that thing,
in this case, the number of mRNA in the cell.
Questions about why that is?
What happened?
How we calculate it?
AUDIENCE: Could you explain why [INAUDIBLE]?
PROFESSOR: Sure.
So this is basically f of n.
And this is basically this g of n.
But remember here n is the number of proteins
or the number of mRNA.
So then that's in the context of the master equation,
then m and n are there.
You get n by the current number of m.
Does that make sense?
Yes?
AUDIENCE: I'm confused how you changed m0 to the e
to the minus lambda.
PROFESSOR: OK.
Well let's just do it.
So mn, this is the probability that we observe n mRNA.
And we know that the sum over mn,
so all these probabilities from n equal to 0 to infinity,
has to be equal to 1.
Something has to happen.
Well let's do this sum.
This is equal to the sum of lambda to the n,
over n factorial m0.
But m0, is this a function of n?
No.
m0 is just, this is just the probability at equilibrium
that you have 0 mRNA.
So we can just pull this thing out.
This is just some number, some probability.
Now the statement is that while this thing,
this is the definition of e to the lambda.
So in general, so e to the x we often write is equal to 1,
plus x, plus x squared over 2, plus dot, dot, dot.
So this thing is indeed just equal e to the lambda.
So what we know is that this is still 1.
So m0 times e to the lambda, is equal to 1,
so m0 is to the minus lambda.
Any other questions about how we got here?
What's going on?
Yes?
AUDIENCE: The plot of the solution
to the adjoining equation, that would
be like the mean value, that would
be the behavior of the mean values?
PROFESSOR: That Is the expected behavior
of the mean value over time.
In this case, fn and gn are both linear functions
of the number of the mRNA.
Which means that in the context of the master equation,
if you ask about the expectation of mn,
this quantity is indeed equal to-- it has the same behavior
as, over time, as the deterministic equations.
So if f and g are nonlinear, then
actually you get a deviation.
But in this case, it is indeed the same.
What it means that if you compare
the stochastic and the deterministic trajectories,
what you would see is that this thing
is going to be a little bit jagged, or whatnot.
And then even at equilibrium it's
going to come up and down a little bit.
I'm trying to add a little bit of jaggedness
because it's discrete.
But the deterministic equation here
is what you would get if you average together
an infinite number of these stochastic trajectories.
Because another one might have come down here.
Does that answer?
AUDIENCE: Is m playing a double role?
Like in that deterministic equation,
m is the concentration of mRNA?
PROFESSOR: I think that I'm-- yeah I think that I should--
my nomenclature I think was not very good.
I've used two different things.
And now that I'm doing this, I think
that I should have-- I should have just called it p of n,
or maybe I should've used n here.
I think I was trying to be consistent with some
of the previous, but I think it was a mistake.
Yes?
AUDIENCE: Are you plotting stochastic?
PROFESSOR: I'm plotting-- OK, so no, I'm not.
So this is if you run an actual stochastic trajectory.
Then at any moment in time, you just
have one-- there's some number of mRNA.
Whereas the sum over the mn's, this
is talking about the probability distribution
of the entire thing.
So really if you started here, the master equation would give
you some distribution for the n's, some distribution for m's.
And so if you looked at these over time,
than the mean of these distributions
is indeed equal to the deterministic behavior.
Yes?
AUDIENCE: Is it possible to recover, like how would we
recover the differential equation from the master
equation?
Is that possible?
Maybe that would help.
PROFESSOR: Yeah.
I think that in the end, there's going
to be a one-to-one relationship from, I guess,
this differential equation to the master equation.
I'm trying to think of any weird case
or something funny's going to happen.
Is something funny going to happen?
AUDIENCE: No.
But like the easy way is just to write them all in terms
of the distribution.
And you can just differentiate the whole sum.
And in that sum, we express the [INAUDIBLE]
with your last equation.
[INAUDIBLE]
PROFESSOR: Right.
But I think this is the much more mathematical way.
I mean because I think that actually, I
mean, from the differential equation, you
actually from the terms here, you can actually
construct the master equation.
And I think by the same way, you can go from the master
equation, and I think that there's
going to be a unique differential
equation that would have gotten you to that master equation.
So I think just from the terms you can do it.
You could also do like moment generating functions
to get to how things change.
But I mean I think that it's really from this, for example,
I think it tells you that that was the differential equation.
Does that--
I mean it's sort of-- the way that we typically
do things these things, is that we have a differential
equation, and then we construct the master equation.
So then we already knew what the differential equation was.
But I think just from the terms in your master equation,
you can say, all right.
This was the differential equation that it started with.
Any other questions about what happened here?
So we have, I think, a fair number, a fair knowledge
of what's going on here now.
We know that the equilibrium distribution
of mRNA in the cell is going to be Poisson.
We also know that the distribution
of the number of mRNA produced per sell cycle is also Poisson.
But it's a different Poisson from the first one.
We know that the number of proteins produced per mRNA
is going to be geometrically distributed.
The one thing that we have not yet done
is to ask about the distribution of protein in the cell.
So let's say something about that.
I'm not going to do the whole derivation.
Because it's harder.
But I encourage you to-- even the continuous version
of the derivation is definitely harder than this.
But then the discrete derivation is even worse.
So what we're going to talk about,
and the way we'll typically maybe
think about this from the standpoint of this class
is the continuous approximation to-- oh,
that might have ended up being useful.
Well it's OK.
Is the continuous approximation to the real answer.
And in particular, just the way that the exponential is
the continuous approximation of the geometric distribution,
in the same way you can think about the equilibrium
distribution of protein in the cell.
In this model is going to be gamma distributed.
But gamma is a continuous distribution.
But it's a continuous analog of the negative binomial.
So let me just make sure I'm-- and Sunney Xie actually has
a nice PRL paper where he derives the gamma distribution.
But even earlier actually Paulson
had derived this negative binomial distribution,
the discrete version of the solution.
So this is the number of protein per cell.
We already know the mean.
So this is approximately distributed as a gamma.
A gamma is a distribution that requires two parameters
to describe.
So a Poisson can be described by single parameter.
Gamma is typically described by two.
And b is going to be the burst size,
whereas a is the mean number of bursts per cell cycle, which
is the same as the mean number of mRNA
produced, so mean number of bursts.
So the gamma of this a, b.
All right.
So the gamma of a is the gamma function.
It's equal to-- now is it a minus 1 factorial?
I always get the-- is it a minus 1 or a plus 1 factorial.
Anybody remember this?
Yeah, a minus 1.
I mean it's like a lot of things.
You look at this equation.
It doesn't really mean a whole lot.
But I think that a reasonable way to think about this
is the gamma is approximately what
you get when you add together a different exponentials
with length scale, given by b.
When you add probability distributions,
you have to do a convolution.
So in some ways, the way to think about it,
and this kind of makes sense.
Because what is happening is that it
takes something of order cell division time
for these proteins to go away.
Because they're stable.
Now each-- and so then what you want to know
is how many proteins are kind of produced
over the course of a cell cycle.
Well that actually you can get at by asking how many bursts
are there going to be.
And then how big are the bursts?
So indeed, the mean here is equal to a times b.
And the variance is equal to a times b squared.
So for example, if you have a single exponential
distribution, with burst size b, then this is what you get.
So this is the probability that you get n proteins.
And this is this function of n.
So for a single burst, this is exponentially distributed.
So this is the continuous version.
Now if we add together multiple of these bursts,
this is really saying that we sample from this distribution,
say twice.
And then we add the resulting value.
So this is a convolution.
You guys will have an opportunity
to practice this on your problem sets.
But what happens is that you end up
getting something that looks like-- it's going to go.
So it increases linearly.
If you added three of them together
this increase is quadratic.
And it kind of goes like that.
So this thing becomes kind of-- it
goes from a distribution where it's
peaked at 0, to something that's peaked at a nonzero value.
Now you can ask, for example, what
happens as for a large a, if you have many bursts,
what does this thing look like?
Oh, I wish I hadn't erased my probability distributions.
So what are the gamma converged to for large a?
A normal distribution.
Right?
So that's the central limit theorem.
If you take any well-behaved probability distribution,
you add it.
You sample from it many times.
Then you end up getting a Gaussian.
If you don't remember that very well,
then this is something to read about over the weekend.
Just like the Poisson is also going
to go to-- for large lambda, the Poisson
also looks like a Gaussian.
Can somebody give an explanation,
an intuitive explanation for why that should be?
Why it-- yes?
AUDIENCE: Because in a Poisson distribution,
you can't have anything negative.
PROFESSOR: OK.
So a Poisson distribution can't have anything--
but now I feel like you're arguing against me.
Because a Gaussian has negative values, right?
AUDIENCE: Right.
So when the mean is really small, only have [INAUDIBLE].
PROFESSOR: OK.
Yeah.
All right.
So what you're saying is that Poisson for small lambda
it can't go negative.
OK.
No I think that that's true.
Yeah, and so somehow the probability distribution
is somehow piling up, as you say.
What are some other ways of thinking about this?
AUDIENCE: [INAUDIBLE].
Because if you have a low lambda that means it's a Poisson.
And then I'm just imagining stretching out.
[INAUDIBLE]
PROFESSOR: OK.
So I think that's fair.
Another way we can think about this,
is let's say that we have some process that's
occurring randomly over some period of time.
And this could be say, mRNA production.
And here this is just the number that we observe here,
this is going to be a Poisson, with some mean lambda.
Now let's just say that I take another one, same process,
same period of time.
How is this guy going to be distributed?
So this also Poisson of lambda.
Now let's say I take this probability distribution,
and I take this probability distribution.
And I convolve them.
I'm going to do the calculation of my head.
I did it.
So for those of you who haven't done convolutions-- whatever.
Yes, what's the new distribution going to be?
Poisson 2 lambda.
And why does that have to be?
AUDIENCE: That line was sort of you put it by n.
PROFESSOR: Yeah.
That's right.
This line, I just kind of like I just made it up.
I could have just said, oh.
Well it's the same process occurring over here.
So we have to have the mean.
It's still is going to be a Poisson process.
And the mean has to be the-- well we just
had twice the length.
And indeed, for independent probability distributions,
means always add.
So this all consistent will all the things we know.
So this has to be a Poisson of 2 lambda.
If I add another segment on here,
it has to Poisson of 3 lambda.
But what you see is that we see that Poisson of n lambda, which
is the sum over many Poissons.
Poissons are well-behaved probability distributions.
You add them together, you're going
to have to get a Gaussian.
So you can see that the Poisson has to become
Gaussian for large lambda.
And indeed it does.
So there's a comment about this in the--
AUDIENCE: It's a little bit more complicated than this
because obviously you always just divide
from lambda [INAUDIBLE].
Like you would have to say that Poisson lambda is just
like a combination of S--
PROFESSOR: OK.
You're saying that if I do this calculation backwards,
I'm going to get into trouble.
Because if I try to break them--
AUDIENCE: So if you require lambda to be-- you
have to have like a significant probability of getting at least
one candidate, right?
PROFESSOR: So I'd say lambda has to be much, much larger than 1.
So once you're at lambda of 100, it looks like a Gaussian.
And in Sunney's paper, he had a comment about this.
Does anybody remember what it was?
Was mRNA production really well described
as-- they mention that actually there
is some violation of this model in the data.
AUDIENCE: Does it go into eukaryotes?
PROFESSOR: Oh, as soon as you go into eukaryotes,
this is why I stay away from them.
But even in their data, in E. coli,
they actually observed a deviation.
So what they found is that there was a cell cycle dependence
to this bursting rate, i.e. the mRNA production
over the course of the cell cycle.
And presumably their conclusion of this
was that you have this guy.
And then he turns into, gets longer.
And then eventually he septates, and then you get two cells.
What he found is that these longer cells had actually
a larger rate of mRNA synthesis than the smaller cells.
And actually this makes sense.
Because here you maybe have just one copy of the genome.
Whereas here you might have-- you're making a second copy.
So you might have two copies of that gene.
So it may make sense that this bursting rate should grow.
But does that mean that you should not
expect it to be a Poisson distribution for the number
of bursts per cell cycle?
No.
It actually is still-- it still is described by a Poisson.
Because you can just say, this is the cell cycle.
And here this is Poisson of sum lambda 1.
Here is a Poisson of sum lambda 2.
So there could be a different rate
over the course of the thing.
But you still have just two Poissons.
You still get another Poisson.
So adding Poissons, gives you backup Poisson.
They don't have to have the same mean lambda.
I just want to make one comment about what
you have to do once you start thinking about eukaryotes.
And the basic-- so you can see the gamma distribution
can either be peaked at 0, or it can be peaked at nonzero value.
So most, for like highly expressed proteins,
you'll see that it looks something like this.
Now for eukaryotes, you also have
to consider there's some rate that you
go between an active and inactive promoter.
And this actually makes things much more complicated.
So there's a rate going to inactive,
a rate going to active.
And so now if you look at, for example,
the mRNA number per cell, you'll see
that it is no longer a Poisson.
And I encourage you, if you're curious about such things,
to come up and look at this.
The solution for the steady state distribution
has been solved analytically.
For example, Arjun Raj, who is the author of the review
that you guys just read, derived this equation here,
which I don't know if you can see.
But even from a distance, you can
see that this is the solution.
And this is just for the mRNA distribution.
This is not even getting to the level of the protein.
And it involves many gamma functions, as well as
a confluent hypergeometric function of the first kind,
which is a disaster.
But he went to Courant.
He was an applied mathematician.
So this is, I guess, this is what
you can do after doing a PhD in applied mathematics.
The point though is that it ends up being very complicated.
And you can get hugely varying distributions for the mRNA.
And indeed this is seen in individual cells.
If you look at mammalian cells, just at the mRNA level,
you can have some cells that have hardly any mRNA, some
that have a huge number.
The protein distributions actually
end up being more regular than the mRNA distributions.
Because of this difference in lifetime.
So the mRNA numbers may fluctuate wildly.
But the protein numbers will fluctuate less,
because they last longer.
So then you do some averaging over this crazy mRNA business.
Now in the last-- yeah, go ahead.
AUDIENCE: In terms of timescale, like all
this is switching to the active and inactive promoter,
like to the other--
PROFESSOR: Ah, yes.
That's a good question.
I think that people argue very much about this.
This is kind of minutes.
This can be hours.
And this is maybe in between those timescales
would be typical.
And when I say hours, especially like in mammalian cells,
they might only divide once a day or so.
So then this gets to be many hours.
And then I'd say minutes is kind of the--
So there were many biological examples that
were discussed in that review.
And I'm not going to talk about all them.
But I think that it's a nice review.
Because it goes over some of the papers that you've read,
or that we've talked about over the course of the semester.
It also illustrates some different biological context
in which noise may play a role.
But I want to mention one study that
was done by actually again, Arjun together
with Hedia Maamar, in collaboration with Dave Dubnau,
where they were studying this process of competence.
So in B. subtilis, during sometimes particularly
of starvation, or other forms of unhappiness,
they kind of pick up DNA from outside.
So they'll import DNA.
Some of it may just be consumed.
But some of it could actually be incorporated into the genome.
Now what they found is that this competence process is
mediated by this protein comK.
And there was a positive feedback loop,
where this guy ends up positively activating itself.
And this helps lead to bistability in this network.
Only a small fraction of the cells
kind of get into this high feedback state.
Only a small fraction of them activate competence and then
uptake DNA.
And what they were able to show in that study
was that it was sort of noise-induced.
That they were able to vary both the transcription
rate and the translation rate, in a way
so as to reduce the noise.
The mean is the same.
So if you in the context of this model, what they did
is they varied transcription rate,
and they varied translation rate,
each say, by a factor of 2.
So they got the same mean, but then different noise.
And we're out of time.
I would have you vote.
But can anybody remember?
If you want to decrease the noise
in the number of the proteins, which of these
do you want to go up, and which do you want to go down?
And which one is going up?
Which one's going up let's say?
AUDIENCE: Sm.
PROFESSOR: Sm.
Right, so if you want to reduce the noise,
but keep the mean constant, you increase
the rate of transcription, and you decrease
the rate of translation.
Because the noise is really driven by this protein
bursting behavior here.
And that's precisely what they did.
They changed those two quantities.
They got the same mean, lower noise,
and then they reduced the amount of competence in that sur--