All rightso, moving on from spatial scales
ah, let us just quickly take a look at temporal
scales.
So, let us see.
Again I cannot read most of these things ah.
Over here is evolutionary timescale so, 10
to the power 15, 10 to the power of 18.
So, this is diversification of humans and
chimpanzees over there ah.
You come down, this is 72 hours which is the
cell line doubling time ah.
This is an E. coli doubling time of 20 minutes.
So, when that is basically the cell division
time or the cell lifetime ah, protein half-life
sort of the order of minutes ah.
This over here is the scale of around protein
foldings ah.
Let us see, these are enzyme turnover rates
which go from milliseconds to seconds and
over here are very sort of chemical timescales
side chain rotation, hydrogen bond rearrangements
and so on, covalent bond vibrations which
occur in thescale of nanoseconds to picoseconds
even femtoseconds I guess.
So, all of in principle all of these timescales
taken together would constitute.
Well, let us forget about evolution, but at
least as far if I cut it off as the level
of a cell this whole range of timescales constitute
processes that are integral to the proper
functioning of the cell . So, if the cell
takes is in non equilibrium at some time scale
over here and you are looking at processes
which happen over nanoseconds and so on, we
could use equilibrium approximations to deal
with those processes .
ok.
So, here are someexamples of different time
scales again taken from a random assortment
of systems ah.
So, here is drosophila development.
Drosophila is the common fruit fly.
It is again it is a model system in biology
just like E. coli; drosophila is another model
system . So, you start from an egg, you go
to a larva to a pupa to an adult fly.
This whole process takes around 9 days, 9
to 10 days.
If you look at the early development when
this egg is just sort of starting to differentiate,
you are starting to get all these wings and
this vertebrae divisions forming that takes
place over a timescale of hours.
So, this thing forms roughly over 10-11 hours
and this whole development from the egg to
the adult fly takes place over a period of
days.
So, these are completely different timescales
and again the physics that you would use to
describe these sort of processes would again
be tend to be very different ah.
Here is a nice experimental moviewhich shows
the development of drosophila.
This particular one shows the early development.
So, this is in this stageover herewhere you
so, drosophila is slightly special in that.
In this initial stage, you do not have cell
divisions; you just have nuclei which are
not separated by cell membranes.
So, here is my drosophila embryo .
You have nuclei over here andthese nuclei
sort of divide.
So, this one nuclei maybe becomes two nuclei,
but there is no cell membrane surrounding
this nuclear initially.
After some point the cell membrane forms and
you get cellularization .
So, this first video is in this stage where
you just have these different nucleis which
are dividing.
So, this is at that stage ah.
So, this is I think 1 hour 35 minutes past
the fertilization of embryo.
These blue things are individual nuclei . 
You that was a cell division cycle and you
will see that the number of nuclei has increased,
the system gets denser and denser right.
Also very interestingly you see that this
cell division sort of propagates as a wave.
So, the color so, the colored the cells depending
on when cell division was happening.
So, at the time of cell division, they colored
the cells by this magenta color and you will
see that the cell division sort of progresses.
So, this is one end this is called the anterior
end of the embryo, that is the posterior end
of the embryo.
So, the cell divisions are wave sort of progresses
from this anterior end to the posterior end
and this every time the cell division happens,
the number of nuclei is becomes twice what
it was earlier .
So, this is one very nice example of what
I meant when I said that this whole field
of mathematical or quantitative biology has
become possible due to this extremely extremely
difficult sophisticated and yet very illuminating
experiments.
So, here you can track each individual nuclei
of this embryo right.
This whole, this will become the whole organism.
You have the positions; you have the velocities
of each individual nuclei.
So, you could do a very microscopic level
modeling in trying to understand what sort
of processes are causing this cell division
to happen oh sorry this development of the
drosophila embryo to happen . So, that is
the clock over there.
So, this is very early times.
If you go to slightly later times, this is
again I cannot see this is around the 3 hour
mark I think and these are two different views
the dorsal view and the ventral view of the
same organism as a different shapes.
So, here is here, at this stage the cells
have already formed these.
Now these individual dots are the cells ah,
they are no longer thannuclei.
You have formed the cell membrane and as time
progresses, you will see different features
sort of starting to become clearwithin this
embryo.
So, you can see the segmentation patterns
starting to formin this ventral view right.
So, over here this let it be played.
This is the other ventral this is the dorsal
view sorry where you see from the bottom and
if you let it play go on to develop more.
So, again again in this case, you have exactinformation
about each individual cell of this whole organism;
the positions the velocities is a function
of time.
You can see how these different cells flow
from one point to another.
So, this is basically at roughly this larva
this pupa stage rather no, not the pupa stage
the larva stage . This is a very fascinating
experiments relatively recent over the last
5-6 years where has been possible to sort
of image each individual cell or each individual
nucleiat that resolution and trying to understand
what sort of processes are going on alright
ah.
Coming back to this array of timescales: so,
if you look at cell division ah, what time
were you asking about , Samarth?
Self assembly.
Self assembly okay, we will come to that ok
ah.
So, if you think about cell division like
E. coli cell division that takes place over
a timescale of minutes.
So, around 30 minutes is when the mother cell
is going to divide into two daughter cells.
If you look at cell movements, this E. coli
moving with the help of its flagella and this
is what a typical trajectory of an E. coli
would look like that happens over a timescale
of seconds.
So, it moves when it moves like this in a
directed fashion, all the flagella are bunched
up together.
They move beating and synchronously, the cell
moves.
It reaches one point the flagella sort of
let go of one another, it is sort of tumbles
around for a bit until it chooses a new direction
and again it movesin that direction.
If we look at the tracks, it just looks like
random walk trajectories and this happens
this process happens over a time scale of
seconds .
If you look at protein synthesis ah, that
takes place over roughly a second.
So, this is 0.1 seconds, 0.5 seconds one.
So, this is the ribosome which comes and attaches
to the mRNA.
The transfer RNAs come and feed in the correct
amino acids as the amino acids get fed in
the protein gets better spit spit out and
that happens over time scale of around a second
roughly.
If you look at transcription, so RNA polymerase
coming on to the DNA and producing the same
RNA.
This growing mRNA transcript that happens
over pointtenths of a second so, 0.4 seconds
is what is given here, but roughly of that
order .
If you look at ion channels, remember ion
channels are these objects which are embedded
within this lipid bilayer ah.
They open and close and when they are opened
they allow ions to pass through that happens
over a timescale of milliseconds; 0.001 seconds.
So, this is very fast process compared to
these slower scale processes of synthesis
and so on that we are talking about .
You could also talk about faster processes
where you are leaving more getting more into
the chemistry aspect of it.
For example, enzyme catalysis that takes place
over a time scale of around microseconds ah.
So, both of these are proteins substrate and
enzyme they come together to do whatever they
are supposed to do and this process takes
place roughly around microseconds ok ah.
It turns out, I do not have capsid assembly
over here ah.
I think capsid assembly typical time scales
is roughly of the order of minutes.
If I am not mistaken, I will check once more,
but roughly I think of order of minutes of
I thought I had it over here, but apparently
I do not.
And in fact, you can show that if you if you
just took this protein subunits which make
up this viral capsid and you let them be.
So, you take these protein subunits and you
let you just put them together and you let
they will also self assemble actually into
and they will form whatever like a nice capsid
like this ah.
Turns out that the timescales of that are
much slower and in order to get it to the
biologically relevant timescales, you have
to introduce this electrostatic forces.
. So, if you introduce electrostatic forces
between these charged proteins in this viral
capsid proteins and the DNA that this capsid
is going to encapsulate which is negatively
charged.
You can show that you will get to the right
timescales roughly; this process of cell division
ah.
So, here is a movie again forgive the poor.
So, here is a process colony of E. coli cells
dividing.
It started off from two cells; they divide
and divide and divide until it sort of fills
the petri dish that you had .
You can take the frames of this movie and
you can analyze that and you can plot the
area that is covered as a function of time
ah.
So, this is the this is the frames of that
movie, you sort of calculate what area is
covered by these cells and you plot that area
as a function of time.
This x axis is time in minutes and here is
the sort of plot that you get.
You can calculate roughly a doubling time
which is that how fast does this area dub
double and from this plot, it comes to roughly
around 45 minutes.
So, this is a sort of geometric growth right
. You get you started off with 2 2 bacteria
that becomes 4 that becomes 8 and so on.
So, I could write if I would write an evolution
equation for the number of bacteria, how that
changes as a function of time . So, I have
N number of bacteria and some time.
Again I will try to be very naive and try
to write the simplest thing that I canandI
will say thatwell let me say that dN dt will
grow depending on how many bacteria you have
to start off with right because its doubling.
So, let me say it grows as something like
this r times N right.
If I wrote an equation like this dN dt is
equal to r N and I gave you this curve of
this experimental curve which says that my
doubling time is roughly around 45 minutes.
Could you estimate what sort of an r this
implies, this data implies?
How would you go about doing that?
What is the solution of this equation ? What
is N of t ? Exponential.
So, what is this r then ? 
The doubling time.
So, r is basically log2 divided by the doubling
time.
So, you can try even if you do a very simple
equation like this, you can try to say that
well you can explain this data that I have
that I observe in this experiment.
If I provided, I take this rate to be log2
to over this doubling time that I have observed.
This is called Neidhardt’s equation, it
is in this bacterial growth paper it is of
course, a very naive model it is of course,
at some point this model will fail ah.
When will this model fail?
Large number of.
When it when the numbers become very large
and whatever agar or whatever food that you
have put in the petri dish, it cannot sustain
an infinitely growing colony.
So, at some point this rate of growth must
slow down right.
So, it cannot keep growing like this .
So, at initial times it might grow very fast,
but as you sort of reach the limits of the
population that you can sustain that growth
is naturally has to slow down.
Can I write down a simple equation that will
correct for that?
You have come across an equation like that?
Yes, minus minus N square something something,
that is about it.
What is this sort of an equation?
Logistic.
Logistic equation so, I can write down a logistic
sort of growth model for this.
Typically it is written like r N 1 minus N
over K where K is called the carrying capacity
or the maximum population that you can sustain
right.
So, basically when N reaches K, you will see
that dN dt goes to 0.
It will not grow any more.
When N is much much smaller than K, then this
dN dt is like rN which goes back to this Neidhardt’s
equation.
This is called the logistic equation and then
you can see how this equation will look like.
So, if you look plot the number as a function
of time, this is what will look like initially
it will grow very fast, but then after some
time after some time it will as it reaches
the carrying capacity, the growth will first
slow down and then entirely stop.
This N of t will as t goes to infinity.
This N will just tend to K . Talking from
a non-linear perspective, this equation has
two fixed points.
One fixed point is at 0 if you did not start
off with any bacteria of course, you would
remain at no bacteria .
But provided you started off with somethingso,
there is an unstable fixed point t . The moment
you perturb away from this fixed point, it
will go to this stable fixed point which is
this N star equal to K.
So, that is the logistic equation, you can
solve the logistic equation we are going to
get N as a function of time and then try to
see whether that fits your experimental data
or notthis is roughly.
So, the basic idea is.
So, this is what we will start off with ah.
We will start off at looking at these biological
processes growth, movement and so on.
So, from next class , I think we will start
with diffusion and movement and we will try
to see what is the sort of simplest model
that we can write down.
We will try to analyze that model and see
what are the shortcomings of that in, what
regimes are those model correct in what regimes
are they not and then can we do anything better
than this.
So, for example, this rN was the simplest
possible thing that we could write.
It works in a certain regime when number of
cells is small, it will fail at a certain
regime and the number reaches close to the
carrying capacity and then we would need to
read just your model ok.
So, that is the sort of spiritand it is a
very trivial example of course.
We will try to do slightly more complicated
things as the course goes on ah.
So, this sort of glossary of terms ah; what
are the molecules, what are the numbers and
so on.
Those are roughly taken from these first two
references Rob Phillips, chapters 2 and 3
and Nelson's chapter 2 ah.
If you are interested even more, these two
books Albert's and Lodish; these are sort
of the bible for molecular biology ah.
These have all this information in a lot lot
more lot greater detail and depth.
So, for those of you are interested, you can
take a look at these books over a long period
of time.
These are really really thick books.
So, yeah okay.
So, I think, I will stop here today ah.
I will start off with movement and diffusion
starting with next class and how to do modeling
of diffusive processes in biology, good.
So, I will see you again on Tuesday then and
we will sort of get into it properly.
