Welcome to this lecture on ‘Culture Growth’. It is popularly referred to as ‘cell growth’
and it actually represents the growth of a population of cells or a culture. So these
terms will be used interchangeably; cell growth is a very ‘common term’, rather misleading
because we are not looking at the growth of a single cell here. You already know from
the previous lectures, that the cell goes through a cell cycle, during which its size
changes, its volume and other things change as it goes through the cell cycle; and therefore
you can probably look at it as some sort of a growth, from our perspective. But that is
not what we are going to talk about in this particular lecture.
It is referred to as cell growth, but actually means the growth of a population of cells
or the growth of culture. What we mean is actually this, a single cell divides to become
two cells, it is either mitosis, of all the somatic cells, which yields two daughter cells
or the meiosis, of germ cells, which yields two gametes. We, let us limit ourselves to
somatic cells to understand the population growth here, and that is typically where it
is used, predominantly in the production, from a production perspective.
And therefore let us look at mitosis of one cell giving two cells, and each one of those
giving two cells and so on and so forth. It is this process that we are interested in,
in this particular lecture. The cell number, or the cell mass in a given volume, right,
that is, what we are normally interested in, the cell number, number of cells in a given
volume, or conversely, the cell mass or alternatively, the cell mass in a given volume increases
in time because of this multiplication process. In other words, the cell number concentration
or the cell mass concentration increases with time. When you normalize it with respect to
volume, we call it concentration usually. Cell number concentration or cell mass concentration
increases with time and to know how it increases in a precise fashion is important. In other
words, quantification is important; quantification is important in biology also. More and more,
that is being realized more and more even by classical biologists nowadays.
I will give you a very simple example to understand why quantification is important. Suppose you
are making a dish, cooking a dish, okay? There is always a recipe that we average people
follow, we are not, let us assume that we are not great chefs who have an, an inherent
feel for what they cook, and therefore they do not need all these things. That is inside
them. Let us say that we are average people, we
are following a recipe to cook a dish. The, if the recipe says you add rice, you add milk,
you add sugar, and so on and so forth, we will be completely lost, right? We need to
know, okay let us make it a little more specific. Let us say that we are making cup of tea.
If it says boil water, how much water do you boil? If it says add tea leaves or tea dust,
how much tea dust do you add? Do you add an entire box, to make one cup of tea? All these
are very common questions that come about even while making tea, right?
You need to know that you take, let us say, a cup of water, then you add, let us say,
a teaspoon of tea powder, and if you are adding any masala to it, may be about a quarter teaspoon
of it and so on and so forth. Therefore every single thing needs to have a certain measure
associated with it for us to successfully make a dish, okay? So the quantification,
one glass of water, one teaspoon of tea leaves and quarter teaspoon of chai masala and so
on and so forth are very essential aspects to get the dish right. And therefore, quantification
is very important in many different things, especially if you want to do things reproducibly.
In biology too, this aspect is gaining more and more importance, and here, let us see
how to quantify this population growth process, cell growth. For dynamic systems, dynamic
means, which change with time. Biological systems are dynamic systems, including the
cell, they change with time all the all the time.
They go through life, right? So they, they are subject to changes at every single point
in time. The time rates are convenient to use for quantification, this is something
that has not been realized much, and I think it is best to present it during a first course
in biology; that the time rates are rather important (because) I mean are very important,
it is, it does not make much sense to deal with just masses in volumes and so on and
so forth, when we are trying to quantify a dynamic system, such as a cell.
Only rate, time rates would provide us with easy answers to aspects that we are usually
interested in, especially when we are trying to use biological systems for various different
ends, as well as understand biological systems quantitatively so that it can be reproducibly
used later on. To understand this, let me start at the very basics, and, so that we
are all at the same level, everybody understands the need and everybody takes a certain view
to that. Let us say that we are filling a water tank; water tanks are quite common in
Chennai in summer, they provide water for use when water scarcity sets in. The typical
volume, or let me say an average, kind of an average volume is about twelve thousand
litres. These were kind of old tankers, twelve thousand litres. Nowadays you have eight thousand
litres or sixteen thousand litres, depending on the size; twelve thousand is somewhere
in the middle.
So I have chosen this for our representation. The volume here is twelve thousand litres.
So what is the mass? You know that the density of water is one gram per centimeter cubed,
one, one gram per ml, or ten power three kilogram per meter cubed, okay? So if you are dealing
with twelve thousand litres, that is twelve meter cubed, right, and therefore, the mass
is volume into density, which would be twelve thousand kilograms. This is rather straightforward
for people who have some background in physics, maths and so on.
Now let us ask the question, how long would it take to fill the tank, and let us call
that time t. How long would it take t to fill this tank?
This can be easily answered if we know the input rate of water into the tank. If the
input rate happens to be ten kilogram per second, the time taken would be thousand two
hundred seconds, or twenty minutes. If the input rate is twenty kilogram per second,
the time taken would be six hundred seconds or ten minutes. If the input rate is fifty
kilogram per second, the input, the time would be 240 seconds to fill he tank, or four minutes,
okay? The, in other words, if we know the rate of
water input, okay, then the time is nothing but the mass, total mass divided by the rate.
It becomes very straight forward. For further discussion, let us choose this time or this
rate, twenty kilogram per second; ten minutes is the usual time that it takes to fill tankers
here. So that is what we are going to choose for further discussion.
Remember, input rate is twenty kilogram per second. Now, let us complicate things slightly.
The first one is straight forward, it did not really need a certain different view to
answer, only thing is that if you had known that view, the rate view, it was a very straight
forward answer. Even otherwise you would have got into that answer by some roundabout means.
Now let us complicate the question that we are asking or the situation that we are in.
Suppose, there is a hole in the tanker, which oozes water at the rate of five kilogram per
second. Now, how long would it take to fill the tank, right? If we need to answer such
questions, if we start looking at mass volume, there is going to be total confusion, okay
you can try that. Many people do this; even engineers who have gone through courses in
material balances don’t internalize it, don’t internalize the concept of rate and
kind of intuitively get into mass and volume and so on so forth, and get confused.
Whereas, if you know the net rate, that is the rate of input minus the rate of output,
water is coming in at twenty kilogram per second, going out at five kilogram per second;
so the net rate is fifteen kilogram per second, and it is a one step answer to find the time,
it is mass by the net rate. Twelve thousand kgs by fifteen kilogram per second, that is
eight hundred seconds or thirteen point three minutes; divide by sixty, thirteen point three
minutes. So if you know the rate, if you focus on the rate, it is a one step answer to the
kind of information that we engineers or quantifiers are interested in. Biology is no different.
Let us complicate things even further. Now suppose that in addition to the leak, there
is some mechanism inside the tank itself that is generating water at one kilogram per second,
okay? Fictitious, but let us consider this. Some reaction inside the water, inside the
tank that is generating water, and some other reaction in which water is used up inside
the tank at a rate of point two five kilogram per second, and all of them simultaneously
occur, okay, which is usually the killer for any intuitive kind of an approach with mass
and volume, okay? All these processes simultaneously occur, there is filling in, there is water
oozing out through the hole, there is water being generated and there is water being consumed.
Okay? And all of these are simultaneously occurring.
If this is the case, how long would it take to fill the tank? And this becomes very straight
forward, I mean if you take the mass and if you keep thinking about the mass and volume
and so on so forth; you are welcome to do it, but most likely you will probably never
arrive at the answer. Whereas, if you take the rate route, the net rate is nothing but
the rate of input minus the rate of output, plus the rate of generation minus the rate
of consumption, okay? That happens to be fifteen point seven five kilogram per second, twenty
is input, five is the output, one is the generation and point two five is the consumption; all
rates that comes out to be 15.75 kilogram per second as the net rate.
Now, if this is the net rate, this is the rate at which water should get accumulated
inside the tank, right? Or in other words, the rate of change of water mass with time
inside the tank, and we are going to call the tank as our system. Some of you would
be familiar with this term system, something on which we focus our attention, and we, we
are considering the the tank as our system. That is the case, it is a straightforward
calculation one step, total mass is, mass that it can hold is twelve thousand, the rate,
the net rate at which it is getting filled is fifteen point seven five, and therefore
the time to take is twelve thousand by fifteen point seven five, seven sixty one point nine
seconds, divide by sixty, twelve point seven minutes. Okay?
So, rate is a fundamental parameter in terms of usefulness. In most engineering calculations,
and that is something that needs to be internalized. This is true for all dynamic systems, the
cell biological systems or all dynamic systems. So, we need to look at rates of things happening,
their rates of processes happening there, for us to be able to meaningfully quantify
anything to do with this cell, okay, this I would like all of you to internalize because
this is pretty much the key for any meaningful analysis. I see a lot of people even after
lot of experience have not internalized this particular concept.
Now to quantify the population or culture growth, we are going to use something called
a growth rate, which is represented as r subscript x, rx. The rx is nothing but the time rate
at which the cell concentration increases, okay? Cell, see cell (con), talking of cell
concentration makes sense, okay, because whether you have a one litre system, or whether you
have a ten thousand litre system, this concentration can easily be considered the same or can be
scaled up, can be considered to be equal in between these two systems. If you are looking
at total cells in the one litre, it will be something; in the ten thousand litre, it will
be ten thousand times that, okay? So it is not kind of scalable. That is the
reason why we normalize the cell number or the cell mass with respect to volume, we call
that cell concentration, that remains the same across scales. It is scalable. For certain
purposes, even cell concentration becomes a problem, we will, we will look at it if
we have a need. Our first approximation that works well in many useful situations, especially
when you have single cells, is that the growth is directly proportional to the cell concentration
at that time. Okay? The, in other words, the rate of growth, growth rate is directly proportional
to the cell concentration at that time, and this is nothing but the standard first order
representation. Rate is proportional to the concentration,
okay, to the broth. And, the constant of proportionality, let us call it as mu, it is usually represented
as mu, where mu is called the ‘specific growth rate’, okay?
It is all it is growth rate with respect to cell concentration ‘rx by x’, if you want
to look at it that way; therefore that has been normalized with respect to cell concentration;
therefore it becomes specific, and (there) therefore, it is called specific growth rate.
Let me tell you another important principle that will be useful for analysis, I will just
mention it to you and then leave it there, may be we will use one aspect of it.
This is a very general principle. This is a principle of mass balance or material balance.
As you would have guessed this based on the law of conservation of mass, which essentially
says that the total mass is a constant as or the mass of a species is a constant as
long as we do not deal with nuclear reactions or travel at speeds close to that of light.
Right? I mean total species in in a certain formulation as a constant, we look at total
mass is a constant as long as we do not deal with nuclear reactions where there is mass
to energy conversion, or if we do not travel at speeds close to that of light, there is
mass dilation and so on. Hence we will not consider.
As long as we are not dealing with these things; we are quite comfortable not dealing with
these things for our purposes, then, if we focus our attention on something which is
represented by these dotted these dashes, this could be anything, this could be a bioreactor
(())(, this could be the entire building, this could be an entire city and so on and
so forth. All it says is that we are going to concentrate on that particular volume.
There are inputs to this, there are output streams from this, and therefore, if we follow
the mass of a species; we are looking at a species here, not total mass here; mass of
a species, only the following can happen to the species.
You can think about it, all all you want, if you can come up with something else, please
let me know, you would be a genius or (whate), I mean, see whether you can come up with something
else. This species is input into the system, may be at a rate of rin; the species is output
from the system at the rate of rout; the species is consumed in the system at the rate of r,
I am sorry, this is, this must be r (cons)(consum) consumed, con, and species is generated in
the system at the rate of rgen, okay? This , these have been interchanged, please write
this as consumed con, and this is generated. Therefore the net rate is rate of input minus
rate of output plus the rate of generation minus the rate of consumption of that particular
species. The net rate is the rate at which the species mass gets accumulated in the system,
and let us represent it by by ddt of m, okay? This is an accumulated mass; and if the species
happens to be the cells x, then rate of input of cells minus rate of output of cells mass
in terms of mass, plus the rate of generation of cells, minus the rate of consumption of
cells equals the rate of accumulation of the cell mass. Remember, we are dealing with mass
rates in all these cases, not concentrations.
The batch operations are very common in the industry. A batch operation just means that,
this is the representation of a bioreactor, you have a vessel, you have a highly controlled
vessel, specialized vessel, you have a broth here with cells, you have a stirrer to keep
the cells in suspension, and for other means, you have various probes here, probably the
hbo probe, temperature probe and may be an air inlet here to provide oxygen to the cells.
This is a, this is a typical stirred reactor, stirred bioreactor. The batch operation of
the bioreactor is something like this. We dump everything in, okay? Medium cells
and so on and so forth, and at time zero, the operation starts after we have dumped
everything here. Then, wait for the process to go to completion, then dump everything
out and proceed with further processing. In such an operation, rather such an operation
is called a batch operation. There are other kinds of operation, we will not get into that.
And, very many biological processes happen this way, specially in biological and pharmaceutical
industries, okay? The workhorse of a biological or a pharmaceutical industry is still a batch
operation, because it is rather easy compared to the other processes to carry out, given
the various constraints.
So, we are going to consider batch growth, growth in a batch culture, in a batch operation.
If we plot the cell concentration with time, the way the cell concentration varies with
time during growth in a batch, that is we have dumped all the cells and provided the
medium and so on so forth, staggering the process and we are monitoring how the cell
concentration varies with time. The variation is typically like this. You have a certain
time when there is not much of an increase in cell concentration. After a certain time,
there is a significant increase in cell concentration, and then there is, it kind of flattens out,
there is not much of an increase in cell concentration after it reaches its maximum value.
The period during which, the initial period during which there is no change, not much
of a change in cell concentration is called the ‘lag phase’. The period where there
is a significant increase is called the ‘log phase’, and the later phase where it tapers
off is called the ‘stationary phase’. By the way, this is a typical batch growth
curve. You may not always find it. Depending on the situation, the log phase may start
immediately after inoculation. Or, it may take a very long time in lag phase, and then
grow very quickly and then reach stationary phase, and so on and so forth, okay?
There are variations to the theme, this is a typical theme. And after it has reached
a certain stage, if you are following the total cell concentration, it will probably
remain (station), remain the same for a large period of time and then probably go down depending
on the situation. Whereas if you are following the viable cell concentration, it is going
to go through the same lag phase, log phase, stationary phase, and then there will be a
decrease in the viable cell concentration. Now, let us do a mass balance on cells in
the bioreactor broth. A bioreactor broth is what you are going to consider as a system.
The bioreactor broth is nothing but this, right? This is what we are going to take as
our system, this is what we are going to concentrate on, this is what we are going to focus our
attention on. If we do that, the basic material balance equation, this equation can be blindly
written. We are following cells, therefore the rate of input of cells minus the rate
of output of cells plus the rate of generation of cells minus the rate of consumption of
cells equals the rate of accumulation of cells in the broth, bioreactor broth. Since this
is a batch system, and we are going to consider only the growth now, let us not consider the
death process here and so on and so forth, we will consider till somewhere here.
There is no input, because the batch operation, as I had said, we dump everything in, and
start the time after everything is inside, okay? So there is no input stream into the
system that we are considering. Similarly, there is no output stream. After the process
is complete, we take everything and and go for further processing. And our period of
interest now is the time when from start, after everything has been dumped in, to end
before everything has been taken out. Therefore during that time of interest, there is no,
there are no inputs, there are no outputs, and therefore the rates of input and output
of cells is zero.
The rate of consumption of cells is also zero because the cells are not getting consumed
by some means. They are only being generated. This is the case, then this, this and this
(goes) go to zero, rg is the only term that remains and rg equals dmdt. We can write mass
as nothing but concentration times volume, because concentration is nothing but mass
per volume, right? So, concentration, given as x, x into the broth volume, this gives
you the mass of cells, ddt of xV equals rg. In the case of a batch, the broth volume remains
a constant, and therefore it can be taken out of the derivative here, and you get V
dxdt. If rx is the rate on a volume basis, which
it usually is, then, we need to write rxV equals rg. rg, remember, is on a mass basis,
right? It is mass per time, whereas rx is cell concentration per time. So cell concentration
needs to be converted to cell mass, which can be done by multiplying it by the volume,
rx into V equals rg, equals V dxdt. Or, rx, the V V gets cancelled out, rx equals dxdt.
One of the most common models for growth rate that we have already seen is rx equals mu
x, okay? Therefore, if you substitute rx equals mu x, we get mu x equals dxdt. Let us write
as dxdt equals mu x.
And one by x dxdt equals mu, it is easy to see why it is called the specific growth rate.
This we have already seen earlier, where even in a more generic representation of rx. This
model can be used to describe batch growth, but only in parts. The lag phase, the log
phase, the stationary phase is what we had seen in a typical batch growth. In the lag
phase, there is no increase in cell concentration over time, therefore the specific growth rate
is zero. In the log phase, this you you will see immediately after the specific growth
rate happens to be a constant. And, in the stationary phase, again, the specific growth
rate is zero, because there is no increase in cell concentration with time. Right?
Among the various stages of growth, this phase, the specific growth rate and this phase happens
to be the maximum, and therefore this is also called the maximum, the, the growth rate here,
the specific growth rate here is also called the maximum specific growth rate in a batch,
okay, that is another term that is used, and usually it is a constant here. But this model
does not describe what is going on here when there is a variation, does not describe what
is going on here, and so on and so forth. Therefore it does not describe the phases
in between, okay? This we need to keep in mind.
But probably we do not need that, okay, all this is dependent on the need. Therefore we
will realize the limitations of its representation, and use it appropriately. In other words,
we cannot use it to represent things changes here, and changes here. Let us begin, or let
us consider the log phase where mu is a constant. And we can write dxdt equals mu x. If we solve
this simple first order differential equation, with the initial condition that a time t equals
zero, the beginning of the log phase, the cell concentration is x0, okay?
It is not the beginning of the batch, it is a beginning of the log phase, time t0, and
the cell concentration is x0 at that time. If we do that, and we will solve this dx by
x equals mu dt, integrating both sides, we get lon x equals mu t plus c. This constant
can be evaluated using our initial condition, that at t equals t0, x equals zero, we substitute
that lon of x0 minus mu t0 equals c, and therefore the solution becomes lon of x by x0 equals
mu into t minus t0, or x equals x0 exponential mu into t minus t0. Okay?
And that is the reason why it is called logarithmic growth phase or the exponential growth phase.
Logarithmic in this form, exponential in this form, and that is the reason why it is called
so. This equation can directly be used to find the time that is needed to reach a desired
cell concentration, starting from a certain initial cell concentration. And that is a
very important design aspect. We would like to know what is the maximum cell concentration
that needs to be reached, how long do you need to operate the bioreactor to reach that
and so on and so forth apriori. Okay? And if we know the specific growth rate apriori,
this is a straight forward simple calculation to find out the time t that is needed to reach
a given maximum cell a given cell concentration, okay? It is always important.
And of course, we are assuming that the cell concentration is less than the maximum cell
concentration that is possible to achieve under those set of conditions. I think that
is where we will sign off here, , we have seen culture growth or population growth,
we have seen the need for quantification, insistence and the importance of sticking
to time rates in trying to, when, when we quantify systems, dynamic systems, including
biological systems, that is a cell, and then we looked at batch growth and quantification
of batch growth through specific growth rate and in application of that, to find out the
time that is required to reach a certain desired cell concentration starting from a certain
cell concentration. Batch growth happens to be the work horse
of many industries, many biological industries, now even nowadays. See you later in another
module. Bye.