Download 1.1.8 Case study: Bacterial reproduction 1.1.9 Mathematical corner

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In this Case Study, we will continue to look at Antiderivatives as solutions of differential equations.
In this case, however, we will see a different class of differential equations which are called “autonomous”
and occur commonly in studies of population dynamics. Here, we will focus on populations of bacteria,
which reproduce by cellular division.
1.1.8
Case study: Bacterial reproduction
Cellular division is a process by which a ”parent” cell first duplicates all of its chromosomes and then
divides itself into two identical ”daughter” cells. It is the method of reproduction used by unicellular organisms, the most common example being bacteria. It is also used by individual cells within a multicellular
organism, such as the human body (skin cells and liver cells, for example).
The timescale for a cell cycle, i.e. from the start of chromosome duplication to the full cellular division, can vary significantly. One of the fastest ones is for the cells of fly embryos, which replicate every 8
minutes or so, while cells in the human liver can take a year to do so. Typically, for most cells in mammal
bodies, division takes between 1 or 2 days.
Let’s consider, for example, E-coli bacteria. In ideal conditions, they divide roughly every 20 minutes.
What is the approximate rate of change of the number of cells? What does this depend on?
Population models where the rate of change of the population is proportional to the number of individuals
in the population are quite common: in general, they are described by the following differential equation:
This equation, however, is different from the ones we studied in the previous lectures... Indeed:
1.1.9
Mathematical corner: Autonomous differential equations (part I)
Definition:
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CHAPTER 1. FUNCTIONS
Different population models, as we shall see later, lead to different autonomous ODEs for the function
N (t). Since knowing an equation is nowhere near as useful as knowing its solution, the rest of this lecture
will be dedicated to learning some methods for finding solutions of autonomous equations (with application
to biology).
1.1.10
Case study: Bacterial reproduction (part II)
We would now like to solve the differential equation we created for the E-coli bacteria growth, to find out
what N (t) is. Unfortunately, the method we learned in the previous lectures for “time-only” equations
cannot be used for autonomous equations.
Luckily, in this simple example, one can actually “guess” what N (t) is fairly easily. Indeed, for a given
number “a”, what kind of function satisfies f ′ (t) = af (t)?
Going back to our bacteria population, we find that
This implies that the bacteria population is expected to grow exponentially! All that remains to do is
to find what the constant “C” is...
1.1.11
Mathematical corner: Autonomous differential equations (part II)
The simplest autonomous differential equation is:
2.5. COMPOSITION OF FUNCTIONS
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The general solution for this equation is:
Important Note: In autonomous equations, the arbitrary constant is not “added”, by contrast with
time-only equations!
If additional conditions are given, then we can find a constant C which satisfies this equation:
The most common additional condition is f (0) = f0 (i.e. f (0) is given). In that case, we simply have
Examples:
• What is the solution of f ′ (t) = 3f (t) with f (0) = 1?
• What is the solution of f ′ (x) = 3f (x) with f (1) = 1?
• What is the solution of g ′ (t) = − g(t)
a with g(0) = 10?
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CHAPTER 1. FUNCTIONS
1.1.12
Case study: Bacterial reproduction (part III)
Let’s now look at real data! The figure on the handout shows real photographs made every 10 minutes
of E-coli bacteria growing in a petri dish. The number of bacteria in the petri dish was counted in each
photograph. The data is collected in the file “Bacteria.dat” , and shown in the figure below.
Note that, due to the difficulty in counting the number of bacteria accurately, the data is quite “noisy”.
So how can we determine whether this population actually grows exponentially or not?
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N(t)
N(t)
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10
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1
0
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t (min)
(a)
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0
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t (min)
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(b)
Exponential growth appears to describe correctly the evolution of this E-coli population, and validates the
use of a simple model in which the population growth rate is proportional to the number of individuals.
Mathematically speaking:
Combining the data with the mathematical model, we can find what the timescale “T” in this expression
is. Indeed,
Question #1: How is T related to the “doubling-time” of the population?
2.5. COMPOSITION OF FUNCTIONS
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Question #2: After a certain time, we see that the bacteria stop growing exponentially. Their growth
appears to slow down. What could this be due to?
1.1.13
Case study: Bacterial populations: more complex growth models.
We will study some of these effects now, and see if they can reproduce this “slow-down” of the population
growth. We will switch to another application, which is very common in the industry, and see whether a
similar model can be applied to the E-coli experiment above.
The effect of death/harvesting. Yeasts are other kinds of unicellular organisms that also reproduce
through cellular divisions. They are used in a wide range of agro-alimentary industries (production of
bread, yogurth, beer, wine, nutritional supplements, etc...), and cultivated for that purpose. Yeast cells are
grown in vats, and regularly harvested.
The equation describing the evolution of a harvested yeast population is:
As before, we are interested in finding out what the evolution of the yeast population is, as a function
of time. This can help us determine, for example, what a reasonable harvest rate is for sustainable yeast
cultures. This time, however, the solution of the differential equation is not quite as obvious as before....
1.1.14
Mathematical corner: The graphical method for solving autonomous
differential equations
If we want to find out “approximately” what the solution of an autonomous differential equation looks
like, there is in fact a very simple graphical method to do so.
• Step 1:
• Step 2:
• Step 3:
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CHAPTER 1. FUNCTIONS
Examples:
1.1.15
Case study: Bacterial populations: more complex growth models (part II)
Let’s use this graphical method to find out what the evolution of the yeast population is under harvesting.
This method suggests that, for a given harvesting rate “h”, two outcomes are possible:
2.5. COMPOSITION OF FUNCTIONS
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This is an interesting result, but how can we be sure about it? Can we prove this mathematically? We
are back to the problem of solving this differential equation – but at least, now we know what the expected
solution looks like!
In fact, insight on how to solve the differential equation specific to this particular problem can come
from noting that there is, actually, a different way of solving the original problem of E-coli growth.
The solution obtained in this manner is exactly the same as before (see Section 1.1.11)! However, the
method is very much more general. We can in fact use it for our yeast population...
The graphical method suggests that the overall evolution of the population depends quite sensitively on the
initial number of yeast cells – is this indeed the case? The previous “general solution” contains the usual
“arbitrary constant C” that is found by fitting the solution to initial conditions. Let’s do so here, assuming
that at time t = 0, there are exactly N0 yeast cells in the vat.
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CHAPTER 1. FUNCTIONS
We therefore find that
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The following figure shows the real solutions of the differential equation for different initial conditions.
Here, for simplicity, I chose T = 1 and h = 1.
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N(t)
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-1
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1.5
t
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In final conclusion, however, we see from the general solution that we can only have exponential behavior in the “growth with harvesting model”. This implies that a model of the kind studied here cannot
explain the slow-down of growth in the original E-coli model. Later on this quarter we will study a related
model, called the “Logistic” model, and see that it provides a much better fit to the observed results. In
the meantime, let’s generalize the findings of this lecture.
1.1.16
Mathematical corner: The mathematical method for solving autonomous
equations (part I)
In this lecture we learned a general, albeit not necessarily simple, method for solving autonomous equations. The method has 4 steps. Consider the autonomous equation and conditions:
Solution method:
• Step 1:
2.5. COMPOSITION OF FUNCTIONS
• Step 2:
• Step 3:
• Step 4:
These steps are best illustrated in the following examples:
• f ′ (t) = f 2 , with f (0) = 3
• f ′ (t) = ef , with f (0) = 1
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• f ′ (t) = f 1/2 , with f (0) =
1
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Note, however, that many equations do not have a “simple/easy-to-guess” solution.. for example, the
famous “Logistic Equation”, is harder to solve:
In future lectures, we will learn to deal with harder problems. In the meantime, lets complete the list of
“simple” antiderivatives:
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