BI 559 Lecture 3: diffusion and math review
Today:
Math review: exponential growth
Let’s say you open a bank account with 10% interest compounded once a year.��What does that mean?
Each year, you get 10% return, i.e. your money gets multiplied by 1.1:
initial amount of money
rate of return
Imagine you keep your money there for 2 years:
With exponential growth, in each increment of time, the population is multiplied by a certain number instead of adding a number!!!
Example: interest on a bank account
money after 1 year
Math review: exponential growth
Instead of computing the return every year, what if we compute half of it every 6 months so that the return can earn more return?
For mathematical simplicity, I’ll assume an interest rate of 100%:
initial
1 year
$$$
$1
$2
initial
1 year
$$$
$1
$2.25
6 months
$1.50
We made more money!
Computed annually:
Computed twice a year:
Math review: exponential growth
What about 1/3rd computed every 4 months?
initial
1 year
$$$
$1
$2
Computed annually:
initial
1 year
$$$
$1
$2.25
6 months
$1.50
Computed twice a year:
initial
1 year
$$$
$1
$2.37
4 months
$1.33
Computed three times a year:
$1.77
8 months
The more often we compute the interest, the more money we get!
Math review: exponential growth
| |
2 | 2.2500 |
3 | 2.3704 |
4 | 2.441 |
5 | 2.488 |
6 | 2.522 |
10 | 2.594 |
100 | 2.705 |
1000 | 2.717 |
| ? |
Exponential growth contrasted with linear growth
For something growing exponentially, with every interval of time, the amount of it is multiplied by a number instead of adding a number!
Linear growth: every regular time interval, a constant amount is added
If the shoveler shovels at a steady rate, the volume of the pile will grow linearly.
The size of the pile is a straight line on a linear axis!
Exponential growth contrasted with linear growth
If somehow the snow pile could grow exponentially, then its size would be multiplied by a certain number every regular time interval
Exponential growth is not a line on a linear axis!
A more useful way to plot exponential curves is using a logarithmic axis
Logarithmic axes
Linear axis
Logarithmic axis
Each tick is adding a number
Each tick is multiplying by a number
Exponential growth is linear on any log axis!
Base 2
Base 10
What changes is just the numerical value of the slope!
For a base-2 log axis, the slope is the inverse of the doubling time
1 log unit
A logarithm is the inverse of an exponential
Math review: derivatives
(the great Eric Koston)
Derivatives
Derivatives: rates of change
Time derivative: reaction rates, speed, etc
Spatial derivative: gradients
*iGEM
How do we define the derivative?
What is the rate of change of concentration?
Decaying chemical:
Note: we will not do a lot of math manipulations. We mainly need to get the concepts!
What does the derivative of this function look like?
time, t
rate of change, dN/dt
?
0
Let’s compute the derivative in python.
Let’s manually compute the derivative
What does the derivative of this function look like?
time, t
rate of change, dN/dt
?
0
What does the derivative of this function look like?
We will often look at derivatives of experimental data
Let’s compute the derivative of the bacterial growth data
Notation we will use
Common derivatives and rules (polynomial, exponential, chain rule)
Derivatives act linearly
Derivatives: rates of change
Very important note on units!
units of concentration
units of time
Derivatives: rates of change
Very important note on units!
units of concentration
units of time
Becomes very important when dealing with units of constants that appear in equations!
Derivatives: rates of change
Very important note on units! The numerator and denominator of derivatives have units!
Say we have an equation
(units)
Integrals
Integrals are the inverse of derivatives
constant
“indefinite” integral:
“definite” integral:
The definite integral corresponds to the area under the curve
Integrals we may care about in BI559
A key application of differential calculus
CILSE
Star Market
A key application of differential calculus
Boston
Manchvegas, NH
A key application of differential calculus
Tehran
Perth
Tehran
Perth
Isfahan
Isfahan
Tehran
Perth
Isfahan
Plane tangent to Tehran
Isfahan and Tehran are approximately connected by a straight line/plane.
There are many planes that go through Tehran.
How do you find the right one?
Use the derivative to make a linear approximation of the Earth’s curved surface!!!
To see how we do this, let’s take a function that’s curved, but not as complicated as the Earth’s surface
We want a tangent line at x = 3.5
What is the equation for that line?
(near x = 3.5)
Now let me do something weird!
We can approximate a non-linear function as a line with a slope equal to the derivative of the function!
We want a tangent line at x = 3.5
What is the equation for that line?
Taylor Series
constants!
→ this is a polynomial
In this course, we only care about the first two terms:
The derivative lets us approximate the function as a line near x = a. This makes math much easier.
!
Looks . . .
Kind of linear, no?
I’m going to plot another curve here.
Let’s zoom back out.
Why should we care about derivatives!?
Proportional to cell density
Bacillus subtilis growing in a simple nutrient medium
Let’s do another computation on our growth data
Why should we care about derivatives!?
derivative
Low cell density > little growth
High cell density > little growth
intermediate cell density > high growth
Low cell density: little growth
High cell density: little growth
intermediate cell density: high growth
Population Density
Population Growth Rate