BI 559 Lecture 4: differential equations & statistics
BI 559 Lecture 4: differential equations & statistics
Today
Probability & statistics—why?
Throughout the course, we will study groups of microbes like the one on the left.
Statistics lets us describe the properties of these groups.
Questions we will be interested in:
By knowing something about the math underlying these statistical distributions, we will gain insight into the biology of the above scenarios!
↑ B. Subtilis cells
Last time: why should we care about derivatives?
derivative
Low cell density > little growth
High cell density > little growth
intermediate cell density > high growth
Last time: why should we care about derivatives?
derivative
low growth
low growth
high growth
Can we write an equation that relates the density of bacteria to the growth rate?
exponential doubling time
saturating cell density
“differential equation”
What differential equations will we look at?
rate of population growth
constant
population size/density
rate of change of protein concentration
translation rate
protein conc.
mRNA conc.
degradation rate
DNA
mRNA
protein
How do we solve differential equations? Python of course!
Generally, we will have an equation and an initial condition, for example the initial condition may be the bacterial population size or density at the beginning of an experiment.
Next: probability and statistics
Two key statistical measures of a population: mean and variance
Mean: the center of the data
Variance: how spread out the data is
Mean and standard deviation
protein concentration
time
Protein 1
Protein 2
The two proteins have different mean concentrations in the cell, but the same standard deviation.
Probability distributions
mean = 7.26 lbs
std. dev ≈ 1 lbs
Graphs or formulas that describe how a certain property of a population is distributed—the probability of finding a certain value in the population!
On the right is an example of a very common distribution: the normal distribution.
(also called a Gaussian or sometimes a bell curve)
The Poisson distribution: independent random events
Example: chocolate chip cookies
Chocolate chip cookies
Our experiment:
Chocolate chip cookies
90 total chocolate chips
300 total chocolate chips
Chocolate chip cookies
90 chocolate chips, 20 cookies → 4.5 chips/cookie
300 chocolate chips, 21 cookies → 14.3 chips/cookie
Chocolate chip cookies
Attempt to count chips in each cookie
1
2
3
Et cetera . . .
Flip over; repeat
Note: do miss some chocolate chips that are on the interior of cookies. Experimental limitation!
Chocolate chip cookies
Chocolate chip cookies
Not too bad, right!?
What is the formula for the Poisson distribution?
First, you need to know the mean, for example the mean number of chips per cookie.
Let’s look at the distribution for several values of λ.
Note: for the Poisson distribution, the mean is equal to the variance!
Now, some quick history . . .
mean = 10
Prussia
Duchy and state on the Baltic Sea in what is now Germany and Poland.
Ruled by House of Hohenzollern
Immanuel Kant
J.S. Bach’s Brandenburg Concertos
Ladislaus Bortkiewicz
convert to a probability distribution
Ladislaus’ data
Ladislaus’ data
convert to a probability distribution
Poisson distribution for the same mean!
Why is this important for us?
We saw in lecture 2 that for many genes, we see a very small number of mRNA copies per cell in bacteria.
mRNA
Say there are on average 5 copies of a particular mRNA in a cell.
This is just a population average—what do we expect the distribution of numbers of mRNAs in single cells to be?
Do they all have 5 copies?
Do half have 10 copies and half have 0?
We can predict this distribution with Poisson statistics!
How many mRNAs are in single cells?
mRNA
An mRNA with an average of 5 per cell is a rare “event” where we can predict the mRNA distribution with Poisson statistics.
We will use 5 as our average in the Poisson formula:
We can use this formula to plot a full probability distribution!��Let’s do that.
How many mRNAs are in single cells?
On average, cells have 5 copies of this mRNA.
But a few cells will have only 0 or 1!
And a few have 10 or 11!
Let’s look at what we’d expect a population of 20 of these cells to look like!
How many mRNAs are in single cells?
Multiply probability by population 20
(population size)
probability
Expected number
How many mRNAs are in single cells?
mRNA
A gene regulatory system that results in low average copy numbers makes the cells very susceptible to “stochasticity”.
Noise
protein concentration
time
“Long-tailed” distributions
The “tail” of a distribution refers to the probability density of values far above the mean.
A distribution is “long-tailed”/”heavy-tailed”/etc if there is a non-zero probability of values very far above the mean.
These values are still extremely rare, so the probability densities are very low, but importantly they are not zero.
This is a short-tailed distribution!
There is 0 probability of observing
mean
mean
Intuition for long-tailed distribution
Short-tailed: height of humans
Some humans are taller than others, but if you pick one of the 7 billion humans at random, you will never find one that is 20 feet tall.
The heights are distributed narrowly around the mean
Long-tailed: height of buildings
Most buildings are houses or similar-sized buildings, but if you pick one at random, you might get the Burj Khalifa in Dubai, which is immensely bigger than your house
The heights have a long tail
Baltimore, MD
Will come into play when we look at rates of mutation in bacteria!
Friday: our first paper presentation!
Wednesday: start bacterial growth!