Lecture 7: Density-dependent growth Part 1
Today’s goals:
Quick review of last time
This network of metabolism and gene expression takes in nutrients
E. coli
and turns them into more biomass
The network is too complex to understand. But however it operates, it obeys these simple relationships!
Quick review of last time
Incorporating these measurements into a proteome allocation model allows you to precisely predict how the output of this network will change if you alter its inputs!
Simple relationship between nutrient conditions, metabolic proteins, and growth
Simple relationship between translational capacity, ribosome proteins, and growth
Today: back to our thought experiment
Thought experiment
Liquid broth
?
+
etc . . .
etc . . .
What does exponential growth look like?
Exponential curves
Exponential growth not limited to microbes!
Collared Doves introduced to Great Britain after 1952
Also elephants . . .
Darwin, On the Origin of Species
Then why is the earth not overrun with elephants?
Why can’t exponential growth continue unabated?
M9 minimal medium
Why can’t exponential growth continue unabated?
It takes nutrients to build a cell.
Finite amount of nutrients, finite amount of cells!
Or high density of cells accumulates too much toxic byproducts
As always, let’s look at data!
Straight line on a log axis → what’s that?!
So what happens next?
OD remains constant for a while!
“Stationary phase”
The Bacterial Growth Curve!
A quick note:
For a system with a defined volume, like a bacterial culture tube, the population size and population density are directly proportional to each other!
Forgive me, I may use “population density” and “population size” independently!
Growth Phases:
Time
We have two parameters now:
We saw that the “quality” of the nutrient medium determines doubling time.
What determines the maximum cell density?
Let’s look at some experiments.
Experiment: what factors limit maximum cell density?
As we often do:
What do we see??
Cornelis Bernardus van Niel:�
total biomass grown
Looks like total growth is directly proportional to initial carbon concentration!
total growth
initial nutrient concentration
proportionality constant that indicates how efficiently a nutrient is assimilated into biomass!
Total bacterial growth with acetate as a limiting nutrient (Athiorhodaceae) | ||||
[acetate] (mg/ml) | 0.5 | 1.0 | 2.0 | 3.0 |
total growth (mg/ml) | 0.18 | 0.36 | 0.7 | 1.12 |
| 0.36 | 0.36 | 0.35 | 0.38 |
Total growth determined here by centrifuging final culture and measuring dry weight.
Efficiency specifically identified by isolating carbon
Total bacterial growth with acetate as a limiting nutrient (Athiorhodaceae) | ||||
[acetate carbon] (mg/ml) | 0.144 | 0.29 | 0.58 | 0.87 |
total carbon growth (mg/ml) | 0.1 | 0.2 | 0.39 | 0.625 |
| 0.69 | 0.69 | 0.67 | 0.72 |
Total growth determined here by centrifuging final culture and measuring dry weight.
E. Coli grown with mannitol as the only source of carbon:
Time
We’re looking for a growth model that is exponential at early times/small population size
and then levels off.
We’ve seen how properties of the environment (nutrient quality and concentration) determine the properties of this growth curve.
Can we find a model for this growth dynamics?
“Density-dependent growth”
Time
We see that for this growth, the per capita growth rate changes as the culture density changes!
For this reason, this kind of growth is called “density-dependent growth”
How can we find a model for density-dependent growth?
For exponential growth, our key observation was:
One cell grows a little.
A lot of cells grow a lot.
Leading to a model where growth is proportional to population size!
Another way to visualize this model is by plotting the equation itself.
Population density vs. growth rate in exponential growth
How can we find such a relation for density-dependent growth?
As always, let’s first look at experimental data.
derivative
Low cell density > little growth
High cell density > little growth
intermediate cell density > high growth
Population Density
Population Growth Rate
These are both functions of time, but to get our model we just want to relate the density and the growth rate to each other independent of time. Let’s plot them against each other.
Low cell density: little growth
High cell density: little growth
intermediate cell density: high growth
Population Density
Population Growth Rate
↑ Our equation should look like this! ↑
Time
small N:
as N approaches some saturating value:
Something that is ~1 when N is small and 0 as N approaches some value
Time
Simplest:
Our model!! :
There are two phases for this kind of growth
There are 3 approaches to an equation:
“Just tell me the solution.”�
“How do I solve it? Why do I believe it’s true?”�
“How much can I say about the solutions to the equation without ever solving it?”
Razvan Fetecau, my undergrad differential equations professor
Sounds funny, but this is a powerful approach!!
Back to our thought experiment
Let’s say we took an extremely dense bacterial culture, near saturation. What is its growth rate?
+
Now we add some cells to it.
What happens? Does its growth rate go way up?
Back to our thought experiment
+
Now we take media with no cells. What’s its growth rate?
What happens when you add a few cells to this?
!
One more experiment
We take our extremely dense culture again.
But now we dilute it with fresh media containing no cells.
What happens to the growth rate?
What is significant about these two situations in our model?
Tons of cells
No cells
“Steady states”
unstable
stable
How do you find steady states?
Look at the equation for your model!
Let’s do it!
µ
µ
µ
growth rate vs. N
etc..
µ
growth rate vs. N
etc..
µ
µ
What is the solution to this equation?
Let’s check the behavior we want to model:
Small t (small N):
t becomes large: