Lecture 6: Exponential Growth Pt. 2
“The Rules of the Game”�
Today:
Microbes grow exponentially when provided with sufficient nutrients
proportional to biomass density
Microbes grow exponentially when provided with sufficient nutrients
proportional to biomass density
What determines doubling time?
How does cell physiology determine doubling time?
The field of “growth physiology”
Another way to think about it: how does a cell organize this network to expand exponentially?
Need to build cells:
Hypothesis: ribosomes underlying everything. Measure ribosomes during growth
If the doubling time is a simple, measurable parameter that describes this entire complex network under different growth conditions, are there simple quantitative rules the network obeys??
Our map for today
Our goal today is to understand quantitatively how the allocation of biomass to ribosomes relates to cell doubling time.
Total cell biomass
bacterial cell
Experiment
Extract RNA and measure total
Extract protein and measure total
In exponentially growing E. coli, ~85% of RNA is ribosomal RNA (Bremer, et al. E coli and Salmonella, 1996).
So RNA/protein ratio gives an estimate of the cell’s ribosomal content.
What do we see in this experiment?
Fire up jupyter for some exciting in-class code!
What do we see in this experiment?
We will focus on the results from a paper that attempted to answer this question
The basic experiment
What does the data look like?
measured per capita growth rate
constants
When thinking about parameters like this, it’s useful to consider the extremes. Let’s do that!
What if we inhibit translation??
Moving along this curve corresponds to nutrient conditions changing, but the translational apparatus of the cell being unperturbed.
Antibiotics inhibit translation
chloramphenicol antibiotic
ribosome
E. coli strain Xac
M63+glycerol
M63+glucose
cAA+glycerol
cAA+glucose
Now inhibit translation with the antibiotic chloramphenicol (Cm)
+2 µM Cm
Growth rate decreases.
But the RNA/protein ration increases.
E. coli strain Xac
M63+glycerol
M63+glucose
cAA+glycerol
cAA+glucose
Now inhibit translation with the antibiotic chloramphenicol (Cm)
+2 µM Cm
+4 µM Cm
Growth rate decreases.
But the RNA/protein ration increases.
E. coli strain Xac
M63+glycerol
M63+glucose
cAA+glycerol
cAA+glucose
Now inhibit translation with the antibiotic chloramphenicol (Cm)
+2 µM Cm
+4 µM Cm
+8 µM Cm
E. coli strain Xac
M63+glycerol
M63+glucose
cAA+glycerol
cAA+glucose
Now inhibit translation with the antibiotic chloramphenicol (Cm)
+2 µM Cm
+4 µM Cm
+8 µM Cm
+12 µM Cm
Whoa!!!
Another linear relationship has been revealed!
As translation is inhibited, the cell compensates by making more ribosomes, but still grows more slowly.
Let’s look at the parameters of this line.
Growth rate decreases.
But the RNA/protein ration increases.
E. coli strain Xac
M63+glycerol
M63+glucose
cAA+glycerol
cAA+glucose
+2 µM Cm
+4 µM Cm
+8 µM Cm
+12 µM Cm
E. coli strain Xac
M63+glycerol
M63+glucose
cAA+glycerol
cAA+glucose
+2 µM Cm
+4 µM Cm
+8 µM Cm
+12 µM Cm
What about inhibiting translation under nutrient conditions besides the very fast cAA+glucose?
For each condition, as ribosome translation is inhibited through the addition of Cm, we observe a linear relationship:
This is a remarkable number of conditions under which to observe such a simple relationship!!!
How does one account for this?
The authors propose an allocation model
Each cell has a finite pool of nutrients from which to make proteins and, hence, more cells.
How does it allocate those resources to different kinds of proteins?
Ribosome-associated proteins: R-proteins
Non-ribosome proteins whose abundances are not affected by translation inhibition: Q-proteins
Non-ribosome proteins whose abundances are affected by translation inhibition: P-proteins
Using this model and the linear relationships we’ve seen, what can we say about protein allocation and growth rate?
Bear with me, we’re doing some math again!
A proteome allocation model
As antibiotics inhibit translation, the fraction of R proteins increases and P decreases. Q stays the same.
At some point, R will reach a maximum and ~0% of the proteome will be P. This means:
Here is an equation relating proteome fractions. How can we relate it to the experimental data?
A proteome allocation model
constant conversion factor
total protein mass
A proteome allocation model
The goal is to relate these fractions to growth rate, which will show general rules that E. coli use for gene expression to grow exponentially.
We also know from experimental data:
As translation is kept the same, but nutrient conditions are varied
As nutrients are kept the same but translation is inhibited
A proteome allocation model
The goal is to relate these fractions to growth rate, which will show general rules that E. coli use for gene expression to grow exponentially.
We now have these rules that relate growth rate, ribosome content, nutrient conditions, and translational capacity.
These relatively simple equations show us how E. coli efficiently allocates these resources for exponential growth!
We can also write the first equation in terms of proteome fraction:
Relates growth, nutrients, and protein allocation
Relates growth, ribosomes, and protein allocation
Protein synthesis must balance nutrient influx
constant!
High-level model
Molecular interpretation
intracellular signaling molecule
Not tested in this paper!
Predictions of the allocation model
The protein allocation model, combined with the experimental data, suggests that per capita growth rate is related to the fraction of the cell mass in ribosomes is a particular quantitative way.
A powerful way to test this model would be to artificially limit how much mass can be dedicated to ribosomes and observe the predicted quantitative reduction in per capita growth rate.
To test this prediction, the authors used a strain in which the expression of a particular protein can be induced to unnecessary levels.
IPTG – non-digestible sugar
IPTG-inducible gene
The more IPTG provided—the more the gene is expressed!
New experiments to test this prediction!
(different colors are different nutrient conditions)
IPTG-inducible β-galact.
What have we learned here?
M63+glycerol
M63+glucose
cAA+glycerol
cAA+glucose
+2 µM Cm
+4 µM Cm
+8 µM Cm
+12 µM Cm
What have we learned here?
Mechanical and electrical analogies
“Rules of the game”
Mechanical and electrical analogies
We don’t understand the complex molecular details, but by looking at quantitative metrics and performing careful experiments, you can find phenomenological rules
What is missing from this model?