A ribosome's speed sets the shortest possible cell cycle

One number from lecture 2, taken seriously, gives a hard lower bound on how fast anything alive can divide. Nothing known beats it by much. Extension mini-essay · example written by the lecturer · lectures 1–2 · ~750 words · this is the length and the shape yours should have

Why this file exists This is the worked example referenced from the course organization page. It takes one thing from the lecture, computes with it, arrives somewhere the lecture did not go, and says plainly where the estimate breaks. It is one type of extension, the one argument type. Yours can equally be one example, one scenario, one opinion or one perspective.

1The claim

Growth is protein synthesis. Protein synthesis is ribosomes. So the ribosome sets a speed limit that no amount of clever biology can evade. Lecture 2 counted what is inside a cell and how fast things move in it. I want to push one of those numbers somewhere it was not pushed: the elongation rate of a ribosome, about 20 amino acids per second. My claim is that this single number, combined with the fact that a ribosome is itself made of protein, produces a hard floor on the doubling time of any cell built the way ours are. The floor is about four minutes, and the fastest bacterium anyone has measured sits within a factor of two of it. The argument is short enough to fit in a paragraph, which is what makes it worth writing down.

2Four numbers

Everything below comes from these, and every one is a lookup with a source.

Table 1. The inputs. All for *E. coli*, all order-of-magnitude.
Quantity	Symbol	Value	Source
Translation elongation rate	k	20 aa/s	BNID 100059¹
Protein content of one ribosome	m_R	≈ 7,500 aa	55 r-proteins, ≈ 0.85 MDa¹
Total protein per cell	P	≈ 9 × 10⁸ aa	3 × 10⁶ proteins × 300 aa¹
Ribosomes per cell (moderate growth)	r	≈ 2 × 10⁴	BNID 101441¹

3The estimate

Production must equal growth, and production is ribosomes times their speed. In balanced exponential growth every component doubles on the same clock, so the total protein obeys

dP/dt = \lambda P

. But the left-hand side is something we can count directly: it is the number of ribosomes times how fast each one works,

rk

. Setting them equal,

\lambda = \pf{rk} / P = (2\times 10^{4})(20\,\mathrm{s}^{-1}) / 9\times 10^{8} = 4.4 \times 10^{-4}\,\mathrm{s}^{-1}

(1)

which is a doubling time of $\ln2/\lambda \approx 1.6 \times 10^{3}\,\mathrm{s} \approx 26 \text{ minutes}$ . That is the right answer: E. coli in good medium doubles in twenty to thirty minutes. One division, four lookups, no fitting.

Now rewrite (1) in the form that matters. Let $\phi_{R} = r\cdot m_{R}/P$ be the fraction of the cell's protein that is ribosome. Then $r = \phi_{R}P/m_{R}$ and equation (1) becomes

\lambda = (\pf{k}/m_{R}) \cdot \pf{\phi_{R}} = \epsilon \phi_{R}, \,\, \epsilon = 20/7500 = 2.7 \times 10^{-3}\,\mathrm{s}^{-1}

(2)

The growth rate is proportional to the ribosome fraction, with a proportionality constant that is nothing but a ribosome's speed divided by its own size. For our numbers $\phi_{R} = 17\%$ , which is the right ballpark for a cell growing at that rate. Equation (2) is the growth law, and it fell out of a lookup table.

4The floor

Set the ribosome fraction to one, which is the most extreme cell imaginable, and read off the answer.

Equation (2) is a hyperbola in disguise: $T_{d} = \ln2 \cdot m_{R}/(k \phi_{R})$ . The only free parameter left is $\phi_{R}$ , and it cannot exceed 1: a cell cannot be more than entirely ribosome. So

T_{d}^{min} = \ln2 \cdot m_{R}/k = 0.69 \times 7500/20 = \pf{260\,\mathrm{s} \approx 4.3 minutes}

(3)

Figure 1. Equation (2) plotted as doubling time against ribosome fraction,

T_{d} = 260\,\mathrm{s} / \phi_{R}

. The curve is steep on the left, which is why slow-growing cells can afford a small ribosome fraction, and flat on the right, which is why buying speed gets progressively more expensive. The dashed floor is the

\phi_{R} \to 1

limit: 4.3 minutes, unreachable but never violable. Drawn from equation (2) with

k = 20

aa/s and

m_{R} = 7500

aa.

So here is the payoff. Vibrio natriegens, the fastest-growing bacterium on record, doubles in 9.8 minutes.² Our floor says 4.3, and a cell that is 43% ribosome by protein mass would sit exactly there. The fastest thing alive is running at roughly half the speed that the ribosome's own arithmetic permits, and the gap is spent on everything that is not a ribosome. That is a remarkably tight bracket for four numbers off a lookup table, and it says the ribosome really is the binding constraint rather than one constraint among many.

What the 9.8 minutes actually is

I first wrote "about ten minutes" from the paper's title, because that was all I had. The full text arrived later and it is worth saying what changed, because the number turns out to carry conditions that the title does not.

Eagon grew P. natriegens in brain heart infusion broth with 1.5% sea salt at 37 °C, on a rotary shaker, and followed growth turbidimetrically at 650 nm. The 9.8 minutes is not a fit to the whole curve. It is computed from one 15-minute window, between 3.50 and 3.75 hours, in which the culture went through 1.53 generations. He calls it the generation time "during its most active period of multiplication", which is the honest description of a best window rather than a sustained rate.

And in the same figure, with a larger inoculum, he gets 14.1 minutes. He notes it himself and offers carry-over of inhibitory products as one explanation. So the paper's own spread is a factor of 1.4, on a measurement of optical density rather than of cells.

None of this breaks the argument, and one part of it helps. A best-window rate is the right quantity to compare against a floor, because a floor is about what is possible rather than what is typical. But the honest bracket is now 4.3 against 9.8–14.1, which is a factor of 2.3 to 3.3 rather than a clean factor of two. I would not have known that from the title.

5Where it is wrong

Three things I glossed, in decreasing order of how much they matter.

Not every ribosome is translating. A fraction is assembling, idle, or stalled. Roughly 80% active is the usual figure, which slows real cells by about 20% and raises the true floor accordingly. This is also why the measured growth law has an intercept: there is a floor of non-ribosomal protein the cell cannot give up.
Protein is not everything. The cell also makes rRNA, membrane, DNA and metabolites. rRNA is the big omission, since a ribosome is roughly two-thirds RNA by mass, and RNA polymerase has its own speed limit. Including it makes the floor worse, not better, so the bound survives.
The numbers are all "about". $k$ is quoted between 12 and 21 aa/s, and $m_{R}$ and $P$ are good to maybe 30%. The floor is therefore 4 minutes to within a factor of two, which is exactly the claim an order-of-magnitude argument is entitled to make and no more.

Where this goes

Equation (2) is the growth law $\lambda = \epsilon \phi_{R}$ , which lecture 14 develops properly: what sets $\phi_{R}$ , how the proteome repartitions under a nutrient shift, and the reformulation in which the growth rate is the largest eigenvalue of the self-replication network. What I did here is get the answer the cheap way first. That is the habit lecture 1 was arguing for.

†Sources

Two, and note honestly what has been checked at what depth.

R. Milo & R. Phillips, Cell Biology by the Numbers, and the BioNumbers database. book.bionumbers.org Translation elongation rate (BNID 100059, "rate of translation by ribosome at 37 °C as a function of growth rate", range 12–21 aa/s, from Young & Bremer 1976 and Dennis & Bremer 1974), ribosomes per cell (BNID 101441), protein copy number (BNID 100088) and mean protein length (BNID 100017). Every entry was opened and the value read off, not recalled. BNID 100059's own note writes the balance as $dP/dt = N_{r} \beta_{r} c_{p}$ , which is equation (1) with the active-ribosome fraction $\beta_{r}$ that §5 flags as the leading correction.
R. G. Eagon, "Pseudomonas natriegens, a marine bacterium with a generation time of less than 10 minutes", Journal of Bacteriology 83, 736–737 (1962). PubMed PMC279347 The original measurement of the fastest known bacterial doubling time. The organism was later reclassified as Vibrio natriegens. Now read in full. The 9.8 minutes is on page 737, equation (4), from the 15-minute window between 3.50 and 3.75 hours of curve A in Figure 1. Medium, temperature and the 14.1-minute figure from the larger inoculum are on the same two pages. An earlier version of this essay cited only the title and said so; the box in §4 records what reading the paper changed, which is the more useful thing to have written down.

The code behind the numbers

k, mR, P, r = 20.0, 7500.0, 3e6*300, 2e4
lam = r*k/P # 4.4e-4 /s
Td = log(2)/lam/60 # 26 min
phiR = r*mR/P # 0.17
eps = k/mR # 2.7e-3 /s
floor = log(2)/eps/60 # 4.33 min

Five lines. Every figure in this essay comes from them, and Figure 1 is $T_{d}(\phi ) = \log(2)*mR/(k*\phi )/60$ plotted over $\phi \in [0.05, 0.5]$ . This is what "one thing you made" means: not that the computation was hard, but that you ran it.