Lecture notes. Written from the material of MIT 8.591J 2014, licensed CC BY-NC-SA 4.0. These are notes, not a transcript: the material has been reorganised and rewritten. This adaptation carries the same licence, and the original is linked above.
5. Cost-Benefit Optimization and Statistical Evidence¶
What this covers¶
This chapter works through one lecture built around a close reading of a single paper: Dekel & Alon, "Optimality and evolutionary tuning of the expression level of a protein" (Nature, 2005), which argues that the expression level of the lac operon in E. coli has evolved to balance the metabolic cost of making the proteins against the benefit of digesting lactose. The lecture uses that paper for two purposes at once: to work through the biology of measuring a cost and a benefit for gene expression, and to build a general statistical toolkit for deciding how much a plotted curve should convince you. The two threads are not separable in what follows, because the second is what the lecture uses to interrogate the first. It assumes you already know what the lac operon does — lacZ, lacY, the repressor, and lactose's role in inactivating it — and have seen a Gaussian distribution before.
Two views of "why does this protein have this expression level"¶
The paper opens with a sentence it is hard to disagree with: different proteins have different expression levels. The question is why. One view treats expression levels as evolved solutions to a cost-benefit optimization problem: over evolutionary time, in some ancestral environment we cannot observe directly, a given level of expression won out because it balanced what the protein cost the cell against what it bought. A second, more skeptical view grants that selection could optimize such a thing, but doubts that we can ever reconstruct the ancestral environment well enough to claim it did, for any particular protein. This is a real disagreement among people who study this, and the lecture does not ask you to resolve it. It asks something narrower and more answerable: given the specific evidence in this paper, how far should it move you toward the optimization reading for this one operon?
For the lac operon specifically, cost and benefit have concrete referents. The cost is whatever finite resource — energy, protein-synthesis machinery, anything the cell has only so much of — goes into making LacZ (\(\beta\)-galactosidase, which cleaves the lactose disaccharide into its two monosaccharides) and LacY (the membrane protein that actively imports lactose; the operon's third gene, LacA, has no settled story attached to it and nobody in the field talks about it much). Making these two proteins is a cost precisely because the same resources could have gone to making something else. The benefit, when lactose is actually present, is the sugar the cell gets to eat once it has been imported and cleaved.
Isolating the cost: the IPTG trick¶
Measuring the cost alone, with no benefit mixed in, takes a specific experimental move. Left alone, lactose itself controls expression: it inactivates the lac repressor, which then stops blocking the lac promoter. To decouple the amount of protein made from the presence of any actual sugar to metabolize, the experiment instead adds IPTG, a chemical that inactivates the repressor directly without being a substrate for the pathway. Titrating the IPTG concentration titrates lac operon expression continuously between zero and the maximum the wild-type promoter allows — and it does this with no lactose in the medium, so any change in growth rate can only be a cost, never a benefit.
But cells still need something to eat, so the medium carries a small amount of glycerol — a second-rate carbon source that keeps the population alive without feeding it well. Two things make glycerol the right choice rather than a strong carbon source like glucose. First, if glucose were abundant, adding lactose later would show essentially no growth benefit, since the cells would already be doing fine — you'd have destroyed your own ability to see a benefit in a later experiment. Second, glucose triggers catabolite repression: a preferred carbon source represses CRP-dependent alternative metabolic machinery wholesale, so you would need an additional mutation just to unmask the effect you were trying to measure in the first place.
Plotting relative growth-rate deficit (roughly, percentage decrease in growth rate relative to an uninduced cell) against lac expression level (normalized to the wild-type promoter's maximum, induced with IPTG, at \(1\)) gives a cost curve that is convex — it grows faster than linearly. At the wild-type maximum, the growth deficit is only about 4%. There is no data beyond expression level \(1\), because IPTG cannot push a fixed promoter sequence past what that promoter allows; going further needs a stronger promoter, which is exactly what a later experiment in this chapter supplies via evolution rather than titration.
Reading an error bar¶
Before returning to that cost curve, the lecture takes a long detour through a question that applies to reading almost any plotted curve in a paper: if you measure some quantity \(y\) at several values of \(x\), plot the mean at each \(x\) with an error bar, and there is some true underlying curve \(y(x)\) that you don't get to see directly, what fraction of your error bars should actually contain that true curve?
Two different quantities are easy to confuse here. The standard deviation of your measurements at a given \(x\) tells you how noisy an individual measurement is — how good an experimentalist you are, essentially. What you actually want to know is how well you know the true mean at that \(x\), and that is the standard error of the mean, $\(\text{SEM} = \frac{\text{SD}}{\sqrt{n}},\)$ where \(n\) is the number of independent measurements averaged at that point (with the usual small bookkeeping question of whether the sample SD itself divides by \(n\) or by \(n-1\)). In the paper, each growth-rate point comes from 48 independent cultures grown in a checkerboard layout on a microtiter plate, and the error bars plotted are the SEM across those 48, not the raw spread — and the raw spread, from the paper's own supplementary histograms, is large: a standard deviation of several percent on a signal that is itself only a few percent.
Why does averaging help, and why is the result Gaussian even before you know the shape of the underlying measurement noise? Because you are adding up many noisy measurements and dividing by \(n\): by the central limit theorem, the distribution of that sample mean tends toward a Gaussian centered on the truth, with width shrinking as \(\text{SD}/\sqrt{n}\), regardless of the exact shape of the individual measurement errors.
That has a sharp, slightly counterintuitive consequence. If the noise really is Gaussian and there is no hidden systematic bias, an error bar drawn at \(\pm 1\) SEM should miss the true value about a third of the time — that's just what "one sigma" means for a Gaussian. And this fraction does not depend on \(n\): as you take more measurements, the error bars shrink, but the sample mean also gets closer to the truth, and the two effects cancel exactly, leaving the same roughly two-thirds coverage regardless of how many measurements went into each point. So seeing a chunk of your error bars fail to touch the plotted curve is not, by itself, evidence of sloppy measurement or a wrong model — it is what a Gaussian is supposed to do.
The peril of fitting¶
That whole argument assumed the plotted curve is the actual, God-given truth. Suppose instead the curve is a fit to the very data whose error bars you're asking about. Now the picture changes, because fitting — typically, minimizing the sum of squared deviations from the data — pulls the curve toward the points by construction. Fewer than a third of the error bars will now typically miss it, and how much fewer depends on how many free parameters the fit used.
Push this to the extreme and the danger becomes obvious: with as many free parameters as data points, you get a perfect fit — the curve passes through every single point exactly, and no error bar, however small, can possibly miss it. Concretely: measure 15 points and fit a 14-degree polynomial (15 free coefficients), and it will thread through all 15 points precisely, not merely within their error bars. That is not evidence the model is right. It is solving 15 equations in 15 unknowns, nothing more — and it is an easy trap to fall into without noticing, because a fit that looked mediocre with two parameters can suddenly look spectacular with three or four, for reasons that have nothing to do with whether the extra parameters mean anything. The general habit worth building: whenever a fitted curve looks convincing, ask how many free parameters it used relative to how many independent data points it was fit against. That ratio tells you how much of the apparent success is structural rather than a discovery about the system.
This lands directly on the cost curve above. The paper compares a straight line against a quadratic — one extra free parameter — and finds the quadratic fits better. That should not surprise anyone: an extra parameter almost always fits at least as well, whether or not the true relationship is really superlinear. Read honestly, the cost data argue for a superlinear cost rather weakly: draw a straight line through the same four or five points and it looks basically fine too.
Back to the cost curve: two models, one range of data¶
The paper actually carries two different nonlinear cost curves, not one. Besides the quadratic — a purely phenomenological curve with no mechanism attached — there is a second, mechanistic curve built from the assumption that the cell has a finite total amount of some protein-making resource; push expression high enough and that resource runs out, driving growth to zero. Over the range where there is actual cost data (expression from \(0\) to \(1\), relative to the wild-type maximum), the quadratic and the finite-resource curve are numerically almost indistinguishable — which means the cost data alone cannot tell you which is right, only that both beat a straight line by about the same small amount. The two curves diverge sharply only outside that range, for expression levels above wild type — levels that IPTG titration cannot reach, because titrating an existing promoter has a ceiling, but that a stronger, mutant promoter could, given enough evolutionary time. That divergence, and not the cost data itself, is what a later experiment in this chapter is actually built to test.
Measuring the benefit¶
The benefit side is measured differently, and less directly: relative growth rate is plotted against external lactose concentration, not against expression level, because holding expression level fixed while varying lactose independently is the harder experiment. At zero lactose, induced cells sit at the same roughly 4-4.5% deficit found on the cost curve — paying the cost, getting no benefit. As lactose concentration rises, growth rate climbs, crossing from deficit to advantage and saturating by full induction at roughly a 10-11% advantage.
In the paper's model, that saturation comes from the import step obeying Michaelis-Menten kinetics: $\(\text{import rate} = V_{\max}\,\frac{[\text{lactose}]}{K + [\text{lactose}]},\)$ which saturates in the lactose concentration. But this is a narrower claim than it looks: it is saturation with respect to how much substrate is around, not with respect to how much LacY the cell is making. Make more importer and \(V_{\max}\) itself goes up — so a benefit curve that saturates as a function of lactose concentration does not, by itself, pin down whether benefit saturates as a function of expression level. Many different benefit-versus-expression shapes are consistent with the lactose-concentration data actually shown; the data neither confirms nor rules out a saturating benefit function.
That distinction matters because of the earlier fitting argument. Net growth is \(b(x) - c(x)\), benefit minus cost as a function of expression \(x\). Two straight lines subtracted give another straight line — no interior optimum, just a function pinned to one boundary or the other. An interior optimum needs real curvature somewhere. If, as later, more direct measurements suggest, cost is actually close to linear over the relevant range (see below), the paper's superlinear-cost story stops being able to supply that curvature — which means whatever nonlinearity produces an optimum has to live on the benefit side instead. A saturating benefit — diminishing returns as expression rises, the first slice of pizza being better than the fifth — is a generic feature to expect of almost any benefit function, not something special to lac.
Why the cost curve may be the wrong place to look for the nonlinearity¶
Later, independent measurements complicate the paper's cost story directly. Other work (from Terry Hwa's group) has measured relative growth rate as a function of expression of a variety of deliberately non-useful proteins — \(\beta\)-galactosidase or \(\beta\)-lactamase expressed with no matching substrate to act on — across many conditions, and finds this cost curve is close to linear over a wide range, only becoming severe (growth driven toward zero) around something like 30% of total cellular protein — much further out than anything the Dekel-Alon cost data covers. This is one of a family of phenomenological "growth laws" that Hwa's group has documented for E. coli, and it undercuts the superlinear-cost claim on its own terms: measured more directly and over a wider range, cost looks like a line, not the accelerating curve the paper needed.
The evolution experiment: what "500 generations" actually costs¶
The paper's strongest evidence is not the cost curve at all — it is a laboratory evolution experiment. Populations were grown continuously for about 500 generations at several different fixed external lactose concentrations, and the operon's evolved expression level was measured against the lactose concentration of its environment. Here the finite-resource cost model's prediction tracks the data, while the straight-line and quadratic cost models predict a much larger response at high lactose — expression evolving out to several times the wild-type level — that is not observed. This match is real evidence, because it operates exactly in the regime (expression above wild type) where the cost models genuinely diverge, unlike the cost data itself.
How long should such an experiment take? Naively, at a roughly 20-minute doubling time, \(E.\ coli\) manages something like 75 generations a day, which would put 500 generations at under a week. That is not what happens, because populations are grown by daily batch dilution: each day the saturated culture is diluted 100-fold into fresh medium, grown back up to saturation, and diluted again the next day. A hundred-fold dilution corresponds to only about 6.6 generations per day (since \(2^{6.6} \approx 100\)), not 75, and the reason is that most of the day is not spent dividing at all.
The point matters beyond the arithmetic, because whatever is under selection in this protocol need not be division rate alone. A mutation could spread by shortening the lag phase (dividing sooner after dilution, so a genotype gets a head start on its neighbours), or by doing specifically well during the long saturated stretch — the "growth advantage in stationary phase" (GASP) phenotype, where surviving cells can consume the contents of others that lyse during that stretch — or simply by dying more slowly while saturated. Richard Lenski's long-running E. coli evolution experiment (running since the late 1980s, now tens of thousands of generations) has documented several of these, lag-time reduction among the earliest.
What evolved, and what didn't¶
Two results from the evolved lines are worth holding onto separately, because they point in opposite directions.
What did not change: the intrinsic catalytic efficiency of LacZ — its activity normalized by the amount of LacZ protein actually present — showed no significant improvement across 500 generations. Read one way, this is consistent with the enzyme having already been optimized for lactose cleavage over millions of years of prior evolution, leaving little for a few hundred more generations in the lab to improve. But the argument carries a real caveat: E. coli's natural habitat for eating lactose is the gut, whose conditions (pH, and much else) differ from a lab flask, so "already optimal" risks conflating optimal-for-the-ancestral-environment with optimal-here — an easy story to tell after the fact and a hard one to have predicted beforehand.
What did change, and how far: the experiment kept IPTG present throughout, which decouples the amount expressed from the repressor's own response to lactose and lets expression level itself sit under direct selection at each fixed lactose concentration. Lines evolved at higher lactose concentrations evolved somewhat higher LacZ activity; a line at an intermediate concentration barely moved; a line evolved with no lactose at all saw expression drop substantially. But across every line, evolved activity never rose above roughly 1.2-1.3 times the wild-type level, even under sustained selection pressure for more. Something is genuinely constraining upward evolution of expression beyond wild type — consistent with, though not by itself proof of, a real nonlinearity somewhere in cost or benefit.
A separate comparison sharpens why cost matters even when it's small. Consider a population grown for 500 generations with neither lactose nor IPTG: the operon is never transcribed, so there is essentially no growth cost to carrying it — only the marginal cost of replicating that stretch of DNA each division, which is tiny. Even so, in that lineage the operon was actually deleted — close to a kilobase of DNA, including the promoter, was lost, leaving cells unable to grow on lactose even if it were added back later. Contrast that with the IPTG-selected lines, where the operon really was costing about 5% of growth rate whenever expressed, so a mutant that lost the ability to express it had a genuine ~5% advantage and could sweep the population outright. Both outcomes are selection against an unused gene; the difference is how much advantage was on offer, and in both cases it was apparently enough.
Where the paper's argument stands¶
Put together, the picture is more mixed than the paper's headline suggests. The hypothesis is a good one, and the laboratory evolution data are real: expression really does track the lactose concentration a population was evolved in, roughly as the finite-resource model predicts, and that match is genuine evidence, because it lands in the one regime where the candidate cost models actually disagree. But the cost curve itself — the piece meant to justify which cost model to trust in the first place — is weak evidence on its own terms: a quadratic beating a line is exactly what an extra free parameter buys you regardless of the truth, and the two nonlinear candidates the paper compares are nearly indistinguishable over the range they actually measured. Independent, more direct measurements of cost (Hwa's growth laws) subsequently found cost to be close to linear over a much wider range than this paper probed, which removes the curvature the paper leaned on. If cost really is close to linear, the interior optimum this whole framework needs has to come from curvature in the benefit function instead — plausible on general grounds (diminishing returns are generic), but not something the paper's own benefit data, measured against lactose concentration rather than expression level, actually establishes. None of this makes the evolution experiments wrong; it specifically undermines the mechanistic cost-function story used to explain and, above all, to extrapolate them outside the range where data exist.
Sources¶
- MIT OpenCourseWare, 8.591J Systems Biology (Fall 2014), lecture recording transcript
9ygxpwvwydy(docs/computational-biology/mit-ocw/8591j-2014/recordings/recordings/9ygxpwvwydy.md). No slides accompanied this recording in the supplied material; the whole chapter is built from spoken content, 00:00 through 1:19:29. - The cost-benefit framing, the IPTG protocol, glycerol/glucose choice, and the cost curve (00:00-16:00).
- The standard-error-of-the-mean and error-bar digression (16:00-32:05).
- The curve-fitting and overfitting digression (32:05-41:00).
- Returning to the two nonlinear cost models and their range of agreement (41:00-48:00).
- The benefit curve and Michaelis-Menten import saturation (48:00-1:01:00).
- The evolution protocol, daily batch dilution, and what else might be under selection (1:01:00-1:13:21).
- Evolved lacZ activity by lactose concentration, the deletion result, and the closing critique (1:13:21-1:19:29).
- Referred to in the lecture but not contained in the supplied material, and not reproduced here beyond the professor's verbal description:
- U. Dekel and U. Alon, "Optimality and evolutionary tuning of the expression level of a protein," Nature 436, 588-592 (2005) — the paper itself, including its Figure 2A (the cost curve and its three fitted models) and Figure 4 (evolved lacZ activity vs. lactose concentration), and its supplementary growth-rate histograms.
- Unnamed work from Terry Hwa's group on phenomenological bacterial growth laws, cited for the finding that the cost of non-useful protein expression is close to linear up to roughly 30% of total protein.
- Richard Lenski's long-term E. coli evolution experiment (Michigan State, running since the late 1980s), cited for lag-time reduction and other stationary-phase adaptations.
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