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.
10. Single Molecules and Protein Bursts¶
What this covers¶
How do you watch a single protein molecule fluoresce inside a living cell, and how do you turn that observation into a count of how many protein molecules a gene made in one go? The lecture works through both questions using a single 2006 paper from Sunney Xie's lab, which was the first to get single-molecule fluorescence measurements working inside live bacteria rather than in a purified, in-vitro sample. It assumes you can already think in orders of magnitude, that you know what a diffraction limit and a Stokes shift are, and that you have at least seen the exponential, geometric and Poisson distributions before (they get a proper treatment the following lecture).
Why live-cell single-molecule fluorescence was thought impossible¶
Before this paper, single-molecule biophysics — both fluorescence detection and mechanical manipulation of single molecules — was almost exclusively an in-vitro technique: purified components on a glass slide, not inside a living cell. That gave real insight into the dynamics of molecular motors, transcription and translation, but doing the analogous experiment inside a living cell was widely doubted to be possible at all. Two papers from the same lab, one in Science and one in Nature, appeared within the same month in January 2006 and demonstrated two independent ways of getting single-molecule dynamics out of living cells. The lecture works through one of them in detail; the other used a microfluidic trap to enclose single cells in a small volume and run a more conventional enzymatic assay inside it, and was shown to work in both E. coli and yeast.
Four separate obstacles stand between "this works in a test tube" and "this works in a living cell", and the paper's design is a response to each of them:
- Photodamage. Enough laser power kills the cell outright; the question is whether you can dial the power down far enough to still see something.
- Many molecules. A live cell contains many copies of most things, but a single-molecule measurement needs to isolate one at a time — either temporally or spatially.
- Diffusion. Molecules move on the timescale of the measurement.
- Autofluorescence. The cell itself is weakly fluorescent across a broad range of wavelengths, because it is full of many different molecules, each with its own absorption/emission profile. That broadband background — not the tightly-defined signal from an introduced fluorescent label — is the noise you are trying to detect a single molecule's signal above.
The photon budget is not actually the bottleneck¶
A natural worry is that a single fluorescent molecule just does not emit enough light to detect. It does not, in fact, run out: a single molecule can put out of order \(10^4\) photons per second, for anywhere from a few tenths of a second up to tens of seconds before it stops fluorescing permanently (the paper's illumination conditions, deliberately intense, push this down to around \(250\ \text{ms}\); more on that below). Modern camera quantum efficiency — the fraction of photons hitting the camera that get registered — is close to \(0.9\), effectively \(1\) for these purposes. So the number of photons collectable from a single molecule is not small. The actual problem is detecting that signal against the autofluorescence background, which the paper's figures show is comparable in scale to the single-molecule signal itself.
Localising a single molecule far below the diffraction limit¶
Start with scale. A typical globular protein — GFP-sized, say — is a few nanometers across. Illuminate it and image it, and the spot you actually see on the camera is nothing like a few nanometers wide: it is a diffraction-limited spot, a fundamental optical limit set by the wavelength of the light, of order \(\lambda/2\) (up to numerical-aperture factors). With excitation at \(514\ \text{nm}\) and fluorescence emission at a longer wavelength (emission is always redder than excitation — the emitted photon carries less energy than the absorbed one, and energy goes as \(1/\lambda\)), the resulting spot is of order \(300\ \text{nm}\) across: roughly a hundred times the size of the protein producing it.
That gap between the actual size of the object and the size of its image matters in two very different ways, and it is easy to conflate them.
Resolving two molecules apart is genuinely limited by the spot size. If two molecules sit closer together than about \(\lambda/2\), their two diffraction-limited spots overlap almost completely and the combined pattern looks like a single, slightly brighter spot — not obviously two molecules. You need real spatial separation, on the order of the wavelength, before the sum of the two spots visibly splits into two peaks.
Locating a single, known-to-be-single molecule is a completely different question, and it is not limited by the spot width in the same way. If you know there is exactly one molecule producing that \(300\ \text{nm}\) spot, the question is only: where is the center of that spot? And locating the center of a measured distribution is a statistics problem, not an optics problem. If you sample a distribution \(N\) times, your uncertainty in its mean falls as
where \(\sigma\) is the width of the distribution — it does not go to zero as you take more samples, but the uncertainty in where the mean is does. Here \(N\) is the number of photons collected, and \(\sigma\) is the \(\sim 300\ \text{nm}\) spot width. Collecting \(N \sim 10^4\) photons — not an unusual number, as established above — gives \(\sqrt{N} \sim 100\), so
A few thousand photons buy you nanometer-scale localization of a spot that is itself three hundred nanometers wide. This is genuinely surprising the first time you see it, and it survives a check that seems like it should break it: cameras do not record a continuous position, they bin light into discrete pixels, and a \(100\ \text{nm}\) pixel at the sample plane (a physical pixel of order ten microns, divided by roughly \(100\times\) magnification) is not small compared to the spot. Doing the calculation carefully shows the pixelation costs you very little — you still land close to the \(1/\sqrt{N}\) estimate. (There is a further, genuinely technical wrinkle: in the presence of camera read noise, larger pixels can sometimes be better than the naive argument suggests, for reasons that trade off against each other in practice.)
This is exactly the trick behind "super-resolution" localization microscopy, and it long predates the live-cell paper — it was popularized for single-molecule biophysics in work such as Ahmet Yildiz's FIONA method (fluorescence imaging with one nanometer accuracy), which attached single fluorophores to the heads of molecular motors and watched them step. The trick only works this well while the molecule stays still during the exposure — those experiments slowed the motors down (limiting ATP, so steps happened every second or ten) precisely so the assumption "the molecule is not moving" would hold during each snapshot, at a cost of only collecting maybe \(10\)–\(15\%\) of the photons actually emitted (the rest go in directions the objective does not catch).
The general strategy behind localizing two or more molecules that sit within a single diffraction-limited spot — the various "super-resolution" schemes — is to force the fluorophores to turn on and off one at a time, localize each one separately while it is the only one lit, and then assemble the individual localizations. The differences between named methods (STORM, from Xiaowei Zhuang's lab; PALM, from Eric Betzig; FIONA and its playful cousins) are essentially all about the mechanism used to get molecules to blink on and off one at a time; the localization argument once only one is on is the same argument as above.
Diffusion sets the real limit on exposure time¶
Locating a single molecule this precisely assumed it stayed put during the exposure. Inside a cell it does not, and a back-of-the-envelope calculation shows why this is a serious constraint rather than a minor correction.
At the scale of a protein moving through a viscous fluid, inertial forces are negligible — this is the low-Reynolds-number regime (developed further later in the course, in the context of how bacteria swim), and the object's response to a pushing force is not Newton's second law but something closer to Aristotelian physics: velocity proportional to force, not acceleration proportional to force. The relevant relation is the Stokes–Einstein one,
where \(kT\) is thermal energy (about \(4.1\ \text{pN·nm}\) at room temperature) and \(\gamma\) is a drag coefficient that scales with the size of the object. For a protein-sized object in the cytoplasm — whose effective viscosity is taken to be roughly ten times that of water, since it is packed with other macromolecules — this gives \(D \sim 10\ \mu\text{m}^2/\text{s}\). The characteristic distance travelled by diffusion in time \(t\) is
The paper's camera exposure was \(\Delta t = 0.1\ \text{s}\). Plugging in gives \(\ell\) of order one micron — comparable to the entire length of an E. coli cell. In other words, over the course of a single exposure, an untethered protein would be expected to explore a large fraction of the cell's volume, and any attempt to localize it as a single sharp spot would simply fail.
Shortening the exposure does not obviously help: the number of photons collected scales linearly with exposure time, so cutting the exposure by a factor of ten also cuts the signal by a factor of ten, and the signal was already only barely clear of the autofluorescence background (this is visible directly in the paper's Figure 1). There is also a hard floor on how fast you can pump more photons out by raising the laser intensity, set by the finite cycling time of the excitation– emission process itself. The solution actually used was not to fight diffusion optically but to remove it physically: anchor the fluorescent protein to the cell membrane, which slows its motion enough that it stays effectively put during a \(0.1\ \text{s}\) exposure. In the paper this was done by fusing the fluorescent protein (Venus, chosen for faster maturation than ordinary GFP) not directly into the membrane, but onto Tsr, a membrane protein (the same one that appears later in the course as part of the E. coli chemotaxis receptor system) — a fusion the authors checked did not itself change the amount of fluorescence produced, ruling out the confound that some other membrane-targeting sequences introduced.
How do you know you are looking at one molecule?¶
Everything above assumed you could tell a single-molecule spot from a many-molecule spot in the first place. A diffraction-limited spot of \(300\ \text{nm}\) could in principle contain one molecule or a thousand — \(10\times10\times10\) molecules packed into a \(30\ \text{nm}\) cube is still far smaller than the spot — so spot size alone tells you nothing. The evidence has to come from the dynamics of the spot's intensity over time.
A single fluorescent molecule eventually bleaches: some side reaction (with oxygen, for example) irreversibly knocks it out of the excitation–emission cycle for good. Each cycle carries some small, roughly constant probability — of order \(1\) in \(10^5\) — of triggering that irreversible step, which means the time to bleach is exponentially distributed, with some characteristic lifetime \(\tau\). For a single molecule this shows up as intensity holding roughly steady and then dropping to background in a single step. For many independent molecules bleaching in the same spot, the population-level intensity instead decays smoothly, because at any moment a roughly constant fraction of the surviving molecules bleach — the same logic as radioactive decay of a bulk sample. A single abrupt bleaching step, rather than a gradual exponential decay, is treated as the gold standard for having isolated a single molecule. A weaker, secondary piece of evidence used in the paper was that the intensity of the spot matched what had been measured for isolated Venus molecules on a slide — weaker because the same molecule can genuinely fluoresce at different intensities in different environments, so a match is supporting rather than decisive.
(Bleaching, in this irreversible sense, should be kept separate from blinking, where a fluorophore drops into a dark state temporarily and later recovers — a different, reversible phenomenon not part of this argument.)
The imaging cycle: deliberately bleaching to create a clean baseline¶
The measured bleaching lifetime under the paper's illumination was \(\tau = 250\ \text{ms}\) — short, reflecting fairly intense laser illumination. This short, controllable bleaching time is put to direct use in the experimental design. Every three minutes, the cell is illuminated for a total of \(1.2\ \text{s}\), but only the first \(0.1\ \text{s}\) of that window is the actual image used for analysis. The remaining second-plus of illumination is a deliberate, intentional bleaching step, whose purpose is to guarantee that essentially nothing fluorescent from this round survives into the next one.
How good is that guarantee? The probability that a fluorophore present at the start of the \(1.2\ \text{s}\) illumination is still unbleached at the end of it is the exponential survival probability
So after each three-minute cycle's illumination, on the order of \(1\%\) of whatever fluorescence was there survives — close enough to zero that any spot seen in the next cycle's \(0.1\ \text{s}\) snapshot can be attributed with confidence to Venus molecules that matured since the previous snapshot, rather than to leftover molecules from before. This is what turns "count the spots you see" into "count the new protein molecules produced in the last three minutes."
What counts as a "burst"¶
The paper reports the number of protein molecules produced in each burst of gene expression, and getting from the raw three-minute snapshots to a burst size is a small but easy-to-get-wrong step. Consider a worked example, reading off the zoomed-in trace of new molecules over time (Figure 3B in the paper): at one sampling instant the count is \(0\); three minutes later, one new molecule has appeared; three minutes after that, two more new molecules appear; only after that does the count of new molecules return to \(0\).
The size of the first burst here is not \(1\) — the count at the first nonzero sample — because that sample could simply have caught the process in the middle. It is not obviously \(2\) either. The correct reading is that a burst is the run of consecutive nonzero samples between two zero readings, summed together: \(1 + 2 = 3\) molecules made up this one burst. Some bursts recorded this way run as large as \(10\)–\(15\) molecules.
This matters because bursts are not instantaneous: a typical burst lasts on the order of five to seven minutes, while bursts under these repressed conditions occur at a rate of roughly one per cell cycle (the cell cycle itself measured at \(55\) minutes). Comparing a burst's duration to the typical spacing between bursts gives a rough estimate that around \(15\%\) of the events counted as a single burst could actually be two bursts that overlapped in time and were merged by the counting procedure.
What a burst is made of¶
The paper argues for a specific mechanistic picture behind each burst. The promoter sits under Lac repressor control; the repressor occasionally falls off the operator, and while it is off, RNA polymerase can bind and transcribe. The claim, supported by the data, is that each burst typically corresponds to exactly one mRNA transcript — one repressor-unbinding event usually yields one round of transcription, which is then translated repeatedly (with a geometrically distributed number of protein molecules per burst) before the mRNA itself degrades.
Establishing "typically one mRNA per burst" did not come from directly counting mRNA molecules one at a time; it came from combining two separately measured quantities. RT-PCR (reverse transcribing the mRNA to DNA and amplifying it) gave the average number of mRNA copies per cell. That average is related to the burst-level quantities by
with an extra factor to convert the burst rate (naturally measured per cell cycle) into the same time units as the mRNA lifetime — i.e. using the \(55\)-minute cell-cycle length as the unit conversion. Solving this relation for "mRNA per burst" using the measured average mRNA copy number gives an answer consistent with \(1\). This is also why occasional bursts genuinely produced by two close-together transcription events are hard to distinguish from single-mRNA bursts: the mRNA lifetime here is short (about \(1.5\) minutes), so two mRNAs made within that short a window would be essentially indistinguishable from one in the data, but this happens rarely enough (few bursts per hour) not to change the "typically one mRNA per burst" conclusion.
The number of bursts observed per cell cycle was found to fit a Poisson distribution with mean around \(\lambda \approx 1.2\),
and the number of protein molecules within a single burst was found to fit a geometric distribution. Both of these are exactly the distributions predicted by the simplest possible kinetic model of gene expression — constant-rate mRNA production and degradation, constant-rate translation and protein degradation — which the course develops in the following lecture. The point made explicitly in the lecture is that this agreement is worth checking rather than assuming: "even things that we assume to be true, we should still check to see if they are." Here, the simple model's prediction and the single-molecule measurement agreed.
Sources¶
All material in this chapter is drawn from the transcript of a single MIT 8.591J (Fall 2014)
lecture: docs/computational-biology/mit-ocw/8591j-2014/recordings/recordings/dp4nqipuh6w.md
(converted from dp4nqipuh6w-captions.srt; timestamps below refer to this transcript). No slides,
written notes or problem set were supplied for this lecture, and there was no board content
captured — anything drawn or written on the board during the lecture (the bleaching-curve sketches,
the diffusion and localization scratch work, the illumination-cycle timeline) has been
reconstructed here from the spoken description only.
- Overview of live-cell single-molecule fluorescence and the two January-2006 papers from Sunney
Xie's lab: [00:00]–[03:32]. The paper analyzed in the rest of the lecture is referred to
throughout but its authors are inaudible in the recording (
[INAUDIBLE]); the companion paper using a microfluidic trap in E. coli and yeast is likewise not named in the audio. Neither paper was supplied as a source file — a reader wanting the primary text should locate both via the OCW course page linked in this transcript's front matter. - The four challenges (photodamage, many molecules, diffusion, autofluorescence) and the photon budget: [03:32]–[12:42].
- Diffraction limit, localization precision, the standard-error argument, CCD pixelation: [12:42]– [30:25]. FIONA is attributed to Ahmet Yildiz's group; that paper is referred to but not supplied.
- Super-resolution naming aside (STORM/Xiaowei Zhuang, PALM/Eric Betzig, FIONA and related acronyms): [30:25]–[35:56]. These are named in passing, not treated in technical depth, and no paper for them was supplied.
- Diffusion back-of-envelope calculation and the membrane-anchoring solution (Venus–Tsr fusion): [35:56]–[45:06].
- Single-step bleaching as evidence for a single molecule, and the bleaching mechanism: [45:06]– [51:55].
- The illumination/bleach/image cycle and the survival-probability calculation: [51:55]–[1:03:46].
- Defining a "burst" from Figure 3B, and the mRNA-per-burst and Poisson/geometric burst statistics: [1:03:46]–[1:19:54]. Figure references (Figure 1, Figure 3B) are to the paper's own figures, not reproduced here.
← 9. Robustness in Bacterial Chemotaxis · Contents · 11. Clonal Interference and Fitness Landscapes →