The cell weighed by the light it delays

The specimen that is invisible until its light is delayed found that an unstained cell under an ordinary microscope is nearly invisible, because it absorbs almost no light. It changes the light’s phase instead, by a fraction of a wavelength, and Frits Zernike’s phase-contrast microscope turned that change into brightness by delaying the undiffracted light by a quarter wave. His method made the invisible visible. It did not make it measurable: the brightness of a phase-contrast image is a nonlinear function of the phase, surrounded by halos, and no one can read a number off it.

That essay ended with the question of what the phase alone can say about a cell. The answer, worked out in Britain in the same years that Zernike’s microscope won him the Nobel Prize, is that it says how much the cell weighs.

The index of a solution #

The refractive index of a solution rises above that of the pure solvent in proportion to how much is dissolved in it. For proteins in water the rise is remarkably uniform: about 0.18 for each gram of dry protein per millilitre, whatever the protein. That number, the refractive increment, is nearly the same for albumin, haemoglobin and the thousands of proteins in a cell, because a protein’s polarisability, which sets its refractive index, comes from the sum of its amino acids and its amino acids are a narrow family. The reason the increment is so universal lies in where refractive index comes from. A molecule in a light wave is polarised, its electrons displaced slightly by the field, and the index of a substance is set by how polarisable its molecules are per unit volume. Polarisability is close to additive: a protein’s is nearly the sum of its atoms’ and bonds’, so a gram of any protein, whatever its sequence, carries about the same polarisability and raises the index by about the same amount. Everything a scatterer removes tied each particle’s forward scattering to what it takes out of a beam, and the same forward scattering, added up over all the particles in a dilute medium, is the medium’s refractive index above that of the medium without them. The refractive increment is that sum for molecules dissolved in water, strictly at concentrations low enough that each scatters as though its neighbours were absent. Inside a cell the proteins are crowded, a third of the volume or more, and that the increment stays nearly constant even there is an empirical fact, checked by measuring the index of concentrated protein solutions, as much as a consequence of the theory. A cell is about seventy per cent water and most of the rest protein, so its contents raise its refractive index above the medium’s by a few hundredths, and that small excess, multiplied by the cell’s thickness of a few micrometres, delays the light by a few tenths of a wavelength.

The delay at any one point of the cell’s image is

$$\varphi = \frac{2\pi}{\lambda}\int (n - n_m)\,dz = \frac{2\pi\alpha}{\lambda}\int C\,dz,$$

the integral of the concentration $C$ of dry matter along the path through the cell, times the increment $\alpha$. Integrate it once more, over the whole image of the cell, and the double integral of the concentration over the cell’s area and through its thickness is simply the total dry mass:

$$m = \frac{\lambda}{2\pi\alpha}\int \varphi\,dA.$$

Robert Barer in Oxford and, independently, Hugh Davies and Maurice Wilkins in London — the same Wilkins who later worked on the X-ray diffraction of DNA — pointed this out in 1952, and Barer used interference microscopes to weigh living cells within the decade.

Shape drops out #

The first figure shows what makes the method a balance rather than a picture. A cell of a fixed dry mass is drawn in three shapes: rounded up, as cells are when they divide; spread on its surface, as they are most of the time; and flattened. The phase delay at the centre differs sixfold between the rounded cell and the flat one, because the rounded cell is six times thicker. But the integral over the image is the same for all three. A thicker cell delays the light more at each point and covers fewer points; a flatter one delays it less and covers more. The integral sees only the total amount of dry matter the light has crossed.

This is the property Zernike’s images lacked. A phase-contrast image of a rounded cell is bright and haloed, of a flat one faint, and nothing in the brightness says that the two contain the same amount of protein. The phase itself, measured quantitatively, says so exactly. Measuring it requires an interference measurement rather than Zernike’s quarter-wave trick: the light that has passed through the cell is made to interfere with a reference, and the fringes, as the fringe and the spectrum are one measurement put it for a spectrometer, record the delay directly. Barer’s interference microscopes did it by hand, point by point; since the 2000s cameras and holographic reconstruction do it for every pixel of an image at video rate, under the name quantitative phase imaging. Many such instruments use light of short coherence, for the reason the depth a broad spectrum can see gave for optical coherence tomography: a broad spectrum interferes only over a short range of path differences, which suppresses the speckle and the stray fringes that a laser’s long coherence creates from every dust grain and surface in the instrument.

Blur drops out too #

There is a third thing the integral forgives, and it is the one that surprises microscopists. Any real microscope blurs its image: the image that is a diffraction pattern twice found that every point of an object is spread into a small disc by the finite aperture, so the phase at each pixel of the image is a local average of the true phase over a fraction of a micrometre. Fine detail is lost. But blurring is a redistribution, not a loss: smoothing a function by averaging it over a small neighbourhood leaves its integral unchanged, provided nothing is pushed outside the region summed. A slightly defocused phase image, or one taken through a lens too weak to resolve a cell’s organelles, gives the same dry mass as a sharp one, so long as the whole cell and the light it scatters are collected. The resolution decides how finely the mass can be mapped inside the cell; it does not decide the total. That is part of why the method works on inexpensive instruments and on cells imaged at low magnification for days at a time, where a sharp image would be a luxury.

Water drops out #

The second figure takes the argument further. A cell placed in a dilute medium takes up water by osmosis and swells, here to twice its volume, and its contents are diluted to half their concentration. Its refractive index excess halves. The light now crosses twice the thickness of cell at half the excess, and the integrated phase does not change. The water has added volume and nothing else, because the water the cell took up has exactly the refractive index of the water around it, and the measurement is of the excess.

That is the deeper reason the method weighs dry mass and not total mass. Every quantity a microscope can see without a phase measurement — the cell’s outline, its area, its apparent volume — changes when the cell gains or loses water, which cells do constantly in response to the salt concentration around them, as the swelling a membrane cannot stop found for any semi-permeable bag. Dry mass is what a cell has made: its proteins, nucleic acids, lipids and sugars. A cell’s growth is a growth in dry mass, and until this measurement it could only be inferred from volume, which water makes unreliable.

Reading the delay off the light #

A camera cannot see a delay. It records how bright the light is at each pixel, and a transparent cell changes the brightness hardly at all; the delay is in the phase of the light, which a detector averages away. Every quantitative method therefore turns phase into brightness first, and they differ in how.

The oldest is interference with a reference. Split the light before it reaches the cell, send one part round the specimen and recombine the two on the camera: the image is crossed by fringes, and wherever the cell delays the light the fringes shift sideways by the delay’s fraction of a cycle. Barer’s interference microscopes read those shifts by eye. Digital holography reads them by computer, tilting the reference so that the fringes are fine and regular and recovering the phase at every pixel from their local position by a Fourier transform of the recorded frame — one exposure per measurement, fast enough to follow a beating heart cell. Phase-shifting methods take three or four frames with the reference stepped by a quarter of a cycle between them and solve for the phase pixel by pixel, more slowly and more precisely.

A second family needs no reference beam at all. A phase delay that curves across the image focuses or defocuses the light slightly, so the brightness changes a little as the camera is moved through focus, at a rate proportional to how sharply the delay curves. Recording two or three images at different focus settings and solving the equation that links the change of brightness to the curvature of the phase recovers the phase map, with ordinary microscope illumination.

Every method meets the same complication. Phase is measured modulo one cycle, so a delay of 7 radians reads as 0.72, and the true map must be unwrapped by following it continuously from the cell’s edge inward. A spread cell, a few micrometres thick, delays light by a cycle or less and causes no trouble. A rounded cell about to divide can delay it by more, and an unwrapping error across part of it adds or removes a whole cycle there — a mass error of tens of per cent, made in a single step, which is why careful measurements check that the mass varies smoothly through division rather than jumping.

How accurate a balance #

The balance has two kinds of error, and they are of very different sizes. The first is in the conversion. The method assumes one refractive increment for everything in the cell, and the third figure shows how far that is from true for each class of molecule. Proteins sit close to the usual value; nucleic acids a little below it; carbohydrates and lipids further below. A cell’s dry matter is mostly protein, with a fifth or so of nucleic acid and smaller fractions of the others, so the assumed increment is right to a few per cent for a typical cell and the absolute dry mass is known to that accuracy. A cell unusually rich in fat or starch, a seed cell or an adipocyte, would be weighed less accurately.

The second error is noise, and it is much smaller. Each pixel of a phase image carries some noise, from the camera, the light source and vibration, typically some milliradians. Summing over a cell’s pixels adds the noise in quadrature, so the uncertainty in the total grows only as the square root of the number of pixels. The fourth figure computes it: for a large animal cell, at ten milliradians a pixel, a tenth of a picogram out of a few hundred, a part in a thousand. Changes smaller than the composition error can therefore be measured far more precisely than the mass itself, which is what matters for growth: the question is usually not how much a cell weighs but how fast its weight changes, and a constant error in the increment cancels from a rate.

That relative precision is what makes the comparison with the weight a frequency can find apt. There a vibrating cantilever was weighed by the shift of its resonant frequency as mass landed on it, and cantilevers with a fluid channel inside them now weigh single cells flowing through, to a fraction of a picogram. The two methods measure different things — the cantilever weighs a cell’s buoyant mass, its mass minus that of the fluid it displaces, and the phase weighs its dry mass — and the difference between them carries information about a cell’s density.

Growth, weighed #

The last figure shows what the precision is for. A cell doubles its mass between one division and the next, and whether it grows at a constant rate or at a rate proportional to its size was argued for half a century, with consequences for how cells keep their size from drifting. The two laws are hard to tell apart: over a whole cycle they differ by at most a few per cent of the mass. Measurements of volume could not resolve that, because volume fluctuates with water uptake by more than the difference. Measurements of dry mass, by phase and by vibrating channels, could, and in the 2010s they found that many animal cells grow at rates that rise with their size and that they adjust those rates during the cycle — evidence that cells sense and regulate their mass, not only their volume or the time since they last divided.

Because the measurement needs no stain and no contact, it can follow a single cell for days, and the same cell’s mass can be traced through several divisions. It can also be made at every point of the image, so that the dry mass of a nucleus, a chromosome or a lump of aggregated protein inside a living cell can be followed separately. And because a drug that stops a cell from growing stops its dry mass from rising within hours, long before the cell dies or changes shape, phase imaging is used to test how cancer cells respond to treatment, by weighing them.

A cousin in a spectroscope #

The arithmetic has an exact counterpart in another part of optics. The hooks that weigh an absorption line found that the refractive index of a vapour near one of its absorption lines measures the number of absorbing atoms, through the strength of the line, and that Roshdestvensky’s hook method reads it from the bending of interference fringes. Both are weighings by refractive index. The hook method counts atoms in a vapour through the index they add; phase imaging counts the dry matter of a cell through the index it adds. In each case the measured quantity, a phase, is linear in the number of particles present, and so it can be summed, which a brightness cannot.

Where the balance stops being exact #

The formula treats the cell as a thin object in a single plane of focus, with each ray crossing it in a straight line, and treats the refractive increment as a constant. A thick cell refracts and diffracts the light passing through it, so that the measured phase at a point is not exactly the integral along a straight path, and very thick or very dense samples — tissue slices, clusters of cells — need tomographic reconstruction rather than a single image. The phase is measured modulo a full wave, so cells thick enough to delay light by more than a wavelength need unwrapping, which can fail at sharp edges. And the medium must have a known, uniform refractive index; cells in a medium whose index varies, or near the edges of a dish, carry errors from the background.

The method also weighs everything that is not water, including the salt dissolved in the cell’s water, which raises the index slightly. Since the medium contains salt too, at a similar concentration, the effect largely cancels, but a cell whose internal salt concentration differs from the medium’s carries a small error from it.

What a mass map does not show #

The figures draw delays and masses, and leave out what the material is. A phase image says how much dry matter is at each point and nothing about which molecules make it up; a cell full of protein and a cell full of an equal mass of glycogen look the same, apart from the small difference in their increments. Identifying the material needs a second measurement — fluorescence, Raman scattering, or the variation of the phase with wavelength, which differs slightly between classes of molecule. Nor do the figures show the dynamics a phase image records: the fluctuations of a cell’s membrane, the movement of organelles, the flicker of a red blood cell’s edge, all of which appear as changes in phase and are used to measure the mechanical properties of cells. Those are motions; the mass is what stays when they are averaged out.

Still open: what a cell’s mass is telling it #

The measurement is solved; the biology it opened is not. How a cell senses its own mass, so that it can grow faster or slower and divide at the right size, is not understood, and the dry-mass measurements that showed cells do it have also shown how much variation there is between cells of one type and between the stages of one cell’s life. Whether the regulation acts on mass itself, on the concentration of some particular protein, or on the cell’s density, and how it fails in cancer cells that grow without regard to size, are questions that phase imaging is now being used to ask.

The habit worth carrying away is to integrate a quantity that is linear in what is wanted. Light crossing a cell is delayed by $(2\pi\alpha/\lambda)\int C\,dz$, so the delay summed over the cell’s image is $(2\pi\alpha/\lambda)$ times its dry mass — 536 rad·μm² for 300 pg in any shape, unchanged when the cell swells with water — and a phase noise of ten milliradians a pixel weighs a mammalian cell to a tenth of a picogram, limited in absolute terms only by a few per cent of uncertainty in what its dry matter is made of. A measurement that is linear in an amount can be added up; one that is not, like the brightness of a phase-contrast image, can only be looked at.