The post How to Solve Equations That Are Stubborn as a Goat first appeared on Quanta Magazine

]]>Math teachers have stymied students by sticking goats in strangely shaped fields for hundreds of years, but one particular grazing goat problem has gotten the goat of mathematicians for more than a century. Until last year they were only able to find approximate answers to the problem, and it took a new approach with some very advanced mathematics to finally produce an exact solution. Let’s take a look at how a question you might find on a math test can turn into a problem that stumps mathematicians for over a century.

The simplest kind of grazing goat problem has the hungry animal attached to the side of a long barn by a fixed length of rope.

Usually in these problems we want to find the area of the region the goat has access to. What does that region look like?

With the leash pulled taut the goat can make a semicircle and can reach anything inside it. The area of a circle is *A* = *πr*^{2}, so the area of a semicircle is *A* = $latex \frac{1}{2}$*πr*^{2}. If, for example, the rope has length 4, then the goat could graze in a region with area *A* = $latex \frac{1}{2}$*π* × 4^{2} = 8*π* square units.

This straightforward setup doesn’t pose much of a challenge to the math student or to the goat, so let’s make it more interesting. What if the goat is tied to the side of a square barn?

Let’s say the rope and the side of the barn both have length 4, and that the rope is attached to the middle of one side. What’s the area of the region the goat has access to now?

Well, the goat still has access to the same semicircle as in the first problem.

But the goat can also continue around the corner of the barn. Once it’s at the corner, the goat has two more units of rope to work with, so it can sweep out another quarter circle of radius 2 on either side of the barn.

The goat can access the semicircle of radius 4 plus two quarter circles of radius 2, for a total area of *A* = $latex \frac{1}{2}$*π* × 4^{2} + $latex \frac{1}{4}$*π* × 2^{2} + $latex \frac{1}{4}$*π* × 2^{2} = 10*π* square units.

You can make the problem more challenging by changing the shape of the obstruction. I’ve seen goats attached to triangles, hexagons and even concave shapes.

You can also make a new math question from an old one by reversing it: Instead of starting with rope length and finding the area, you can start with the area and find the rope length.

For example, let’s stick with our square barn and ask a new question: How long would the rope have to be for the goat to have access to a total of 50 square units of area? Reversing a math problem can breathe new life into an old idea, but it also makes this problem much more challenging.

First, notice that the shape of the region depends on the length of the rope. For example, if the rope is shorter than 2 units in length, the goat can’t get around the corner of the barn, so the region will only be a semicircle.

If the rope is longer than 2 units, the goat can get around the corner, as we saw above.

And if the rope is longer than 6 units, the goat can get behind the barn, creating another set of quarter circles to consider. (If the rope gets much longer, there will be overlap. See the exercises at the end of the column for an example of this.)

We want to find the rope length that gives us 50 square units of total area. The way to do this mathematically is to set our area formula equal to 50 and solve for *r*. But each kind of region has a different area formula. Which one do we use?

Figuring this out requires a little casework. If r ≤ 2 the area of the region is *A* = $latex \frac{1}{2}$*πr*^{2}. The biggest area would occur when *r* = 2, which yields a total area of *A *= $latex \frac{1}{2}$*π* × 2^{2} = 2*π* ≈ 6.28. This is less than 50, so we know we need more than 2 units of rope.

If 2 < *r* ≤ 6, this gives us the semicircle plus the two quarter circles we encountered before. The radius of the semicircle is *r*, and the radius of the quarter circles is *r* – 2, since two units of rope are needed to get to the corner and whatever rope remains acts like the radius of the quarter circle centered at the corner.

The area of this semicircle is $latex \frac{1}{2}$*πr*^{2}, and the area of each quarter circle is $latex \frac{1}{4}$*π*(*r* – 2)^{2}. Adding this up gives us a total area of

$latex \begin{aligned}A&=\frac{1}{2} \pi r^{2} +\frac{1}{4}\pi (r-2)^{2} + \frac{1}{4}\pi (r-2)^{2}\\

\\A &=\frac{1}{2} \pi r^{2} + \frac{1}{2}\pi (r-2)^{2}.\end{aligned}$

We get the biggest possible area when *r* = 6, which gives an area of *A* = $latex \frac{1}{2}$*π* × 6^{2} + $latex \frac{1}{2}$*π* × 4^{2} = 26*π* ≈ 81.68 square units. Since 50 < 26*π*, that means the *r* that will give us 50 square units of area must be less than 6.

Knowing that *r* must be between 2 and 6 units settles the question of which area formula we should use: When 2 < *r* ≤ 6, the area is *A* = $latex \frac{1}{2}$*πr*^{2} + $latex \frac{1}{2}$*π*(*r* – 2)^{2}. To find the exact value of *r* that gives us 50 square units of area, we set up this equation:

50 = $latex \frac{1}{2}$*πr*^{2} + $latex \frac{1}{2}$*π*(*r* – 2)^{2}.

Notice that this is another way in which our reversed question is more complicated than the original: Instead of just computing the area the goat can reach, we need to solve an equation to figure out the length of the rope. To do that, we need to isolate *r*. We have to use arithmetic and algebra to get *r* by itself on one side of the equation, and that will tell us exactly what *r* must be.

Our equation may look a little intimidating at first, but it’s just a quadratic equation in *r*. There’s a standard procedure for solving such equations: We rearrange it in the form *ar*^{2} + *br* + *c* = 0 and then use the quadratic formula. A little algebra and arithmetic does the trick.

50 = $latex \frac{1}{2}$*π*^{2} + $latex \frac{1}{2}$*π*(*r* – 2)^{2}

$latex \frac{100}{\pi}$ = *r*^{2} + (*r* – 2)^{2}

$latex \frac{100}{\pi}$ = 2*r*^{2} – 4*r* + 4

0 = 2*r*^{2} – 4*r* + 4 – $latex \frac{100}{\pi}$.

This may not be the most beautiful mathematical expression in the world, but it’s just a quadratic equation, so we can apply the quadratic formula to solve exactly for *r*. This gives us an answer of

*r* = 1 + $latex\sqrt{\frac{50}{\pi} – 1}$ ≈ 4.86.

Because we were able to isolate *r* in our equation, we now know exactly how long the rope must be to get an area of 50 square units. (Notice that the value of r we found is between 2 and 6, as expected.)

As challenging as this reversed goat grazing problem was compared to the initial ones we looked at, mathematicians discovered that the problem becomes even more challenging when you stick the goat inside the barn. So challenging, in fact, that they couldn’t solve it exactly.

Let’s put the goat inside our square barn with side length 4 and attach the rope to the middle of a wall. How long does the rope need to be for the goat to have access to half the area inside the barn?

As above, part of the challenge is that the shape of the region depends on the value of *r*. To get half the area of the square we need *r* to be longer than half the side of the barn but shorter than the full side, which gives us a region like this.

Finding a formula for the area of this region isn’t so easy. We can imagine the region as one sector of a circle of radius *r* plus two right triangles, and then use some high school geometry to get a formula. But as we’ll soon see, the mixing of circles and triangles is going to cause some trouble.

Let’s start with the triangles. The Pythagorean theorem tells us that the length of the missing leg in each right triangle is $latex\sqrt{r^{2}-4}$. This makes the area of one of the triangles $latex \frac{1}{2}$ × 2 × $latex\sqrt{r^{2} – 4}$ = $latex\sqrt{r^{2} – 4}$, so the two triangles together have an area of 2 $latex\sqrt{r^{2}-4}$.

Now for the circular sector.

The area of a sector is *A* = $latex\frac{1}{2}r^{2}$*θ*, where *θ* is the measure of the central angle (in radians, not degrees). We need a formula for the area in terms of *r*, so we need to express the angle *θ* in terms of *r*. To do this, we’ll use the law of cosines, an underappreciated theorem from high school trigonometry.

Applying the law of cosines to the isosceles triangle with sides *r*, *r* and 4 gives us

4^{2} = *r*^{2} + *r*^{2} – 2*r*^{2}cos*θ*,

which we can solve for cos*θ*:

cos*θ* = $latex \frac{2 r^{2}-16}{2 r^{2}}$ = $latex \frac{r^{2}-8}{r^{2}}$.

To isolate *θ*, we need to take the inverse cosine, or arccosine, of both sides of the equation. This gives us

*θ* = arccos $latex \left(\frac{r^{2}-8}{r^{2}}\right)$.

Now we have the angle *θ* in terms of *r*, so we can now express the area of our sector in terms of *r* and *r* alone.

*A* = $latex \frac{1}{2}$*r*^{2}*θ*

*A* = $latex \frac{1}{2}$*r*^{2}arccos $latex \left(\frac{r^{2}-8}{r^{2}}\right)$.

Our final area formula is the sum of the sector area and the area of the two triangles, which is

*A* = $latex \frac{1}{2}$*r*^{2}arccos $latex \left(\frac{r^{2}-8}{r^{2}}\right)$ + 2 $latex\sqrt{r^{2}-4}$.

We now have a formula for the area of the region accessible to the goat inside the square entirely in terms of *r*. Now we just need to find the value of *r* that gives the goat access to half the square. The entire square has area 16, so all we have to do is plug *A* = 8 into our equation and solve for *r* and we’ll be finished.

8 = $latex \frac{1}{2}$*r*^{2}arccos $latex \left(\frac{r^{2}-8}{r^{2}}\right)$ + 2 $latex\sqrt{r^{2}-4}$.

There’s just one small problem: It’s not possible to solve for *r* in this equation.

That is, it’s not possible to solve exactly for *r* in this equation. We can use a calculator to approximate the value of *r* that makes this equation true (*r* ≈ 2.331), but we can’t isolate *r*in our equation. The mixing of trigonometric functions and polynomial functions in our equation creates obstacles we can’t get around.

We could try to get the *r*’s out from inside the arccosine function, but to do that we’d have to put the other *r*’s inside a cosine function. Either way we’d be dealing with an equation that involves a transcendental function, like an exponential or trigonometric function. Transcendental functions can’t be simply expressed in terms of the usual algebraic operations like addition and multiplication, and so in general transcendental equations can’t be solved exactly.

This issue lies at the heart of a famous grazing goat problem posed in the 19th century where the goat was placed inside a circular barn. As in our square barn problem, the goal was to determine how long the rope had to be for the goat to have access to half the region.

The region accessible by the goat takes the shape of a “lens” — two circular segments stacked together.

It’s possible to use high school geometry to find the area of this lens in terms of the rope length r, but the formula is much more complicated than it is for the square. And when you set this equal to half the area of the circular barn, you run into the same problem we ran into inside the square: You just can’t isolate *r*. You can approximate it, but you can’t solve for *r* exactly.

This sort of obstinacy is no more appealing in an equation than it is in a goat. For over 100 years, mathematicians tried to find an exact solution to this goat-in-a-circle puzzle, but it wasn’t until last year that a German mathematician finally figured it out. He used complex analysis — mathematics far removed from the geometry of circles and squares most goat problems rely on — to solve explicitly for . And while using something as advanced as a contour integral to find the length of a goat’s leash may seem like overkill, there’s always mathematical satisfaction in doing what couldn’t be done before. And there’s always the possibility that these new methods, even if they arise from studying a silly problem about goats, might lead to insights beyond the barnyard.

1. If the goat is attached to the middle of the side of a square barn with side length 4 by a rope of length 8, outside the barn, what’s the area of the region the goat has access to?

2. If the goat is attached to the corner of a square barn with side length 4 by a rope of length 8, outside the barn, what’s the area of the region the goat has access to?

3. Suppose the goat is inside an equilateral triangle of side 4 attached to a vertex. How long would the rope have to be for the goat to have access to half the triangle?

4. If the goat is attached to the middle of the side of a square barn with side length 4 by a rope of length 10, outside the barn, what’s the area of the region the goat has access to?

Click for Answer 1:

Click for Answer 2:

Click for Answer 3:

Click for Answer 4:

The post How to Solve Equations That Are Stubborn as a Goat first appeared on Quanta Magazine.]]>The post Quantum Double-Slit Experiment Offers Hope for Earth-Size Telescope first appeared on Quanta Magazine

]]>Such precise observations are currently impossible. But scientists are proposing ways to quantum mechanically link up optical telescopes around the world in order to view the cosmos at a mind-boggling level of detail.

The trick is to transport fragile photons between telescopes, so that the signals can be combined, or “interfered,” to create far sharper images. Researchers have known for years that this kind of interferometry would be possible with a futuristic network of teleportation devices called a quantum internet. But whereas the quantum internet is a far-off dream, a new proposal lays out a scheme for doing optical interferometry with quantum storage devices that are under development now.

The approach would represent the next stage of astronomy’s obsession with size. Wider mirrors create sharper images, so astronomers are constantly designing ever-bigger telescopes and seeing more details of the cosmos unfold. Today they’re building an optical telescope with a mirror nearly 40 meters wide, 16 times the width (and thus resolution) of the Hubble Space Telescope. But there’s a limit to how much mirrors can grow.

“We’re not going to be building a 100-meter single-aperture telescope. That’s insane!” said Lisa Prato, an astronomer at Lowell Observatory in Arizona. “So what’s the future? The future’s interferometry.”

Radio astronomers have been doing interferometry for decades. The first-ever picture of a black hole, released in 2019, was made by synchronizing signals that arrived at eight radio telescopes dotted around the world. Collectively, the telescopes had the resolving power of a single mirror as wide as the distance between them — an effectively Earth-size telescope.

To make the picture, radio waves arriving at each telescope were precisely time-stamped and stored, and the data was then stitched together later on. The procedure is relatively easy in radio astronomy, both because radio-emitting objects tend to be extremely bright, and because radio waves are relatively large and thus easy to line up.

Optical interferometry is much harder. Visible wavelengths measure hundreds of nanometers long, leaving far less room for error in aligning waves according to when they arrived at different telescopes. Moreover, optical telescopes build images photon-by-photon from very dim sources. It’s impossible to save these grainy signals onto normal hard drives without losing information that’s vital for doing interferometry.

Astronomers have managed by directly linking nearby optical telescopes with optical fibers — an approach that led in 2019 to the first direct observation of an exoplanet. But connecting telescopes farther apart than 1 kilometer or so is “extremely unwieldly and expensive,” said Theo ten Brummelaar, director of the CHARA Array, an optical interferometric array in California. “If there was a way of recording photon events at an optical telescope with some kind of quantum device, that would be a great boon to the science.”

Joss Bland-Hawthorn and John Bartholomew of the University of Sydney and Matthew Sellars of the Australian National University recently proposed a scheme for doing optical interferometry with quantum hard drives.

The principle behind the new proposal traces back to the early 1800s, before the quantum revolution, when Thomas Young devised an experiment to test whether light is made of particles or waves. Young passed light through two closely separated slits and saw a pattern of regular bright bands form on a screen behind. This interference pattern, he argued, appeared because light waves from each slit cancel out and add together at different locations.

Then things got a whole lot weirder. Quantum physicists discovered that the double-slit interference pattern remains even if photons are sent toward the slits one at a time; dot by dot, they gradually create the same bands of light and dark on the screen. However, if anyone monitors which slit each photon goes through, the interference pattern disappears. Particles are only wavelike when undisturbed.

Now imagine that, instead of two slits, you have two telescopes. When a single photon from the cosmos arrives on Earth, it could hit either telescope. Until you measure this — as with Young’s double slits — the photon is a wave that enters both.

Bland-Hawthorn, Bartholomew and Sellars suggest plugging in a quantum hard drive at each telescope that can record and store the wavelike states of incoming photons without disturbing them. After a while, you transport the hard drives to a single location, where you interfere the signals to create an incredibly high-resolution image.

To make this work, quantum hard drives have to store lots of information over long periods of time. One turning point came in 2015, when Bartholomew, Sellars and colleagues designed a memory device made from europium nuclei embedded in a crystal that could store fragile quantum states for six hours, with the potential to extend this to days.

Then, earlier this year, a team from the University of Science and Technology of China in Hefei demonstrated that you could save photon data into similar devices and later read it out.

“It’s very exciting and surprising to see that quantum information techniques can be useful for astronomy,” said Zong-Quan Zhou, who co-authored the recently published paper. Zhou describes a world in which high-speed trains or helicopters rapidly shuttle quantum hard drives between far-apart telescopes. But whether these devices can work outside laboratories remains to be seen.

Bartholomew is confident that the hard drives can be shielded from errant electric and magnetic fields that disrupt quantum states. But they’ll also have to withstand pressure changes and acceleration. And the researchers are working to design hard drives that can store photons with many different wavelengths — a necessity for capturing images of the cosmos.

Not everyone thinks it’ll work. “In the long run, if these techniques are to become practical, they will require a quantum network,” said Mikhail Lukin, a quantum optics specialist at Harvard University. Rather than physically transporting quantum hard drives, Lukin has proposed a scheme that would rely on a quantum internet — a network of devices called quantum repeaters that teleport photons between locations without disturbing their states.

Bartholomew counters that “we have good reasons to be optimistic” about quantum hard drives. “I think in a five-to-10-year time frame you could see tentative experiments where you actually start looking at real sources.” By contrast, the construction of a quantum internet, Bland-Hawthorn said, is “decades from reality.”

The post Quantum Double-Slit Experiment Offers Hope for Earth-Size Telescope first appeared on Quanta Magazine.]]>The post How Gravity Is a Double Copy of Other Forces first appeared on Quanta Magazine

]]>Albert Einstein first spoke of gravity in terms of bends in space-time in his general theory of relativity. Most theorists assume that gravity actually pushes us around through particles, called gravitons, but attempts to rewrite Einstein’s theory using quantum rules have generally produced nonsense. The rift between the forces runs deep, and a full unification of the two grammars seems remote.

In recent years, however, a baffling translation tool known as the “double copy” has proved surprisingly adept at turning certain gravitational entities, such as gravitons and black holes, into dramatically simpler quantum equivalents.

“There’s a schism in our picture of the world, and this is bridging that gap,” said Leron Borsten, a physicist at the Dublin Institute for Advanced Studies.

While this unproven mathematical relationship between gravity and the quantum forces has no clear physical interpretation, it’s allowing physicists to pull off nearly impossible gravitational calculations and hints at a common foundation underlying all the forces.

John Joseph Carrasco, a physicist at Northwestern University, said anyone who spends time with the double copy comes away believing “that it’s rooted in a different way of understanding gravity.”

On one side of the fundamental physics divide stand the electromagnetic force, the weak force and the strong force. Each of these forces comes with its own particle carrier (or carriers) and some quality that the particle responds to. Electromagnetism, for instance, uses photons to push around particles that possess charge, while the strong force is conveyed by gluons that act on particles with a property called color.

Physicists can describe any event involving these forces as a sequence of particles scattering off each other. The event might start with two particles approaching each other, and end with two particles flying away. There are, in principle, infinitely many interactions that can happen in between. But theorists have learned how to make frighteningly accurate predictions by prioritizing the simplest, most likely sequences.

On the other side of the divide stands gravity, which rebels against this kind of treatment.

Gravitons react to themselves, generating looping, Escher-like equations. They also proliferate with a promiscuity that would make a bunny blush. When gravitons mingle, any number of them can emerge, complicating the prioritization scheme used for other forces. Just writing down the formulas for simple gravitational affairs is a slog.

But the double copy procedure serves as an apparent back door.

Zvi Bern and Lance Dixon, later joined by Carrasco and Henrik Johansson, developed the procedure in the 2000s, advancing older work in string theory, a candidate quantum theory of gravity. In string theory, O-shaped loops representing gravitons act like pairs of S-shaped strings corresponding to carriers of other forces. The researchers found that the relationship holds for point particles too, not just hypothetical strings.

In the sum of all possible interactions that could happen during a particle scattering event, the mathematical term representing each interaction splits into two parts, much as the number 6 splits into 2 × 3. The first part captures the nature of the force in question; for the strong force, this term relates to the property called color. The second term expresses the movement of particles — the “kinematics.”

To perform the double copy, you throw away the color term and replace it with a copy of the kinematics term, turning 2 × 3 into 3 × 3. If 6 describes the outcome of a strong-force event, then the double copy tells us that 9 will match some comparable graviton event.

The double copy has an Achilles heel: Before executing the procedure, theorists must rewrite the extra kinematics term in a form that looks like the color term. This reformatting is hard and may not always be possible as the sum is refined to include ever more convoluted interactions. But if the kinematics oblige, getting the gravity result is as easy as changing 2 × 3 to 3 × 3.

“Once you’ve realized this relationship, then gravity comes for free,” Borsten said.

The procedure doesn’t make much physical sense, as gravitons are not literally pairs of gluons. But it’s a powerful mathematical shortcut. Since developing the double copy, Bern has taken advantage of the massively discounted lunch to challenge the conventional wisdom that all particle theories of gravity give nonsensical, infinite answers.

Bern, Carrasco and others have spent years grinding away at an exotic theory called supergravity, which balances gravitons with partner particles in a mathematically pleasing way. Using the double copy, they’ve completed increasingly precise supergravity calculations. While supergravity is too symmetric to reflect our world, its simplicity makes it the lowest apple on the tree of possible particle theories of gravity. Bern and company hope to extend their computational successes to more realistic theories.

Encouraged by the double copy’s success in dealing with gravitons — the smallest possible ripples in space-time — Donal O’Connell of the University of Edinburgh and Ricardo Monteiro and Chris White at Queen Mary University of London have used it to reimagine the most extreme trick in gravity’s repertoire.

Black holes famously warp space-time intensely enough to trap light, and spinning black holes drag the warped space-time fabric around with them. The equations are super complicated. If you look at the equations for a spinning black hole, “your eyes will probably bleed,” O’Connell said.

The researchers split the black hole’s warped space-time into two pieces: flat space-time and a term representing a strong deviation from flatness. Then, Monteiro explained, they asked themselves whether the deviation term is the double copy of something.

It is. Stationary and spinning black holes alike act as if they are double copies of charged particles, the group reported in 2014. “You reduce this complicated thing to something unbelievably simple,” O’Connell said.

Black holes are not literally two copies of electrons. But their mathematical relationship is loosening the stranglehold that Einstein’s relativity theory exerts on the gravitational realm. “My secret master plan is to show that you can compute anything using the double copy that you can compute with the classical Einstein equations,” O’Connell said.

Recently, double copy practitioners have jumped into gravitational wave astronomy, the new discipline that detects distant objects and events by means of the ripples they kick up in space-time. In just a few years, Bern and his colleagues have used the shortcut to make gravitational wave predictions that already rival state-of-the-art calculations from general relativity.

The double copy has revealed a hidden, simpler side of gravity, but even theorists who have devoted their careers to exploring the relationship wonder where the simplicity is coming from.

“Is it telling us something important and primal, or is it some trick?” said Carrasco.

Researchers note that electromagnetism, the weak force and the strong force each follow directly from a specific kind of symmetry — a change that doesn’t change anything overall (the way rotating a square by 90 degrees gives us back the same square).

Intriguingly, when rewritten with the double copy, gravity appears to obey a symmetry similar to those of the three quantum forces. It’s “like there’s a fourth, mother symmetry,” O’Connell said, “a symmetry underlying the whole lot.”

The path to a complete theory of quantum gravity is long and uncertain, and the double copy might not lead all the way there. But its ability to cut through the verbiage that fills calculations gives theorists hope that the two dueling formulations of modern physics aren’t the final story. “This is a striking example that there are languages to learn that aren’t manifest in the way we traditionally talk about theories,” Carrasco said.

The post How Gravity Is a Double Copy of Other Forces first appeared on Quanta Magazine.]]>The post Researchers Read the Sugary ‘Language’ on Cell Surfaces first appeared on Quanta Magazine

]]>His ecological view of that interaction is rooted in his previous field work studying wild chimpanzee behavior in the dense West African forests. During those treks, he began to ask himself: “Why is it that humans and chimpanzees, who share so much of the same DNA, don’t deal with diseases the same way?”

“Diseases that give you and me the sniffles will actually kill chimpanzees,” he explained. But the opposite is also true. Chimps are not susceptible to influenza A, and HIV infections turn lethal in humans but stay mild in chimps. The malaria parasites that kill humans can’t infect chimps, and vice versa. This odd selectivity is not peculiar to primates — there are countless examples of pathogens devastating certain host species but not others.

Seeking an answer, Gagneux pivoted to the study of the glycomolecules, or glycans, in that “rainforest canopy” that shrouds cells. Glycans are a spectacularly diverse group of complex sugars (polysaccharides). They can exist on their own — cellulose is a plant glycan made up of long chains of glucose — or they can be anchored to other biomolecules like proteins and lipids, whose chemical properties they modify. Their structure can be linear (as in cellulose), but they can also be very highly branched, adding to their variety and complexity.

Their endless variation among cells and species is central to why pathogens devastate certain host species but not others. It helps to explain the “spillover” of certain infectious agents, like SARS-CoV-2, from one species to another, leading to global pandemics. But it’s also a key to cellular behaviors even within species, such as the interactions between human sperm and the egg and uterine cells.

Now scientists may be verging on a breakthrough in the understanding of glycans and glycobiology. After analyzing a comprehensive data set of glycan structures and their known interactions, researchers at Harvard University and the Massachusetts Institute of Technology found a shared structural “language” that all organisms use when making glycans, like a municipal building code that ensures consistent, compatible architecture. The researchers have released a set of online tools that anyone can use to analyze glycan structures and functions.

The shift in Gagneux’s interests happened when he met Ajit Varki, now a physician-scientist and co-director of UCSD’s Glycobiology Research and Training Center. Gagneux said that Varki, who became his mentor, had “just stumbled across the first biochemical difference between humans and chimpanzees.” Varki and his team had found that, more than 2 million years ago, a mutation in humans’ ancestors inactivated a gene that modifies sialic acids in all other primates and most other mammals. As a result, hundreds of millions of sialic acid glycans that are present in other primate cells are missing from human ones.

To Varki, glycans are still one of the greatest enigmas of the biological universe. They’re “actually so prominent, they’re a major component of biomass on the planet.” In fact, glycans make up most of the organic matter by mass: Cellulose and chitin, the major building material of arthropod exoskeletons and fungal cell walls, are nature’s two most abundant organic polymers. And yet in contrast with the overabundance of glycans, “this whole field has been left behind,” Varki said.

Daniel Bojar, a bioinformatics researcher at the University of Gothenburg and the Wallenberg Center for Molecular and Translational Medicine in Sweden, agrees that our knowledge of glycans pales in comparison to what we know about the other major biopolymers: DNA, RNA and proteins. Glycans, he explained in an email, “are a mysterious, omnipresent entity in biology that we either conveniently ignore or struggle to make sense of.”

According to Varki, the current state of glycobiology harks back to the late 20th century, when major changes were happening in biology. Glycans were heavily researched through the 1970s and the first half of the 1980s. “Glycans were very prominent, with one Nobel Prize every decade. There were very prominent people in many fields studying glycans,” he said.

But as Varki wrote in a 2017 review, “The field of glycosciences originated in ‘descriptive’ carbohydrate chemistry and biochemistry and remained in these domains for a long time,” instead of probing harder questions about the synthesis and functions of glycans.

Meanwhile, major technical advances were accelerating the study of nucleic acids and peptides, long linear molecules directly specified by genetic code templates. In contrast, the complex branching structures of glycans arise through a series of chemical reactions that add and modify sugar residues. There was no corresponding improvement in resources for studying them.

As a result, by the mid-1980s, “DNA, RNA and proteins, all the molecular biology, came and took off and left the glycans behind at the station,” Varki said. That development was disheartening for Varki, who was looking for his first independent research position around that time. But despite the challenges, he told himself, “I’m going to stick with studying these things,” even when many other researchers were giving up on them.

Gagneux said that “quite a lot of molecular biologists are borderline annoyed by glycans,” which are tiny and translucent. “You can only see them if you start throwing things at them that stick to them,” such as lectins, which are proteins that can tag short, distinctive saccharide sequences. Yet neglecting to study these critical components could mean missing game-changing information about some of humankind’s biggest challenges and questions.

Richard Cummings, a professor of surgery at Beth Israel Deaconess Medical Center and Harvard Medical School, describes his “life’s work” as focused on “understanding the structure of complex carbohydrates, these glycomolecules how they’re made.” Glycomolecules, he said, are “the most complex structures that the human body makes.”

Cummings is a co-director of the worldwide Human Glycome Project. He and other researchers on that project, which was only launched in 2018, aim to “sequence and identify all of the glycans and carbohydrate structures — glycomolecules — in humans,” he noted. In contrast, the Human Genome Project launched in 1990 and formally concluded in 2003, illustrating just how big the gap has grown between knowledge of the human genome and the glycome.

Yet it is critical that researchers determine which roles specific glycans play in illness and disease if they hope to develop more effective strategies for preventing and treating these conditions.

Some of that research is already proving fruitful. Huge strides have been made in the study of a growing group of rare genetic metabolic disorders stemming from defects in glycosylation, according to Varki. “After a slow start in the early 1990s an international effort of many investigators has now resulted in a veritable explosion in discoveries of human genetic disorders of glycosylation,” he wrote in his 2017 review article.

Researchers have already turned to glycomolecules to gain new insights about conditions as diverse as cystic fibrosis, cancers, sickle cell anemia, HIV and COVID-19. For instance, in 2020, Cummings and his colleagues published a *Molecular Psychiatry *review article covering 25 years of post-mortem brain studies on abnormal glycosylation in people with schizophrenia.

Cummings, who also directs the National Center for Functional Glycomics and the Harvard Medical School Center for Glycoscience, studies the function of glycomolecules in human biology and how mutations or alterations in those functions can cause pathologies. He also investigates how bacteria, parasitic worms and viruses such as influenza infect and sicken humans.

“It turns out in almost any of these cases, it is through interactions of glycomolecules that microorganisms and parasites cause human disease,” Cummings said. Linking that knowledge to new treatments or preventive measures often remains a grand challenge.

One hurdle for glycobiology, Gagneux noted, is that even closely related species with high levels of genetic similarity, like chimps and humans, have glycans that can vary significantly because of constant, ongoing coevolution. Each species faces its own evolutionary pressures from diseases that leave a mark on its library of glycans: The host glycome evolves to evade or counter pathogens’ attacks, and the pathogens’ glycomes evolve to escape the immune defenses of their potential hosts.

“It gives rise to this molecular arms race that happens differently once you go separate evolutionary ways,” Gagneux said. For instance, even if you inject humans with chimp malaria parasites, they don’t get sick. (“Believe it or not, this was done in the ’50’s,” he said.) That’s partly because the chimp malaria parasites can’t find the blend of sialic acids they seek on human red blood cells.

On the other hand, chimps are highly resistant to cholera because the *Vibrio* bacterium that causes the disease makes a toxin that targets only the sialic acids on the cells lining the human gut, punching holes through their membranes. Because of host-pathogen coevolution, “there’s a lot of diversity” in the glycome, Cummings said.

That diversity was apparent when scientists at MIT and the Wyss Institute for Biologically Inspired Engineering at Harvard used glycan-focused machine learning models to analyze a data set of more than 19,000 unique glycans. This included “6,969 eukaryotic, 6,119 prokaryotic, and 152 viral glycans,” they wrote in their 2020 *Cell Host & Microbe* study.

“Because we included all species for which we could find glycans, this dataset constituted a comprehensive snapshot of currently known species-specific glycans,” the researchers wrote.

Bojar, who was a postdoctoral fellow at the Wyss Institute and MIT at the time, is the study’s first author. He and his colleagues observed 1,027 unique simple sugars (monosaccharides) and chemical bonds in the glycan sequences. They treated these as “glycoletters” — “the smallest units of an alphabet for a glycan language,” they wrote. They then began looking through the data set for patterns of “glycowords,” defined as sequences five glycoletters long (that is, three monosaccharides linked by two bonds).

To that end, they trained a bidirectional recurrent neural network on sequences from their database and used it to create a model for a glycoletter-based language. Such neural networks are commonly used to learn and train language models. “You can kind of think about it as reading a sequence of text forward and then reading it backward,” said Rani Powers, a senior staff scientist at the Wyss Institute and a researcher on the study. “You want to keep the context of what is essentially the sentence in this case, rather than just pulling out all of the words or all of the letters out of context.”

In theory, the glycoletters in the data set could have formed nearly 1.2 trillion different glycowords. Yet, surprisingly, the researchers’ results indicated that only 19,866 distinct glycowords were present across all the available sequences. Notwithstanding the immense complexity and diversity of glycans, and the differences in glycans that are characteristic of various species, the evidence suggested that all organisms follow very similar rules in assembling them and use essentially the same biomolecular language to define their structure.

The researchers discovered that by fine-tuning their models, they could predict with high accuracy the taxonomic groups of the organisms from which glycans came. Furthermore, they were able to train the models to predict with about 92% accuracy whether glycan sequences in a reference data set were immunogenic to humans.

The results are “very exciting,” and the further application of sophisticated computational tools to understanding glycans could turn out to be “important and revelatory,” said Lara Mahal, a glycomics researcher at the University of Alberta who was not involved with the study. (She is working on a different project with Bojar.) “It helps reduce the complexity of glycans into clear patterns from which we can gather important information, for example on the pathogenicity of glycans,” she added.

The Wyss and MIT researchers hope that other teams will use the tools for glycomic design and analysis that they have developed and posted free online. According to Bojar, their most immediately useful application may be in the pharmaceutical industry, for glycoengineering therapeutic monoclonal antibodies. Antibody proteins latch onto specific antigen targets on pathogens. But it is the glycans linked to the proteins that determine how the antibodies interact with the rest of the body’s defenses and help to direct what kind of immune response follows. In the future, Bojar said, the tools might be able to suggest glycans that would improve the performance of antibodies, for example by limiting their side effects or more precisely calibrating their half-life in the body.

Mahal noted that she is already using the tools to learn more about the specificity of the assays used to identify the glycans on cells. “These new computational technologies combined with high-throughput analysis will revolutionize our understanding of the glycome and its role in disease,” she said.

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]]>Marletto grew up in Turin, in northern Italy, and studied physical engineering and theoretical physics before completing her doctorate at the University of Oxford, where she became interested in quantum information and theoretical biology. But her life changed when she attended a talk by David Deutsch, another Oxford physicist and a pioneer in the field of quantum computation. It was about what he claimed was a radical new theory of explanations. It was called constructor theory, and according to Deutsch it would serve as a kind of meta-theory more fundamental than even our most foundational physics — deeper than general relativity, subtler than quantum mechanics. To call it ambitious would be a massive understatement.

Marletto, then 22, was hooked. In 2011, she joined forces with Deutsch, and together they have spent the last decade transforming constructor theory into a full-fledged research program.

The goal of constructor theory is to rewrite the laws of physics in terms of general principles that take the form of counterfactuals — statements, that is, about what’s possible and what’s impossible. It is the approach that led Albert Einstein to his theories of relativity. He too started with counterfactual principles: It’s impossible to exceed the speed of light; it’s impossible to tell the difference between gravity and acceleration.

Constructor theory aims for more. It hopes to provide the principles behind a vast class of theories of physics, including the ones we don’t even have yet, like the theory of quantum gravity that would unite quantum mechanics with general relativity. Constructor theory seeks, that is, to provide the mother of all theories — a complete “Science of Can and Can’t,” the title of Marletto’s new book.

Whether constructor theory can really deliver, and how much it truly differs from physics as usual, remains to be seen. For now, *Quanta Magazine* caught up with Marletto via Zoom and by email to find out how the theory works and what it might mean for our understanding of the universe, technology, and even life itself. The interview has been condensed and edited for clarity.

The standard laws of physics — such as quantum theory, general relativity, even Newton’s laws — are formulated in terms of trajectories of objects and what happens to them given some initial conditions. But there are some phenomena in nature that you can’t quite capture in terms of trajectories — phenomena like the physics of life or the physics of information. To capture those, you need counterfactuals.

The word “counterfactual” is used in various ways, but I mean a specific thing: A counterfactual is a statement about which transformations are possible and which are impossible in a physical system. A transformation is possible when you have a “constructor” that can perform a task and then retain the capacity to perform it again. In biology, we call that a catalyst, but more generally we can call it a constructor.

In the current approach to physics, some laws already have this counterfactual structure — the conservation of energy, for example, is the statement that it is impossible to have a perpetual motion machine.

Yes. Counterfactuals do appear in existing laws, but these laws are regarded as second class. They are not incorporated wholeheartedly. Constructor theory puts counterfactuals at the very foundation of physics, so that the most fundamental laws can be formulated in these terms.

For example, consider quantum gravity. Some people say: “Why do we even need to quantize gravity given that we don’t even have experimental evidence for it? We could have a classical theory of gravity and a quantum theory of everything else.” Well, constructor theory provides us with a robust and general theoretical foundation for an experimental test that would prove that gravity must be quantum.

This test was proposed by Vlatko Vedral and me, and independently by Sougato Bose and collaborators. It goes like this: You measure the properties of two quantum masses that interact with each other through gravity only. If they develop entanglement, then you can conclude something very strong about the mediator that causes the entanglement, which is gravity. It allows you to conclude that the mediator cannot be classical — it’s got to have some quantum features.

Now, as Vlatko and I proved, the most general way to get to this conclusion is to use a constructor-theoretic principle called “interoperability,” which implies that if entanglement can be generated locally through a medium, that medium has to be quantum. It doesn’t matter in what way gravity is quantum — whether it’s loop quantum gravity or string theory or something else — but it has to be a quantum theory. It’s a test you can devise at this level of generality only by thinking in terms of constructor-theoretic principles.

There are lots of counterfactuals in quantum theory, but that’s true in classical physics too! Quantum and classical information are two aspects of the same set of information-theoretic properties. What makes quantum information different is that it has two additional counterfactual properties.

First, it has at least two information variables — for example, position and velocity — for which it’s impossible to copy both simultaneously with arbitrarily high accuracy. Second, it must be possible to reverse any transformations on those variables.

So quantum theory has more counterfactuals, but you still need counterfactuals to fully express classical information theory and even classical thermodynamics. Concepts like work and heat can’t be captured fully with trajectories and laws of motion, because in the standard conception they are considered emergent and approximate. In constructor theory we can talk about them using exact statements about possible and impossible transformations.

I think you have to turn it the other way around. Take thermodynamics. When you say that a perpetual motion machine is impossible, that statement is not something you have to prove exhaustively by checking every possible model of a perpetual motion machine, using different initial conditions and different dynamics for each one. If you had to do that, it would be a very exhausting task!

What you do is state the law in terms of possible and impossible tasks, and then work out the consequences. For instance, if you have this general statement that perpetual motion machines are impossible, you can combine it with other statements about other tasks being possible or not and work out that a heat engine is possible. And that gives you a lot of predictive power. That’s the logic. You take these statements as fundamental.

First, let me clarify that whether you’re using constructor theory or using the current approach, we are just dealing with some guess as to what the actual laws are. And these guesses can always be wrong. But if you buy constructor theory, the key is that just conjecturing dynamical laws will be insufficient to capture all of physical reality. You need additional principles, given by constructor theory.

So I think David is stressing that constructor theory is not just a list of things that are possible and things that are impossible. It’s the explanatory theory of why a certain pattern of possible and impossible tasks best captures what we know at the moment about physical reality. Then, if you want to question that explanatory theory, you can. But the conjecture is that whatever explanation you would come up with to improve on that, it would itself have to be expressed in terms of possible and impossible tasks.

In short, yes, though this is something that we haven’t really developed yet. John von Neumann, the great physicist and mathematician, conjectured a machine that was supposed to be more general than Turing’s universal computer. Von Neumann called it a universal constructor. He realized that if you think of some tasks that, for instance, living systems can do, like creating copies of themselves, a universal Turing machine cannot do that. My Mac cannot create another Mac out of some boring raw materials, even though I wish it could!

So von Neumann asked: What do I have to add to a Turing machine for it to become a more powerful machine that could construct itself? It turns out you have to add a number of things: a set of implements that allow the machine to grab the raw materials and assemble them, the ability to read instructions for assembly, et cetera.

The universal constructor is an analogue of the Turing machine in the sense that it’s supposed to be able to perform all physically allowed tasks. And we don’t know if one is possible under the laws of physics that we have. And the reason we don’t know, even 70 years after von Neumann first suggested this, is that nobody took the original proposal and connected it to physics.

Once constructor theory can define the universal constructor in physical terms and understand the principles that allow you to say that the universal constructor is possible, then we will have an answer to your question — we’ll know what are the elementary gates or elementary possible tasks that the universal constructor can appeal to when it’s trying to perform a complicated task.

I think this is really cool. By studying the properties of something that sounds very technological — like a computer or a constructor — you end up actually studying the deepest features of the laws of physics. It’s something that fascinated me when I started studying quantum information.

Initially I thought quantum information is just some quirky application of quantum physics to computer science — but it’s not true. It’s the best tool we have to understand quantum theory itself. Measurement, EPR, entanglement: All of these things that were very puzzling even to the founding fathers of quantum theory have been worked out properly by people working in quantum information, and at the same time they were working out how to build a universal quantum computer.

It’s very cool that you can do something that’s very useful in technology, like cryptography, but at the same time you’re studying the foundations of entanglement and superpositions and so on. In constructor theory, we’re trying to follow the same kind of logic to an even more general level.

Yes.

It’s more fruitful to think in terms of repertoires of a given programmable machine, where the repertoire is the set of tasks or transformations that the machine can perform when given the appropriate input program. In that sense, being a quantum computer or being capable of building a quantum computer given sufficient materials are essentially the same thing — because once a machine can build another machine, then the first machine has the second one’s repertoire in its own repertoire. The universal constructor has all the physically allowed computations in its own repertoire, which means it is a universal computer, too.

Yes. The physics of life would be considered a subpart of this more general theory of the universal constructor. And you could imagine how a better understanding of the constructor-theoretic foundations of the laws of physics could give you ways of programming the universal constructor to perform tasks that are relevant to that field.

So for instance, for quantum biology, you could think of the universal constructor as being programmed to mimic what happens in a plant cell when photosynthesis occurs and then thinking of ways of improving on that. You can imagine all sorts of programmable nanomachines that are specific instances of the universal constructor programmed for specific tasks. And underlying all those should be this set of principles that we will uncover by studying constructor theory. That’s the vision.

DNA is a replicator and contains the instructions for building the cell, and then the cell is the vehicle which is capable of reading those instructions, constructing a new instance of itself, copying the instructions and inserting them into the new cell. And in constructor theory you can explain why this is the only viable mechanism possible if you want reliable self-reproduction, under laws of physics that aren’t especially designed for life. So it’s not just that it’s one of the ways life can work under our laws of physics, but it’s the only way it can work. That’s a feature of living systems regardless of whether they are built with the chemistry we have on Earth.

Ultimately what we need is a theory of what makes life distinct from nonlife, and so far there isn’t a quantitative, predictive answer. What are the laws underlying this phenomenon? What’s lacking isn’t the biology — the biologists have already done their job; they’ve worked it out beautifully with evolutionary theory. Now physicists need to solve the problem within the boundaries of fundamental physics. And I’m hoping constructor theory can provide the tools to tackle that problem.

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