Tied Up in Knots

Tied Up in Knots

Anything that can tangle up, will, including DNA

Knotted threads secure buttons to shirts. Knots in ropes attach boats to piers. You can find knots in shoestrings, ties, ribbons, and bows. But even without Boy Scouts or sailors, knots would be everywhere.

 

CATALOG OF COILS. Ropes knotted up spontaneously when tumbled in a box. Each knot (off-white) example is paired with the corresponding idealized knot (gold). Raymer/UCSD

 

Call it Murphy's Law of knots: If something can get tangled up, it will. "Anything that's long and flexible seems to somehow end up knotted," says Andrew Belmonte, an applied mathematician at Pennsylvania State University in University Park. Belmonte has plenty of alarming anecdotal evidence. "It certainly happens in my house, with the cords of the venetian blind." But the knot scourge is a global one, as anyone who owns a desktop computer can confirm after peeking at the mess of connection cables and power cords behind the desk.

Now, scientists think they may have found out how and why things find their way into knotty arrangements. By tumbling a string of rope inside a box, biophysicists Dorian Raymer and Douglas Smith have discovered that knots—even complex knots—form surprisingly fast and often. The string first coils up, and then its free ends swivel around the other coils, tracing a random path among them. That essentially makes the coils into a braid, producing knots, the scientists say.

The results' relevance may go well beyond explaining the epidemic of tangled venetian blind cords. That's because spontaneous knots seem to be prevalent in nature, especially in biological molecules. For example, knottiness may be crucial to the workings of certain proteins (see "Knots in Proteins"). And knots can randomly form in DNA, hampering duplication or gene expression—so much so that living cells deploy special knot-chopping enzymes.

Raymer's interest in knots began as an answer waiting for a question. Two years ago, he was an undergraduate student working in Smith's lab at the University of California, San Diego (UCSD). Raymer fancied taking a class about the abstract theory of knots, offered by UCSD's math department. Smith told him that he should take it only if he could find a practical use for it—some kind of knot experiment.

Raymer never took the class, but he and Smith did come up with a simple idea for an experiment. They put a string in a cubic container the size of a box of tissue. By tumbling the box 10 times "like a laundry dryer," as Raymer puts it, the researchers hoped to observe knots forming spontaneously on occasion. They didn't have to wait for long: Knots formed right away. "The first couple of times, it was pretty amazing," Raymer says.

The researchers repeated the procedure more than 3,000 times, and knots formed about every other time. Longer strings, or more-flexible strings, tended to knot more often.

The researchers took pictures, planning to gather precise statistics of the types of knots that were forming. Raymer soon realized that, to make sense of the mess, he'd need to teach himself the mathematics of knots after all.

 

Ready-made tools

The theory of knots began in earnest in the 1860s, under the stimulus of the British physicist William Thomson, later known as Lord Kelvin. Kelvin suggested that atoms of different elements were really different kinds of knotted vortices in the ether. So to lay the foundations of chemistry, he believed, it was imperative to classify knots. Ultimately, physicists discovered that the ether didn't exist. But mathematicians took an interest in knots for knots' sake, as part of the young branch of mathematics called topology.

Topology studies shapes. Specifically, it studies shapes' properties that are not affected by stretching, moving, twisting, or pulling—anything that doesn't break up the object or fuse some of its parts. The proverbial example is that, to a topologist, a coffee mug is the same as a doughnut. In your imagination, you can squash the mug into a doughnut shape, and it will retain the property of having a hole, namely its handle.

A sphere is different. You can stretch a sphere into a stick and bend the stick so its ends touch. But turning that open ring into a doughnut will involve fusing the ends, and that's forbidden.

In topology, a knot is any curved line that closes up on itself, possibly after a circuitous path in three dimensions. A circle is regarded as the "trivial" knot. Two loops are considered to be the same knot if you can turn one into the other by topological manipulation, which in this case means anything that does not break the curve or force it to run through itself.

Topologically, a knotted string is not a real knot, as long as its ends are free. That's because either of the ends can always thread back through any entanglement and undo the knot. An open string, no matter how garbled, is the same as a straight segment. (Mathematicians usually think of strings as being stretchable and infinitesimally thin, so in topology there is no issue of a knot being tight.)

Strictly speaking, then, the string in Raymer and Smith's box was never knotted. But it was still a mess. When the researchers joined the string's ends, they made it into a closed loop, often something that even a mathematician would call a knot.

Raymer soon realized that telling different knots apart, or recognizing when two knots are the same is a tricky business. Topologists usually work with two-dimensional drawings of knots called knot projections. From different points of view, the same curve will look different and so will its projections. Topologists' best tools for distinguishing knots are algebraic expressions called knot polynomials. These are sums of multiples of a variable, such as x, raised to different powers. The variable has no meaning per se, and all the information is in the numbers by which it's multiplied. But the x's make it easier to calculate a knot polynomial starting from a knot projection.

James Alexander, a Princeton University mathematician, invented the first knot polynomial in the 1920s. Two topologically equivalent knots always will give the same Alexander polynomial, no matter how different their projections look. So if two knots have different polynomials, they're certainly nonequivalent. The converse, however, is not true: Some distinct knots have the same Alexander polynomial. That means that the Alexander polynomial is not a fail-safe way of distinguishing knots.

In the early 1980s, Vaughan Jones of the University of California, Berkeley rekindled mathematicians' interest in knots when he defined a new kind of knot polynomial, a discovery that earned him the Fields Medal, the most coveted prize in mathematics. The Jones polynomials distinguish knots with greater, if not complete, accuracy than the Alexander polynomials. That made the Jones polynomials Raymer's choice to catalog his knots.

Tie land

Raymer wrote a computer program to calculate Jones polynomials from the pictures he had taken each time he opened the box. The program found that the humble box had produced at least 120 distinct types of knots. Some were pretty complex.

KNOTS HAPPEN. Computer models showed how a coiled string in a tumbling box (top) will tangle up when one of its ends is free to swivel around the other coils. Raymer/PNAS

The most basic measure of knot complexity is the minimal crossing number, the number of overpasses needed to draw the simplest possible projection of the knot. For the trivial knot, that number is zero. The simplest true knot, the trefoil requires that just three crossings be drawn. A few of the knots from the tumbling box required as many as 11, Raymer and Smith report in the Oct. 16 Proceedings of the National Academy of Sciences

Raymer says he and Smith were surprised, because previous knot experiments—physicists have tried a few in recent years—had seen only some of the simplest knots. For example, in 2001 Belmonte and his collaborators showed that a hanging chain (not from Belmonte's venetian blinds) tended to knot up when shaken. In 2006, a team led by physicist Jens Eggers of the University of Bristol in England got a ball chain to form knots by setting it on a vibrating dish.

De Witt Sumners, an applied mathematician at Florida State University in Tallahassee, says he was not surprised that knots would form in a box. In computer simulations, mathematicians have found that random motion creates paths that almost always tie themselves up. Together with Stu Whittington of the University of Toronto, Sumners demonstrated mathematically in 1988 that if you wait long enough, these random walks will get knotted virtually 100 percent of the time.

Sumners suspects that with longer tumbling, Raymer and Smith would have gotten knots almost always, instead of just every other time. "They should have spun longer," to see the full effects, Sumners says.

In their paper, on the other hand, Raymer and Smith propose a theoretical explanation for the mess in their box that differs from the most general type of random walk. Because their string tended to coil up whether or not it formed knots, they created a mathematical model of a bundle of coils as a series of parallel, horizontal strands. In a computer simulation, Raymer and Smith allowed one of the strands—representing one of the free ends of the string—to cross over or under one of the others in the bundle. After several such steps, the strands had braided, which often meant that the string as a whole was now knotted.

This simplified model didn't reproduce the exact results of their experiment, but it did predict that specific knots had about the right odds of forming within the allowed time.

Jam-packed

Belmonte calls the braid model "very obvious, but maybe not universal," meaning that different physical phenomena probably tie knots in different ways. In bacterial DNA, for example, one way that knots can form is by genetic recombination. That's when, to facilitate the reshuffling of genes, enzymes cut DNA at two places and reattach the ends in a different order. Bacterial genomes are circular, so recombination can produce veritable knotted loops.

 

LOOP THE LOOP. This electron microscope image shows bacterial DNA tied in a trefoil, the only knot with just three crossings. Stasiak/Nature

 

In the late 1990s, biochemists discovered enzymes that seem able to detect when DNA has a knot. The enzymes then undo the knot by brute-force cut and paste.

Keeping DNA tidy may be crucial to some of the cell's most important functions. That's because copying DNA and reading out the information it contains are performed by other enzymes, called polymerases, which walk along DNA. "When [a polymerase] comes to a knotted area, it will be stuck," Belmonte says.

Scientists have discovered similar knot-busting enzymes in cells that have open-string chromosomes, such as in humans. The presence of such enzymes suggests that knotting may be an issue for human chromosomes as well. And scientists have also found knots in mitochondria, cellular organelles that contain loop DNA.

Another place where DNA knots can form is inside viruses, says Andrzej Stasiak, a structural biologist at the University of Lausanne in Switzerland. Viruses build containers called capsids in which the viruses tightly pack their DNA for traveling from one host cell to the next. In some viruses, the capsid keeps DNA at a pressure of more than 60 atmospheres.

Stasiak says that the packing process probably produces coiling similar to that seen by Raymer and Smith. Their coil-and-braid model could help explain why the DNA of some viruses often ends up being knotted.

But even if Raymer and Smith's results don't prove to be directly relevant to the molecules of life, they are "a very good beginning" for a general study of physical knots, according to Belmonte. "Now we can at least ask these questions: Are there universal laws of knots?" 

Modern Beetles Predate Dinosaurs

Modern Beetles Predate Dinosaurs

RNPS PICTURES OF THE YEAR - A boy watches beetle sumo competition of the IWBC (Insect World Battle Championships) in a miniature ring in Tokyo August 26, 2007.

Wait, don't squash that beetle! Its lineage predates dinosaurs.

New research hints that modern-day versions of the insects are far older than any tyrannosaur that trod the Earth.

Today's plethora of beetle species were thought to have blossomed 140 million years ago, during the rise of flowering plants. But the new study of beetle DNA and fossils, published in the Dec. 21 issue of the journal Science, pushes their appearance back to 300 million years ago.

That beats the arrival of dinosaurs by about 70 million years.

"Unlike the dinosaurs which dwindled to extinction, beetles survived because of their ecological diversity and adaptability," said the study's lead scientist Alfried Vogler, an entomologist at Imperial College London and the Natural History Museum in London.

Today, 350,000 species of beetles dot collections around the world, and millions more are estimated to exist but haven't been discovered — which means they make up more than one-fourth of all known species of life forms. The reason for this tremendous diversity has been debated by scientists for many years but never resolved.

Vogler thinks beetles' head start on our planet with its ever-changing environments was the secret to their success.

"The large number of beetle species existing today could very well be a direct result of this early evolution," Vogler said, "and the fact that there has been a very high rate of survival and continuous diversification of many lineages since then." 

To reach this conclusion, Vogler and his team teased out evolutionary data from the DNA of 1,880 modern beetle species, then compared it to fossil records dating back 265 million years to build detailed evolutionary trees. The new genetic maps suggest that a common ancestor to beetles crept up well before its descendants showed up in the fossil record.

"With beetles forming such a large proportion of all known species, learning about their relationships and evolution gives us important new insights into the origin of biodiversity and how beetles have triumphed over the course of nearly 300 million years," Vogler said.

Brain Cells, Doing Their Job With Some Neighborly Help

Brain Cells, Doing Their Job With Some Neighborly Help 

A region deep in the brain called the hippocampus tracks, sorts and stores the onslaught of information pouring through the senses every waking minute. A large question in neuroscience is one a kindergartner would ask: How?

How does a dollop of tissue containing a small fraction of the brain’s neurons possibly absorb and hold so much, even temporarily?

A study published last week in the journal Nature provides the first step toward an answer, as well as a showcase for some of the most advanced methods available to study the brain.

Researchers at the Howard Hughes Medical Institute in Maryland stimulated not a single cell but a single dendritic spine, one of the hairlike growths that sprout from a cell’s branching arms.

Brain cells communicate with their neighbors by sending a chemical burst from the tips of these spines, across a space called the synapse to the tip of a spine on the next cell. If the chemical bath is strong enough, the receiving spine bulges forward — strengthening the connection between the spines. This is thought to be the fundamental process underlying learning.

But the researchers, Christopher D. Harvey and Karel Svoboda, found something unusual when they stimulated a single spine. Not only did the spine bulge, but it also somehow made its neighbors more sensitive to chemical signals — standing ready, in effect, to digest any spillover of information. Imagine every neighbor on the block calling up to offer a corner of his basement for storage, just in case.

The combined effect of these helpers multiplies the capacity of any single brain cell, the authors concluded. Neuroscientists had theorized that this effect, called clustered plasticity, might help account for the tremendous capacity of the brain, but they had not seen it in action.

“The traditional view was that each synapse functioned independently, and the strength of individual connections modulated memory storage,” said Mr. Harvey, a graduate student at the Cold Spring Harbor Laboratory on Long Island. “What we’ve shown is that neighboring synapses may function together, which leads to the idea that information is stored in a clustered manner, with related things concentrated in the same neighborhood.”

The ability to watch a synapse in action is itself a scientific accomplishment. The average human brain has about 100 billion neurons, and about 1,000 times that many synapses. To zero in on a single one, the researchers used mice that were genetically engineered so their brains produced a fluorescent protein that glowed only in specific cells of the hippocampus. Peering through a high-powered microscope at a slice of this tissue, the researchers could zero in on a single synapse.

Using a laser, they triggered a burst of glutamate, a brain chemical, into the synapse. The entire slice of brain tissue was soaked in a form of inert glutamate, and the laser activated the chemical in precisely the area the scientists were focusing on.

Previous efforts to observe this cell-to-cell communication in action used electrical stimulation, which sends a brush fire of activity through a neighborhood of cells, swamping any of the subtle effects that happen naturally.

In this study, “glutamate is delivered to individual spines located relatively deep within the brain tissue, under conditions that mimic” processes in the body, wrote Dr. Bernardo L. Sabatini, a neurobiologist at Harvard, in an editorial titled “Neighborly Synapses” accompanying the study.

After firing the synapse, the researchers found that receptors on neighboring cells remained extra sensitive to stimulation for 10 minutes. That makes sense, Mr. Harvey said, given the brief impressions that people need to process when walking into an unfamiliar room, say, or navigating a party.

“An hour or more would be too long,” he said. “There would be all sorts of information pouring in, unrelated stuff. And a few seconds would be too short, not enough time to take in something you were paying attention to. Ten minutes is just about right.”

Conserving Cuba, After the Embargo

Conserving Cuba, After the Embargo

  

 

 

Through accidents of geography and history,

Cuba is a priceless ecological resource. That is why many scientists are so worried about what will become of it after Fidel Castro and his associates leave power and, as is widely anticipated, the American government relaxes or ends its trade embargo.

Cuba, by far the region’s largest island, sits at the confluence of the Atlantic Ocean, the Gulf of Mexico and the Caribbean Sea. Its mountains, forests, swamps, coasts and marine areas are rich in plants and animals, some seen nowhere else.

And since the imposition of the embargo in 1962, and especially with the collapse in 1991 of the Soviet Union, its major economic patron, Cuba’s economy has stagnated.

Cuba has not been free of development, including Soviet-style top-down agricultural and mining operations and, in recent years, an expansion of tourism. But it also has an abundance of landscapes that elsewhere in the region have been ripped up, paved over, poisoned or otherwise destroyed in the decades since the Cuban revolution, when development has been most intense. Once the embargo ends, the island could face a flood of investors from the United States and elsewhere, eager to exploit those landscapes.

Conservationists, environmental lawyers and other experts, from Cuba and elsewhere, met last month in Cancún, Mexico, to discuss the island’s resources and how to continue to protect them.

Cuba has done “what we should have done — identify your hot spots of biodiversity and set them aside,” said Oliver Houck, a professor of environmental law at Tulane University Law School who attended the conference.

In the late 1990s, Mr. Houck was involved in an effort, financed in part by the MacArthur Foundation, to advise Cuban officials writing new environmental laws.

But, he said in an interview, “an invasion of U.S. consumerism, a U.S.-dominated future, could roll over it like a bulldozer” when the embargo ends.

By some estimates, tourism in Cuba is increasing 10 percent annually. At a minimum, Orlando Rey Santos, the Cuban lawyer who led the law-writing effort, said in an interview at the conference, “we can guess that tourism is going to increase in a very fast way” when the embargo ends.

“It is estimated we could double tourism in one year,” said Mr. Rey, who heads environmental efforts at the Cuban ministry of science, technology and environment.

About 700 miles long and about 100 miles wide at its widest, Cuba runs from Haiti west almost to the Yucatán Peninsula of Mexico. It offers crucial habitat for birds, like Bicknell’s thrush, whose summer home is in the mountains of New England and Canada, and the North American warblers that stop in Cuba on their way south for the winter.

Zapata Swamp, on the island’s southern coast, may be notorious for its mosquitoes, but it is also known for its fish, amphibians, birds and other creatures. Among them is the Cuban crocodile, which has retreated to Cuba from a range that once ran from the Cayman Islands to the Bahamas.

Cuba has the most biologically diverse populations of freshwater fish in the region. Its relatively large underwater coastal shelves are crucial for numerous marine species, including some whose larvae can be carried by currents into waters of the United States, said Ken Lindeman, a marine biologist at Florida Institute of Technology.

Dr. Lindeman, who did not attend the conference but who has spent many years studying Cuba’s marine ecology, said in an interview that some of these creatures were important commercial and recreational species like the spiny lobster, grouper or snapper.

Like corals elsewhere, those in Cuba are suffering as global warming raises ocean temperatures and acidity levels. And like other corals in the region, they reeled when a mysterious die-off of sea urchins left them with algae overgrowth. But they have largely escaped damage from pollution, boat traffic and destructive fishing practices.

Diving in them “is like going back in time 50 years,” said David Guggenheim, a conference organizer and an ecologist and member of the advisory board of the Harte Research Institute, which helped organize the meeting along with the Center for International Policy, a private group in Washington.

In a report last year, the World Wildlife Fund said that “in dramatic contrast” to its island neighbors, Cuba’s beaches, mangroves, reefs, seagrass beds and other habitats were relatively well preserved. Their biggest threat, the report said, was “the prospect of sudden and massive growth in mass tourism when the U.S. embargo lifts.”

To prepare for that day, researchers from a number of American institutions and organizations are working on ecological conservation in Cuba, including Harte, the Wildlife Conservation Society, universities like Tulane and Georgetown, institutions like the American Museum of Natural History and the New York Botanical Garden, and others. What they are studying includes coral health, fish stocks, shark abundance, turtle migration and land use patterns.

Cuban scientists at the conference noted that this work continued a tradition of collaboration that dates from the mid-19th century, when Cuban researchers began working with naturalists from the Smithsonian Institution. In the 20th century, naturalists from Harvard and the University of Havana worked together for decades.

But now, they said, collaborative relationships are full of problems. The Cancún meeting itself illustrated one.

“We would have liked to be able to do this in Havana or in the United States,” Jorge Luis Fernández Chamero, the director of the Cuban science and environment agency and leader of the Cuban delegation, said through a translator in opening the meeting. “This we cannot do.” While the American government grants licenses to some (but not all) American scientists seeking to travel to Cuba, it routinely rejects Cuban researchers seeking permission to come to the United States, researchers from both countries said.

So meeting organizers turned to Alberto Mariano Vázquez De la Cerda, a retired admiral in the Mexican navy, an oceanographer with a doctorate from Texas A & M and a member of the Harte advisory board, who supervised arrangements for the Cuban conferees.

The travel situation is potentially even worse for researchers at state institutions in Florida. Jennifer Gebelein, a geographer at Florida International University who uses global positioning systems to track land use in Cuba, told the meeting about restrictions imposed by the Florida Legislature, which has barred state colleges from using public or private funds for travel to Cuba.

As a result of this move and federal restrictions, Dr. Gebelein said “we’re not sure what is going to happen” with her research program.

On the other hand, John Thorbjarnarson, a zoologist with the Wildlife Conservation Society, said that he had difficulty obtaining permission from Cuba to visit some areas in that country, like a habitat area for the Cuban crocodile near the Bay of Pigs.

“I have to walk a delicate line between what the U.S. allows me to do and what the Cubans allow me to do,” said Dr. Thorbjarnarson, who did not attend the conference. “It is not easy to walk that line.”

But he had nothing but praise for his scientific colleagues in Cuba. Like other American researchers, he described them as doing highly competent work with meager resources. “They are a remarkable bunch of people,” Dr. Thorbjarnarson said, “but my counterparts make on average probably less than $20 a month.”

American scientists, foundations and other groups are ready to help with equipment and supplies but are hampered by the embargo. For example, Maria Elena Ibarra Martín, a marine scientist at the University of Havana, said through a translator that American organizations had provided Cuban turtle and shark researchers with tags and other equipment. They shipped it via Canada.

Another thorny issue is ships.

“If you are going to do marine science, at some point you have to go out on a ship,” said Robert E. Hueter, who directs the center for shark research at the Mote Marine Laboratory in Sarasota, Fla., and attended the Cancún meeting.

But, he and others said, the United States government will not allow ships into American ports if they have recently been in Cuban waters in the previous six months, and the Cuban government will not allow American research vessels in Cuban waters.

One answer might be vessels already in Cuba, but nowadays they are often tied up in tourism-related efforts, Cubans at the Cancún meeting said.

And even with a ship, several American researchers at the conference said, it is difficult to get Cuban government permission to travel to places like the island’s northwest coast, the stretch closest to the United States. As a result, that region is the least-studied part of the Cuban coast, Dr. Guggenheim and others said.

Another big problem in Cuba is the lack of access to a source of information researchers almost everywhere else take for granted: the Internet.

Critics blame the Castro government, saying it limits access to the Internet as a form of censorship. The Cuban government blames the embargo, which it says has left the country with inadequate bandwidth and other technical problems that require it to limit Internet access to people who need it most.

In any event, “we find we do not have access,” Teresita Borges Hernández, a biologist in the environment section of Cuba’s science and technology ministry, said through a translator. She appealed to the Americans at the meeting to do “anything, anything to improve this situation.”

Dr. Guggenheim echoed the concern and said even telephone calls to Cuba often cost as much as $2 a minute. “These details, though they may seem trite,” he said, “are central to our ability to collaborate.”

Dr. Gebelein and several of the Cubans at the meeting said that some American Web sites barred access to people whose electronic addresses identify them as Cuban. She suggested that the group organize a Web site in a third country, a site where they could all post data, papers and the like, and everyone would have access to it.

For Dr. Guggenheim, the best lessons for Cubans to ponder as they contemplate a more prosperous future can be seen 90 miles north, in the Florida Keys. There, he said, too many people have poured into an ecosystem too fragile to support them.

“As Cuba becomes an increasingly popular tourist resort,” Dr. Guggenheim said, “we don’t want to see and they don’t want to see the same mistakes, where you literally love something to death.”

But there are people skeptical that Cuba will resist this kind of pressure. One of them is Mr. Houck.

The environmental laws he worked on are “a very strong structure,” he said, “But all laws do is give you the opportunity to slow down the wrong thing. Over time, you can wear the law down.”

That is particularly true in Cuba, he said, “where there’s no armed citizenry out there with high-powered science groups pushing in the opposite direction. What they lack is the counter pressure of environmental groups and environmental activists.”

As Mr. Rey and Daniel Whittle, a lawyer for Environmental Defense, put it in the book “Cuban Studies 37” (University of Pittsburgh Press, 2006), “policymaking in Cuba is still centralized and top down.” But, they wrote, “much can be done to enhance public input in policymaking.”

Mr. Rey said in the interview that Cubans must be encouraged to use their environmental laws. By “some kind of cultural habit,” he said, people in Cuba rarely turn to the courts to challenge decisions they dislike.

“There’s no litigation, just a few cases here and there,” Mr. Rey said. “In most community situations if a citizen has a problem he writes a letter. That’s O.K., but it’s not all the possibilities.”

Mr. Rey added, “We have to promote more involvement, not only in access to justice and claims, but in taking part in the decision process.”

“I know the state has a good system from the legislative point of view,” Mr. Rey said. But as he and Mr. Whittle noted in their paper, “the question now is whether government leaders can and will do what it takes to put the plan on the ground.”

Inside Movie Animation: Simulating 128 Billion Elements

Inside Movie Animation: Simulating 128 Billion Elements

 
Simulations based on physics help make this Disney character incredibly rat-like.

Ever wonder how animated films such as The Incredibles get hair, clothing, water, plants, and other details to look so realistic? Or how, like the lion in The Chronicles of Narnia, animated characters are worked into live-action films? If not, the animators would be pleased, since they don't want special effects to distract from the story. Behind the scenes, though, is a sophisticated combination of artistry, computation, and physics.

Traditionally, animation was hand drawn by artists who needed"some of the same magical eye that the Renaissance painters had, to give the impression that it's realistically illuminated," says Paul Debevec, a computer graphics researcher at the University of Southern California. Over the past decade or so, the hand-painted animation has faded as physically-based simulations have increasingly been used to achieve more realistic lighting and motion. Despite this movement toward reality in animated films, the physics of the real world remains a slave to expediency and art: Simplifications and shortcuts make the simulations faster and cheaper, and what the director wants trumps physical accuracy.

In one dramatic scene in the movie 300, which came out early in 2007, several ships collide violently -- their hulls splinter, masts break, sails tear, and the ships sink. Stephan Trojansky, who worked on 300 as visual effects supervisor for the German-based company ScanlineVFX, said just creating the ocean in that scene involved simulating 128 billion elements. “We probably created the highest fluid simulation detail ever used in visual effects,” he said.

"For the fracturing and splintering of the ships," he added, "we developed splintering technology. Wood doesn't break like a stone tower. It bends. To get realistic behavior, you have to take into account how the ship is nailed together. The physics involved is mainly equations that define where the material will break."

Animations of both fluids and solids—and of facial expressions and clothing, among other things—use various computational methods and a host of equations. But there is a tradeoff in the push for more realistic animations – moving closer to reality requires more and more computer power, and becomes increasingly expensive. There are three commonly used methods of computer animation -- break the object being simulated into discrete elements, use sample points from the object, or create fixed cells in space.

Mark Sagar, of WETA Digital, a visual effects company in Wellington, New Zealand, specializes in simulating faces. One technique is motion capture, in which markers are placed on an actor's face, their positions are noted for different expressions, and the positions are then mapped onto an animated character. "For King Kong we mapped the actor's expressions onto a gorilla," said Sagar.

Simulating the face involves interpreting movement in terms of muscle, Sagar said. "We approximate the detailed mechanical properties of live tissue and its layers and layers. You have motion data and start working out what the driving forces are.” Modeling realistic stretching of the skin requires a lot of finite elements—each a small patch of tissue,” he said. "You compute and solve for forces at each point and then sum until you get a balanced equation. It's not sophisticated from an engineering standpoint but produces high-quality results."

Realistic motion is often too complicated for animators to do by hand, said Michael Kass, a researcher at Pixar Animation Studios. "The results can be awful and very expensive." In the original 1995 Toy Story, he said, "if you see a wrinkle in clothing, it's because an animator decided to put in a wrinkle at that point in time. After that we [at Pixar] decided to do a short film to try out a physically based clothing simulation."

The movement of clothing is computed as a solution to partial differential equations, he said. "You start with individual threads. What are their basic properties? Then you consider the bulk properties when [they're] woven. The main physical effects are stretching, shearing, and bending. To a certain degree, you can take real cloth and get actual measurements."

While animating clothing still presents problems, he said, “it's now part of a standard bag of tricks. Our simulations have become accurate enough that we can design garments with commercially available pattern-making software and then have them move largely as a tailor would expect in our virtual simulations."

Animating hair "is in many ways easier than clothing because it's like individual threads,” Kass said. “The difference is that clothing doesn't move like clothing unless the threads interact. In a real head of hair, the threads do interact, but you can get convincing motion without taking that into account."

Illumination is another area in which physics plays a key role in animation. For a long time, says Cornell University's Steve Marschner, "rendering skin was hard. It would look waxy or too smooth." The fix, he says, was to take into account that skin is translucent, which he and colleagues "figured out from looking at a different problem—rendering marble."

As with simulations of fluids, cloth, rigid bodies, and so on, incorporating translucency to model skin involves old physics. "In some cases we have to create the models from the ground up. But sometimes we find somebody in another branch of physics who has solved a similar problem and we can leverage what they've done." For skin translucency, "we were able to adapt a solution from medical physics, from a calculation of radiation distributions inside the skin that was used for laser therapy in skin diseases."

"One of the coolest things you see in a movie is when there is some sort of otherworldly beast or digital character that is sitting in the scene, roaming around, and it looks like it was really there," says Debevec. "The only way you can do that is by understanding the physics of light transport, respecting how light works in the real world, and then using computers to try to make up the difference from what was really shot."

For example, he says, in Narnia "they filmed a lot of it with the children dressed up in their knight costumes and left an empty space for the lion." Then, to get the digital lion just right, "Rhythm and Hues Studios used radiometrically calibrated cameras to measure the color and intensity of illumination from every direction in the scene." The measurements, he adds, "are fed into algorithms that were originally developed in the physics community and have been adapted by the computer graphics community as a realistic way to simulate the way light bounces around in the scene.”

Similar methods are used for creating digital doubles—virtual stunt characters that fill in for live actors. For that, Debevec said, "film studios sometimes bring actors here to our institute, where we've built devices to measure how a person or object, or whatever you stick in [the device], reflects light coming from every possible direction.” The resulting data set, he says, can be used to simulate a virtual version of the person. "There are about 40 shots of a digital Alfred Molina playing Dr. Otto Octavius in Spider-Man 2. It looks like him, but it's an animated character. The reflection from the skin looks realistic, with its texture, translucency, and shine, since it's all based on measurements of the real actor."

"We rarely simulate more than two indirect bounces of illumination, whereas in reality light just keeps bouncing around," Debevec continued. "With no bounces, things look way too spartan and the shadows are too sharp. One bounce fills in perhaps three-quarters of the missing light, and with two bounces you're usually past 95%. That's good enough." Another shortcut, he adds, is to focus just on the light rays that will end up at the eye. "We try to figure out the cheats you can make that give you images that look right."

"There is a long tradition of cheating as much as possible," said Marschner, "because setting up an exact simulation is either not possible or too expensive." “We use physics to get realism,” Trojansky said. "But I am a physics cheater. I use it as a base, but I am interested in the visual effect."