MINNEAPOLIS — As a physics professor, I don’t usually take requests to create a new mathematical formula on demand, but when the call came from Hollywood for an equation for cellular regeneration, my only concern was whether I would have it ready by Monday.
The equation I provided appears in Sony’s "The Amazing Spider-Man," the latest take on the Spidey Saga.
The request came from Andy Siegel, the prop master of the film. He needed a mathematical expression that would be distinctive and memorable, so that the audience would recognize it at various points throughout the film. In addition to being a mild-mannered physics professor at the University of Minnesota, I had volunteered as one of the science consultants on the film. For this opportunity I would like to thank the Academy — the National Academy of Sciences, that is.
Hollywood has been calling on academia more and more for assistance in creating believable fake realities, a relationship that dates back to the early days of cinema. Rocket scientist Wernher von Braun served as a science consultant for Fritz Lang’s 1929 "Woman on the Moon," notable for the first time that a countdown was invoked as a prelude to a rocket launch, in both cinema and the real world.
Matchmaker for the movies
As the number of science-fiction and superhero films and television programs has increased of late, so have the requests for technical advice from the scientific community. In 2008, the National Academy launched the Science and Entertainment Exchange, a program that serves as a matchmaker for scientists and the creators of TV shows and films.
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Whether it is providing the physics equations on Sheldon Cooper’s white board in CBS' "The Big Bang Theory" or outfitting Tony Stark’s laboratory in "Iron Man," we scientists have been asked to add our 2 cents to the creative process. Hollywood creators appreciate our contributions, for they realize that when the audience is questioning the physics of what they are watching or the authenticity of the laboratory set, that's a moment when they are not paying attention to the story.
The goal is not to ensure that everything on the screen is 100 percent scientifically accurate — which would, after all, defeat the purpose of the escapist fantasy we have paid our money to watch — but rather to get it just right enough to maintain the audience’s suspension of disbelief.
Being a physics professor who is also a fan of superhero comic books (I created a class at Minnesota titled: “Everything I Know About Physics I Learned from Reading Comic Books”) I recognize that one must invoke a "miracle exception from the laws of nature" in order to justify "spider-powers" or men mutated into giant lizards. When Andy Siegel asked me to create an equation that would relate to cell regeneration and human mortality — an equation called the "Decay Rate Algorithm" in the film — I nevertheless wanted to ground the formula in real science. Naturally, I thought of the Gompertz equation.
The Gompertz equation?
Have you ever wondered what determines the average person’s lifespan? If you are careful and avoid fatal accidents, what are your chances of dying, once you make it to, say, 25 years of age? A natural first guess would be that in any given year, there’s a certain constant probability of dying.
Think of it as the "Death Lottery," the competition where no one wants to "win" the jackpot, but we all do eventually. Unlike a real lottery, your chance of dying is not the same regardless of your age. If it were, there would be some people who could go quite a long time without passing away, just as I’m able to not win the Lotto week after week. It’s easy to show that if your chance of dying remained unchanged as you get older, then with billions of people on the planet, there would be millions who are 350 years old or older. Of course this is not the case, and in fact, few ever make it to over 110, That suggests that the odds change as we get older, and in fact the game is rigged so that we all "win" in the end.
As we get older, the probability of dying is not the same as when we are younger, but actually increases. It would not be hard to win the Lotto if the odds of matching the winning numbers got better for you the longer you played. The mathematical expression for the probability of surviving to a given age, incorporating a mortality rate that increases with age, was first identified by Benjamin Gompertz in 1825 and is named after him.
Why does the mortality rate rise as we age? My colleague at Minnesota, Professor Boris Shklovskii, has proposed an explanation that starts with single defective cells that malfunction (say, by forming tumors or producing defective proteins that can lead to Alzheimer’s). The cells become fatal when allowed to multiply, growing to a critical population size. The body’s immune response, upon encountering a rogue cell, will destroy it — thus preventing it from reaching a lethal stage.
The defective cells may be thought of as criminals, and the body’s defenses as patrolling police officers. When we are in our youthful prime, it is equivalent to having many cops on the street. Crime has a difficult time getting a toehold. A continuous series of budget cuts would reduce the number of available police officers, so eventually crime would win out. Similarly, as we age, either an accumulation of mutations or a natural degradation of the organism’s ability to maintain its previous immunological state increases the chances that the populations of defective cells could grow to a lethal size.
Science as a superpower
To me, the above argument illustrates the power and beauty of science. Patterns in natural phenomena lie waiting to be discovered, whether in the laboratory, in observatories or actuarial tables. Our efforts to account for and explain these patterns deepen our understanding of nature.
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In some cases, this understanding turns out to be very useful. Efforts to elucidate the mechanisms underlying the sequence of wavelengths of light emitted by atoms at the start of the 20th century led physicists to develop quantum mechanics, which enabled the construction of the laser and the transistor, and eventually, cell phones, laptop computers and magnetic resonance imaging. Patterns first, products (sometimes) later.
Armed with this background, I was ready to provide an equation to Andy Siegel, but I did not want to just simply send him the actual Gompertz equation, as there would seem to be little reason for the professional scientists in the film to not already know it. I therefore combined expressions from “The Reliability Theory of Aging and Longevity” by Leonid Gavrilov and Natalia Gavrilova into a single formula, and added extra terms (“mathematical glitter,” if you will) so that it would appear sufficiently complex. (The actual Gompertz equation can be written in a simple and compact manner that did not meet the visual needs of the filmmakers.)
Behind Spider-Man is Peter Parker, and behind Peter Parker is some real and interesting science, in the guise of the Decay Rate Algorithm. We may not be able to cling to walls, or transform into giant lizards, but our ability to use our intelligence to understand the world around us is our true superpower!
More reality checks on superhero stories:
- Your quantum superpowers
- How far off are real 'superhero' powers?
- Who needs superpowers? Real people take over comics
James Kakalios is the Taylor Distinguished Professor in the School of Physics and Astronomy at the University of Minnesota, and the author of "The Physics of Superheroes" (2nd edition, Gotham, 2009) and "The Amazing Story of Quantum Mechanics" (Gotham, 2010).
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