Pi helps us describe almost everything, not just circles.

Most people know of π, or ‘pi’, as the number they learned in high school that has to do with circles: it is the ratio of a circle’s diameter to its circumference (π=C/d), the area of the circle is πr2 (especially hilarious because pie are round, not squared), etc. Some of us even remember it as an irrational number, meaning you cannot write it down as a simple fraction, and maybe some people, certainly not me, still have it memorized as starting with 3.14159265. What is less appreciated, however, is that this number has utility far beyond allowing us to calculate the area of a circle.

In biophysics, and in science in general, we use statistics to compare our data with our hypotheses. Many of the phenomena we measure fall along (or can be manipulated to fall along) a normal distribution. A normal distribution is a common continuous probability distribution characterized by the familiar “bell curve” shape, or Gaussian, which corresponds to the Gaussian distribution shown in the image below. When the mean, μ, is zero and the variance, σ2, is one, this function (the blue curve) is e^(-x2) and the area under the curve is the square root of pi! When the mean and variance are other values, the curve can be described more fully with the equation:

Where a = 1 / (σ (2π)1/2) a , b = μ, and c = σ.

pi day graph

 

Normalized Gaussian curves with expected value μ and variance σ2. The corresponding parameters are a = 1 / (σ (2π)1/2) a , b = μ, and c = σ.

 

How was the Gaussian distribution first determined, you may ask? While pi itself is thought to be first measured by the ancient Babylonians between 1900-1680 B.C., the Gaussian distribution originated in the 18th century when Abraham de Moivre started calculating gambling odds extremely precisely. De Moivre studied a very simple system at first: flipping a coin. He would calculate the probability of getting a certain number of heads from a certain number of coin flips. He found that as the number of events (coin flips) increased, the more his probability distribution approached a smooth curve. Thus he went about finding a mathematical expression for this curve, which resulted in the “normal curve”.

Independently, two mathematicians Adrain and Gauss in 1808 and 1809, respectively, developed the formula for the normal distribution and showed that errors observed in astronomical data fell along this distribution. Small errors in measurements occurred more frequently than large ones. The distribution was also independently discovered by Laplace, who elegantly showed how pi enters into the Gaussian distribution (which is summarized nicely here: http://www.umich.edu/~chem461/Gaussian%20Integrals.pdf). Laplace also introduced the Central Limit Theorem, which proves that with a large enough number of samples the mean will be normally distributed, regardless of the underlying original distribution. This is why the normal distribution ends up popping up in so many places.

In biophysics, every time we think about mean and variance, calculate a p value (which assumes a normal distribution), do image processing, or try to understand the probabilities of a particular event, we owe a debt to pi. Not only do we use the Gaussian for statistics, but we also often use it in fields where we need to apply a potential or some external force either experimentally or in simulation. Basically, pi underlies all of the fundamental biological process we study on a daily basis. Thanks pi!

By Sonya Hanson, postdoc at Memorial Sloan Kettering Cancer Center

 

References:
https://en.wikipedia.org/wiki/Gaussian_function (Including public domain figure)
http://onlinestatbook.com/2/normal_distribution/history_normal.html
https://www.amazon.com/Cartoon-Guide-Statistics-Larry-Gonick/dp/0062731025

 

Advertisements

Pi Is Encoded in the Patterns of Life

 

-By Santiago Schnell

Every March 14th, mathematical scientists like me are commissioned to write articles about the ancient and mysterious number: Pi. It is denoted by the Greek letter “π” and used in mathematics to represent a constant, approximately equal to 3.14159. Pi was originally discovered as the constant equal to the ratio of the circumference of a circle to its diameter.  The number has been calculated to over one trillion digits beyond its decimal point. Calculations can continue infinitely without repetition or pattern, because Pi is an irrational number.  Mathematicians called it irrational, because Pi cannot be expressed as a ratio of integers.

To children and adults alike, Pi is perplexing… a constant with an infinity number of digits and no pattern. We all learn about Pi in geometry class at high school. However, Pi doesn’t seem to have a practical utility outside of the world of geometry. So why does Pi – a geometrical constant – deserve a celebration? If we celebrate Pi, why don’t we celebrate any other number?  Well, Pi is different from all other numbers. It is a universal constant encoded in most processes occurring in the universe, including those in the life sciences!

Now you are probably wondering how Pi appears in biological processes. The answer to this question lies in the interdisciplinary field of biophysics: biology + physics. Biology studies life and living organisms. Biologists investigate how organisms grow, get food, communicate, sense and response to the environment, reproduce, and evolve.  On the other hand, physics studies the nature and properties of matter and energy. Physicists search for the mathematical laws of nature and the universe. Biophysicists look for patterns in life and analyze them with mathematics to gain novel insights about how organisms work.

Let’s now consider one of the patterns observed in the life sciences.  The appearance of an organism’s body plan – a process called morphogenesis – is one of the most striking features of living creatures.  In animals, the embryo grows from an almost uniform group of cells into a patterned structure with a brain, backbone, and limbs. In 1952, the mathematician and father of computer science, Alan Turing, proposed a mathematical model describing the simple biophysical principles of pattern formation during morphogenesis. He proposed that an embryo becomes patterned into different anatomical features by chemicals (termed morphogens by Turing), which diffuse through tissues. In the simplest case, the formation of the pattern results from the reaction of two morphogens, an activator and inhibitor. The activator self-amplifies and can only diffuse locally. It also stimulates the growth of the inhibitor which, in turn, suppresses the activator, and diffuses long distances. Mathematical analysis and computer simulations of this seemingly simple system reveal that Turing’s model produces a bewildering array of patterns, including spots and stripes.  The activator morphogen forms local patches of spots or stripes, while the inhibitor prevents the patches growing too close to each other. Turing’s model is supported by experimental evidence as one of the candidate mechanisms driving the formation of patterns during the growth of organisms. In fact, it can explain the formation of stripes and spots in animal fur coats, pigmented markings in tissues, limb structure and the development of the small finger-like protrusions in the animal gut which significantly increase the intestinal surface area used to absorb food.

figure1_option1

Pi, the number with no pattern, plays a role in the formation of pattern.

 

How does Pi, the number with no pattern, play a role in the formation of pattern? Close your eyes, and imagine the stripes of a zebra. Those stripes have a size and spacing that is encoded by a constant: Pi! The same goes for the spots of a leopard. In fact, it seems that Pi encodes the size and spacing of many patterns, not confined to the field of biology.

figure2

Computer simulations of Turing’s model produces a bewildering array of patterns, including spots and stripes.

Pi is also intimately woven into periodic processes. It appears in the governing biophysical laws of cell division timing, heart beats, breathing cycle, and circadian rhythms controlling sleep-wake cycles. However, this is another interesting and exciting topic at the interface between physics and biology, which we will need to leave for next year.  You’ll just have to wait π x107 seconds!

About the author

Santiago Schnell

1/29/2010 Photo in the lab of disease research for the Brehm Center.

Santiago Schnell is Professor of Molecular & Integrative Physiology and Computational Medicine & Bioinformatics at the University of Michigan Medical School.  He is one of the scientists, who discovered that Turing’s model explains the formation of the small finger-like protrusions in the mammalian gut. Santiago Schnell is member of the Biophysical Society, Fellow of the Royal Society of Chemistry, and President of the Society for Mathematical Biology.