Have you ever wondered why tigers have stripes, giraffes have spots, and humans don't have either? If you have, how did you go about answering that question? Since leopards, tigers, and humans are all different species, it might make sense that our appearance would be different. But what exactly makes us look different? Maybe, you might say, it is all genetically programmed. Maybe nature has a library with volumes of recipe book: one for humans, one for zebras, one for raccoons, and one for giraffes. However, most animals, like humans, begin their journey through life as a single cell. Is it really possible that each species has developed unique biological machinery that mimics specialized biological recipe books? Remember, evolution is slow and most animals share the same basic biological building blocks so maybe species specific “recipe books” are improbable. So what determines whether you get spots, stripes, a combination, or nothing?

In the past fifty years, a lot of work has gone into understanding the mechanisms underlying pattern formation which might explain pattern formation beyond just the ones found on animal coats. In fact, patterns are so abundant in nature that they can be found in just about everything from hurricanes, seashells and snowflakes to how neurons make connections in the brain.

What is the first thing that comes to mind when you think of patterns? Whether it is “repetition,” “regularity,” or perhaps “symmetry,” our intuition of patterns is that they are spatially inhomogeneous (i.e. you would probably say that a portrait with randomly arranged black and white balls does not have patterns, but a painting with vertical white and black stripes does.)

Patterns are pretty, but to really understand what patterns in nature are, we want to gain a deeper understanding of the mechanisms underlying pattern formation. The first person to propose a mathematical model to describe pattern formation was Alan Turing, who is more famously known for his work in computer science. He proposed that pattern formation could be understood using reaction-diffusion equations, which are a very important set of partial differential equations that we will be studying closely for a big portion of these notes.

These notes were created to introduce the reader with some of the key ideas underlying the mathematics of patterns, and also discuss how the theory was developed hand in hand with new experimental discoveries. We will talk about the following examples in more depth as they can better illustrate some of the subtler properties of the mathematical models. These notes will talk about some of the influential models in terms of these topics:

Our understanding of pattern formation is inextricably linked to research in developmental biology, which deals with one of the most interesting “patterns”: the form and structure of the body. The origin of form, also referred to as morphogenesis, captivated many researchers who sought to develop models that could explain how multicellular organisms emerge from single fertilized eggs. For a long time, this line of research emphasized the evolutionary aspect of form, i.e. the slow changes in traits due to random mutations in the genetic material from generation to generation and the process of natural selection. The emphasis shifted away from evolution and towards the study of morphogenesis itself after 1915, when D'Arcy Thompson published his influential book, On Growth and Form [(darcy_1915)]. D'Arcy highlighted the importance of physical laws and first principles that underlie morphogenesis. His approach was perhaps too theoretical, as he solely presented examples where mechanical phenomena could be connected to forms that he observed in nature without presenting any experimental data. However, it inspired many researchers to focus their attentions to the principles underlying morphogenesis, or the origin of form.

One of the fundamental questions in development is how different types of cells throughout the body - all of which arise from a single precursor and contain identical genetic material -ultimately have distinct shapes and function. An idea that is widely accepted is that the cell fate (i.e. the type of cell it becomes) will be determined by the location of the cell within the embryo [(malacinski_1984)]. In other words, we imagine that there exists a blueprint that maps out where different classes of cells will be, but each cell only has local information. How does the cell figure out where it is on the blueprint? One possibility is that a pre-pattern is set up that will tell the cell where it is located almost like a GPS signal.

Pre-patterning is thought to involve many chemicals that may vary across species, but for the sake of these notes we will focus on a single maternal gene called bicoid, which plays a key role in determining what cells become part of the head in fruit flies [(frohnhofer_1986)], [(little_2011)], [(johnston_1982)]. The story of how bicoid was isolated, and the subsequent experiments that were conducted to determine its function are extremely interesting, and a good introduction to this set of related experiments can be found in Chapter 2 of Principles of Development [(wolpert_2010)]. It is through this process that reaction-diffusion equations were first suggested as a simple but powerful model that could lead to complicated pattern formation.

As experimentalists sought to find and measure the chemicals that could explain how organisms develop, others were developing the mathematical tools that could explain the mathematical principles of pattern formation. One of the major breakthroughs came from Alan Turing who proposed that complex patterns could be formed by relatively simple partial differential equations. The idea that such simple equations could lead to complicated behaviors was surprising and compelling. Even though there are debates as to whether Turing patterns really do explain patterns in nature, they are still fundamental to the modern understanding of pattern formation.

Once we discuss the historical and biological context through which reaction-diffusion equations were first proposed as plausible mathematical models of pattern formation, we will move on to study some of the properties that these equations have. With these models, the exact pattern that will form depends greatly on the spatial domain in which they form. In order to study the influence of spatial domain on reaction-diffusion equations we consider an example of plankton in the ocean. Plankton are tiny marine creatures that can only survive and grow in waters with the right temperature, nutrition etc. Their name comes from the Greek planktos meaning “errant”, which may describe how plankton drift with the currents of water due to their inability to swim against the current. As such, their motion is well-approximated by diffusion. By imagining the ocean as having plankton-friendly waters intermixed with volumes of water where plankton will not survive, we follow the derivation of Kierstead and Slobodkin to study the effect of the spatial domain on the long-term population of the plankton [(kierstead_1953)].

Our whirlwind tour of the key ideas in pattern formation will culminate with the application of reaction-diffusion equations to animal coat patterns. This has been done to model a variety of animals including fish [(kondo_2009)], leopards, [(murray_1988)], and even shell patterns [(meinhardt_1987)]. We will take all the building blocks of reaction-diffusion equations and show how they can be used to model animal coat patterns, and we will also discuss some of the key concepts in numerical methods that are needed to run these simulations.

I hope you will enjoy this introduction to the world of pattern formation, its relevance to different areas in biology, and the different elements ranging from the theoretical to the experimental and applied that are necessary to build our understanding of such ubiquitous natural phenomena.

You can now proceed to the next chapter: Reaction Diffusion I

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[(:murray_2003» author: Murray, J.D. ref-author: Murray title: Mathematical Biology II: Spatial Models and Biomedical Applications edition: Third Edition year: 2003 publisher: Springer )]

[(:malacinski_1984» author: Malacinski, G.M. and Bryant S.V. ref-author: Malacinski and Bryant title: Pattern Formation: A Primer in Developmental Biology publisher: Macmillan Publishing Company year: 1984 )]

[(:yoon_1998» author: Yoon H.S. and Golden J.W. ref-author: Yoon and Golden title: Heterocyst Pattern Formation Controlled by a Diffusible Peptide journal: Science volume: 282 year: 1998 )]

[(:frohnhofer_1986» author: Fronhofer H.G. and Nusslein-Volhard C. ref-author: Fronhofer and Nusslein-Volhard title: Organization of anterior pattern in the Drosophila embryo by the maternal gene bicoid journal: Nature volume: 324 year: 1986 )]

[(:johnston_1982» author: Johnston D.S. and Nusslein-Volhard C. ref-author: Johnston and Nusslein-Volhard title: The Origin of Patterns and Polarity in the Drosophila Embryo journal: Cell volume: 68 year: 1992 )]

[(:wartlick_2009» author: Wartlick O., Kicheva A., and Gonzalez-Gaitan, M. ref-author: Wartlick, Kicheva and Gonzalez-Gaitan title: Morphogen Gradient Formation journal: Cold Spring Harb Perspect Biol year: 2009 volume: 1 number: 3 )]

[(:little_2011» author: Little, SC, Tkacik G, Kneeland TB, Wieschaus EF, Gregor T ref-author: Little et al title: The Formation of the Bicoid Morphogen Gradient Requires Protein Movement from Anteriorly Localized mRNA journal: PLoS Biol volume: 9 number: 3 year: 2011 )]

[(:wolpert_2010» author: Wolpert L., and Tickle, C. ref-author: Wolpert and Tickle title: Principles of Development edition: Fourth Edition year: 2010 publisher: Oxford University Press )]

[(:darcy_1915» author: Thompson, D'Arcy ref-author: Thompson title: On Growth and Form: The complete revised form publisher: Dover Publications year: 1992 )]

[(:kierstead_1953» author: Kierstead, H, and Slobodkin LB. ref-author: Kierstead and Slobodkin title: The size of water masses containing plankton blooms journal: J. Mar. Res volume: 12 number: 1 year: 1953 )]

[(:murray_1988» author: Murray, J.D. ref-author: Murray title: Mammalian coat patterns: How the leopard gets its spots journal: Scientific American volume: 256 number: 3 year: 1988 )]

[(:kondo_2009» author: Kondo S., Iwashita M., and Yamaguchi M. ref-author: Kondo et al title: How animals get their skin patterns: fish pigment pattern as a live Turing wave journal: Int. J. Dev. Biol. volume: 53 year: 2009 )]

[(:meinhardt_1987» author: Meinhardt H. and Klingler M. ref-author: Meinhardt and Klingler title: A model for pattern generation on the shells of molluscs. journal: J. Theor. Biol. volume:126 year: 1987 )]