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 are patterns? ======
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.
===== Overview ======
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:
==== Developmental Biology and Patterns====
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.
==== Bicoid ====
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.
==== Reaction-Diffusion Equations ====
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.
==== Patterns, space, and plankton ====
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)].
==== Animal coats ====
{{ dynsys:pattern:tiger2.gif }}
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: [[.:reactdiffuse1|Reaction Diffusion I]]
* [[dynsys|Home]]
* [[dynsys:pattern|Pattern formation in nature]]
* [[dynsys:reactdiffuse1|Reaction-diffusion I]]
* [[dynsys:reactdiffuse2|Reaction-diffusion II]]
* [[dynsys:turinginst1|Turing I]]
* [[dynsys:turinginst2|Turing II]]
* [[dynsys:turinginst3|Turing III]]
* [[dynsys:numerics|Numerical methods and Matlab]]
* [[dynsys:vids|Videos]]
* [[dynsys:about|About]]
===== References =====
[(: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
)]