Examples for Pattern Formation in Biological Systems · 2009-06-10 · Activator-Inhibitor...

Artificial Life & Computational Intelligence, Wed. 10. June 2009

Examples for Pattern Formation in Biological Systems

Nils GoerkeDepartment of Computer Science VIUniversity of Bonn

Pattern Formation

• Cellular Automata (CA)

• Langton's Loop, Langton's Ant• Self Organising Criticality (SOC)

• Lindenmeyer systems

• Iterated functions

• Differential equations• Population dynamics

Pattern Formation• Iterated functions

• Linear and exponential growth

• Fibonacci sequence

• Logistic growth

• Predator-prey system

• Lotka-Volterra equations

• Activator-inhibitor equations

• Reaction-diffusion systems

• Plant morphogenesis, phyllotaxis

• Golden section, Golden angle

Iterated Functions

Iterated functions are closely related to sequences.

Sequence of values:

{ x(1), x(2), x(3), x(4), x(5), ... }

Sequences can be generated using recursive defined or iterated functions:

x(i+1) = f ( x(i) )

In iterated functions the next value x(i+1) is a function of the predecessing value x(i).

Iterated functions need an intitial value x(0) .

Iterated Functions, examples

Examples for iterated functions:

x(i+1) = x(i) + 2 with x(0)=3{ 3, 5, 7, 9, 11, .... }

x(i+1) = a x(i) with x(0)=100.0, a=0.9

{ 100.00, 90.00, 81.00, 72.90, 65.61, 59.049, ... }

x(i+1) = (1+) x(i) with x(0)>0, and 0 1

Iterated Functions, examples 2

The definition of iterated functions can contain logic functions as well: ½ x(i) if x(i) is even

x(i+1) =

3 x(i) +1 else{Can you predict the final value of x(i) when the starting value x(0) is given ?

Play this iterated function as a two person game: start with x(0) and iterate the function by turns:You lose, if you end up with x(i)=1.

Linear Growth

Implementing linear growth of a population of individuals is easy with iterated functions:

x(i+1) = x(i) + c with x(0)>0, and c >0

time titeration i

* * * * * * *

Linear Growth

Linear growth of a population of individuals is easily implemented using a constant increase c:

x(i+1) = x(i) + c with x(0)>0, and c >0

time titeration i

* * * * * * *

Exponential Growth

To obtain exponential growth, the increase is proportional to the number of individuals (birthrate).

x(i+1) = x(i) + rr x(i) with x(0)>0 = ( 1 + r ) x(i) with b=1+r = b x(i)

Depending on b a different behaviour results: 0.0 < b < 1.0 exponential decay

b = 1.0 constant, stable population b > 1.0 exponential growth

Exponential Growth

To obtain exponential growth, the increase is proportional to the number of individuals (birthrate).

x(i+1) = b x(i) with x(0)>0

0.0 < b < 1.0 b = 1.0 b > 1.0

Fibonacci SequenceLeonardo of Pisa, also known as Fibonacci, has mentioned in 1202 AD a recursive sequence that has been inspired by a growing population (e.g. number of pairs of rabbits, unbounded reproduction).

x(i+1) = x(i) + x(i-1)

x(0)=0, x(1)=1

{ 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... }

The sequence (although named after Fibonacci) has been reported to be known before that time,e.g. 200 BC, by Indian mathematicians. From Wikipedia, (9.6.2009)

http://en.wikipedia.org/wiki/Fibonacci_number

Fibonacci Sequence

From Ron Knott's web pages on Mathematics, Fibonacci Numbers and Nature , 9.6.2009http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html

Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on.

Logistic GrowthUnbounded reproduction is not realistic.Several circumstances exist (e.g. restricted resources) that yield to a maximal, bounded growth rate.

x(i+1) = x(i) + g x(i) with a growth rate g

The growth rate g shall now reflect restricted resources.The available ressources are those ressources that have been left over by the population x(i). Thus g is modeled as a growing rate a that is bounded by

g = a ( M – x(i) ) with M maximum of the ressources

Logistic GrowthThus a modified growth formula can be used:

x(i+1) = x(i) + g x(i) with g=a( M-x(i) ) = x(i) + a ( M – x(i) ) x(i)

This iterated function implements the differential equation published by P.-F. Verhulst (1838) to model realistic growth of populations:

From Wikipedia, (9.6.2009)http://en.wikipedia.org/wiki/Logistic_function

Logistic GrowthIterating the equation with a small initial value of x(0)

x(i+1) = x(i) + a( M-x(i) ) x(i)

The population starts to increase with an almost exponential growth of x(i). In this phase the increase is getting larger from step to step. This continues until x(i) reaches a value of x(j)=½ M. After that, the increase is getting smaller and smaller; until the value of x(i) converges to the maximal possible value of x(i) = M.

Logistic GrowthSolving the Verhulst differential equation yields the logistic function with it`s typical sigmoid shape.

From Wikipedia, (9.6.2009)http://en.wikipedia.org/wiki/Logistic_function

In the field of neural networks one can find this function denoted as:Sigmoid function or Fermi function.

Predator-Prey SystemConsider a situation with more than one species that interact with each other:

Predator and Preyone species (prey) is chased by the other species (predator).

Both, predator and prey are growing, and both are interacting. The dynamics can be modeled by the following equations:

N(i+1) = N(i) + aN(i) – b N(i)P(i) )

P(i+1) = P(i) + cP(i)N(i) – d P(i)

Predator-Prey System

N(i+1) = N(i) + aN(i) – b N(i)P(i) ) prey

P(i+1) = P(i) + cP(i)N(i) – d P(i) predator

The number N(i) of prey individuals: N(i) is increasing with the growth factor of a (birth)and is decreasing due to limited ressources bN(i)and due to the number of predators P(i) (hunt).

The number P(i) of predators: P(i) is increasing proportional to the number of predators P(i) (birth) and the number of prey N(i) (food), and is decreasing due to limited ressources defined by d.

Lotka-Volterra EquationsThe prey-predator equations written as system of two coupled differential equations: Lotka-Volterra (1925, 1926)

∂N/∂t = aN – bPN = N (a – bP) (prey)∂P/∂t = cPN – d P = P (cN – d) (predator)

Several observations from real animal data indicate that the Lotka-Volterra model is describing population dynamics sufficiently:Hudson-Bay Company, 1845 – 1935, lynx, hares.

Lotka-Volterra Equations

From Wikipedia, (9.6.2009)http://en.wikipedia.org/wiki/Lotka-Volterra_equation

The prey-predator equations from Lotka and Volterra establish a periodic oscillation over time.

Activator-Inhibitor EquationsThe prey-predator equations are an example of a complete family of differential equations called:

Activator – Inhibitor Systems

Typical Activator – Inhibitor systems have two components that interact with each other. One component is called the Activator, making the system grow.The other component is called the Inhibitor, which is the component that restricts the systems and prevents the growth from getting exponential..

Activator-Inhibitor Equations

The Activator X enforces it's own growth, and enforces the growth of the inhibitor.

X(i+1) = ... + aX(i) ... Y(i+1) = ... +cX(i) ...

The Inhibitor Y constrains it's own growth, and constrains the growth of the activator.

X(i+1) = ... – bY(i) ... Y(i+1) = ... – dY(i) ...

X(i+1) = X(i) +aX(i) – bY(i) activator

Y(i+1) = Y(i) +cX(i) – dY(i) inhibitor

Activator-Inhibitor EquationsActivator-Inhibitor characteristics can be found in a lot of systems from biology, physics, medicine and technics.

Very often the Activator-Inhibitor principle is combined with a spatial component.Not only the conditions at one spatial position are considered, but also the conditions from the direct heighborhood are taken into account via a diffusion process.

Reaction-Diffusion SystemThe differential equations of a Reaction-Diffusion system try to model the local influence from the neighborhood via a diffusion processes and the activity within one compartment using the reaction terms.

dV/dt = μ ∂2V/∂x2 + fV(V,W)

dW/dt = σ ∂2W/∂x2 + fW

Example for a one-dimensional Reaction-Diffusion systemwith the diffusion terms: μ ∂2V/∂x2 and σ ∂2W/∂x2 and the reaction terms: f

V(V,W) and f

W(V,W)

Reaction-Diffusion SystemCellular automata can be seen as discretised versions of Reaction-Diffusion systems.

The rule of a cellular automaton takes the state of the direct neighborhood into account, thus the diffusion part of a Reaction-Diffusion system can be easily implemented.

0 1 000 111 90D

Reaction-Diffusion System

http://www.uni-muenster.de/Physik.AP/Purwins/DE/ElNw-de.html

An real implementation of an Activator-Inhibitor Reaction-Diffusion system is a field (1-dim, or 2-dim) of nonlinear electrical oscillators.

Spatial structures emerge as result of the coupling.

128 oscillators are connected to a one- dimensional field.

Spatial structures are the result of the coupling.

Spatial structures are the result of the coupling.http://www.uni-muenster.de/Physik.AP/Purwins/DE/ElNw-de.html

Belousov-Zhabotinsky-Reaction

http://www.scholarpedia.org/article/Belousov-Zhabotinsky_reaction

Example for an Activator-Inhibitor Reaction-Diffusion System from chemistry.

The Belousov-Zhabotinsky reaction yields a spatio-temporal moving pattern.

Belousov-Zhabotinsky-Reaction

http://www.uni-muenster.de/Physik.AP/Purwins/DE/BZ-Reaktion-de.tml

Patterns in Mixtures of Granulated Material

http://www.uni-muenster.de/Physik.AP/Purwins/DE/Streifen_im_Salz-Mohn-Gemisch-de.html

A mixture of granulated material with different density consisting of Salt (white) and Poppy Seed (black) can establish a spatial structure.

The nonlinear combination of different densities, the movement and the friction cause this effect.

Growth PatternsGrowth patterns of plants are often determined by activator-inhibitor reaction-diffusion systems.

Growth PatternsThe growing plant produces ressources (activator)

Growth PatternsAs soon as enough ressources (activator) are available, a leaf is growing in that position.

Growth PatternsThe growing leaf is producing an inhibitor, to prevent that a second leaf will grow near by.

Inhibitor distribution

Activator distribution

Growth PatternsWith the growing plant, the concentration of activator and inhibitor is decaying.

Growth PatternsAfter a while, enough ressources are available again to produce a new leaf.

Growth PatternsBut the inhibitor is still present; thus the position of the next leaf will be a far avay from the fist one.

Growth PatternsAgain, the growing leaf will produce inhibitor.

Growth PatternsThe next leaf to grow is influenced by the inhibitor of both existing leaves, leading to a position in between.

Growth PatternsThe next leaf to grow is influenced by the inhibitor of all the existing leaves, leading to a positions in between.

Growth PatternsAfter a while the angle between subsequent leafs has become almost the golden angle of 137.51 deg..

Growth Patterns

Growth patterns of plants are often determined by activator-inhibitor reaction-diffusion systems.

Fibonacci Spirals in Plant Morphogenesis

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#pinecones

From Ron Knott's web pages on Mathematics, Fibonacci Numbers and Nature , 9.6.2009http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#pinecones

The number of spirals that can be found in either direction are typically consequitive Fibonacci numbers.

Self Similarity in Plant Morphogenesis

Pattern Formation• Iterated functions

• Linear and exponential growth

• Fibonacci sequence

• Logistic growth

• Predator-prey system

• Lotka-Volterra equations

• Activator-inhibitor equations

• Reaction-diffusion systems

• Plant morphogenesis, phyllotaxis

• Golden section, Golden angle

Vielen Dank für Ihre die Aufmerksamkeit

Thank you for your attention.

Examples for Pattern Formation in Biological Systems · 2009-06-10 · Activator-Inhibitor...

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