Modeling Plant Form

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Modeling Plant Form

• Is plant form an emergent property of simple
  module systems?

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L-Systems

• L-systems are basically a way to rewrite
  something following a set of rules
• For instance you have two letters a and b.
• The rules for rewriting are a-ab and b-a
• If we start with a b and start rewriting we get

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The Turtle interpretation of strings

• So we have a turtle with a string on its back,
  the turtles state is a triplet (x,y,a). This
  represents the turtles Cartesian coordinates and
  the angle (a) at which it is traveling.
• Now, d step size and angle increment
• So we can tell the turtle where to go if we give
  it directions. We will use the following
  symbols
• F Move forward by one step length d
• Turn counterclockwise by angle
• - Turn clockwise by angle

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Lets put our turtle to work

• Given the axiom w F-F-F-F and the production
  successor p F-F-FFF-F-F
• We can rewrite the phrase n times and tell out
  turtle to walk.

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Now lets make it a little bit more complex

• Edge rewriting productions substitute figures for
  polygon edges
• Fl and Fr represent the turtle obeying the move
  forward command, but now Fl and Fr edges by
  lines forming left or right turns.
• These curves can be space-filling and self
  avoiding (FASS).

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FASS curves generated from edge-rewriting
L-systems
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• Node rewriting substitutes polygons for nodes on
  the curve
• Now we need more things Entry and exit points
  (Pa and Qa) and an entry vector and an exit
  vector (pa and qa)

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• You can also consider an array of m x m square
  tiles.
• Each m x m contains a small box inside of it
  called a frame. Each frame bounds an open
  self-avoiding polygon.
• Now when we connect many tiles we will get a
  macrotile

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3-D
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Axial Trees

• All of the previous examples were all a single
  line, but trees are not!
• An axial tree starts from a base node
• At each of its nodes there is at most one
  outgoing straight segment
• All other edges are lateral segments
• A terminal segment is an apex
• An axis must
• The first segment in the sequence originates from
  the base or a lateral segment at a node
• Each subsequent segment is straight
• The last segment is not followed by any straight
  segment
• So each axis is a mini axial tree!
• An axis with all of its descendants is a branch

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Axes and branches are ordered as order 0 If they
originated At the base and you Can guess the rest
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Lets build a tree

• We need to have a rewriting mechanism that acts
  on axial trees
• Our rewriting rule, or tree production, must
  replace an edge with an axial tree

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Bracketed system
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Examples of bracketed system
Note The system for adding Leaves to this bush
is Biologically whack
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Stochastic L-Systems

• Since all plants dont look the same we will add
  in some randomization.

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Context-sensitive L-Systems

• We can make an L-System that show signal
  propagation so we can send signals from the
  leaves down or from the roots up.

Plants Really Use Signals!
Removing P2 makes Permanent signal
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Parametric L-Systems

• Will help us show time, angles, and irrational
  line lengths (if d 1, you cannot express
  sqrt(2).
• Is easier than trying to add stuff to
  non-parametric model.

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Now for the real stuffLets try to simulate
herbaceous plants

• Emphasis on space-time relation between plant
  parts
• So there can be flowers and buds on the tree at
  the same time
• Inherent capability of growth simulation
• Our model is good for growing and we can simulate
  plants at different times and watch how they grow
• Lets only do herbaceous plants because
• The model assumes that the plant controls its own
  development (endogenous interaction).
• Herbaceous plants have a lot of directions from
  their parents (lineage interaction).
• Woody plants are much more sensitive to their
  environment, competition among branches and
  trees, and accidents (exogenous interaction).

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A glimpse at the models

• http//algorithmicbotany.org/vmm-deluxe/QT/Greenas
  h/apexview.qt
• http//algorithmicbotany.org/vmm-deluxe/QT/Bluebel
  l/field.qt
• We can use confocal microscopes to get a real
  idea of how plants develop and then write a
  computer model that fits the behavior
• We can also use empirical data on plant
  development
• Other models try to use known mechanisms to
  explain the emergence of plant forms

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Three Main Type of Models

• Partial L-Systems Your basic model that is
  supposed to show us the possible structures of
  plants
• L-System Schemata Topology and temporal aspects
  of plants expressed, could help us understand
  mechanisms
• Complete L-Systems Geometric aspects added in
  (growth rates of internodes, values f branching
  angles, appearance of organs)

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Partial L-System
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Examples of cool things in L-system Schemata
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Examples of cool things in L-System Schemata
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Examples of cool things in L-System Schemata
Plants actually use signals and feedback loops a
lot (WUS acts on SAM)!

• This says that the apex (a) produces internodes
  (I) and leaves (L) p2. The time in between
  growth is m p1.
• After delay (d) a signal (s) p3 an p4. The
  signal is sent down the main axis with delay (u)
  steps per internode (I) p5 and p7.
• p6 removes the signal from the node by using an
  empty string (e)
• When the signal reaches the apex (a), the a is
  transformed into a flowering state (A), which
  turns into a flower (K) p8 and p9.
• Note u

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COMPLETE MODELSMUAHAHA

• These are good enough to make images
• We can tell the model when to make branches using
  subapical growth
• Plants actually grow like this!

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I like flowers!

• There are a few different types of flowers we can
  make
• Monopoidal branching - lateral buds make flowers
  and can not make any more branches (raceme
  inflorescence)

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I still like flowers!

• In sympodial branching the apex produces a flower
  bud (which cannot branch further) and two new
  lateral apices (cyme florescence).

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I hope you arent allergic to pollen

• In polypodial branching, the apex makes three
  active apices, and at some point they change into
  buds (panicle inflorescence).

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But I want more!
Leaf model created trying to represent known
biology (auxin), not bad right? -

• Modeling exogenous effects are improving
• http//algorithmicbotany.org/vmm-deluxe/QT/OpenLsy
  s/two.qt
• How leaves develop
• How flowers develop
• How roots develop
• A photosynthesis model ---
• Clovers sense different wavelengths of light to
• perceive self-shade (light reflected off leaves
  is far-red)
• A model that makes branches fall off when
• The amount of energy leaves get from
• Photosynthesis isnt enough to maintain
• Leaves and branch (self-thinning) ---

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Other models

• Large trees dont exhibit the recursive branching
  described in models because of exogenous factors.
  One group decided to model tree branching as a
  function of branch competition for space.

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By changing values for the number of attraction
points, the kill distance, influence distance,
and the distribution of attraction points
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Resource Acquisition Model

• Colasanti and Hunt wanted to see if their model
  could produce properties on different levels
• S-shaped growth curve for individuals
• Equilibrium between shoots and roots
• Plasticity in root and shoot foraging
• Self thinning according to geometric power laws
• Competitive exclusion
• They used two binary trees
• One for roots and one for shoots

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Waitwhats a binary tree

• Modules linked together.
• Each module is linked to one parent module and
  potentially two offspring modules
• A module knows the identity and state of its
  parent and offspring modules, but not the state
  of the whole plant
• Base module has no parent and end module has no
  offspring
• Spatial area made into cells, these cells can
  have resource units (light units for
  shoots/mineral nutrient units for roots)
• The module can transport the units to base module
• New growth requires a light unit and a mineral
  unit
• They mutated the plant by giving it a competitive
  advantage for resources at the expense of extra
  energy

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Their Results

• Success.
• S-Shaped growth curve
• Self-thinning
• Plasticity in roots and shoots of modified plants
• When resources are high, modified plants did well
• When resources are low, regular plants did better
• Could always make it better

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Conclusion

• These models show that a very simple module
  behavior can account for many aspects of trees
  and herbaceous plants
• By comparing these models to nature, we can learn
  more about the actual mechanisms in nature
• Nature is math-y and pretty (or is math pretty
  and nature-y?)
• Now when you see a tree, a bush, a leaf, a
  flower, or a root systemthink about L-Systems
  and how cool nature is

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References

• S. Wolfram, A New Kind of Science. Chapter 3, 6,
  8.5, 8.6, 8.7
• P. Prusinkiewicz and A. Lindenmayer, The
  Algorithmic Beauty of Plants
• R. L. Colassanti and R. Hunt, Resource Dynamics
  and Plant Growth A Self-Assembling Model for
  Individuals
• Runions et al., Modeling Trees with a Space
  Colonization Algorithm
• Runions et al., Modeling and visualization of
  leaf venation patterns
• O. Prusinkiewicz and Anne-Gaëlle Rolland-Lagan,
  Modeling plant morphogensis
• P. Prusinkiewicz, Simulation Modeling of Plants
  and Plant Ecosystems