Transcript Modeling Plant Form

Modeling Plant Form
Is plant form an emergent
property of simple module
systems?
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:
The Turtle interpretation of strings
 So we have a turtle with a string on its back,
the turtle’s state is a triplet (x,y,α). This
represents the turtle’s Cartesian coordinates
and the angle (α) at which it is traveling.
 Now, d = step size and ƒ =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 ƒ
 - = Turn clockwise by angle ƒ
Let’s put our turtle to work
 Given the axiom w = F-F-F-F and the production successor p =
F->F-F+FF-F-F+
 We can rewrite the phrase n times and tell out turtle to walk.
Now let’s 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).
FASS curves generated from
edge-rewriting L-systems
 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)
 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
3-D
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
Axes and branches
are ordered as order
0 If they originated
At the base and you
Can guess the rest
Let’s 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
Bracketed system
Examples of bracketed system
Note: The system for adding
Leaves to this bush is
Biologically whack
Stochastic L-Systems
Since all plants don’t look the same we
will add in some randomization.
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.
Removing
P2 makes
Permanent
signal
Plants
Really
Use
Signals!
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.
Now for the real stuff…Let’s 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
 Let’s 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).
A glimpse at the models
 http://algorithmicbotany.org/vmmdeluxe/QT/Greenash/apexview.qt
 http://algorithmicbotany.org/vmm-deluxe/QT/Bluebell/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
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)
Partial L-System
Examples of cool things in Lsystem Schemata
Examples of cool things in LSystem Schemata
Examples of cool things in L-System Schemata
Plants actually use signals and feedback loops a lot
(WUS acts on SAM)!
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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<m or the signal is slower than growth!
COMPLETE MODELS…MUAHAHA
 These are good enough to make images
 We can tell the model when to make branches using subapical
growth
 Plants actually grow like this!
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)
I still like flowers!
 In sympodial branching the apex produces a flower bud (which
cannot branch further) and two new lateral apices (cyme
florescence).
I hope you aren’t allergic to pollen
 In polypodial branching, the apex makes three active apices, and
at some point they change into buds (panicle inflorescence).
Leaf model created trying to represent
known biology (auxin), not bad right? ->
But I want more!
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Modeling exogenous effects are improving
http://algorithmicbotany.org/vmm-deluxe/QT/OpenLsys/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 isn’t enough to maintain
Leaves and branch (self-thinning)
--->
Other models
 Large trees don’t 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.
By changing values for the number of attraction
points, the kill distance, influence distance, and
the distribution of attraction points…
Resource Acquisition Model
 Colasanti and Hunt wanted to see if their
model could produce properties on different
levels:
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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
Wait…what’s 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
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
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 system…think about L-Systems and how cool
nature is
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