Modeling Plant Form Is plant form an emergent
- Slides: 37
Modeling Plant Form Is plant form an emergent property of simple module systems?
L-Systems ü L-systems are basically a way to rewrite something following a set of rules ü For instance: you have two letters a and b. ü The rules for rewriting are a->ab and b->a ü If we start with a b and start rewriting we get:
The Turtle interpretation of strings ü 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. ü Now, d = step size and ƒ =angle increment ü So we can tell the turtle where to go if we give it directions. We will use the following symbols: ü F = Move forward by one step length d ü + = Turn counterclockwise by angle ƒ ü - = Turn clockwise by angle ƒ
Let’s put our turtle to work ü Given the axiom w = F-F-F-F and the production successor p = F>F-F+FF-F-F+ ü We can rewrite the phrase n times and tell out turtle to walk.
Now let’s make it a little bit more complex ü Edge rewriting productions substitute figures for polygon edges ü Fl and Fr represent the turtle obeying the “move forward” command, but now Fl and Fr edges by lines forming left or right turns. ü These curves can be space-filling and self avoiding (FASS).
FASS curves generated from edge -rewriting L-systems
ü Node rewriting substitutes polygons for nodes on the curve ü Now we need more things: Entry and exit points (Pa and Qa) and an entry vector and an exit vector (pa and qa)
ü You can also consider an array of m x m square tiles. ü Each m x m contains a small box inside of it called a frame. Each frame bounds an open self-avoiding polygon. ü Now when we connect many tiles we will get a macrotile
3 -D
Axial Trees ü All of the previous examples were all a single line, but trees are not! ü An axial tree starts from a base node ü At each of its nodes there is at most one outgoing straight segment ü All other edges are lateral segments ü A terminal segment is an apex ü An axis must: ü The first segment in the sequence originates from the base or a lateral segment at a node ü Each subsequent segment is straight ü The last segment is not followed by any straight segment ü So each axis is a mini axial tree! ü 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 ü We need to have a rewriting mechanism that acts on axial trees ü 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 ü Since all plants don’t look the same we will add in some randomization.
Context-sensitive L-Systems ü 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 P 2 makes Permanent signal Plants Really Use Signals!
Parametric L-Systems ü Will help us show time, angles, and irrational line lengths (if d = 1, you cannot express sqrt(2). ü Is easier than trying to add stuff to non-parametric model.
Now for the real stuff…Let’s try to simulate herbaceous plants ü Emphasis on space-time relation between plant parts ü So there can be flowers and buds on the tree at the same time ü Inherent capability of growth simulation ü Our model is good for growing and we can simulate plants at different times and watch how they grow ü Let’s only do herbaceous plants because: ü The model assumes that the plant controls its own development (endogenous interaction). ü Herbaceous plants have a lot of directions from their parents (lineage interaction). ü Woody plants are much more sensitive to their environment, competition among branches and trees, and accidents (exogenous interaction).
A glimpse at the models ü http: //algorithmicbotany. org/vmmdeluxe/QT/Greenash/apexview. qt ü http: //algorithmicbotany. org/vmm-deluxe/QT/Bluebell/field. qt ü We can use confocal microscopes to get a real idea of how plants develop and then write a computer model that fits the behavior ü We can also use empirical data on plant development ü Other models try to use known mechanisms to explain the emergence of plant forms
Three Main Type of Models ü Partial L-Systems: Your basic model that is supposed to show us the possible structures of plants ü L-System Schemata: Topology and temporal aspects of plants expressed, could help us understand mechanisms ü 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)! ü ü ü This says that the apex (a) produces internodes (I) and leaves (L) [p 2]. The time in between growth is m [p 1]. After delay (d) a signal (s) [p 3 an p 4]. The signal is sent down the main axis with delay (u) steps per internode (I) [p 5 and p 7]. [p 6] 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) [p 8 and p 9]. Note: u<m or the signal is slower than growth!
COMPLETE MODELS…MUAHAHA ü These are good enough to make images ü We can tell the model when to make branches using subapical growth ü Plants actually grow like this!
I like flowers! ü There a few different types of flowers we can make: ü Monopoidal branching - lateral buds make flowers and can not make any more branches (raceme inflorescence)
I still like flowers! ü 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 ü 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! ü ü ü Modeling exogenous effects are improving http: //algorithmicbotany. org/vmm-deluxe/QT/Open. Lsys/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 ü 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 ü Colasanti and Hunt wanted to see if their model could produce properties on different levels: ü ü ü 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 ü They used two binary trees ü One for roots and one for shoots
Wait…what’s a binary tree ü Modules linked together. ü Each module is linked to one parent module and potentially two offspring modules ü A module “knows” the identity and state of its parent and offspring modules, but not the state of the whole plant ü Base module has no parent and end module has no offspring ü Spatial area made into cells, these cells can have resource units (light units for shoots/mineral nutrient units for roots) ü The module can transport the units to base module ü New growth requires a light unit and a mineral unit ü They mutated the plant by giving it a competitive advantage for resources at the expense of extra energy
Their Results ü Success. ü S-Shaped growth curve ü Self-thinning ü Plasticity in roots and shoots of modified plants ü When resources are high, modified plants did well ü When resources are low, regular plants did better ü Could always make it better
Conclusion ü These models show that a very simple module behavior can account for many aspects of trees and herbaceous plants ü By comparing these models to nature, we can learn more about the actual mechanisms in nature ü Nature is math-y and pretty (or is math pretty and nature-y? ) ü 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 ü S. Wolfram, A New Kind of Science. Chapter 3, 6, 8. 5, 8. 6, 8. 7 ü P. Prusinkiewicz and A. Lindenmayer, The Algorithmic Beauty of Plants ü R. L. Colassanti and R. Hunt, Resource Dynamics and Plant Growth: A Self-Assembling Model for Individuals ü Runions et al. , Modeling Trees with a Space Colonization Algorithm ü Runions et al. , Modeling and visualization of leaf venation patterns ü O. Prusinkiewicz and Anne-Gaëlle Rolland-Lagan, Modeling plant morphogensis ü P. Prusinkiewicz, Simulation Modeling of Plants and Plant Ecosystems
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