Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
MGS « encore un Modèle Général de Simula3on »
and Spa&al Compu&ng
1
Jean-‐Louis Giavi:o, Antoine Spicher, Olivier Michel
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
Chronology and warnings
• MGS: start September 2000 • Spa3al Compu3ng
– Dagsthul 2006 – SC workshop since 2008
• The tutorial is – “unconven3onal programming” oriented – examples and applica3on based: unconven-onal algorithmic, biology, abstract morphogenesis
– no complexity and few theory
2
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
A topological manifesto Spa3al compu3ng proposes to celebrate
corporeality of data rather than trying to deny it. Simon Greenworld (MIT medialab)
• The logical approach in computer science
computa3on = deduc3on (the Curry-‐Howard isomorphism)
• Other paradigms can be fruiZull : topology
computa&on = moving in some space • Try to perceive space (and 3me) in programs
(rather than logical opera3ons) purposes: technical, heuris-c, didac-cal
3
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
Material
• The MGS home page (new home page for the end of the year) h:p://mgs.spa3al-‐compu3ng.org
• MGS download (new home page for the end of the year) h:p://www.spa3al-‐compu3ng.org/mgs
• This MGS tutorial h:p://www.spa3al-‐compu3ng.org/mgs:tutorial
• MGS publica3on h:p://mgs.spa3al-‐compu3ng.org/PUBLICATIONS (old) h:p://repmus.ircam.fr/giavi:o/publica3ons h:p://www.lacl.fr/~michel/doku.php?id=research:start h:p://www.lacl.fr/~michel/www/bib h:p://www.lacl.fr/~aspicher/publica3ons.html
• The Spa3al Compu3ng home page
h:p://www.spa3al-‐compu3ng.org
4
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
A complete schedule
5
A. Mo3va3on & Applica3on domains 1. (DS)2 and morphogenesis 2. Interac3on-‐Based Compu3ng 3. Spa3al Compu3ng
B. MGS …
C. Applica3ons …
D. Implementa3on …
E. Current Fron3ers …
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
Lecture I (Tuesday, September 4)
A. Mo3va3on & Applica3on domains 1. Modelling Morphogenesis 2. (DS)2: dynamical systems with a dynamical structure
B. MGS 1. Collec3on 2. Transforma3on 3. A Func3onal Language
C. Applica3ons 1. Algorithmic Examples 2. Monoidal Collec3ons 3. GBF : Cellular and Laice Gaz Automata 4. Delaunay : the growth of a meristem
6
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
Lecture II (Wednesday, September 5)
A. Mo3va3on & Applica3on domains 1. Spa3al Compu3ng 2. Interac3on-‐Based Compu3ng
B. MGS 1. Abstract Combinatorial Complex 2. Formaliza3on
C. Applica3ons 1. Remeshing 2. Growing a « T » 3. Self-‐Assembly of Musical Spaces 4. Growing an ontology with the Li:le Riding Hood 5. Growing an analogy with Paul Ricœur
7
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
Lecture III (Thursday, September 6)
B. MGS 1. Formalisa3on 2. Stochas3c strategies 3. Discrete Differen3al Operators
C. Applica3ons 1. A generic diffusion operator 2. Four models for the Paris iGEM’07 project
D. Implementa3on 1. Incremental HOAS implementa3on of a DSL 2. Beyond algebraic data type : a Generic Advanced API for Containers 3. Generic pa:ern matching
E. Current Fron3ers From Global to Local , Mul3-‐level and par3al processes, Hybrid Modeling, Time
8
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org 9
Modeling and Simula&on of Morphogenesis
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
Environment characterized by its effects on the system
System described by a state (determined by observation)
10
Modelling morphogenesis: the approach of A. Turing
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
statet-‐1 statet statet+1 N
statet-‐dt statet statet+dt R
Specifying a dynamical system (for simula3on)
11
statet
Specifica&on of • structure of state • structure of &me • evolu&on func&on
H H
H*
∫H(t)dt
state
time
evolution
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org 12
Morphogenesis as a Dynamical System
Modelling a dynamical system • state, including space (e.g. fields) • time • evolution function
C : continuous, D: discrete
PDE Coupled ODE
Iteration of functions
Cellular automata …
state C C C D …
time C C D D …
space C D D D …
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org 13
Modelling morphogenesis: the approach of A. Turing
• Uniform maDer, con&nuous-‐oriented system descrip&on è SPACE based One choice is to ignore cells completely, e.g., Physiome models 3ssues as con3nua with bulk mechanical proper3es and detailed molecular reac3on networks, which is computa3onally efficient for describing dense 3ssues and non-‐cellular materials like bone, extracellular matrix , fluids, and diffusing chemicals, but not for situa-ons where cells reorganize or migrate.
versus
• Cell-‐oriented discrete system descrip&on è ENTITY based Mul3-‐cell simula3ons are useful to interpolate between single-‐cell and con3nuum-‐3ssue extremes because cells provide a natural level of abstrac3on for simula3on of 3ssues, organs and organisms. Trea3ng cells phenomenologically reduces the millions of interac3ons of gene products to several behaviors: most cells can move, divide, die, differen3ate, change shape, exert forces, secrete and absorb chemicals and electrical charges, and change their distribu3on of surface proper3es.
(CompuCell3D manual)
Aggregate-‐ vs. En&ty-‐based models
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org 14
Modeling morphogenesis: paDerning vs. development
4 Spicher, Michel & Giavitto
data) and the content (chemical data) are clearly decoupled, making these modelsusually fall into two categories2:
• formalisms that focus on pattern formation in an initially homogeneous but staticsubstrate, and
• generative formalisms that specify the creation and the evolution of a dynamicshape, irrespective of the processes that may take place within the shape.
Examples of the first category are given by reaction-diffusion systems, activator-inhibitor models, or random Boolean networks. They rely on various model of dy-namical systems (Table 1). Examples of the second category include Lindenmayersystems (L-systems) [36], membrane computing [46], graph grammars [49], self-assembly systems, and others (Table 2).
Table 1 Formalisms that can be used to specify structured dynamical systems according to (a) theunderlying space in which the patterning process takes place, (b) a continuous or discrete repre-sentation of time, and (c) the state variables of the system’s components. “Numerical Solutions”refer to explicit numerical solutions of partial differential equations (PDE) or systems of coupledordinary differential equations (ODE).
C: continuousD: discrete PDE Coupled ODE Numerical
SolutionsCellular
Automata(a) Space C D D D(b) Time C C D D(c) States C C C D
Table 2 Generative formalisms that can be used to specify the evolution of a shape, accordingto the topology connecting the components of the shape. In a multiset, all elements are consideredconnected to each other. In a sequence, elements are ordered linearly: this case includes lists and ex-tends to tree-like structures (lists of lists). Uniform structures involve a regular neighborhood, e.g.,a rectangular lattice where each element has exactly four neighbors. Group-Based Fields (GBF,Section 2.1.4) are a powerful tool relying on mathematical groups to represent such structures.The first four cases describe spatial structures that can be accurately depicted by a graph. Beyondgraphs, nD combinatorial structures are used to define arbitrary connections between componentsof various dimensions. The MGS language (next section) can handle all these types of shapes.
Topology: multiset sequence uniform arbitrary graph nD combinatorialstructures
Formalism: membranesystem
L-systems GBF map L-systems,graph-grammars
MGS
2 This distinction is somewhat contrived. For instance, cellular automata have been devised tostudy self-reproduction of distinct entities, but these entities are represented by specific patterns ina predefined medium.
generative formalism
C : continuous, D: discrete
PDE Coupled ODE Iteration of functions
Cellular automata …
state C C C D …
time C C D D …
space C D D D …
(vs. dynamical systems)
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org 15
Modeling morphogenesis: pa:erning in a predefined medium
Alexander Gavrilovitch Gurwitsch (1875-‐1954) Morphogene3c Fields
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org 16
© H. Meinhardt, c.1970 Fixed shape, diffusion and reac3on
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org 17
Modeling morphogenesis: pa:erning in a predefined medium
Compatible with • the notion of morphogenetic field • cell fate But • there is evidence for
feedback loops between the shape and the process inhabiting the shape
from E. Haenkel (cited by C. Goodman-‐Strauss): example of a nega3ve curvature surface. Curvature can be controlled while the surface is growing along a ‘front’
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org 18
The medium/process problem
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
• (px, py) • (vx, vy) • (px’, py’) • (vx’, vy’)
19
The medium/process problem
a falling ball
at any -me a state is a posi-on and a speed
A dynamical system (DS)
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org 20
The medium/process problem
a falling ball a developing embryo
the structure of the state (chemical and mechanical state of each cell)
is changing in -me A dynamical system with a dynamical structure
(DS)2
• (px’, py’) • (vx’, vy’)
at any -me a state is a posi-on and a speed
A dynamical system (DS)
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org 21
The interplay between state and form
a developing embryo
the state as well as the structure of the state is changing in 3me (chemical and mechanical state of each cell as well as the arrangement of the cells)
A dynamical system with a dynamical structure (DS)2
Dynamics ON form Dynamics OF form F
orm
State
“Strange loop”
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
Waddington : epigene3c landscape
Link Li & Mhairi Towler, 2012
Conrad Waddington (1905-‐1975)
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org 23
Such cycles are not new in biology
Tibor Ganti’s CHEMOTON (1971)
metabolic cycle
template molecules
membranogenic molecule
… a “chemical automaton”… (also called fluid automata)
Varela’s AUTOPOIESIS (1974)
“an autopoietic unity is able to self-generate owing to a reaction network taking place within its
own boundary” (the reaction network makes all components of the unit, including the boundary)
M. Eigen & P. Schuster: hypercycle, W. Fontana & L. Buss: organization …
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
• Complex systems • Whose structure evolves over the 3me
– The structure constrains the behavior of the system that modifies the structure … – The phase space cannot be defined a priori – The evolu3on func3on cannot be specified globally
• Examples – Biology
• Molecular biology • Developmental biology
– Physics • Soz ma:er mechanics,
mul3-‐scale systems • General rela3vity
– Urbanism City growth,
traffic control, … – Computer science
Internet, sensor network, reconfigurable nanobots, knowledge network, …
DS² : Dynamical System with a Dynamical Structure
© David S. Goodsell 1999
© Tecplot
© L. Sanders - EUROSIM
© C. Harrison - Clusterball project 24
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
What has changed since Turing’s &me
25
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org 26
What has changed since Turing’s &me
P. Prusinkiewicz, c.2003 Diffusion and reac3on in a deformable surface (E. Coen’s expanding canvas metaphor). Spring-‐mass system. No topological change.
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
Sozware as Science ?
• Intelligibility The en-re process should be accessible for analysis into a finite, not very large number of stages, each stage being represented as a monotonic func-on of some definite ini-al condi-ons and a single variable such as -me, or distance, etc. (Gurwitsch, 1944) à compress behavior or shape in few rules
• Simula3on is only a first step: models must enable reasoning à stay close to mathema3cal formalism
27
M.-P. C
ani & F. B
ertails
A program is a formal object (and some form of reasoning on it is possible) but a 106 lines of codes is not an explanation !
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
(…) An invagina3on of a germ layer may be explained on a basis of a pressure difference between the two surfaces (or sides) or by cell movements, and so forth. This can be considered as an 'explana3on' un3l we ask about the origin of pressure differences, or the mechanisms involved in cell movement, etc. However, ques3ons of this kind become trivial when a larger process, rather than its individual components becomes the main problem. Suppose for a moment that each element in the succession A, a, C… can be explained separately, e.g. A as a swelling, a as a chemical reac3on, etc. Interes3ng as they may be, these explana3ons are of subordinate importance when related to the main ques3on: Why indeed is a regular (emphasis added) succession of these obviously quite different processes taking place at all? Most biological problems are of this kind and all of embryogenesis is just such a single problem. Here we require a peculiar or, maybe, original explanatory principle... A process may become accessible to explana3on only insofar as one can succeed in subs3tu3ng [understanding of] a purely phenomenological mul3plicity and diversity of events [for understanding] of a less diverse and less arbitrarily created picture correctly reflec3ng reality. The main aim of such a construc3on would be as follows. The en3re process should be accessible for analysis into a finite, not very large number of stages, each stage being represented as a monotonic func3on of some definite ini3al condi3ons and a single variable such as 3me, or distance, etc. If this cannot be realized, we consider a given set of events as scien3fically inaccessible. On the other hand, even a par3al success of such an enterprise is an obvious step forward."
(Gurwitsch, 1944) cited by Beloussov in “Life of Alexander G. Gurwitsch and his relevant contribu3on to the theory of morphogene3c fields”, Int. J. Dev. Biol. 41, 771-‐779 (1997)
28
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
COMBINING PATTERNING AND DEVELOPMENT
29
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
Pa:erning vs. Growth
• Dynamics ON form pa:erning, mo3f forma3on in a predefined form – Diffusion, reac3on-‐diffusion, transport (con3nuous models) – cellular automata (discrete models)
• Dynamics OF form growth, deforma3on of a shape – Deforma3on of elas3c bodies (con3nuous models) – Lindenmayer systems (discrete models)
30
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
The interplay between state and form a developing embryo
the state as well as the structure of the state is changing in 3me
(chemical and mechanical state of each cell as
well as the arrangement of the cells)
A dynamical system with a dynamical structure (DS)2
Dynamics ON form Dynamics OF form F
orm
State
31
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
(DS)2 versus DS
• Processes in the form are topologically (geometrically) meaningful e.g. growth rate
• Topological (geometric) informa3on is meaningful e.g. domain of diffusion, informa3on transfert
32
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
Topological informa3on is meaningful
A basic example [T. Gross, Adapta3ve Networks]: – Infec3ous disease spreading across a network – Node are either suscep-ble or infected
– Suscep3ble nodes linked to an infected node become infected with probability α
– Infected nodes recover with probability β
33
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
Topological informa3on is meaningful
The disease persists in the network depending: – of the probability α
– of the probability β
– of the mean excess degree d the average number of links that one finds connected to a node that is reached by following a random link: a global topological informa3on
• d > threshold: every node will be infected once in a while (in a large network)
• By watching the behavior of a single node (local observa-on) for a sufficiently long period of 3me, we can es3mate if the mean excess degree (global topological informa-on) exceeds the threshold
• Simple mechanism can put the system at the cri-cal point! (no special reason to be in a generic state)
34
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
The interplay between form and topology is meaningful
• A suscep3ble node is allowed to cut a link with an infected node with a certain probability decreasing when the delay between its own infec3ons increase
• Now there is a feedback loop between topologies and states of the network the dynamics of the prevalence of the disease depends on the network topology, but the evolu3on of the network topology also depends on the prevalence of the disease
Simple mechanism can put the system at the cri@cal point! no special reason to be in a generic state
e.g. synchroniza3on of growth rate in flat leaf
(and it can be worst if there is no 3mescale separa3on)
35
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
OK: the coupling is important. What can we do?
1. “Dynamics ON form” toward DS2 – parameterize (control) the shape by some quan33es
(e.g. curvature for a manifold or adjacency matrix for a graph)
– link these quan33es with processes in the shape
• growth depending on concentra3on
36
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
Differen&al geometry is not enough
• Encoding a shape into con3nuous parameters is difficult
• It does not handle topological changes very well
apical elonga3on depends of 3me
• Ok, some3mes they are some tricks (e.g. level set) but they are very smart tricks
37
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
OK: the coupling is important. What can we do?
1. “Dynamics ON form” toward DS2 – parameterize (control) the shape by some quan33es
(e.g. curvature for a manifold or adjacency matrix for a graph)
– link these quan33es with processes in the shape
• growth depending on concentra3on
2. “Dynamics OF form” toward DS2 – enhance the form by parameters
– put a dynamic on these parameters example: module in Lindenmayer systems
38
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
Lindenmayer systems
39
The growth of Anabaena Catenula • two cell types : vegeta3ve and heterocyst • asymetric division : one is smaller than the other • polarized cell (lez/right orienta3on)
A a B b
a -> aB A -> bA b -> aB B -> bA a a B a B b A a B b A a B b A a B b A a B b A a B b A a B b A ...
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org 40
A good example of PaDerning+Development in a declara&ve formalism: Lindenmayer systems with modules
• The structure of a tree can be coded by a string of parenthe3sed symbols
• A symbol is an elementary part of the plant
• The symbol between [ and ] represents a sub-‐tree
• Addi3onal conven3ons are used to represent main axis, orienta3on, depth, etc.
• A rule s0 → s1 s2 s3 …
represents the evolu3on of s0 P. P
rusi
nkie
wic
z
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org 41
Diffusion and reac3on in a linear growing medium M. Hammel and P. Prusinkiewicz (1996)
The following rules state that a differen3ated cell (heterocyst) returns to a vegeta3ve state if the concentra3on of the ac3vator is too low. In addi3on, if the cell is large enough, it con3nues to grow.
e / (D(e) & (e.a < thr) | (e.x >= shorter*gr)) => {type ="C", a=e.a/gr, h=e.h/gr, x=e.x*gr, p=e.p};
The following rule specifies when a cell with a lez polarity divides. Only vegeta3ve cells can divide (hence the predicate C in the rule guard) and it must be large enough. The volume of the two daughter cells remains the same, so there is no varia3on in the concentra3on.
e / (C(e) & (e.x >= lm) & (e.p == L))
=> {type="C", a=e.a, h=e.h, x=e.x*shorter, p=L},
{type="C", a=e.a, h=e.h, x=e.x*longer, p=R};
Linear cell structure
morphogene concentration
time
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org 42
• Rewri3ng system – Used to formalize equa3onnal reasoning – A genera3ve device (grammar) – Replace a sub-‐part of an en3ty by an other – Set of rewri3ng rules α à β
• α: pa:ern specifying a sub-‐part • β: expression evalua3ng a new sub-‐part
• Example: arithme3c expressions simplifica3on
Rewri&ng systems (and abstract transi3on systems)
x
+
0
x
y
+
x x
+
y
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
A non-‐standard interpreta3on
43
e1 + e2 → … e1 can be a cell and e2 a signal e1 and e2 can be interac3ng cell
+ is the possibility of interac-on between en33es (or some other rela3onships)
→ is the passing of 3me, a local evolu3on, a transi3on, the concre3za3on of the interac3on
Examples: if e is a cell and i a biochemical signal
e + i → e’ growth (evolu3on of e on signal i) e + i → e+i’ quorum sensing e + i → e’ + e’’ division e + i → . apoptose
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org 44
Rewri3ng systems in simula3on
Complex systems Rewri&ng techniques
Modelling Specifica3on State (space) Data structure
hierarchical and tree organiza3ons arbitrary complex organiza-ons
formal trees (or terms) ?
Evolu&on func&on Set of rules interac@ons à evolu3on
local evolu&on laws α : pa:ern à β : expression rewri3ng rules
Simula3on Applica3on Trajectories
Time management Deriva3ons Rule applica3on strategy
discrete, event-‐based, synchronous vs. asynchronous
maximal parallel, sequen3al, determinis3c, stochas3c
ßà
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org 45
• local evolu-on rules mandatory when you cannot express a global func3on/rela3on because the domain of the func3on/rela3on is changing in 3me
• interac3on based approch the l.h.s. of a rule specifies a set of elements in interac-on, the r.h.s. the result of the interac3on
• the phase space is well defined but not well known a genera3ve process enumerates the elements but membership-‐test can be very hard
• various kind of -me evolu-on (for the same set of rules)
• demonstra-on by induc-on on the rules or on the deriva3on (e.g. growth func3on in L system)
Proper3es
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org 46
How to extend to arbitrary spa3al structure?
• Anabaena was « easy » because of the linear uniform structure • How to handle the complex spa3al structure of a cell?
David S. Goodsell
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org 47
Three challenges and some tools
compilation, engineered emergence
validation, analysis, collective
properties
limits, non-standard analysis
approximation, numerical resolution,
partitionning
DNA
cellular machinery
Cellular machinery
DNA
global local
discrete
software/data hardware/programme
population
individual Molecules, Compartments,
Cells
concentration, time, fields
continuous
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
Acknowledgements MGS: Antoine Spicher, Olivier Michel, Julien Cohen S&S Bio : Hanna Klaudel, Franck Delaplace, Hugues Berry, Przemek Prusinkiewicz, Annick Lesne… Spa3al Compu3ng: Jacob Beal, Fréderic Gruau, René Doursat…
Examples: Pierre Barbier de Reuille, Christophe Godin, Samuel Bo:ani, the Paris iGEM’07 team…
Some figures are borowed from Olivier Michel, Antoine Spicher, Pierre Barbier de Reuille, Franck Delaplace, Hugues Berry (INRIA), the iGEM Paris 2007 and many others.
48
MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org
• Antoine Spicher • Olivier Michel • PhD and other students J. Cohen, P. Barbier de Reuille, E. Delsinne, V. Larue, F. Le3erce, B. Calvez, F. Thonerieux, D. Boussié and the others... • Collabora&ons • A. Lesne (IHES, stochas3c simula3on) • P. Prusinkiewicz (Calgary, declara3ve modeling) • P. Barbier de Reuille (meristeme model) • C. Godin (CIRAD, biological modeling) • H. Berry (LRI, stochas3c simula3on) • G. Malcolm (Liverpool, rewri3ng) • J.-‐P. Banâtre (IRISA, programming) • F. Delaplace (IBISC, synthe3c biology) • P. Di:rich (Jena, chemical organiza3on) • E. Goubault (CEA, topological formaliza3on) • F. Gruau (U. PXI, language and hardware) • P. Liehnard (Poi3er, CAD, Gmap and quasi-‐manifold)
49
Thanks http://mgs.spatial-computing.org