Title: Model Reduction for Parameter Estimation
1Model Reduction for Parameter Estimation
- Eric Mjolsness
- Scientific Inference Systems Laboratory (SISL)
- University of California, Irvine
- www.ics.uci/edu/emj
- and Caltech Biological Network Modeling Center
(BNMC) - in collaboration with Rebecca Castaño, Dasha
Chudova, Michael Duff, Victoria Gor, Henrik
Jönsson, Tobias Mann, George Marnellos, Elliot
Meyerowitz, John Reinitz, Bruce Shapiro, David
Sharp, Padhraic Smyth, Yuanfeng Wang, Barbara
Wold, Guy Yosiphon, Li Zhang - Parameter Estimation In Systems Biology (PESB)
- Pascal Workshop, Manchester, UK
- March 28, 2007
2Topics
- A long-running thread in parameter estimation
- Biological applications
- transcriptional regulation
- development
- Perspectives
- a near-universal bio. modeling language and
semantics - and its implications for
- parameter estimation and model reduction
3Transcriptional Gene Regulation Networks
- Gene Regulation Network MSR91 model
E.g. Drosophila A-P axis
Drosophila gap gene expression patterns. Reinitz,
Mjolsness, Sharp, Journal Experimental Zoology
271(47-56) 1995. Fitting method demonstrated in
Mittenthal and Baskin, The Principles of
Organization of Organisms, Addison Wesley 1992.
Mjolsness et al. J. Theor. Biol. 152 429-453,
1991
4GRN Parameter Optimization
- Simulated Annealing 1990, 92
- Lam/Delosme SA for real-valued params
- Gap genes JEZ 271(47-56) 1995
- 33 real-valued parameters
- Genetic Algorithm
- Distributed over islands with migration, for
diversity - SA, GA compared in G. Marnellos thesis 1997
- GA won on evolution (life history) problems
- SA won on development problems
- Other apps to GRNs and signaling Gor, Zhang
- Then many others. Recently
- Kozlov BGRS 2006 differential evolution
- Tomlin 2006 Adjoint method BP/cont. time
5GRN ANN Equations 91
- Model statement and its derivation from stat
mech - Mjolsness Sharp and Reinitz, J. Theor. Biol.
152 429-453, 1991 -
Key properties (1) additivity, (2) saturation
above and below, (3) monotonicity.
6Model Reduction Example Gene Regulation
NetworkDerived from Stat Mech
J. Theor. Biol. 152 429-453, 1991
- MSR91 equations are no longer just
phenomenological.
J.Bioinformatics Comp. Biology, in press 2007
7Dynamical Model Reduction via Clustering
- Core/Halo Models
- From Coexpression to Coregulation NIPS 1999
p.928-34 - Identifiability by Gibbs sampling
- Duff et al., ICSB 2005
- Functional Mixture Models Chudova et al. NIPS
2003
8Core/Leaf Model Inference
- 3-node oscillator leaves
- Modeled by SDE
- topologies
- Identifiability
- x25 time points identifiable
- x10 points not identifiable
- x10 points x2 genotypes identifiable (ranked
3) - Duff et al. ICSB2005
9SDE Advantages
- Intermediate cost for stochastic simulation
- Relationship to stochastic optimization
- Derivation from Fokker-Planck equation
- Eg. for GRN, HCA JBCB in press 2007
10Hierarchical Cooperative Activation Alternative
diagram notations
- Bio-like
- Machine learning
11Hierarchical Cooperative Activation Model (HCA)
In Computational Methods in Molecular Biology,
eds. J. M. Bower and H. Bolouri, MIT Press 2001
12How to model transcriptional regulation?
E.g. Drosophila D-V axis
- Robert P. Zinzen, Kate Senger, Mike Levine, and
Dmitri Papatsenko. Current Biology 16, 18, July
11, 2006
13Hard vs. Soft Logic
Hierarchical Cooperative Activation (HCA)
Zinzen et al. modification
Experiment Yuanfeng Wang, UCI Physics
14HCA- Z and ANN-like Equations
A model reduction
- Assume many binding sites per module
- Assume extreme (usually low) occupancy per site
where
15GRSN Gene Regulation Signal Transduction
Network
Marnellos, Mjolsness, Shapiro
L
Drosophila neurogenesis Marnellos, Mjolsness PSB
98 Xenopus ciliated cells PSB 00
cell
ligands
nucleus
Arabidopsis SAM Gor, Mjolsness,Meyerowitz, NASA
Evolvable Hardware 99
receptors
T
transcriptional regulation targets
16Arabidopsis Shoot Apical Meristem (SAM)
17WUS
Fletcher et al., Science v. 283, 1999
Brand et. al., Science 289, 617-619, (2000)
18SAM growth imageryH2B cell nuclei
V. Reddy, Caltech
19CLV3/WUS networks
V. Agrawal, B. Shapiro, Caltech
20CLV/WUS model behavior
Activation domains in Cellerator model WUS
(yellow), CLV3I1 (green), CLV3 (blue and purple),
CLV1 (red and purple).
B. Shapiro, JPL/Caltech
21CLV/WUS Parameter Optimization by SA
14 parameters
Courtesy H. Jönsson 2007 cf. ICSB 2006
22Biological scale hierarchies
Perspective
mutant
- Biology, networks, models
Noun and verb hierarchies
wild type
23Dynamical Grammar Aims
- Biology Model complex systems
- developmental biology (fly embryo, plant
shoot/root) - molecular complexes
- multiple-scale, heterogeneous, variable-structure
systems - Mathematics Capture, unify, extend techniques
- Generalized reactions cover all processes
- Operator algebra, perturbation theory,
Annals of Math. and A. I., 47(3-4), January 2007
24Elementary Reactions
- A? B C with rate kf
- B C? A with rate kr
- Effective conservation laws
- E.g. NA NB, NA NC
25Elementary Processes
- A(x) ? B(y) C(z) with rf (x, y, z)
- B(y) C(z) ? A(x) with rr (y, z, x)
- Examples
- Chemical reaction networks w/o params
- .
- XXX from paper
- Effective conservation laws
- E.g. ? NA(x) dx ? NB(y) dy ,
- ? NA(x) dx ? NC(z) dz
26Elementary process models
- Composition is by independent parallelism
- Create elementary processes from yet more
elementary Basis operators - Term creation/annihilation operators for each
parm value, -
- Obeying Heisenberg algebra
-
- Yet classical, not quantum, probabilities
27SPG Modeling Language Semantics Semantic map Y
G?H from Grammar to Stochastic Process
- Commutative diagrams for composition operations
- Translation of a Rule
G
G
?
?
?
S
H, dp/dt
H, dp/dt
28Time Ordered Product Expansion (TOPE)
- Time Ordered Product Expansion (TOPE) formula
- H0 the easy part (if only recursively)
- Feynman diagrams result (QFT Perturbation
theory, Wicks theorem) - Gillespie stochastic simulation algorithm
- H0 diag( 1 H) H1 H
- Mixed (heterogeneous) ODE/SSA algorithm (novel)
- Implemented in Plenum (Yosiphon)
Annals of Math. and A. I., 47(3-4), January 2007
29Model Reduction for Dynamical Systems
- Diagram
- Objectives
- Thus, parameter estimation can aid model
reduction - Uses of diagram
UCI ICS TR 05-09
30Composition vs. Specializationin a Lattice of
Models
- Orthogonal kinds of model reduction/expansion
(PartOfInA, IsA) - Commutative diagram for model lattice
- Specialization eg. discretized (DBN) vs.
continuous (ODE) vs. quantized (stochastic) vbls,
time, space - heterogeneous dynamics - Initialize param search in specialized model
- high-level vision app NIPS 1990
- Thus, model reduction can aid parameter estimation
31A Parameter Estimation Future
- Parameter estimation ? model reduction
- Multiscale, heterogeneous, variable-structure,
models all incorporated in a lattice - Common (operator algebra) semantics
- Perpetual data assimilation
- Continual influx of data
- Perpetual fitting to an expanding lattice of
models - Specialize to the limit of identifiability
- Model analyses to explain the hits
32Conclusions
- Model reductions for transcriptional regulation
- GRN91, HCA
- Model reduction for large-scale data
- Core/Halo, Functional Mixture, models
- Common framework generalized reactions
- Dynamical grammars ? operator algebra
- Parameter estimation ? model reduction
- Mutually enhancing interaction
33- For further information
- www.ics.uci.edu/emj
- www.computableplant.org
- Funding US National Science Foundation FIBR
program, - NIH BISTI program
- Invitation
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