Signal Transduction, Cellerator, and The Computable Plant Bruce E Shapiro, PhD bshapirocaltech'edu h

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Signal Transduction, Cellerator, and The
Computable PlantBruce E Shapiro,
PhDbshapiro_at_caltech.eduhttp//www.bruce-shapiro
.com/cssb
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Overview

• Cellerator
• Chemical Kinetics
• Signal Transduction Networks
• Modular outlook
• Switches, oscillators, cascades, amplifiers, etc.
• Deterministic vs. Stochastic simulations
• Multicellular systems
• Synchrony, pattern formation
• The Computable Plant project
• Model Inference

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Cellerator
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Short Cellerator Demonstration
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Law of Mass Action

• Canonical form of a chemical reaction
• Ri,Pi Reactants, Products
• sPi,sRi Stoichiometry
• k Rate Constant
• Example
• Law of Mass Action The rate of the reaction is
  proportional to the product of the concentrations
  of the reactants.

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Law of Mass Action (2)

• Formal statement (for a single reaction)
• Interpretation of

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Law of Mass Action (3)

• Add rates for multiple reactions

Oregonator
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Cellerator Input for Oregonator
Rate Constants
stnBrO3Br?HBrO2HOBr, k1,
HBrO2Br?2HOBr, k2, BrO3HBrO2?HBrO22Ce,
k3, 2HBrO2 ? BrO3HOBr, k4, Ce?Br,
k5 interpretstn, frozen? BrO3
Hold BrO3 concentration Fixed
Stoichiometry
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Cellerator Output for Oregonator
List of Differential Equations and Variables
Brt-k1BrtBrO3tk5Cet-
k2BrtHBrO2t, Cet-k5Cet2k3BrO
3tHBrO2t, HBrO2tk1BrtBrO3t-
k2BrtHBrO2t k3BrO3tHBrO2t
-k4HBrO2t2, HOBrt2k2BrtHBrO2t
k4HBrO2t2k1BrtBrO3t, Br, Ce, HBrO2,
HOBr
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Cellerator Simulation
s predictTimeCoursestn, frozen?BrO3, timeSpa
n?1500, rates? k1?1.3, k2? 2106,
k3?34, k4?3000., k5?0.02, BrO3t
?.1, initialConditions?HBrO2?.001, Br?.003,Ce?.
05,BrO3?.1 0, 1500, Br ?
InterpolatingFunction0., 1500., ltgt, Ce
? InterpolatingFunction0., 1500., ltgt,
HBrO2 ? InterpolatingFunction0., 1500.,
ltgt, HOBr ? InterpolatingFunction0., 1500.,
ltgt
Input
Output
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Plot Results of Simulation
runPlots, plotVariables ? Br, PlotRange ?
400, 1400, 0, 0.002, TextStyle ?
FontFamily -gt Times, FontSize -gt 24,
PlotLabel ? "Br Concentration"
Optional Input
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Basic Syntax

• Format of rate constants varies for different
  arrows
• Modifiers are optional
• Different rate laws for different arrow/modifier
  combinations
• We will focus on reaction
• Generate differential equation by entering
• interpretnetwork

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Basic Mass Action Reactions
Cellerator Syntax
We will generally omit explicitly writing the
rate constants in the remainder of this
presentation.
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Catalytic Mass Action Reactions
becomes
becomes
becomes
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Cascades
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Michaelis-Menten Kinetics

• Catalytic Reaction
• Mass action
• Steady-state assumption
• where E0 is total catalyst (bound unbound)

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Michaelis-Menten Kinetics (2)

• Solve for where
• Therefore
• where vkE0
• If then hence

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Michaelis-Menten in Cellerator
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Comparison of models
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GTP A molecular switch
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GTP A reaction schema
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GTP Cellerator schema
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GTP Cellerator Simulation
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RASGTP Switch
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Cascades
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MAPK Mitogen Activate Protein Kinase
Heat Shock, Radiation, Chemical, Inflamatory
Stress
Cell Growth and Survival
lab of Jim Woodget, http//kinase.uhnres.utoronto.
ca/
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MAPK Cascade
Reactions in solution (no scaffold)
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MAPK in Solution
Kinase Reactions 1st Stage KKKS?KKK-S KKK-S
?KKKS KKK-S?KKKS 2nd Stage (1st Phosphate
group) KKKKK?KK-KKK KK-KKK?KKKKK KK-KKK?KKK
KK 2nd Stage (2nd Phosphate group) KKKKK?KK
-KKK KK-KKK?KKKKK KK-KKK?KKKKK 3rd
Stage (1st Phosphate group) KKK?K-KK K-KK?K
KK K-KK?KKK 3rd Stage (2nd Phosphate
group) KKK?K-KK K-KK?KKK K-KK?KK
K
Phosphatase Reactions 1st Stage KKKPh1?KKK-Ph1
KKK-Ph1? KKK Ph1 KKK- Ph1? KKK Ph1 2nd
Stage (1st Phosphate group) KKPh2?KK-Ph2 KK-
Ph2? KK Ph2 KK- Ph2? KK Ph2 2nd Stage (2nd
Phosphate group) KKPh2?KK-Ph2 KK- Ph2?
KK Ph2 KK- Ph2? KK Ph2 3rd Stage (1st
Phosphate group) KPh3?K-Ph3 K- Ph3? K
Ph3 K- Ph3? K Ph3 3rd Stage (2nd Phosphate
group) KPh3?K-Ph3 K- Ph3? K Ph3 K-
Ph3? K Ph3
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MAPK Cascade on Scaffold

• Scaffold binding significantly increases the rate
  of phosphorylation
• Scaffold has 3 slots one for each kinase
• Each slot can be in different states
• Slot 1 empty, KKK, or KKK bound
• Slot 2 empty, KK, KK, or KK bound
• Slot 3 empty, K, K, K bound
• Enter/leave scaffold in any order
• KKK and either KK or KK must be bound at same
  time produce KK, etc.
• Number of reactions increases exponentially with
  number of slots

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Effect of Scaffold on Simulations
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Reactions in MAP Kinase Cascade

• Phosphorylation in Solution
• Binding to Scaffold
• Phosphorylation in Scaffold

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Effect of Scaffold on MAPK
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Stochastic Comments

• When the number of molecules is small the
  continuous approach is unrealistic
• Differential equations describe probabilities and
  not concentrations
• At intermediate concentrations the continuous
  approach has some validity but there will still
  be noise due to stochastic effects.
• Langevin Approach

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Direct Stochastic Algorithm

• Gillespie Algorithm (1/3)
• At any given time, determine which reaction is
  going to occur next, and modify numbers of
  molecules accordingly

Gillespie DT (1977) J. Phys. Chem. 81 2340-2361.
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Gillespie Algorithm (2/3)
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Gillespie Algorithm (3/3)

• Let t0
• While tlttmax
• Calculate all the aihiki and a0Saj
• Generate two random numbers r1, r2 on (0, 1)
• The time until the next reaction is
  t(1/a0)ln(1/r1)
• Set t t t
• Reaction Rj occurs at t, where j satisfies
• a1a2aj-1 lt r2a0 ajaj1an
• Update the X1,X2,,Xn to reflect the occurance of
  reaction Rj

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Stochastic MAPK Simulation (1/3)
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Stochastic MAPK Simulation (2/3)
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Stochastic MAPK Simulation (3/3)
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Analysis of Multi-step reactions
Simplify to
Adding steps increases sensitivity
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Analysis of multi-stage reactions

• Consider two stages of a cascade with m and n
  steps
• Steady State

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Analysis of multi-stage reactions

• If XltltKx
• Hill exponent is product of m and n
• E.g., a three-step stage followed by a four-step
  stage behaves like a 12-step stage
• By incorporating negative feedback can produce
  high-gain amplification (see refs).

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Oscillators in Nature

• Where they occur (to name a few)
• Circadian rhythms
• Mitotic oscillations
• Calcium oscillations
• Glycolysis
• cAMP
• Hormone levels
• How they occur feedback
• Both negative positive feedback systems
• Some have feed-forward loops also

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Negative feedback canonical model
a0.25,b10,d1,S5
Hoffmann et al (2002) Science 2981241
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2-species ring oscillator
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3-species ring oscillator
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3-species Ring Oscillator
v10KM Robust oscillations
vKM Damped oscillations
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Repressilator
Constructed in E. coli Elowitz Leibler, Nature
403335 (2000)
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Repressilator Model Simulations
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Cell Division - Canonical Model
Minimal Model of Cell Division
Goldbeter (1991) PNAS USA, 889107
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Cell Division - Canonical Model
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Multi-cellular networks
Intracellular Network e.g., of mass action, etc.
Transport, ligand/receptor interactions, etc
Species xi in cell j
Connection matrix
Set of neighbors of cell j
Diffusion Tensor
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Example - coupled oscillators
Two uncoupled Oscillators
Two Coupled Oscillators, a/w.1
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Example - 105 coupled oscillators
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Example - 105 coupled oscillators
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Coupled nonlinear oscillators
Arbitrarily let species X in CMX model diffuse to
adjacent cells
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Coupled CMX Oscillators

• All oscillating at same frequency
• But different phases
• What happens if you have 105 coupled oscillators
  with random phase shifts?

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105 CMX Oscillators uncoupled
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105 CMX Oscillators uncoupled
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105 CMX Oscillators low coupling
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105 CMX Oscillators higher coupling
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105 CMX Oscillators higher coupling
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105 CMX Oscillators Random Period
Uncoupled motion
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105 CMX Oscillators Random Period
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105 CMX Oscillators Random Period
Uncoupled
Coupled Oscillators
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Pattern Formation
Activator-Inhibitor Models

• Single Diffusing Species
• Self-activating (locally)
• Self-inhibitory (externally)

• Two Diffusing Species
• X Activator
• Y Inhibitor

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Two species pattern formation model
Continuous model
Discrete implementation
See Murray Chapter 14 for detailed analysis
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Single species pattern formation model
Continuous (logistic) model
Discrete implementation
Steady State Equation (vKvMKM1, xA)
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Single species pattern formation model
is a steady state only if all neighbors are at x0
Suppose that there are nx neighbors in state x
and all other neighbors are in state x0, 0nx6
Example case when nx 1 (exactly one neighbor at
x, all others at 0) Question what other
combinations are possible?
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Single Species Model - 105 Cells
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Single Species Model - 105 Cells
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Computable Plant Project
Shoot Apical Meristem growing tip of a plant

• Provide
• most of our food and fiber
• all of our paper, cellulose, rayon
• pharmaceuticals
• feed stock
• waxes
• perfumes

Image courtesy of E. M. Meyerowitz, Caltech
Division of Biology
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Computable Plant Project

• NSF (USA) Frontiers in Integrative Biological
  Research (FIBR) Program
• S/W Architecture Production-scale model
  inference
• Models formulated as cellerator reactions or SBML
• C simulation code autogenerated from models
• Mathematical framework combining transcriptional
  regulation, signal transduction, and dynamical
  mechanical models
• Simulation engine including standard numerical
  solvers and plot capability
• Nonlinear optimization and parameter estimation
• ad hoc image processing and data mining tools
• Image Acquisition
• Dedicated Zeiss LSM 510 meta upright laser
  scanning confocal microscope.
• http//www.computableplant.org

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Computable Plant Project
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Model Organism
Arabidopsis Thaliana
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Cell Identification in image z-stack
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Identification of Cell Birth
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Shoot Apical Meristem
Image courtesy of E. M. Meyerowitz, Caltech
Division of Biology
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Meristem Pattern Maintenance Model
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Simulation of Meristem Growth
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Systems Biology Markup Language
http//sbml.org libsbml (C) MathSBML
(Mathematica)
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The Standard Paradigm of Biology
RNA
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Microarrays Produce a lot of data!
Affymetrix GeneChip microarray. Images courtesy
of Affymetrix.
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RNA Fragments are Selectively Sticky
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Affymetrix GeneChip Scanner 3000 with workstation
Data from an experiment showing the expression of
thousands of genes on a single GeneChip probe
array. Images courtesy of Affymetrix.
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Model Inference Fitting A Model to Data

• Cluster to reduce data size
• Use simplest possible mathematical possible to
  determine connectivity
• Fit parameters with some optimization process
  simulated annealing, least squares, steepest
  descent, etc.
• Refine model with biological knowledge
• Refine with better accurate math model
• and repeat until done

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Clustering
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Data clusters in two dimensions
Plot (Xt3 1,Xt2,Xt3,,Xtn) for every
species
y
Concentration at time t2
x
Concentration at time t1
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Data clusters in two dimensions
y
Concentration at time t2
x
Concentration at time t1
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Signal Transduction Network
Clusters (may) correspond to functional modules
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1
3
0 Output
4 Input
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Approximation Models

• Linear
• S-Systems (Savageau)
• Generalized Mass Action

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Approximation Models

• Generalized Continuous Sigma-Pi Networks

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Approximation Models

• Recurrent Artificial Neural Networks
• Recurrent Artificial Neural Networks with
  controlled degradation

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Approximation Models

• Recurrent Artificial Neural Networks with
  biochemical knowledge about some species

Known or hypothesized interactions due to mass
action, Michaelis-Menten, or other reactions (A
priori knowledge or assumptions)
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Approximation Models

• Multicellular Artificial Neural Networks with
  biochemical knowledge about some species

Resources
Diffusion
Geometric Connections
Lower index species Upper Index Cell
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Stripe Formation in Drosophila
Observed
Patterson, JT Studies in the genetics of
drosophila, University of Texas Press (1943)
http//flybase.bio.indiana.edu82/anatomy/D
rosophila
Reinitz, Sharp, Mjolsness Exper. Zoo. 27147-56
(1995)
J Exp Zoology 27147-56
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Some Important Meetings

• ISMB-2004, Scotland, 30 July 04
• Intelligent Systems in Molecular Biology
• 2003 Australia 2005 US 2006Brazil
• ICSB-2004, Heidelberg, Oct 04
• International Conference on Systems Biology
• SBML Forum held as satellite meeting
• 2003US 2002Sweden 2001US 2000 Japan
• PSB-2005, Hawaii, Jan 05 Pacific Symposium on
  Biocomputing
• RECOMB, Spring 05 Research in Computational
  Molecular Biology
• BGRS-04, July 04, Semiannually in Novosibirsk
• Bioinformatics of Genome Regulation and Structure
• Satellite meetings of many major biology and
  computer science meetings SIAM, ACB, IEEE, ASCB
  (US), Neuroscience, IBRO,..

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Collaborators

• Cellerator
• Eric Mjolsness, U. California, Irvine (Computer )
• Andre Levchenko, Johns Hopkins (Bioengineering)
• Computable Plant - Eric Mjolsness, PI
• Elliot Meyerowitz, Caltech (Biology)
• Venu Reddy, Caltech (Biology)
• Marcus Heisler, Caltech (Biology)
• Henrik Jonsson, Lund, Sweden (Physics)
• Victoria Gor, JPL (Machine Learning)
• SBML (John Doyle, PI, Caltech H. Kitano, Japan)
• Mike Hucka, Caltech (Control Dynamical Systems)
• Andrew Finney, University of Hertfordshire, UK