Using Mathematical Models to help understand Planar Cell Polarity in Developmental Biology

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Using Mathematical Models to help understand
Planar Cell Polarity in Developmental Biology

• Keith Amonlirdviman
• Robin Raffard
• Anil Aswani
• Dali Ma
• Jeffrey D. Axelrod
• Claire J. Tomlin

Electrical Engineering and Computer Sciences UC
Berkeley Aeronautics and Astronautics /
Pathology Stanford University
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A very stable and robust system
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Planar Cell Polarity (PCP) in Drosophila wings
proximal
distal
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Signaling Molecules

• System amplifies some global directional cue and
  propagates polarity from cell to cell
• Includes Frizzled (Fz), Dishevelled (Dsh),
  Prickle (Pk), Flamingo (Fmi) and Van Gogh (Vang)
• Dsh and Fz localize on the distal portion of each
  cell
• Pk and Vang localize on the proximal portion of
  each cell
• Hair grows from the distal portion of each cell

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Mutant Wings Domineering non-autonomy

• Loss of Fz disrupts polarity in distal non-mutant
  cells
• Loss of Vang disrupts polarity in proximal
  non-mutant cells
• Disruption of signaling molecules is propagated
  to neighboring cells Suggests diffusible Factor
  X?

Mutant Fz clones
Mutant Vang clones
Vinson and Adler, Nature 329, 549-51, 1987
Taylor, et al., Genetics 150, 199-210, 1998
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Biological Model

• Fz promotes recruitment of Dsh to a membrane
• Dsh stabilizes Fz localization
• Fz promotes the localization of Vang and Pk on
  the membrane of a neighboring cell
• Pk and Vang inhibit the recruitment of Dsh to a
  membrane
• Network amplifies unknown directional cue

Tree et al., Cell 109, 371, 2002
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Directional cue evidence from fat clones?

• In the absence of fat (ft), the feedback loop
  amplifies and propagates polarity across the
  clone
• Polarity does not always propagate correctly,
  resulting in swirled hair patterns

fat clone
fat clone
How do we account for the variability of polarity
defects in ft clones?
Dali Ma
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Biological Model

• Fz promotes recruitment of Dsh to a membrane
• Dsh stabilizes Fz localization
• Fz promotes the localization of Vang and Pk on
  the membrane of a neighboring cell
• Pk and Vang inhibit the recruitment of Dsh to a
  membrane
• Network amplifies unknown directional cue Ft?

?
Ft
Tree et al., Cell 109, 371, 2002
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Biological Model

• Does this explain nonautonomy?
• Some sources of controversy
• Null fz clones are nonautonomous, but dsh clones
  are autonomous
• Some fz alleles show an autonomous polarity
  phenotype
• Increasing Pk actually increases Dsh and Fz
  accumulation

Tree et al., Cell 109, 371, 2002
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Modeling Biological Systems

• Why mathematically model regulatory networks?
• The complex dynamics exhibited by biological
  regulatory networks are often non-intuitive
• Typical characteristics
• No solvable/measurable governing equations
• Systems of interest are robust
• Requires as much system identification as
  simulation
• Limited observability of internal parameters
• Regulatory network not fully understood

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Two modeling approaches that weve taken

• 1. Models which admit mathematical analysis
• Hybrid Discrete state dynamics with continuous
  time linear ordinary differential equations
  (ODEs)
• Admits analysis (e.g., feasible parameter ranges
  / initial conditions, reachable states)

• 2. Models which can be numerically analyzed
• Continuous, non-linear, coupled partial
  differential equations (PDEs)
• Natural representation of biological model
• Results and predictions made through simulation
  (for a given set of parameters and conditions)
• Detailed picture of system behavior implied by
  model

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A Hybrid Model for Differentiation in Xenopus
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Xenopus development

• Two proteins Delta and Notch
• Notch
• Receptor
• Delta
• Ligand
• Delta causes hair to grow

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Xenopus development

• Two proteins Delta and Notch
• Notch
• Receptor
• Delta
• Ligand

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Previous work

• Accepted influence model
• Reaction-diffusion Collier 96, Marnellos 00
• Parameters are unknown.
  

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and may be hard to identify
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Hybrid Model of a Cell

• Idea approximate the nonlinear switch by a
  piecewise linear switch
• For example, using , each biological cell
  has four discrete states

• Notch off/delta off
• Notch on/delta off
• Notch off/delta on
• Notch on/delta on

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Hybrid Systems Theory

analyze directly derive switching logic
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Reach set calculation for piecewise affine systems
are symbolic

• A simple example
• Step 1 Separate partitions into interiors and
  boundaries

diagonal
Interior
Interior
Boundary
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Reach set calculation for piecewise affine systems

• Step 2 Compute transitions between modes. For
  example, in mode 1

Interior
1
2
Interior
3
Boundary
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Reach set calculation for piecewise affine systems

• Step 3 Subpartition modes that have more than
  one exit transition

1
2
3
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Visualization of Reach Sets for 2-cell

• Implementation
• Symbolic manipulations done
  in MATLAB
• Decision procedure on polynomials done in QEPCAD
• Currently 8 continuous, 256 discrete variables

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Simulation using viable parameters and IC
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Planar Cell Polarity (PCP) in Drosophila wings
proximal
distal
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Modeling Planar Cell Polarity

• Discrete model
• Hybrid model
• Continuous model

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A Continuous Model for PCP
Model 4 proteins and introduce 6 complexes
Protein interactions modeled as binding/unbinding
reactions
Dagger denotes a component in a neighboring cell
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Reaction Equations
There are 10 such reaction equations
Reaction-based direct asymmetry signal
Pk and Vang dependent inhibition of Dsh
recruitment
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Model Development
Net production rate Forward reaction rate
Backward reaction rate
Local time rate of change of complex
concentration Reaction production rate
Diffusion rate
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Parameter Selection

• All of the model parameters are unknown and not
  measurable from the available data
• Express PCP phenotypes as feature constraints
• Search for a feasible solution by adjusting model
  parameters using an optimization algorithm

Optimizer
Minimize sum of quadratic penalty functions
enforcing feature constraints
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Governing model
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Parameter ID via Optimization
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Wild-type Numerical Results
Dsh
Periodic infinite array of cells
High Conc.
Dsh
Fz
Pk
DshGFP
Vang
Low
Amonlirdviman et al, Science 307, Jan. 2005
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Loss-of-Fz Numerical Results
Dsh Distribution w/ Resulting Hair Pattern, 14 x
20 periodic cell array
fz clones
Domineering non-autonomy distal of cloned mutant
cells
Mutant Fz clones
Vinson and Adler, Nature 329, 549-51, 1987
Amonlirdviman et al, Science 307, Jan. 2005
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Loss-of-Vang Numerical Results
Dsh Distribution w/ Resulting Hair Pattern, 14 x
20 periodic cell array
vang clones
Domineering non-autonomy proximal of cloned
mutant cells
Mutant Vang clones
Taylor, et al., Genetics 150, 199-210, 1998
Amonlirdviman et al, Science 307, Jan. 2005
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Biological Insights

• Demonstrates sufficiency
• Explains even non-intuitive results

Suppose you overexpress Pk in part of the wing
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Understanding fz Autonomous Alleles

• Suggests mechanism for explaining phenotypes of
  different mutant Fz alleles

fz autonomous allele FzDsh interaction reduced
to 0.01
fz nonautonomous allele All Fz function removed
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Understanding fz Autonomous Alleles

• Propose that Fz autonomous proteins are deficient
  in complexing with Dsh, but retain Vang
  interaction
• Nonautonomous fz alleles lose both interactions
• Hypothesis makes two predictions
• Autonomous Fz protein recruits Vang to
  neighboring membranes
• nonautonomous Fz protein should not
• Both proteins should fail to recruit Dsh

Autonomous
Nonautonomous
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Understanding fz Autonomous Alleles

• VangYFP does not accumulate at boundaries of
  fzR52 (nonautonomous) clones
• VangYFP accumulates at boundaries of fzF31
  (autonomous) clones

fzR52 clone (nonautonomous)
fzF31 clone (autonomous)
Vang
Vang
Wei-Shen Chen
Amonlirdviman et al, Science 307, Jan. 2005
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Understanding fz Autonomous Alleles

• DshGFP is not recruited by fzR52
  (nonautonomous)
• DshGFP is poorly recruited by fzF31 and more
  poorly recruited by fzJ22 (autonomous)

fzR52 (nonautonomous)
fzF31 (autonomous)
fzJ22 (autonomous)
Wild-type
Amonlirdviman et al, Science 307, Jan. 2005
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Lawrence Challenge

• Conditions proposed by Lawrence, Casal, and
  Struhl prior to publication of results from the
  Drosophila abdomen

fz clone / pk background
gtgtfz clone / fz background
gtgtfz / Vang clone
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Lawrence Challenge

• Example of nonautonomy in the absence of a core
  polarity component, pk

fz clone / pk background
Lawrence et al., Development 131, 4651, 2004
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Insights into Nonautonomy

• Demonstrated that the feedback loop can fully
  reproduce characteristic PCP phenotypes
  Unidentified diffusible factors unnecessary
• Showed that the feedback loop model more readily
  accounts for slight nonautonomy of clones of dsh
  and autonomous fz alleles
• Proposed a mechanistic explanation for the
  difference between autonomous and nonautonomous
  fz alleles, motivating experiments supporting
  this hypothesis
• Predicted other phenotypes not used to train the
  model

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Understanding fat clones

• The role of cell geometry
• Polarity defects correlate to irregular cell
  geometry
• Frequency of polarity defects can be modified by
  altering cell shape

fat clone
fat clone
Are polarity defects a consequence of the Fz
feedback loop when confronted with irregular cell
geometries?
Dali Ma
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Building irregular grids
GFP image
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Building irregular grids
Wild-type geometry
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Simulating on irregular grids
Wild-type simulation
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Simulating on irregular grids
Hair polarity plot
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Simulating clones on irregular grids
fz clone simulated on a wild-type geometry
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Summary and current work

• Demonstrated the sufficiency of the model Factor
  X unnecessary
• Begun to derive insights into the nature of
  domineering non-autonomy
• Proposed and conducted experiments exploring the
  interaction of Dsh with different Fz alleles
• Developing analytical tools for parameter
  identification
• Interaction of PCP with other protein networks
  Lymphoma models
• Berkeley Drosophila Genome Project

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Using mathematical modeling to help decode
biological circuits

• Keith Amonlirdviman
• Robin Raffard
• Anil Aswani
• Dali Ma
• Jeffrey D. Axelrod
• Claire J. Tomlin

NIH, NSF, DARPA, Bio-X
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Descent Algorithm in Matlab
Function or gradient evaluation
Distributes computation over 17 nodes
PBS script
.
Computes cost and sensitivity of the cost
Python code
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Hybrid Model for PCP
Example Dsh in proximal compartment
Time rate of change of proximal Dsh concentration

Transport from center compartment if Fz higher
than threshold
-Transport to center compartment if Pk is
higher than threshold
is a discrete switching function that depends
on the Fz / Pk concentrations and on a switching
threshold
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Reaction-Diffusion Model
System of coupled nonlinear partial differential
equations
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Parameter Sensitivity

• Ranges in which each parameter may vary
  individually, keeping all other parameters fixed,
  while satisfying each feature constraint

Jwt
Amonlirdviman, Khare, Tree, Chen, Axelrod,
Tomlin, Science 307, 423, 2005
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Parameter Sensitivity
Ranges in which parameters can vary while holding
all other parameters constant All constraints
enforced
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Parameter Sensitivity
Jdsh
Jfza
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Cell Geometry Analysis
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Cell Geometry Analysis

• Next steps
• Collect statistics on many such clones to look
  for differences between those showing a polarity
  phenotype
• Solve mathematical PCP model on same cell
  geometries to see if we can reproduce hair
  patterns

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Comparing Dynamics w/ Experimental Data
Dshp-d / Dsha-p
( Dshp - Dshd ) / Dsh0
Time
Time hrs

• Plots measure degree of Dsh localization vs. time
• Need more quantitative data to validate/identify
  system parameters

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Experimental Data

• Scanning laser confocal microscopy image of Dsh
  protein distribution in the fly wing after 30
  hours

anterior leading edge

• Localization usually observed qualitatively
• For quantitative measure of localization, cell
  edge locations needed
• Degree of localization measured from image pixel
  intensities

distal tip
proximal root
posterior trailing edge
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Numerical Results
Dsh Distribution w/ Resulting Hair Pattern, 14 x
20 periodic cell arrays
dsh clones
pk clones
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Lawrence Challenge

• One row of nonautonomy pointing away from the
  clone

gtgtfz clone / fz background
Lawrence et al., Development 131, 4651, 2004
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Evidence from the fly eye
Global polarity
fat (ft) four-jointed (fj) dachsous (ds)
Local cell polarity
frizzled (fz) Vang Gogh (Vang)/strabismus
(stbm) dishevelled (dsh) prickle-spiny legs
(pk-sple) starry night (stan)/flamingo (fmi)
Fj
Fj
Wing hair formation Ommatidia orientation Sensory
bristle polarity
Ds
Ft
?
eye
Fz
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and in the wing
(Zeidler et al, Dev Bio 2000 Ma et al Nature
2003.)
Staining for Fj
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Global Directional Cue

• The feedback loop amplifies a global polarity
  asymmetry signal
• Two forms of the input asymmetry
• Reaction-based
• Asymmetric Fz-Dsh interaction
• Diffusion-based
• Reduced Fz diffusion in the distal region

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Adjoint-based Algorithm
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Objective Feature Constraints
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Evidence from the fly eye
Global polarity
fat (ft) four-jointed (fj) dachsous (ds)
Local cell polarity
frizzled (fz) Vang Gogh (Vang)/strabismus
(stbm) dishevelled (dsh) prickle-spiny legs
(pk-sple) starry night (stan)/flamingo (fmi)
Fj
Fj
Wing hair formation Ommatidia orientation Sensory
bristle polarity
Ds
Ft
?
eye
Fz
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Arrows in the direction of higher Ft
concentrations
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Ft
Ft
Ft
Ds
Ds
Ds
Ft
Ds
Ft
Ds
Ft
Ds
Ds
Ft
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Ft
Ft
Ft
Ds
Ds
Ds
Ft
Ds
Ft
Ds
Ft
Ds
Ds
Ft
Arrows in the direction of higher Ft
concentrations
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Arrows in the direction of higher Ft
concentrations
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Entering
Exiting
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Back to the biology
Fj
Fj
Ds
Ft
wing
Fz is directed away from Ft and toward Ds in the
wing!
Boundary conditions contribute to the fat clone
phenotype
Ma et al, submitted, 2007