Y Mori
Adriana Dawes,
Alexandra Jilkine
Veronica Grieneisen,
AFM (Stan) Marée
Yoichiro Mori,
Leah Edelstein-Keshet, UBC
V Grieneisen
A Jilkine A T Dawes AFM Marée

€
What is a reaction-diffusion system?
€
∂ A
∂
∂
∂ t
B t
= f ( A , B ) + D
A
= g ( A , B ) + D
B
∂
2 A
∂ x 2
∂ 2 B
∂ x 2 Reaction
∂ A
= f ( A , B )
∂ t
∂ B
= g ( A , B
€
)
∂ t
Example:
Shankenberg
S
1
⇔ A
S
2
  B
B + 2 A   → 3 A
€
A
Diffusion
∂ A
∂ t
= D
∂ 2
∂ x
A
2 t=0 f ( A , B ) = k
1
€ g ( A , B ) = k
4
− k
2
A + k
3
A 2 B
− k
3
A 2 B
B g=0 t>0 x
Units: [ D ]= L 2 /t f=0
A

1912-1954
• Instrumental in early computer science (“Turing Machine”)
• King’s College Cambridge, 1931-34; PhD Princeton, 1938
• WWII: Bletchley Park, cryptography, deciphering German”enigma” code
• 1952: work on reaction-diffusion equations, pattern formation, and morphogenesis
• 1952: convicted of homosexuality
• 1954: death (suicide?)
A.M. Turing, The Chemical Basis of Morphogenesis, Phil. Trans. R. Soc. London B237, pp.37-72, 1952

Turing (1952):
“The chemical basis of morphogenesis”
Reaction (f,g) Diffusion
+
Activator
D
A
<< D
B
+ inhiBitor
-
-
∂ A
∂
∂
∂ t
B t
= f ( A , B ) + D
A
= g ( A , B ) + D
B
∂
2 A
∂ x 2
∂ 2 B
∂ x 2
With “appropriate” reaction kinetics, an RD system can form spatial patterns due to the effect of diffusion.
A.M. Turing, (1952) Phil. Trans. R. Soc. London B237, pp.37-72, 1952

Ingredients required for “Turing” pattern formation
∂ A
∂
∂
∂ t
B t
= f ( A , B ) + D
A
= g ( A , B ) + D
B
∂
2 A
∂ x 2
∂ 2 B
∂ x 2
• There is a stable spatially homogeneous steady state, f(A,B)=0, g(A,B)=0.
• Range of activator << range of inhibitor
€ e iqx e σ t
€
Close to that state: (
Linearize eqns using Taylor Series for f, g
)
∂ a
∂
∂ t t
∂ b
= c
11 a + c
12 b + D
A
= c
21 a + c
22 b + D
B
∂
2 a
∂ x 2
∂ 2 b
∂ x 2
D a c
11
<<
D b c
22 x
σ q
€
“on center/off surround” lateral inhibition: local excitation, long-range inhibition

Shnakenberg RD system
A 0
€
€
∂ A
∂
∂ t t
∂ B
= f ( A , B ) + D
A
= g ( A , B ) + D
B
∂ 2 A
∂ x 2
∂ 2 B
∂ x 2 f ( A , B ) = k
1
− k
2
A + k
3
A 2 B g ( A , B ) = k
4
− k
3
A 2 B
time
Starting close to the HSS, the system evolves a spatial pattern that persists with time.
200

Contributors to theories of pattern formation and chemotaxis:
Hans Meinhardt Lee Segel

Phyllotaxis (Wikipedia)
Hans
Meinhardt http://www.eb.tuebingen.mpg.de/departments/former-departments/h-meinhardt/home.html

Reaction-Diffusion systems and animal coat patterns
J.D Murray, A prepattern formation mechanism for animal coat markings, J. Theor. Biol. 88: 161-199, 1981

Neutrophils:
Orion Weiner: http://cvri.ucsf.edu/~weiner/
Tumor, Huang et al Nature 2003
Slime mold amoeba
Dictyostelium discoideum (WT)
Sasaki et al (2004) JCB, Volume 167, Number 3, 505-518

Initial step in cell motility is rearrangement of internal biochemistry to pick a “front” and a “back”.
Dictyostelium discoideum (Latrunc)
Sasaki et al (2004) JCB, Volume 167, Number 3, 505-518

• Cells can polarize robustly in response to shallow transient gradients of chemicals (“chemoattractants”)
• Cells remain sensitive: a second stimulus from another direction will cause a cell to repolarize
• Polarization can occur spontaneously (e.g. in neutrophils) even without a directional stimulus.
Top view
Side view
Resting cell Crawling cell
Polarized cell

Back:
Rho
PTEN
Front:
Rac
PI3K,
PIP
2
, PIP
3
What process(es) account for segregation of chemicals?
(“Gradient sensing”)

• Cell biochemistry involves substances that react and diffuse
• Reaction-diffusion systems with Turing-type properties can spontaneously form patterns (including unimodal
“polarized patterns”)
• Thus, it would be an obvious choice to assume that a polarity mechanism might be based on “Turing”.
Q
:
But what substance acts as the “activator”, what is the
“inhibitor” and what else is needed?

Hans Meinhardt (1999)
Orientation of chemotactic cells and nerve growth cones
Signal inhibitor activator
(1) Slightly asymmetric signal (e.g., 2% across diameter) polarizes the cell, regardless of absolute level of signal.
(2) Cell orientation “frozen”, will not respond to later stimuli.
(3) A second global inhibitor can “unfreeze” the polarity, making the cell responsive again.
+
S
-
-
A
+
B
Local
Global
C Local
Meinhardt, H. (1999). J. Cell Sci. 112 , 2867-2874.
http://www.eb.tuebingen.mpg.de/departments/former-departments/h-meinhardt/orient.html

Local Excitation Global Inhibition
Based on essential idea of local activation and global inhibition.
In many cases, activator is purely local (does not diffuse).
Reaction (f,g)
+
+
Activator inhiBitor
-
-
D a
= 0
D b
> 0 .
€

Explains adaptation
Levchenko, Iglesias (2002) Biophys. J. 82:50–63

€
€
€
LEGI Cont’d
External signal S, membrane bound activator E, membrane bound reporter R and global inhibitor I
∂ E
= k e
S − k
− e
E
∂ t
on membrane
∂ R
= k r
E − k
− r
I ⋅ R
∂ t
∂ I in cytosol
∂ t
= − k
− i
I + k i
S + D ∇
2 I
Uniform stimulus, steady state: response (R) independent of S
E
0
= k e k
− e
S ; I
0
= k i k
− i
S ⇒ R
0
= k r k
− r
E
0
I
0
= k e k
− i k
− e k i
€

Fits with experiments on
Dicty amoeba.
Ma, Janetopoulos, Yang, Devreotes, Iglesias (2004) Biophys J 87: 3764-3774

.. need global diffusing inhibitor
Inhibitor has not been identified .

€
RD with mutual inhibition can lead to polarization (Atul Narang)
∂
∂ u t
1
∂ u
2
∂ t
∂ u
3
∂ t
= − R
1 u
1
+ A
12 u
2
+ A
13 u
3
+ D
1
∂
2 u
1
∂ x 2
= ( R
2 r − A
21 u
1
− A
22 u
2
+ A
23 u
3
) u
2
= ( R
3 r − A
31 u
1
− A
32 u
2
+ A
33 u
3
) u
3
+ D
2
∂
2 u
2
∂ x 2
+ D
3
∂ 2 u
3
∂ x 2 u
3
,u
2
= frontness, backness u
1
= inhibitor
Both u
3
,u
2
compete for
(saturated) receptors r , while producing a third competing species, u
1

The Trouble with Turing:
• So far, hard to identify the cellular agent(s) that are the appropriate activator-inhibitor pair, and that are known to have the right type of kinetics.
• Turing systems are unstable to the slightest noise, and will always tend to form non-homogeneous distribution, so a uniform “rest state” is not stable. Cannot explain an unpolarized (unstimulated) resting cell.
• Turing patterns, once created, tend to be rock-stable. Need additional mechanisms to “unfreeze them” so the cell can react to new stimuli.

Biochemically-based model:
Onsum, Rao (2007) PLoS Comput. Biol 3(3): e36

We start with some of the known biochemistry and work our way up form there.

Signaling to actin (KEGG): www.genome.ad.jp/kegg highlights credit: A T Dawes

Schmitz et al (2000) Expt Cell Res 261:1-12

Rho GTPase properties
• Molecular “switches”
• Cycle between membrane (active) and cytosol (inactive)
Fig courtesy: A. Jilkine

Image from: Charest & Firtel (2007) Biochem J. 401
“Molecular switch”
OFF
GAP
GEF
ON

Image from: Charest & Firtel (2007) Biochem J. 401
“Molecular switch”
OFF cytosol
ON
Cell membrane

Distribution in polarized cells
Cdc42 (red) in front
Nabant et al (2004) Science
Rac (green) in front
(FRET, fibroblast)
Krayanov et al (2000), Science .
Rho (green) in back actin (red) in front
(neutrophil)
Bourne lab http://www.cmpharm.ucsf.edu/bourne/
Mutual exclusion: Cdc42 vs Rho
Rho Cdc42, Rac

Cdc42 Rho
This can lead to bistability.
Ferrell (2002) Curr Op Chem Biol 6:140–148

1) Hall (1995):
2) Giniger (2002):
Posisble interactions
3) Evers (2000), Sander (1999)
4) van Leeuwen (1997)
5) Li (2002)
Slide courtesy: A. Jilkine
Cdc42
+
Rac
+
Rho
-
Cdc42
+
Rac
+
Rho
-
Cdc42
+
Rac
-
Rho
-
Cdc42
+
Cdc42
+
Rac
Rac
+
-
+
-
Rho
Rho
-

front back t
Front back

G = conc. active Rho GTPase
G t
= activation - inactivation + diffusion
G t
= I
G
- δ G + D m
Δ G
G i
δ
G
I
G
I
G
nonlinear
(for bistability).

1st attempt: active Rho GTPases
3 PDE’s for active
(membrane) forms:
-
Cdc42
+
Rac
+
Rho
-
∂ G
= I
G
− d
G
G + D m
Δ G , G = C , R , ρ
∂ t
G i
δ
G
I
G
Bistable kinetics
I
G
= Q
G
( C , R , ρ ) G i
,
Q
C
= a
1
I
C
+ ρ n
, Q
R
= I
R
+ α C , Q
ρ
=
I
ρ
+ β R a
2
+ C n
€

Initial model of Rho GTPase dynamics
With Jilkine
Active (membrane) forms:
∂ G
∂ t
= I
G
− d
G
G + D m
Δ G , G = C , R , ρ
3 PDE’s with nonlinear functions I
G
(C,R,p)
Model displays bistable behaviour; travelling waves
€
Cdc42 Rac Rho
-
Cdc42
+
Rac
+
Rho
-

This is not satisfactory.
We want to get a stable polarization.
After the wave sweeps through the whole cell, we cannot distinguish front from back.
Let us try to understand why we get a traveling wave, and try to eliminate it.

€
Keener & Sneyd (1998) Mathematical Physiology; p 270-275
∂ u
∂ t
= f ( u ) + D
∂
∂ x
2 u
,
2
∂ u
∂ x
± ∞
= 0 f(u)
α
With f cubic: f ( u ) = au ( u − 1)( α − u ) 0 < α < 1
1
€ z=x-ct, U(z)=u(x,t)
U is the shape of the wave, z is the position along the wave front

Traveling wave coordinates : z=x-ct, U(z)=u(x,t)
Scaling and transformation of coordinates:
PDE
∂
∂ t
→ − c d dz
∂ u
∂ t
= f ( u ) +
∂ 2 u
∂ x 2
∂
∂ x
→ d dz
→ − c dU dz
= f ( U ) + d 2 U dz 2
ODE
2nd order (nonlin) ODE U zz
− cU z
+ f ( U ) = 0
€
Equivalent ODE system:
U z
= W
W z
= − cW − f ( U )
We can study this qualitatively in the
UW phase plane to get insight into the shape of the wave.
€

€
∂ u
∂ t
= f ( u ) +
(Rescaled)
∂ 2 u
∂ x 2
W
f(u)=u(1-u)(u- α )
W’=0
W
€
U zz
− cU z
+ f ( U ) = 0
U z
= W
W z
= − cW − f ( U )
U’=0
U
Bard Ermentrout’s XPP file
Bistable.ode
# bistable.ode
# classic example of a wave joining 0 and 1
# u_t = u_xx + u(1-u)(u-a)
#
# -cu'=u''+u(1-u)(u-a) f(u)=u*(1-u)*(u-a) par c=0,a=.25
u'=up up'=-c*up-f(u) init u=1
@ xp=u,yp=up,xlo=-.5,xhi=1.5,ylo=-.5,yhi=.5
done
U

W
Each trajectory describes how U (and W ) vary as z increases over range ∞ < z < ∞
If u= density, it has to be positive, so we look for a positive bounded trajectory connecting two steady states.
U U

W
U

W
U

U z

• A reaction-diffusion equation with “bistable” kinetics will admit a traveling wave that looks like a moving “front”, with high level at back, and low level ahead.
• But how fast does the wave move and does it ever stop?
There is a cute trick that allows us to answer this question for arbitrary function f(u).

€
∂ u
∂
− c t
= f ( u ) + dU dz
∂ 2 u
, z = x − ct
∂ x 2
− f ( U ) =
∂ 2 U dz 2 u back c
Multiply by dU/dz ,
− c


 dU dz



2
− f ( U )


 dU dz

 =

1
2 d dz
 dU

 dz



2 integrate,
− c
∞
−∞
∫
 dU

 dz



2 dz − ufront
∫ uback f ( U ) dU =
1
2
−∞
∫
∞ d dz
 dU

 dz



2
= 0
− c
∞
−∞
∫
 dU

 dz



2 dz = ufront
∫ uback f ( U ) dU z u front

In some special cases, e.g. f(u) cubic or piecewise linear, can calculate wave speed fully by this method. (See Keener & Sneyd p 274, Murray p 305)
For f ( u ) = au ( u − 1)( α − u ) 0 < α < 1
Speed of the wave is:
€
Implications: c = a
(1 − 2 α )
2
€ α <1/2 Moves left (c<0) if α >1/2
Wave stops (c=0) precisely for one value of the parameter,
α =1/2

f(u)
1
α
Speed is zero if: i.e.: if : c = −
1
0
∫ f ( U ) dU
∞
∫
−∞
 dU

 dz



2 dz = 0
1
0
∫ f ( U ) dU = 0
€

f(u)
1
α
“Maxwell condition”:
“Equal areas”
1
0
f ( U ) dU = 0
1 α
0
∫ f ( U ) dU = − ∫ f ( U ) dU
α
€
€

• We can apply some of these ideas to studying the polarization of the Rho proteins.
• In particular, we can try to determine how to stop the waves from sweeping across the whole cell.

front back t
Front back

Three forms of the Rho proteins activation inactivation
Diffusion in membrane
Rate off membrane
Rate on membrane
Diffusion in cytosol

k on
G c
=k off
G m

∂ G
∂ t
∂ G i
∂ t
= I
G
G i
− d
G
G + D m
Δ G , G = C , R ,
= − I
G
G i
+ d
G
G + D c
Δ G
ρ
A set of two PDEs for each of the three proteins, Cdc42,
Rac, and Rho.
€

Revised Rho model: include inactive forms
Active (membrane) forms:
Inactive (cytosol) forms:
I
G
= Q
G
( C , R ,
∂
∂
G
∂ t
∂ G i t
= I
G
− d
G
G + D m
Δ G , G = C , R ,
= − I
G
+ d
G
G + D c
Δ G
ρ
ρ ) G i
, Q
C
= a
1
I
C
+ ρ n
, Q
R
= I
R
+ α C , Q
ρ
=
I
ρ
+ β R a
2
+ C n
Inactive forms diffuse faster in cytosol, D c
> D m
€
Result: wave is pinned, stable polarized distribution will form.

• We want to understand why the revised model produces the stable polarization..
•A similar phenomenon occurs in a much simpler (single protein) system that we can more easily study mathematically.

Simplified toy model
€
Unlimited pool of inactive form
∂ A
= f ( A , B ) + D
A
Δ A
∂ t
B
A
Limited inactive form
∂ A
= f ( A , B ) + D
A
Δ A
∂ t
∂ B
= − f ( A , B ) + D
B
Δ B
∂ t
D
A
< D
B
€
B
A

Simplified toy model
Unlimited pool of inactive form
∂ A
= f ( A , B ) + D
A
Δ A
∂ t
B
A
€
Wave sweeps through domain f(A,B)
€ t=0
Limited inactive form
∂ A
= f ( A , B ) + D
A
Δ A
∂ t
∂ B
= − f ( A , B ) + D
B
Δ B
∂ t
D
A
< D
B
Wave is pinned. Stable polarization
B
A
A t=100
0<u<0.6
f ( A , B ) =

B k
 o
+ k 2
γ A 2
+ A 2

 − d

A
A
€

€ f(A;B)
∂ A
∂
∂ t t
∂ B
= D a
= D b
∂ 2 A
∂ x 2
∂
2 B
∂ x 2
+ f ( A , B )
− f ( A , B )
• S-shaped function, f(A, .) with multiple steady states A
+
, A
S
, A
that depend on B .
• Unequal diffusion coefficients:
D a
<< D b
A
A
+
A s
A
-

€ f(A;B)
∂ A
∂ t
∂ B
∂ t
= D a
= D b
∂ 2 A
∂ x 2
∂
2 B
∂ x 2
+ f ( A , B )
− f ( A , B )
D
0 a
<< D
≤ x ≤ L
Reaction terms are equal and opposite b
A
+
A s
A
-
A
∂ A
∂ x x = 0
=
∂ A
∂ x x = L
= 0
∂ B
∂ x x = 0
=
∂ B
∂ x x = L
= 0
€

€ f(A;B)
∂ A
∂
∂ t t
∂ B
= D a
= D b
∂ 2 A
∂ x 2
∂
2 B
∂ x 2
+ f ( A , B )
− f ( A , B )
€
A
+
A s
A
-
A
Sum of A and B are conserved over time,
∫
0
L
( A + B ) dx = C = constant
∂ A
∂ x x = 0
=
∂ A
∂ x x = L
= 0
∂ B
∂ x x = 0
=
∂ B
∂ x x = L
= 0
€

€
∂ B
∂ t
= D b
∂
2 B
∂ x 2
− f ( A , B )
∂ B
∂ x x = 0
=
∂ B
∂ x x = L
= 0
• B diffuses very quickly
( D b
is large)
• B does not enter or leave domain (no flux through boundaries)
This means B is roughly constant over 0 < x < L
€

For constant B , the A eqn is bistable
∂ A
= D a
∂
2 A
∂ x 2
+ f ( A , B )
∂ t
Thus, A will form a travelling wave-front, moving with some speed c
€
A
+ c
A
-

€
A + c = −
f ( A , B )
A − dA
∞
−∞
 dA

 dz



2 dz
A
+
B c
A
-
STOP
0 x L
A +
∫
A − f ( A , B ) dA = 0
€

A f(A,B)
B
A
A
A
Equal areas
STOP
0 x L

membrane cytosol
• Stimulus creates active zone
• +ve feedback reinforces, amplifies, and spreads it
• active zone spreads at expense of inactive GTPase
• Depletion of the “global” inactive form slows down and stops the spread.

Rescale variables:
Assume:
Let:
Pick time scale based on dynamics of (slow diffusing) A: x = XL , t = T τ A u, B v
D a
<< D b
ε
2
=
D a
γ L 2
=
D b
γ L 2
= D = O (1)
τ =

1

 γ
L
D a
2 
1/ 2


=
Range of A diffusion
Domain size
γ
L
D a
= (t decay
2
<< 1
t diffuse
) 1/2
Rescaled equations:
€
∂ u
ε
∂ T
ε
∂ v
∂ T
= ε
2
∂ 2 u
∂ X 2
= D
∂ 2 v
∂ X 2
+ f ( u , v )
− f ( u , v )
Small parameter, ε
€

ε
ε
∂ a
∂ t
∂ b
∂ t
=
=
Asymptotics in 1D for cubic f
ε
2
D b
∂
2 a
∂ x 2
∂
2 b
∂ x 2
+
− f f
( a , b )
( a , b ) a
+
_
D a
ε
2
<<
=
D b
D a
δ L 2
<<
1
•Away from front at x f solution is determined by reaction kinetics.
• Get an ODE for evolution of transition layer at x f
.
• For small ε , prediction of x f total amount C confirmed by based on numerical results (integrating full PDE system from initial conditions)
Slide credit: A Jilkine
0 x f
1 a
-

• Inner solution (close to transition zone)
• Outer solution (farther away)
• Match the two solutions
• Predict shape and speed of wave, as well as evolution for the position of the front
• Test theory with numerical computations, to confirm.

An exact ODE for evolution of the front: f(A;B) mB h
€ dF dt
=
( m + l ) C
2(1 + l + F ( T )( m − l ))
− 2 h
Criterion for wave-pinning: lB
A
2 h (1 + l )
< C < m + l
2 h (1 + m ) m + l
“Cubic function”: f(A,B)=-(A-mB)(A-h)(A-lB)
€

• Test the model with realistic parameter values in the biological ranges appropriate for Rho proteins.
• Consider a variety of “input signals”

# Embed in Reaction-Diffusion XPP code rd2.ode
# (by Bard Ermentrout)
# Kinetic functions used for wave-pinning: f(u,v)=v*(k0+g*u^2/(K^2+u^2))-k1*u
# Parameters for wave-pinning: par k0=0.0,g=1.0,K=0.5,k1=1,du=.01,dv=10,h=0.1
%[1..100] u[j]'=f(u[j],v[j])+du*(u[j+1]+u[j-1]-2*u[j])/h^2 v[j]'=-f(u[j],v[j])+dv*(v[j+1]+v[j-1]-2*v[j])/h^2
%

Initial conditions formUla
U[1..100]
0
U[1..5]
6
V[1..100]
1.2

From XPP window: Viewaxes Array
A plot of v shows it to be uniform in space, as predicted.

Cable100.ani (slightly modified)
# animation for the array
# cable100.ani
vtext .8;.95;t=;t fcircle [1..100]/100;(u[j]+1)/4;.02;[j]/100 fcircle [1..100]/100;(v[j]+1)/8;.02;[j]/100 end

Analysis predicts, and simulations confirm that increasing the total amount (A+B) will shift the position of stalled front.
If total amount too high, wave will not stop at all.
(Possible experimental tests).

Small transient signal applied to one side: polarization occurs.
Small transient graded signal applied: polarization occurs.

Gradient of signal reversed: cell reverses its polarization.
Sufficiently large random noise also leads to robust polarization.

For Turing-type patterns, we need one of two types of interaction sign patterns:
+ +
- -
+ or
+ -
Because of the interconversion kinetics we always have sign pattern:
- +
+ -

Our system:
to a similar Turing model
B
A
Meinhardt’s (Turing) system
B
A
Note conservation

B
Ours
A
Meinhardt’s
B
A

Ours:
Meinhardt’s
A threshold noise level needed to initiate polarization
Sensitive to arbitrarily low level of noise

Ours
Main purpose: robust polarization
Persistent?
√
Noise-sensitive?
above threshold level
Can cell reverse?
√
Reaction speed:
fast
Biological interp?
Rho GTPase, etc ?
LEGI adaptation
none
√
Meinhardt pattern formation
√ acute
slow
?

• stalling of traveling waves and robust stationary fronts in a very simple RD system.
• Essential components: unequal rates of diffusion, finite domain, conservation of total.
• Details of kinetics not important, but need multiple stable steady states (zeros of f ).
• This is NOT pattern arising from a Turing instability, and system does not admit such.
• Has been overlooked in literature about RD systems?

• Some key features of the biology are essential, some can be simplified away.
• For a polarity mechanism that is both robust, amplifies signal, and maintains sensitivity, we do not need to postulate a “global inhibitor”.
• Good candidate proteins are those that have both slow and fast diffusion speeds (in cytosol vs membrane).
• Rho GTPases may have “inherent properties” that allow them (one at a time) to polarize.
• Other biochemical and signaling circuits may refine and adjust these to adapt to specific cell repertoires.

Cdc42 Rac
(WASp)
Arp2/3
(WAVE)
(uncap)
Barbed ends
(PIP2)
Actin filaments
Cell protrusion
Rho
(ROCK)
Myosin
Rear retraction

Stan Marée’s cellular Potts model
Cell interior
Cell exterior
Cell volume too big
Rho,
Myosin contraction
No
Cell retracts
6 Filament orientations
Cell volume
Too small
Pushing barbed ends
Yes
Cell protrudes
Fig: revised & adapted from: Segel, Lee A. (2001) PNAS

Round resting cell, homogeneous biochemistry, homogeneous, isotropic actin filament density
C i
C
R i
R p i p
Small transient signal

•Yoichiro Mori, Alexandra Jilkine, Stan Maree.
•NSF MITACS
• NSERC