Wave-pinning
Essential features
time
The behaviour
“Wave-pinning”
What is the underlying
mechanism?
A Jilkine
Y Mori
Mori Y, Jilkine A, E-K L (2008) Biophysical Journal, 94: 3684-3697.
To investigate this, we study
simple system:
P
P
P
active
+
P
P
Dactive << Dinactive
inactive
Mathematically:
Reaction-diffusion eqns with positive feedback terms
f (u,v)
u
∫ u+v = Constant
u
v
Du << Dv
Simplified 1D geometry:
u
v
Du << Dv
0
x
L
Reduced model
Active form
Inactive form
Du << Dv
Typical kinetics
Neumann BCs
0
L
Moving fronts
f (u,v)
u
If v is fixed, and u eqn is on its own:
RD eqn with bistable kinetics. This is known to have traveling wave solns
(moving fronts).
When coupled with the eqn for v,
get a wave pinning phenomenon:
Rescaled (there is a small
parameter)
Du << Dv
Conditions for Wave-pinning:
1.  For v fixed in some range, vmin< v < vmax,
f(u,v)=0 has 3 roots
Shape of f: f
u-
um
u+
u+
1.  There is a vc in the above range such that 2.  Conservation of
∫
u_
u+v €
f (u,v c ) du = 0
Short time scale:
f
u-
um
To leading order, uo evolves to u- or u+ And v diffuses fast and attains uniform profile
u+
Transition layers form on short
timescale
u(x,t)
u+
u+
u-
u-
0
x
1
u(x,t)
0
x
Focus on single
front solution
(polarized domain)
1
Intermediate time
Outer solution
Away from front u = u- or u+ u
vo
φ(t)
With BCs on [0,1], find that vo
is constant left and at right of
front
Intermediate time scale
Stretched coordinate at the transition layer:
Inner solution
to leading order:
u
u+
V0 becomes spatially uniform.
v
Matching inner and outer sols:
V0=v0 u-
0
φ(t)
1
Front speed
In inner layer, Vo is constant in space, so eqn for Uo
depends only on Uo with Vo a time-varying parameter.
We are back to bistable scalar RD eqn.
Speed of the wave
u
u+
f
u-
u-
0
φ(t)
1
um
u+
Front motion depletes v
Front speed
, and dvo /dt have opposite signs
Stalling of the wave
u
f(u,v)
v
u
Decreasing B
u
u
0
x
1
Equal
areas
STOP
u+
Wave stalls when
= 0
∫
f (u,v) du = 0
u_
(equal areas)
€
f
u-
um
u
u+
STOP
vc
u-
0
φp
1
u+
Where does the wave stop?
u
u
v
u-
0
φ(t)
1
Where does the wave stop?
u
v
v
u-
0
φ(t)
1
Conservation
u
+
v
= constant = K
u
v
u-
0
φ(t)
1
Conservation results in wavepinning
u
+
v
= constant = K
u
v
Wave stops when u-
0
φ(t)
1
The mechanism:
Wave pinning results from the depletion of
the inactive form and the fact that
conservation of total amount of material is
enforced.
Example: Cubic kinetics
Speed can be explicitly computed:
Result:
Dynamics of front and stall position:
Speed of the wave :
Stall position: Restriction for WP to occur:
For wave to stall inside the domain: This can only happen if: How does wave-pinning depend
on parameters ? •  For D> 0, 1< K <3, and ε
sufficiently small, there is a stable
front solution. As ε increases, shape
and stall position change.
•  For ε > εc(D,K) this solution no
longer exists.
wave-pinning loss
Total amount
No Pinning
Pinning
ε
Diffusion Du,
Domain size L
reaction rate Implications
Total amount
No Pinning
If membrane diffusion too
fast, or domain too small
or reaction rate too slow,
no wave-pinning so cell
can’t polarize.
Pinning
ε
Diffusion Du,
Domain size L
reaction rate