VIII Local excitation, global inhibition model
In this lecture we will discuss one of the most frequently used theories to model
biological reactions limited by diffusion. Turing was the first to formulate this problem
mathematically. Gierer and Meinhardt took Turing’s formalisms and applied it to
biological problems. The model described below is therefore often called ‘the TuringGierer-Meinhardt theory’. One of the first models was defined as follows:
∂a
∂ 2a
a2
= ra + ka
− γ a a + Da 2
∂t
∂x
i
2
∂i
∂i
= ki a 2 − γ ii + Di 2
∂t
∂x
[VIII.1]
a and i are the concentrations of an activator and an inhibitor. The activator is produced at
a constant rate ra (for example, a leaky promoter) plus a rate that depends on both a and i.
The activator operates as a dimer whereas the inhibitor operates as a monomer. The
synthesis rate of a is proportional to the saturation function Y to reflects the probability to
have an activator dimer present and an inhibitor monomer absent:
Y=
Kaa2 
K ii 
Kaa2
1
a2


1
=
≈
const
×
−
1 + K a a 2  1 + K ii  1 + K a a 2 1 + K ii
i
[VIII.2]
The approximation above is valid for small concentrations of a and large concentrations
of i. Ka and Ki are the association constants for the activator and inhibitor binding.
Both the activator and inhibitor molecules obey first order decay characterized by γa and
γI, respectively. The last term in equations [VIII.1] reflects the one-dimensional diffusion.
The diffusion coefficient of the activator and inhibitor are Da and Di. The inhibitor is
synthesized at a rate proportional to a2. Apparently the activator enhances the inhibitor
synthesis according to the Hill equation with nH=2. Note that again it is assumed that the
concentration of a is small (with respect to Ka). The system of equations [VIII.1] is an
example of a reaction-diffusion system. Let us analyze [VIII.1] in more detail.
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Dimensionless variables
First we will introduce dimensionless variables to make the algebra simpler. Time will be
normalized to (γa)-1 and the spatial coordinate x will be measured in terms of the distance
that the activator will diffuse in time (γa)-1:
τ ≡ γ at
x
s≡
[VIII.3]
Da / γ a
Substituting this in [VIII.1] yields:
∂a ra ka a 2
∂ 2a
=
+
−a+ 2
∂τ γ a γ a i
∂s
∂i ki 2 γ i
D ∂ 2i
= a − i+ i 2
∂τ γ a
γa
Da ∂s
[VIII.4]
We still have the freedom to the normalize a and i:
A≡
a
ao
[VIII.5]
i
I≡
io
Substituting this in [VIII.4] gives:
∂A
r
k a A2
∂2 A
= a + a o
− A+ 2
∂τ γa ao γaio I
∂s
∂I ki ao2 2 γi
Di ∂ 2 I
=
A − I+
∂τ γaio
γa
Da ∂s 2
[VIII.6]
By choosing
ra
r
= 1 → ao = a
γ a ao
γa
ki ao2 γ i
k a2 k r 2
=
→ io = i o = i a2
γ aio γ a
γi
γ iγ a
[VIII.7]
[VIII.6] finally reduces to:
∂A
A2
∂2 A
=1+ R
− A+ 2
∂τ
I
∂s
2
∂I
∂ I
= Q A2 − I + P 2
∂τ
∂s
(
44
)
[VIII.8]
7.81/8.591/9.531 Systems Biology – A. van Oudenaarden – MIT– November 2004
where
P=
Q=
R=
Di
Da
γi
γa
k aγ i
[VIII.9]
ra ki
P and Q are a measure of the diffusion coefficient and decay rate relative to the activator
properties. R is the ratio of the autocatalytic activator synthesis (ka) and the constant
synthesis (‘leakyness’, ra).
Spatially homogeneous solutions
Version [VIII.8] only contain three parameters (P,Q,R) whereas [VIII.1] contained 9
parameters. For now let’s try to find a solution to [VIII.8] that is spatially homogeneous
( ∂ / ∂s = 0 ). In this case, the steady state solution is:
A = R +1
[VIII.10]
I = ( R + 1) 2
Is this solution stable? The stability matrix evaluated at the fixed point is:
R 
 2 RA
RA 2   R − 1
−
1
−
−
 I
2  = 
( R + 1) 2 
I   R +1


− Q  2( R + 1)Q
−Q 
 2AQ
[VIII.11]
Therefore [VIII.10] represents a stable solution when the trace of this matrix is negative
and the determinant is positive:
R −1
<Q
R +1
Q>0
[VIII.12]
As Q is positive by definition [VIII.9] the first inequality characterizes a unique constant
solution that is stable against small spatially homogeneous perturbations.
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Spatially inhomogeneous solutions
Now consider the full system of equations [VIII.8] and explore stability of the system
around the homogeneous solution [VIII.10]:
A( s,τ ) = A + A' ( s,τ )
[VIII.13]
I ( s,τ ) = I + I ' ( s,τ )
Based on [VIII.8] we can write:
R
∂A' R − 1
∂ 2 A'
A'−
I
=
+
'
∂τ R + 1
∂s 2
(1 + R) 2
∂I '
∂2I '
= 2Q(1 + R) A'−QI '+ P 2
∂s
∂τ
[VIII.14]
To find the solutions of [VIII.14] we start by guessing a trial function:
s
A' ( s,τ ) = Aˆ (τ ) cos( )
l
s
I ' ( s,τ ) = Iˆ(τ ) cos( )
l
[VIII.15]
This trial function makes sense since the second derivative of the cosine is the cosine
itself times a constant. Substituting the trial function gives:
dAˆ  R − 1 1  ˆ
R
Iˆ
− 2 A −
=
dτ  R + 1 l 
(1 + R) 2
dIˆ
P

= 2Q(1 + R) Aˆ −  Q + 2  Iˆ
l 
dτ

[VIII.16]
Again this is system of two linear ordinary differential equations that we can test for
stability:
P  2QR
 R − 1 1 
−
− 2  Q + 2  +
>0
l  1+ R
 R + 1 l 
P   R −1 1 

− Q + 2  + 
− <0
l   R + 1 l2 

[VIII.17]
The latter inequality holds since we impose stability of the homogeneous solution (see
inequality [VIII.12]). The former inequality can be written as:
 Q  4  Q R + 1 2
 l +  −
l + 1 > 0
P
 P R −1
[VIII.18]
For any wavelength l this is satisfied if:
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7.81/8.591/9.531 Systems Biology – A. van Oudenaarden – MIT– November 2004
Q R −1
>
P R +1
[VIII.19]
Both [VIII.19] and [VIII.12] will certainly hold if P < 1. In other words if the activator
diffuses faster than the inhibitor. In the opposite way, spatial nonuniformity and the
possibility of pattern formation requires the inhibitor to diffuse faster than the activator
(long range inhibition, short range activation).
Conditions for inhomogeneous instability
Let us explore the conditions for inhomogeneous instability. To obtain an instable system
we have to obey the reverse of [VIII.19]:
Q
 Q R + 1
f (α ) ≡  α 2 +  −
α + 1 < 0
P
 P R −1
[VIII.20]
where α ≡ l 2 . The function f(α) is a parabola that is concave upward. For f(α) to be
negative the parabola should have two real roots of which at least one is positive. The
roots of f(α) are found by solving:
2
Q R +1 Q
Q
 R +1 Q 
− ± 
−  −4
2α =
P R −1 P
P
 R −1 P 
[VIII.21]
The roots of [VIII.21] are therefore both real if:
2
Q
 R +1 Q 
−  >4

P
 R −1 P 
[VIII.22]
According to [VIII.19] this means that both roots are positive and therefore:
R −1
Q Q
>2
+
R +1
P P
[VIII.23]
This is the instability condition. So if [VIII.23] is true, f(α) will be negative for certain
values of α. Condition [VIII.23] defines a critical value for R. For R>Rc [VIII.23] holds.

1 +  2

Rc =

1 −  2

47
Q Q
+ 
P P 
Q Q
+ 
P P 
[VIII.24]
7.81/8.591/9.531 Systems Biology – A. van Oudenaarden – MIT– November 2004
In the following discussion we will assume strong autocatalytic reaction (R>>1). In that
case [VIII.23] reduces to:
Q
Q
+2
<1
P
P
[VIII.25]
This means that
1/ 2
Q  Da / γ a 

=
P  Di / γ i 
< 0.4
[VIII.26]
The ratio ( Da / γ a )1 / 2 has units of length and can be interpreted as the a typical distance
that activator molecules diffusion before they decay. This distance is often called the
activator range. Equation [VIII.26] tells us that it is necessary for the instability of the
uniform state that the inhibition range is about 2.5 times larger than the activator range.
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