Design principles in pattern formation: Robustness and equivalences

Design principles in pattern formation: Robustness and
equivalences
Michael P.H. Stumpf
Theoretical Systems Biology, Melbourne Integrative Genomics
School of BioSciences &
School of Mathematics and Statistics

Engineering Turing Patterns
Development-
tal Biology
Systems
Biology
Synthetic
Biology
Mathematical
Modelling
James Briscoe
Michael Stumpf Heike Siebert
Mark Isalan
Statistical
Inference
Model
Reduction
Molecular
Biology
Control
Engineering
Multi-scale
Modelling
Genome
Editing
In Silico
Design
Stem Cell
Biology
Bio-
informatics
Natalie Scholes
David Schnoerr
Sean Vittadello

Turing Patterns
Alan Turing
A B
Local Activation & Lateral Inhibition

Turing Patterns are Robust
Gierer-Meinhardt Model
(A)
(B) (D)
(C)
Vittadello et al., Phil Trans Roy Soc B (2021); Leyshon et al., J Theo Biol (2021).

Positional Information ( aka French Flag Model)
n
or
bition
Concentration
T1
T2
Concentration
Space
Positional information creating
the French Flag pattern
B Wolpert – PI
i
Green & Sharpe, Development (2015)
Lewis Wolpert

Why the Acrimony?
Stumpf†, Nature (1966) Raspopovic et al., Science (2014)
†Hildegard F. Stumpf (1924-1967) – No relation.

Turing Patterns vs
Positional Information
The simple dichotomy is wrong. There is
room and need for both mechanisms.
We are interested in
The design principles of Turing Pat-
terns.
The design principles of Positional
Information Mechanisms.
The differences between the two
sets of design principles.
Concentration
Space
Concentration
Space
Concentration
Space
Concentration
Space
Concentration
Space
Auto-activation
Activation
of inhibitor
Lateral inhibition
of activator
Diffusion
of inhibitor
Consistent ‘wavelength’ controlled by global parameters
i
ii
iii
iv
v
Molecular
fluctuations
Concentration
T1
T2
Concentration
Space
the French Flag pattern
T7
Concentration
Space
T6
T5
T4
T3
T2
T1
B Wolpert – PI
i
ii
a periodic pattern
Green & Sharpe, Development (2015)

Turing Generating Mechanisms and Their Properties
We considered
> 5,000 network structures.
> 1011
parameter combina-
tions.
We sampled neighbouring
4–node motifs.
Scholes et al., Cell Systems (2019)

Turing Generating Mechanisms and Their Properties
We need two ingredients:
Non-spatial dynamics allow for a non-trivial stationary state.
Spatial dynamics (≡ spatial dynamics but with diffusion of molecules turned on) are unstable.
C
A
“Fast inhibitor”
“Slow activator”
Non-
homogeneous
initial state
Reaction &
Diffusion
Pattern
formation
time
time
1 A B 2 A B 3 A B 4 A B 5
8 A B 9 A B 10 A B 11 A B 12
15 A B 16 A B 17 A B 18 A B 19
E Non-Competitive
time
activating
inhibiting
D Topology
B
@B
@t
= VB ·
1
1 + (kAB
A )n
+ bB µBB (2)
@A
@t
= VA ·
( A
kAA
)n
1 + ( A
kAA
)n + ( B
kBA
)n
+ bA µAA (3)
@B
@t
= VB ·
( A
kAB
)n
1 + ( A
kAB
)n
+ bB µBB (4)
J =
2
4
@fA
@A D1q2 @fA
@B
@fB
@A
@fB
@B D2q2
3
5 (5)
@A
@t
= ↵ + ·
A2
B
µAA (6)
@B
@t
= · A2
µBB (7)
2
@B
@t
= VB ·
1
1 + (kAB
A )n
+ bB µBB (2)
@A
@t
= VA ·
( A
kAA
)n
1 + ( A
kAA
)n + ( B
kBA
)n
+ bA µAA (3)
@B
@t
= VB ·
( A
kAB
)n
1 + ( A
kAB
)n
+ bB µBB (4)
J =
2
4
@fA
@A D1q2 @fA
@B
@fB
@A
@fB
@B D2q2
3
5 (5)
@A
@t
= ↵ + ·
A2
B
µAA (6)
@B
@t
= · A2
µBB (7)
2

B
2-Node:
3-Node:
0
1
-1
Reduce according to symmetry
Only consider connectednetworks
…
Adjacency matrix
representation
Translating networks into ODE equations
1 1
−1 0
C Estimating Turing Pattern capability
Competitive
Non-Competitive
Graph
representation
Parameter Range
V 0.1-100
Specify parameter boundaries Stability analysis:
A B
A B
1
A B
2
A B
3
A B
4
A
5
A B
6
A B
7
A B
8
A B
9
A
10
A B
11
A B
12
A B
13
A B
14
A
15
A B
16
A B
17
A B
18
A B
19
A
20
A B
21
A
B C
composes the central node and mediates the signals between both HGF and NK4 (see Fig 7 C).
With this model, the network equals the core topology #28 with nodes B and C diffusing. Indeed,
this topology is amongst the top 10 % of the most robust topologies according to our results. Even
so, the classical requirements found for the 2-node system persist HGF and NK4 are more likely to
diffuse at quite similar rates, making this network only implementable if HGF could be modified
to diffuse much more slowly. To decrease the necessity of differential diffusion, and maintain the
original single promoter design, the network atlas reveals that including a positive feedback loop
on top of the receptor would suffice to make equal diffusivity accessible for given network (# 63
BC). This, however, though improving robustness with respect to extracellular parameters does
not improve intracellular robustness significantly and thus in total, robustness is only marginally
improved (1.3 fold). To significantly improve the design (4.6 fold), an additional direct interaction
between HGF and NK4 would have to be engineered (NK4 activating HGF). Overall, this shows
how the network atlas can help decipher engineering options.
2 formulas
@A
@t
= VA ·
1
1 + (kAA
A )n
·
1
1 + ( B
kBA
)n
+ bA µAA (1)
@B
@t
= VB ·
1
1 + (kAB
A )n
+ bB µBB (2)
@A
@t
= VA ·
( A
kAA
)n
1 + ( A
kAA
)n + ( B
kBA
)n
+ bA µAA (3)
@B
@t
= VB ·
( A
kAB
)n
1 + ( A
kAB
)n
+ bB µBB (4)
2@fA
D q2 @fA
3
so, the classical requirements found for the 2-node system persist HGF and NK4 are more likely t
diffuse at quite similar rates, making this network only implementable if HGF could be modifie
to diffuse much more slowly. To decrease the necessity of differential diffusion, and maintain th
original single promoter design, the network atlas reveals that including a positive feedback loo
BC). This, however, though improving robustness with respect to extracellular parameters do
not improve intracellular robustness significantly and thus in total, robustness is only marginal
improved (1.3 fold). To significantly improve the design (4.6 fold), an additional direct interactio
between HGF and NK4 would have to be engineered (NK4 activating HGF). Overall, this show
2 formulas
@A
@t
= VA ·
1
1 + (kAA
A )n
·
1
1 + ( B
kBA
)n
+ bA µAA (1
@B
@t
= VB ·
1
1 + (kAB
A )n
+ bB µBB (2
@A
@t
= VA ·
( A
kAA
)n
1 + ( A
kAA
)n + ( B
kBA
)n
+ bA µAA (3
@B
@t
= VB ·
( A
kAB
)n
1 + ( A
kAB
)n
+ bB µBB (4
J =
2
4
@fA
@A D1q2 @fA
@B
@fB
@A
@fB
@B D2q2
3
5 (5
2 formulas
@A
@t
= VA ·
1
1 + (kAA
A )n
·
1
1 + ( B
kBA
)n
+ bA µAA (1)
@B
@t
= VB ·
1
1 + (kAB
A )n
+ bB µBB (2)
@A
@t
= VA ·
( A
kAA
)n
1 + ( A
kAA
)n + ( B
kBA
)n
+ bA µAA (3)
@B
@t
= VB ·
( A
kAB
)n
1 + ( A
kAB
)n
+ bB µBB (4)
J =
2
4
@fA
@A D1q2 @fA
@B
@fB
@A
@fB
@B D2q2
3
5 (5)
2 formulas
@A
@t
= VA ·
1
1 + (kAA
A )n
·
1
1 + ( B
kBA
)n
+ bA µAA (
@B
@t
= VB ·
1
1 + (kAB
A )n
+ bB µBB (
@A
@t
= VA ·
( A
kAA
)n
1 + ( A
kAA
)n + ( B
kBA
)n
+ bA µAA (
@B
@t
= VB ·
( A
kAB
)n
1 + ( A
kAB
)n
+ bB µBB (
J =
2
4
@fA
@A D1q2 @fA
@B
@fB
@A
@fB
@B D2q2
3
5 (
@A
@t
= ⇢ ·
A2
B
µAA (
so, the classical requirements found for the 2-node system persist HGF and NK4 are more likely
diffuse at quite similar rates, making this network only implementable if HGF could be modifi
to diffuse much more slowly. To decrease the necessity of differential diffusion, and maintain
original single promoter design, the network atlas reveals that including a positive feedback lo
on top of the receptor would suffice to make equal diffusivity accessible for given network (#
BC). This, however, though improving robustness with respect to extracellular parameters d
not improve intracellular robustness significantly and thus in total, robustness is only margina
improved (1.3 fold). To significantly improve the design (4.6 fold), an additional direct interact
between HGF and NK4 would have to be engineered (NK4 activating HGF). Overall, this sho
2 formulas
@A
@t
= VA ·
1
1 + (kAA
A )n
·
1
1 + ( B
kBA
)n
+ bA µAA
@B
@t
= VB ·
1
1 + (kAB
A )n
+ bB µBB
@A
@t
= VA ·
( A
kAA
)n
1 + ( A
kAA
)n + ( B
kBA
)n
+ bA µAA
@B
@t
= VB ·
( A
kAB
)n
1 + ( A
kAB
)n
+ bB µBB
J =
2
4
@fA
@A D1q2 @fA
@B
@fB
@A
@fB
@B D2q2
3
5
Initial conditions:

B
2-Node:
3-Node:
0
1
-1
…
Adjacency matrix
representation
1 1
−1 0
Competitive
Non-Competitive
Graph
representation
Parameter Range
V 0.1-100
A B
A B
1
A B
2
A B
3
A B
4
A
5
A B
6
A B
7
A B
8
A B
9
A
10
A B
11
A B
12
A B
13
A B
14
A
15
A B
16
A B
17
A B
18
A B
19
A
20
A B
21
A
B C
2 formulas
@A
@t
= VA ·
1
1 + (kAA
A )n
·
1
1 + ( B
kBA
)n
+ bA µAA (1)
@B
@t
= VB ·
1
1 + (kAB
A )n
+ bB µBB (2)
@A
@t
= VA ·
( A
kAA
)n
1 + ( A
kAA
)n + ( B
kBA
)n
+ bA µAA (3)
@B
@t
= VB ·
( A
kAB
)n
1 + ( A
kAB
)n
+ bB µBB (4)
2@fA
D q2 @fA
3
2 formulas
@A
@t
= VA ·
1
1 + (kAA
A )n
·
1
1 + ( B
kBA
)n
+ bA µAA (1
@B
@t
= VB ·
1
1 + (kAB
A )n
+ bB µBB (2
@A
@t
= VA ·
( A
kAA
)n
1 + ( A
kAA
)n + ( B
kBA
)n
+ bA µAA (3
@B
@t
= VB ·
( A
kAB
)n
1 + ( A
kAB
)n
+ bB µBB (4
J =
2
4
@fA
@A D1q2 @fA
@B
@fB
@A
@fB
@B D2q2
3
5 (5
2 formulas
@A
@t
= VA ·
1
1 + (kAA
A )n
·
1
1 + ( B
kBA
)n
+ bA µAA (1)
@B
@t
= VB ·
1
1 + (kAB
A )n
+ bB µBB (2)
@A
@t
= VA ·
( A
kAA
)n
1 + ( A
kAA
)n + ( B
kBA
)n
+ bA µAA (3)
@B
@t
= VB ·
( A
kAB
)n
1 + ( A
kAB
)n
+ bB µBB (4)
J =
2
4
@fA
@A D1q2 @fA
@B
@fB
@A
@fB
@B D2q2
3
5 (5)
2 formulas
@A
@t
= VA ·
1
1 + (kAA
A )n
·
1
1 + ( B
kBA
)n
+ bA µAA (
@B
@t
= VB ·
1
1 + (kAB
A )n
+ bB µBB (
@A
@t
= VA ·
( A
kAA
)n
1 + ( A
kAA
)n + ( B
kBA
)n
+ bA µAA (
@B
@t
= VB ·
( A
kAB
)n
1 + ( A
kAB
)n
+ bB µBB (
J =
2
4
@fA
@A D1q2 @fA
@B
@fB
@A
@fB
@B D2q2
3
5 (
@A
@t
= ⇢ ·
A2
B
µAA (
2 formulas
@A
@t
= VA ·
1
1 + (kAA
A )n
·
1
1 + ( B
kBA
)n
+ bA µAA
@B
@t
= VB ·
1
1 + (kAB
A )n
+ bB µBB
@A
@t
= VA ·
( A
kAA
)n
1 + ( A
kAA
)n + ( B
kBA
)n
+ bA µAA
@B
@t
= VB ·
( A
kAB
)n
1 + ( A
kAB
)n
+ bB µBB
J =
2
4
@fA
@A D1q2 @fA
@B
@fB
@A
@fB
@B D2q2
3
5
Initial conditions:
C
qmax
2π /qmax
A
Non-
homogeneous
initial state
Reaction &
Diffusion
Pattern
formation
time
time
1 A B 2 A B 3 A B 4 A B 5 A B 6 A B 7
8 A B 9 A B 10 A B 11 A B 12 A B 13 A B 14
15 A B 16 A B 17 A B 18 A B 19 A B 20 A B 21
E
Competitive
Non-Competitive
time
activating
inhibiting
D Topology
Regulatory function
Function parameters
Diffusion constants
B
@B
@t
= VB ·
( A
kAB
)n
1 + ( A
kAB
)n
+ bB µBB (4)
J =
2
4
@fA
@A D1q2 @fA
@B
@fB
@A
@fB
@B D2q2
3
5 (5)
@A
@t
= ↵ + ·
A2
B
µAA (6)
@B
@t
= · A2
µBB (7)
2
@B
@t
= VB ·
(kAB
)
1 + ( A
kAB
)n
+ bB µBB (4)
J =
2
4
@fA
@A D1q2 @fA
@B
@fB
@A
@fB
@B D2q2
3
5 (5)
@A
@t
= ↵ + ·
A2
B
µAA (6)
@B
@t
= · A2
µBB (7)
2

B
2-Node:
3-Node:
0
1
-1
…
Adjacency matrix
representation
1 1
−1 0
Competitive
Non-Competitive
Graph
representation
Parameter Range
V 0.1-100
A B
A B
1
A B
2
A B
3
A B
4
A
5
A B
6
A B
7
A B
8
A B
9
A
10
A B
11
A B
12
A B
13
A B
14
A
15
A B
16
A B
17
A B
18
A B
19
A
20
A B
21
A
B C
2 formulas
@A
@t
= VA ·
1
1 + (kAA
A )n
·
1
1 + ( B
kBA
)n
+ bA µAA (1)
@B
@t
= VB ·
1
1 + (kAB
A )n
+ bB µBB (2)
@A
@t
= VA ·
( A
kAA
)n
1 + ( A
kAA
)n + ( B
kBA
)n
+ bA µAA (3)
@B
@t
= VB ·
( A
kAB
)n
1 + ( A
kAB
)n
+ bB µBB (4)
2@fA
D q2 @fA
3
2 formulas
@A
@t
= VA ·
1
1 + (kAA
A )n
·
1
1 + ( B
kBA
)n
+ bA µAA (1
@B
@t
= VB ·
1
1 + (kAB
A )n
+ bB µBB (2
@A
@t
= VA ·
( A
kAA
)n
1 + ( A
kAA
)n + ( B
kBA
)n
+ bA µAA (3
@B
@t
= VB ·
( A
kAB
)n
1 + ( A
kAB
)n
+ bB µBB (4
J =
2
4
@fA
@A D1q2 @fA
@B
@fB
@A
@fB
@B D2q2
3
5 (5
2 formulas
@A
@t
= VA ·
1
1 + (kAA
A )n
·
1
1 + ( B
kBA
)n
+ bA µAA (1)
@B
@t
= VB ·
1
1 + (kAB
A )n
+ bB µBB (2)
@A
@t
= VA ·
( A
kAA
)n
1 + ( A
kAA
)n + ( B
kBA
)n
+ bA µAA (3)
@B
@t
= VB ·
( A
kAB
)n
1 + ( A
kAB
)n
+ bB µBB (4)
J =
2
4
@fA
@A D1q2 @fA
@B
@fB
@A
@fB
@B D2q2
3
5 (5)
2 formulas
@A
@t
= VA ·
1
1 + (kAA
A )n
·
1
1 + ( B
kBA
)n
+ bA µAA (
@B
@t
= VB ·
1
1 + (kAB
A )n
+ bB µBB (
@A
@t
= VA ·
( A
kAA
)n
1 + ( A
kAA
)n + ( B
kBA
)n
+ bA µAA (
@B
@t
= VB ·
( A
kAB
)n
1 + ( A
kAB
)n
+ bB µBB (
J =
2
4
@fA
@A D1q2 @fA
@B
@fB
@A
@fB
@B D2q2
3
5 (
@A
@t
= ⇢ ·
A2
B
µAA (
2 formulas
@A
@t
= VA ·
1
1 + (kAA
A )n
·
1
1 + ( B
kBA
)n
+ bA µAA
@B
@t
= VB ·
1
1 + (kAB
A )n
+ bB µBB
@A
@t
= VA ·
( A
kAA
)n
1 + ( A
kAA
)n + ( B
kBA
)n
+ bA µAA
@B
@t
= VB ·
( A
kAB
)n
1 + ( A
kAB
)n
+ bB µBB
J =
2
4
@fA
@A D1q2 @fA
@B
@fB
@A
@fB
@B D2q2
3
5
Initial conditions:
C
qmax
2π /qmax
A
Non-
homogeneous
initial state
Reaction &
Diffusion
Pattern
formation
time
time
1 A B 2 A B 3 A B 4 A B 5 A B 6 A B 7
8 A B 9 A B 10 A B 11 A B 12 A B 13 A B 14
15 A B 16 A B 17 A B 18 A B 19 A B 20 A B 21
E
Competitive
Non-Competitive
time
activating
inhibiting
D Topology
Regulatory function
Function parameters
Diffusion constants
B
@B
@t
= VB ·
( A
kAB
)n
1 + ( A
kAB
)n
+ bB µBB (4)
J =
2
4
@fA
@A D1q2 @fA
@B
@fB
@A
@fB
@B D2q2
3
5 (5)
@A
@t
= ↵ + ·
A2
B
µAA (6)
@B
@t
= · A2
µBB (7)
2
@B
@t
= VB ·
(kAB
)
1 + ( A
kAB
)n
+ bB µBB (4)
J =
2
4
@fA
@A D1q2 @fA
@B
@fB
@A
@fB
@B D2q2
3
5 (5)
@A
@t
= ↵ + ·
A2
B
µAA (6)
@B
@t
= · A2
µBB (7)
2
B
A Defining network structure
2-Node:
3-Node:
0
1
-1
…
Adjacency matrix
representation
1 1
−1 0
Competitive
Non-Competitive
Graph
representation
A B
A B
1
A B
2
A B
3
A B
4
A
A B
6
A B
7
A B
8
A B
9
A
A B
11
A B
12
A B
13
A B
14
A
A B
16
A B
17
A B
18
A B
19
A
A B
21
A
B C
MMP-1 promoter construct. In a previous study, this system was analyzed as a classical 2-node
system and the authors suggested that a single promoter construct driving the expression of both
activating and inhibiting species could suffice to create Turing Patterns [?]. This system would,
however, require differential diffusion (10 fold) and a co-operativity factor of >= 2 for feasible
Turing pattern generation. In reality, the network can be seen as a 3-node network in which c-Met
2 formulas
@A
@t
= VA ·
1
1 + (kAA
A )n
·
1
1 + ( B
kBA
)n
+ bA µAA (1)
@B
@t
= VB ·
1
1 + (kAB
A )n
+ bB µBB (2)
@A ( A
k )n
Turing pattern generation. In reality, the network can be seen as a 3-node network in which
composes the central node and mediates the signals between both HGF and NK4 (see Fig
With this model, the network equals the core topology #28 with nodes B and C diffusing. In
this topology is amongst the top 10 % of the most robust topologies according to our results.
so, the classical requirements found for the 2-node system persist HGF and NK4 are more lik
diffuse at quite similar rates, making this network only implementable if HGF could be mo
to diffuse much more slowly. To decrease the necessity of differential diffusion, and mainta
original single promoter design, the network atlas reveals that including a positive feedback
on top of the receptor would suffice to make equal diffusivity accessible for given network
BC). This, however, though improving robustness with respect to extracellular parameter
not improve intracellular robustness significantly and thus in total, robustness is only marg
improved (1.3 fold). To significantly improve the design (4.6 fold), an additional direct inter
between HGF and NK4 would have to be engineered (NK4 activating HGF). Overall, this
2 formulas
@A
@t
= VA ·
1
1 + (kAA
A )n
·
1
1 + ( B
kBA
)n
+ bA µAA
@B
@t
= VB ·
1
1 + (kAB
A )n
+ bB µBB
@A
@t
= VA ·
( A
kAA
)n
1 + ( A
kAA
)n + ( B
kBA
)n
+ bA µAA
A n
Turing pattern generation. In reality, the network can be seen as a 3-node network in which c-Met
2 formulas
@A
@t
= VA ·
1
1 + (kAA
A )n
·
1
1 + ( B
kBA
)n
+ bA µAA (1)
@B
@t
= VB ·
1
1 + (kAB
A )n
+ bB µBB (2)
@A
@t
= VA ·
( A
kAA
)n
1 + ( A
kAA
)n + ( B
kBA
)n
+ bA µAA (3)
so, the classical requirements found for the 2-node system persist HGF and NK4 are more lik
diffuse at quite similar rates, making this network only implementable if HGF could be mo
to diffuse much more slowly. To decrease the necessity of differential diffusion, and mainta
original single promoter design, the network atlas reveals that including a positive feedbac
BC). This, however, though improving robustness with respect to extracellular parameter
not improve intracellular robustness significantly and thus in total, robustness is only marg
improved (1.3 fold). To significantly improve the design (4.6 fold), an additional direct inter
between HGF and NK4 would have to be engineered (NK4 activating HGF). Overall, this
2 formulas
@A
@t
= VA ·
1
1 + (kAA
A )n
·
1
1 + ( B
kBA
)n
+ bA µAA
@B
@t
= VB ·
1
1 + (kAB
A )n
+ bB µBB
@A
@t
= VA ·
( A
kAA
)n
1 + ( A
kAA
)n + ( B
kBA
)n
+ bA µAA
@B
@t
= VB ·
( A
kAB
)n
1 + ( A
kAB
)n
+ bB µBB
2 3
Turing pattern generation. In reality, the network can be seen as a 3-node network in which
composes the central node and mediates the signals between both HGF and NK4 (see Fi
With this model, the network equals the core topology #28 with nodes B and C diffusing.
this topology is amongst the top 10 % of the most robust topologies according to our result
so, the classical requirements found for the 2-node system persist HGF and NK4 are more l
diffuse at quite similar rates, making this network only implementable if HGF could be m
to diffuse much more slowly. To decrease the necessity of differential diffusion, and maint
original single promoter design, the network atlas reveals that including a positive feedba
BC). This, however, though improving robustness with respect to extracellular paramete
not improve intracellular robustness significantly and thus in total, robustness is only mar
improved (1.3 fold). To significantly improve the design (4.6 fold), an additional direct inte
between HGF and NK4 would have to be engineered (NK4 activating HGF). Overall, thi
2 formulas
@A
@t
= VA ·
1
1 + (kAA
A )n
·
1
1 + ( B
kBA
)n
+ bA µAA
@B
@t
= VB ·
1
1 + (kAB
A )n
+ bB µBB
@A
@t
= VA ·
( A
kAA
)n
1 + ( A
kAA
)n + ( B
kBA
)n
+ bA µAA
A n

Turing Pattern are CommonA
B
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
complexity
activating inhibiting
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
C D Non-Competitive Competitive
ID
8
Level 1 1 2 3
61% of 3-node networks can generate
Turing Patterns.
But only 5 out of 21 2-node networks
can generate Turing Patterns

Turing Pattern are CommonA
B
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
complexity
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
C D Non-Competitive Competitive
ID
8
Level 1 1 2 3
61% of 3-node networks can generate
Turing Patterns.
But only 5 out of 21 2-node networks
can generate Turing Patterns
B
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
complexity
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21

2-Node Turing Pattern Generating
Mechanisms
B
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
complexity
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
C D
complexity
A B A B
Non-Competitive Competitive
ID
8
9
15
16
20
Level 1
Level 2
Level 3
1 2
4 5
3
6
7 8 9 10 11
12 13 17 14 16
15 20 18 19 21
No pattern
No pattern
C D
complexity
A B A B
ID
8
9
15
16
20
Level 1
Level 2
Level 3
1 2
4 5
3
6
7 8 9 10 11
12 13 17 14 16
15 20 18 19 21
No pattern
No pattern

2-Node Turing Pattern Generating
Mechanisms
B
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
complexity
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
C D
complexity
A B A B
ID
8
9
15
16
20
Level 1
Level 2
Level 3
1 2
4 5
3
6
7 8 9 10 11
12 13 17 14 16
15 20 18 19 21
No pattern
No pattern
C D
complexity
A B A B
ID
8
9
15
16
20
Level 1
Level 2
Level 3
1 2
4 5
3
6
7 8 9 10 11
12 13 17 14 16
15 20 18 19 21
No pattern
No pattern
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
complexity
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
C D
complexity
A B A B
ID
8
9
15
16
20
Level 1
Level 2
Level 3
1 2
4 5
3
6
7 8 9 10 11
12 13 17 14 16
15 20 18 19 21
No pattern
No pattern

Turing Pattern are Common
We consider all possible 2– and
3–node networks and identify
shared core topologies that are
capable of generating Turing pat-
terns.
3
activating
inhibiting
-
D
A
B C
3774
A
B C
3772
B C
4564
B C
4147
B C
1291
B C
733
B C
67
4030 1372 475 126
A
B C
4147
A
B C
1291
A
B C
733
A
B C
67
high
low
concentration
A
B
C
Scholes et al., Cell Systems (2019)
A
B C
49
A
B C
50
A
B C
52
A
B C
53
A
B C
65
A
B C
86
A
B C
87
A
B C
93
A
B C
131
A
B C
132
A
B C
16
A
B C
17
A
B C
64
A
B C
67
A
B C
68
A
B C
92
A
B C
119
A
B C
120
A
B C
49
A
B C
50
A
B C
52
A
B C
53
A
B C
65
A
B C
86
A
B C
87
A
B C
93
A
B C
131
A
B C
132
A
B C
16
A
B C
17
A
B C
64
A
B C
67
A
B C
68
A
B C
92
A
B C
119
A
B C
120
A
B C
46
A
B C
47
A
B C
83
A
B C
84
A
B C
125
A
B C
126
A
B
Competitive Turing topologies
Non-Competitive Turing topologies
In-Phase Core Topologies
Out-of-Phase Core Topologies

Turing Pattern are Common, but Robust They are Not
A
B

Turing Pattern are Common, but Robust They are Not
A
B
Ann Babtie Paul Kirk

Evolutionary Robustness of Turing Pattern Generators
B
C
3
4
Top 10 Non-Competitive Networks:
activating
inhibiting
-
D
A
B C
4030
A
B C
1372
A
B C
1528
A
B C
4491
A
B C
3774
A
B C
3775
A
B C
4300
A
B C
3881
A
B C
1331
A
B C
3772
A
B C
5651
A
B C
4564
A
B C
4147
A
B C
1291
A
B C
733
A
B C
67
B C
5703
B C
4491
B C
4030
B C
1372
B C
475
B C
126
A
B C
5651
A
B C
4564
A
B C
4147
A
B C
1291
A
B C
733
A
B C
67
0
high
low
concentration
A
B
C
A
B C
A
B
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
complexity
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
C D
complexity
A B A B
ID
8
9
15
16
20
Level 1
Level 2
Level 3
1 2
4 5
3
6
7 8 9 10 11
12 13 17 14 16
15 20 18 19 21
No pattern
No pattern
B
C D
Top 10 Competitive Networks:
A
B C
1291
A
B C
733
A
B C
725
A
B C
1807
A
B C
4147
A
B C
1261
A
B C
1423
A
B C
726
A
B C
1474
A
B C
2924
B C
5651
B C
4564
B C
4147
0
A
B
C
A
B C
Frequency vs Robustness
There are more non-competitive than competitive networks that can generate Turing patterns.
However, competitive networks tend to be more robust.

Evolutionary Robustness of Turing Pattern Generators
B
C
3
4
activating
inhibiting
-
D
A
B C
4030
A
B C
1372
A
B C
1528
A
B C
4491
A
B C
3774
A
B C
3775
A
B C
4300
A
B C
3881
A
B C
1331
A
B C
3772
A
B C
5651
A
B C
4564
A
B C
4147
A
B C
1291
A
B C
733
A
B C
67
B C
5703
B C
4491
B C
4030
B C
1372
B C
475
B C
126
A
B C
5651
A
B C
4564
A
B C
4147
A
B C
1291
A
B C
733
A
B C
67
0
high
low
concentration
A
B
C
A
B C
A
B
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
complexity
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
A B
1
A B
2
A B
3
A B
4
A B
5
A B
6
A B
7
A B
8
A B
9
A B
10
A B
11
A B
12
A B
13
A B
14
A B
15
A B
16
A B
17
A B
18
A B
19
A B
20
A B
21
C D
complexity
A B A B
ID
8
9
15
16
20
Level 1
Level 2
Level 3
1 2
4 5
3
6
7 8 9 10 11
12 13 17 14 16
15 20 18 19 21
No pattern
No pattern
B
C D
Top 10 Competitive Networks:
A
B C
1291
A
B C
733
A
B C
725
A
B C
1807
A
B C
4147
A
B C
1261
A
B C
1423
A
B C
726
A
B C
1474
A
B C
2924
B C
5651
B C
4564
B C
4147
0
A
B
C
A
B C
Frequency vs Robustness
There are more non-competitive than competitive networks that can generate Turing patterns.
However, competitive networks tend to be more robust.
Evolutionary Change
loss of interactions 3
gain of interaction 3
change of interaction 7
change of regulation 3/7
Simple prototypical Turing pattern generating mecha-
nisms are probably frequently encountered, but easily lost
as well.
Core-motifs can be stabilized by additional interactions
with external nodes.

Positional Information in Mathematical Terms
Schaerli et al., Nature Communications (2014)
There is a notable lack of rigorous mathemat-
ical definitions, especially in the early litera-
ture, related to positional information.

Turing Pattern vs Stripe Formation Mechanisms
10 most robust non-competitive networks
B
C
3
4
activating
inhibiting
-
D
A
B C
4030
A
B C
1372
A
B C
1528
A
B C
4491
A
B C
3774
A
B C
3775
A
B C
4300
A
B C
3881
A
B C
1331
A
B C
3772
5651 4564 4147 1291 733 67
A
B C
5651
A
B C
4564
A
B C
4147
A
B C
1291
A
B C
733
A
B C
67
A
B C
5703
A
B C
4491
A
B C
4030
A
B C
1372
A
B C
475
A
B C
126
A
B C
5651
A
B C
4564
A
B C
4147
A
B C
1291
A
B C
733
A
B C
67
0
A
B
C
A
B C
10 most robust competitive networks
activating
inhibiting
D
3
4
5
6
-
op 10 Competitive Networks:
A
B C
1291
A
B C
733
A
B C
725
A
B C
1807
A
B C
4147
A
B C
1261
A
B C
1423
A
B C
726
A
B C
1474
A
B C
2924
B C
5651
B C
4564
B C
4147
B C
1291
B C
733
B C
67
A
B C
5703
A
B C
4491
A
B C
4030
A
B C
1372
A
B C
475
A
B C
126
A
B C
5651
A
B C
4564
A
B C
4147
A
B C
1291
A
B C
733
A
B C
67
A
B C
5703
A
B C
4491
A
B C
4030
A
B C
1372
A
B C
475
A
B C
126
A
B C
5651
A
B C
4564
A
B C
4147
A
B C
1291
A
B C
733
A
B C
67
A
B C
5703
A
B C
4491
A
B C
4030
A
B C
1372
A
B C
475
A
B C
126
A
B C
5651
A
B C
4564
A
B C
4147
A
B C
1291
A
B C
733
A
B C
67
0
Hierarchy of stripe forming networks

How Different are Models?
Positional Information Model
The morphogen concentrations r(x, t) and s(x, t) can be modelled as
𝜕r
𝜕t
= Dr∇2
r − krr − krs
𝜕s
𝜕t
= Ds∇2
s − kss − krs
Turing–Gierer–Meinhardt Model
This model can be represented mathematically as
𝜕r
𝜕t
= Dr∇2
r +
𝜌r2
s(1 + 𝜇rr2)
− krr + 𝜌r
𝜕s
𝜕t
= Ds∇2
s + 𝜌r2
− kss + 𝜌s

How Different are Models?
Positional Information Model
The morphogen concentrations r(x, t) and s(x, t) can be modelled as
𝜕r
𝜕t
= Dr∇2
r − krr − krs
𝜕s
𝜕t
= Ds∇2
s − kss − krs
with Dr, Ds diffusivities, and kr, ks degradation rates (with Neumann boundary conditions).
Turing–Gierer–Meinhardt Model
This model can be represented mathematically as
𝜕r
𝜕t
= Dr∇2
r +
𝜌r2
s(1 + 𝜇rr2)
− krr + 𝜌r
𝜕s
𝜕t
= Ds∇2
s + 𝜌r2
− kss + 𝜌s
with Dr, Ds diffusivities, 𝜌r, 𝜌s basal production rates, kr, ks degradation rates, and 𝜇r a saturation con-
stant; 𝜌 is the source density (zero flux boundary conditions).

Setting Out the Problem
Model Space Dimension 1
M1
M2
M3
M4
M5
M6
M7
M8
M9
M10
M13
M11
M12
M17
M14
M15
M16
M18
M19
M20
Define a metric space, (ℳ, Δ), via a
distance function, Δ(Mi, Mj), over the
set of models, ℳ={M1, … , Mn}.

Setting Out the Problem
M1
M13
M3
M14
M15
M6
M7
M8
M9
M20
M2
M11
M12
M17
M4
M5
M16
M18
M19
M10
Define a metric space, (ℳ, Δ), via a
distance function, Δ(Mi, Mj), over the
set of models, ℳ={M1, … , Mn}.

Distances between Mathematical Models
To our surprise there was not much useful in the sense of distances between models. There were some
candidates:
We are looking for a distance not based on model outputs.
Candidate a priori Distances
Graph distances: Network representations of mathematical models
fail to capture fundamental aspects of the dynamics.
Semantic distances: This is promising route if we can describe differ-
ent models adequately and consistently.
… : There are a host of other candidates; most of these cannot
treat different models to the same degree.

Simplicial Complexes
dx
dt
= x(𝛼 − 𝛽y)
dy
dt
= y(𝛾x − 𝛿)
Prey Prey
growth
Predation Oscillatory
population
Predator Predator
growth
Predator
death
Prey Y Y Y
Prey growth Y Y Y
Predation Y Y Y
Oscillatory popu-
lation
Y Y Y Y Y Y
Predator Y Y Y Y
Predator growth Y Y Y Y
Predator death Y Y Y
Vittadello & Stumpf, Roy Soc Open Sci (2021).

Model Structures
𝜕r
𝜕t
= Dr∇2
r − krr − krs,
𝜕s
𝜕t
= Ds∇2
s − kss − krs,
v1 ⟷ Morphogen 1 (r)
v2 ⟷ Diffusion 1
v3 ⟷ Degradation 1
v6 ⟷ Influx 1
v9 ⟷ Morphogen 2 (s)
v10 ⟷ Diffusion 2
v11 ⟷ Degradation 2
v13 ⟷ Influx 2
v26 ⟷ Annihilation between morphogens 1 and 2
v40 ⟷ Monotonic gradient
v43 ⟷ Global scale-invariance

Model Structures and Topological Data Analysis

Model Distances
There is no clear way to separate
Turing Pattern models
from
Positional Information models.
Are PI and TP models fundamentally different?
Apparently not.
Can we get PI behaviour from a TP model or
vice versa?
Mathematically: Yes
Biologically: ?

Equivalences ``≡"
A more useful concept to study differences is based on the concept of equivalence.
The key here is to identify which aspects that we may regard as essentially identical.
Then we can see if these equivalences correspond to biologically meaningful concepts or
interventions.

Equivalences ``≡"
A more useful concept to study differences is based on the concept of equivalence.
The key here is to identify which aspects that we may regard as essentially identical.
Then we can see if these equivalences correspond to biologically meaningful concepts or
interventions.
Example equivalences between PI and TPG Models
Positional information Turing Pattern
Influx 1 ⟺ {Basal production 1, Self-activation of morphogen 1}
Influx 2 ⟺ Basal production 2
Monotonic gradient ⟺ Periodic gradient field
Annihilation between
morphogens 1 and 2
⟺
{Activation of morphogen 2 by morphogen 1,
Inhibition of morphogen 1 by morphogen 2}

Patterning Mechanisms are Not that Different
It is possible to define a distance between models using simplicial complexes. This distance takes
into account the structure of the network, but also properties of the nodes.
Based on this distance there is no clear way to separate Turing pattern mechanisms and positional
information mechanisms.
Equivalences allow us to determine systematically where differences between models are only
superficial.
Vittadello &Stumpf, arXiv (2021); Scholes & Isalan, Curr Opin Chem Biol (2017).

Scholes & Isalan, Curr Opin Chem Biol (2017)
Scholes et al., Cell Systems (2019) 9:243–257; Vittadello & Stumpf, Roy Soc Open Sci (2021)
8:211361; Leyshon et al., J Theo Biol (2021) 531:110901; Vittadello et al., Phil Trans Roy Soc
B (2021) 379:20200272.
Natalie Scholes
David Schnoerr
Sean Vittadello

Boundary conditions, not motif structures, are the biggest difference
between PI and TP systems.
B (2021) 379:20200272.
Natalie Scholes
David Schnoerr
Sean Vittadello

Mathematical equivalences identify routes for interconversion between
PI systems and TP systems.
B (2021) 379:20200272.
Natalie Scholes
David Schnoerr
Sean Vittadello

Mathematical equivalences identify routes for interconversion between
PI systems and TP systems.
Patterning systems are robust. The mechanism may change, however.
B (2021) 379:20200272.
Natalie Scholes
David Schnoerr
Sean Vittadello

Design principles in pattern formation: Robustness and equivalences

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