Multistationarity in chemical reaction networks
Badal Joshi
Department of Mathematics
California State University, San Marcos
Partial ordering on the set of multistationary CRNs
Chemical reaction network: (S, C, R)
Focus on the question of sufficient conditions for
multistationarity.
Can we define a sense in which a network N1 ⊂contained N2
such that the positive steady states of N1 lift to the positive
steady states of N2 ?
We will focus on each component:
Reactions
Species
Complexes
Reactions: Non-example
Steady states of N do not lift to steady states of G
k1
N:A�B
k2
k1
G :A�B
k2
Clearly N ⊂subnetwork G .
k
A
0→
A
For network N, A + B = c, a constant and the steady state
within
each stoichiometric
compatibility class “c” is given by
�
�
k2 c
k1 c
,
.
k1 + k2 k1 + k2
G has no steady states.
Reactions: Non-example
k1
N:A�B
k2
k1
G :A�B
k2
k
A
0→
A
SG : stoichiometric subspace of network G .
mG : maximum number of steady states that G admits within
a stoichiometric compatibility class.
mG ≥ 2: G admits MSS/ G is multistationary.
We would “like that” if N ⊂subnetwork G , then mN ≤ mG .
Not true for previous example (mN = 1, mG = 0)
Problem: SN = span{(1, −1)} and SG = span{(1, 0), (1, −1)}
so that SN � SG .
Reactions: subnetwork theorem
Theorem (BJ/Shiu)
Let N be a subnetwork of G such that SN = SG . Then, mG ≥ mN .
Corollary
If G is obtained from N by making some of the irreversible
reactions reversible, then mG ≥ mN .
Reactions: sub-CFSTR theorem
CFSTR: networks with all species in outflow.
Let r ∈ G \ N be a (true or nonflow) reaction. Two cases:
case 1. r does not involve any new species. SN = SG .
case 2. r involves new species. Augment N by adding flow reactions of
� S � = SG .
the new species to get N.
N
Corollary
If N is a sub-CFSTR of G , then mG ≥ mN .
Future work: How far can this theorem be stretched?
SN = SG too strong??
What about the steady states themselves? Analytic results in
parameter space??
Species: embedded network
N ⊂embedded G if N can be obtained from G by “removing
reactions” or “removing species”.
Example of species removal:
k1
A�B
k2
2A
+B
k3
� 3A
k4
Species: embedded network
N ⊂embedded G if N can be obtained from G by “removing
reactions” or “removing species”.
Example of species removal:
k1
A ��
B
k2
✟
2A ✟
+B
k3
� 3A
k4
Species: embedded network
N ⊂embedded G if N can be obtained from G by “removing
reactions” or “removing species”.
Example of species removal:
k1
A�0
k2
2A
k3
� 3A
k4
Non-example
G:
k1
0�A
k2
k3
3A � 2A + B
k4
G has unique mass-action steady state
N ⊂embedded G :
k1
0�A
k2
�
�
k1 k1 k3
,
.
k2 k2 k4
k3
3A � 2A
k4
N does admit multiple positive mass-action steady states.
mN > mG !!
Lifting steady states from embedded networks
Q. When does N ⊂embedded G imply that mN ≤ mG ?
Definition
Fx is a flow-type subnetwork of G if (1) Fx only involves one
species x and (2) x = 1 is an admissible nondegenerate steady
state.
Examples of flow-type subnetworks:
1
0 � A.
2
0 � 2A.
3
A � 3A.
4
0�A
,
2A � 3A.
Lifting steady states from embedded networks
Theorem (BJ/Shiu)
N ⊂embedded G such that:
1
2
SN = R|SN |
If x is a species in G \ N, there exists Fx ⊂subnetwork G .
=⇒ mN ≤ mG .
Lifting steady states from embedded networks
Corollary
N ⊂embedded G such that:
1
N is a CFSTR.
2
G is a fully open CFSTR.
=⇒ mN ≤ mG .
Future work: How far can this theorem be pushed?? For
instance, mG ≥ mN for the following network:
G:
A�B
2A + B � 3A
and N is obtained by removing species B.
Atoms of multistationarity
Definition
G is an atom of multistationarity if:
1
2
mG ≥ 2.
There is no N such that mN ≥ 2 and N ⊂embedded G .
Two reaction bimolecular atoms of multistationarity
A �2A
A�B�0
D
A�D�2A
A�B�0
C�2�
C�3�
C
A�C�2A
A�2C
A�2A
A�B�2C
B
D
E
A�C�2A
A�B�2C
A�D�2A
A�B�2C
A�D�2A
A�B�C�E
D�2�
E
A�2A
A�B�C�E
A�D�2A
A�B�D
D�2�
D
C
A�D�2A
A�B�C�D
A�2A
A�2B
A�2A
A�B�C
D
C
A�A�B
2B�A�D
C�2�
A�C�2A
A�2B
D�2�
A�D�2A
A�B�C
C
D
A�C�A�B
2B�A�C
D
D
A�C�A�B
2B�A
C�2�
A�2A
A�B�C
A�D�2A
A�B�C
C�2�
A�A�B
2B�A
C
C
A�C�A�B
2B�A�D
A�C�2A
A�C�2B
A�D�2A
A�D�B�C
B�2A
2A�A�B
A�A�B
A�B�2A
B�A�B
2A�A�B
B�2A
2A�2B
A�A�B
2B�2A
A�B�C
B�C�2A
C
C
C
C
C
D
B�C�2A
2A�A�B
A�C�A�B
A�B�2A
B�C�A�B
2A�A�B
B�C�2A
2A�2B
A�C�A�B
2B�2A
A�D�B�C
B�C�2A
One reaction atoms
Theorem (BJ)
A one-reaction fully open network
Xi � 0
a1 X1 + . . . + an Xn � b1 X1 + . . . + bn Xn , (ai , bi ≥ 0)
admits
steady states if and only if
� multiple�
ai ≥ 2 or
bi ≥ 2
bi >ai
ai >bi
A 1-rxn network is multistationary if and only if it is
autocatalytic in some set of species which have at least two
reactant molecules.
Examples of atoms:
CFSTR examples of atoms:
1
0�A
2
0 � A, B
,
3
0 � A, B
,
4
A 3-reaction atom (all species are in flow):
,
mA → nA,
(n > m ≥ 2)
A + B → mA + nB,
A → 2A
,
(m, n ≥ 2)
A + B → 0.
A+B →0
B +C →0
C → 2A
Examples of atoms:
1
(all species are in flow):
A → 2C
C →B
2B → 2A
2
(all species are in flow):
A+B →0
A+C →0
B →C +D
D →C +E
Q: Is there a characterization of atoms??
Enumeration of atoms?
Definition
A square network has n reactions and n species. The orientation of
a square network N : X → Y is defined to be
Or (N) = det(X ) det(X − Y ).
Theorem (Craciun/Feinberg, Joshi/Shiu)
If mG ≥ 2, then there exists a square N ⊂embedded G such that
Or (N) < 0.
Q: Is there a characterization of square, negatively oriented
networks? [Conjecture on partial ordering of square, negatively
oriented nets]
Square, negatively oriented networks
1
10
20
30
36
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
1
10
20
30
36
Square, negatively oriented networks
1
200
400
600
800
1000
1
1
20
20
40
40
64
64
1
200
400
600
800
1000
Thank you!