Some results on chemical reaction networks
without the assumption of mass action
Murad Banaji
SIAM Life Sciences, San Diego, 9th August 2012
Murad Banaji
Some results on chemical reaction networks without the assump
Outline
1. Some background
2. Some examples
3. Conclusions
Murad Banaji
Some results on chemical reaction networks without the assump
Why not assume mass-action?
◮
A quote from Wikipedia:
“In more complex environments, where bound
particles may be prevented from disassociation by
their surroundings, or diffusion is slow or anomalous,
the model of mass action does not always describe
the behavior of the reaction kinetics accurately”
◮
We would like to know what effects approximations such as
QSSA could have on allowed dynamics.
Some of what can be said about CRNs actually arises from
the weak law “more substrate ⇒ faster reactions”.
Murad Banaji
Some results on chemical reaction networks without the assump
Some possible behaviours
global stability
oscillation
multistationarity
persistence
chaos
local stability
Murad Banaji
Some results on chemical reaction networks without the assump
Reaction vectors and rate derivatives
A
B
C
A →B+C




−1
1
1
reaction
vector




+ 0 0
reaction rate
derivatives Dv
The final row vector means that we expect the reaction rate v to
increase as [A] increases, but to be unaffected by [B] and [C]. I.e.,
∂v
> 0,
∂[A]
∂v
= 0,
∂[B]
Murad Banaji
∂v
= 0.
∂[C]
Some results on chemical reaction networks without the assump
DSR graph of an irreversible reaction
A→ B+C
A
B
C




−1
1
1








−
0
0




−Dv T
Murad Banaji
Some results on chemical reaction networks without the assump
DSR graph of an irreversible reaction
A
B
C
A→ B+C




−1
1
1








−
0
0




−Dv T
B
C
B
A
Murad Banaji
C
=
+
A
B
C
A
DSR graph
Some results on chemical reaction networks without the assump
DSR graph of a reversible reaction
A
B
C
A⇋ B+C




−1
1
1








−
+
+




−Dv T
B
C
B
A
Murad Banaji
C
=
+
A
B
C
A
DSR graph
Some results on chemical reaction networks without the assump
DSR graphs
A ⇋B+C
A→B+C
B
C
B
C
A
DSR graph
A
DSR graph
Murad Banaji
Some results on chemical reaction networks without the assump
A1
A2
A3
B1
B2
A1 + A2 ⇋ B1
A2 + A3 ⇋ B2
A3 ⇋ 2A1










A3 ⇋ 2A1
A2 +A3 ⇋ B2
A1 +A2 ⇋ B1
Reaction networks: the stoichiometric matrix
−1
0
2
−1 −1
0
0 −1 −1
1
0
0
0
1
0










stoichiometric matrix Γ
Murad Banaji
Some results on chemical reaction networks without the assump
Some important objects associated with a CRN



Γ=


A1 + A2 → B1
A2 + A3 ⇋ B2
A3 ⇋ 2A1

−1
0
2
−1 −1
0 

0 −1 −1 

1
0
0 
0
1
0
− Dv

T


=


A2
B1

− 0 +
− − 0 

0 − − 

0 0 0 
0 + 0
B2
GΓ,−Dv T =
A1
Murad Banaji
2
A3
Some results on chemical reaction networks without the assump
DSR graph for a network
A1 + A2 → B1
A2 + A3 ⇋ B2
A3 ⇋ 2A1
A2
B1
A2
B2
A3
A1
A1
2
A3
Murad Banaji
Some results on chemical reaction networks without the assump
DSR graph for a network
A1 + A2 → B1
A2 + A3 ⇋ B2
A3 ⇋ 2A1
A2
B1
A2
B2
A3
A1
A1
2
A3
Murad Banaji
A2
B1
A1
2
B2
A3
Some results on chemical reaction networks without the assump
Dynamical systems from CRNs
rate of reaction j
vj (x)
ẋi =
Γij vj (x)
rate of production of reactant i by reaction j
X
rate of production of reactant i = the sum
of its rates of production in each reaction.
Γij vj (x)
j
So CRNs give dynamical systems of the form:
ẋ = Γ v (x)
reactant
concentrations
The Jacobian has a
natural factorisation
reaction rates
J(x) = Γ Dv (x)
stoichiometric matrix
Murad Banaji
Some results on chemical reaction networks without the assump
Stoichiometry classes
Consider the CRN:
ẋ = Γ v (x).
The vector field lies in Im Γ, the image of Γ:
Thus all evolution takes place on
invariant affine subspaces,
parallel to Im Γ. The
intersections of these sets with
the nonnegative orthant are
termed stoichiometry classes.
These are termed nontrivial if
they intersect the interior of the
nonnegative orthant.
Murad Banaji
Some results on chemical reaction networks without the assump
Jacobian matrices of CRNs
A lot of interesting claims follow from examination of these alone.
For example we might be able to show that for particular Γ and all
allowed rate derivatives −J(x) = −ΓDv (x):
◮
is nonsingular (no saddle-node bifurcations)
◮
is a P-matrix (no multistationarity)
◮
cannot have imaginary eigenvalues (no Hopf bifurcations)
◮
is stable, and hence D-stable (local stability of all equilibria).
[are there also global implications?]
◮
is “K -quasipositive” for some order cone K (monotonicity of
the local semiflow)
Murad Banaji
Some results on chemical reaction networks without the assump
Questions in matrix theory
“B ∈ Q(A)” will mean “B is in the qualitative class of A” (“B has
the same dimensions and sign pattern as A”).
How do we characterise matrices A such that for each B ∈ Q(A):
◮
AB T is nonsingular?
◮
AB T is in (the closure of) the P-matrices?
◮
AB T is positive (semi)stable?
◮
AB T has no pair of imaginary eigenvalues?
◮
There exists a factorisation A = A1 A2 , a scalar α(B) and a
matrix C satisfying A1 C = 0 such that A2 BA1 + α(B)I + C is
nonnegative?
Murad Banaji
Some results on chemical reaction networks without the assump
Some examples
Murad Banaji
Some results on chemical reaction networks without the assump
Example (1): a Hurwitz Jacobian matrix
Consider the chemical reaction network
A ⇋ B + D, B ⇋ C ⇋ D ⇋ E
A ⇋ ∅, B ⇋ ∅, C ⇋ ∅, D ⇋ ∅, E ⇋ ∅
with stoichiometric matrix

Γ=
1
1
0
1
0
0
−1
1
0
0
0
0
−1
1
0
0
0
0
−1
1
−1
0
0
0
0
0
−1
0
0
0
0
0
−1
0
0
0
0
0
−1
0

0
0
0 
0
−1
ΓDv is Hurwitz for all Dv ∈ Q(−ΓT ). This can be proved without
explicitly checking the Routh-Hurwitz conditions. Saddle-node and
Hopf bifurcations cannot occur, and all equilibria are locally stable.
Murad Banaji
Some results on chemical reaction networks without the assump
Examples (2): ruling out stable oscillation
A1 + A2 ⇋ B1
A2 + A3 ⇋ B2
A3 + A4 ⇋ B3
A4 + A5 ⇋ B4
A5 ⇋ 2A1
A3
B2
B3
A2
A4
B1
B4
A1
2
A5
The system cannot have a nontrivial
attracting periodic orbit. This is proved
by showing that the system restricted to
any stoichiometry class is monotone.
Murad Banaji
Some results on chemical reaction networks without the assump
Examples (3): ruling out multistationarity
P1
E1
S1
P2
E 1S1S2
E 1S1
E2
S2
E 2S2S3
E 2S2
2
E 4S4
S4
E4
E 3S3
E 3S3S4
S3
E3
P3
The system (from Craciun et. al. PNAS 103(23):8697–8702, 2006) does
not allow multiple positive nondegenerate equilibria (MPNE).
Murad Banaji
Some results on chemical reaction networks without the assump
Another example (partial TCA cycle model)
OAA
CIT
MAL
FUM
ISOC
NADH
αKG
∞
Suc
SCoA
Murad Banaji
Some results on chemical reaction networks without the assump
Example (4)
Consider the following system of 4 reactions on 5 chemicals:
A ⇋ B + C,
C +D ⇋A C +E ⇋A
B ⇋ D,
with the DSR graph:
C
D
E
A
B
All orbits on each nontrivial stoichiometry class converge to a
unique (positive) equilibrium.
Murad Banaji
Some results on chemical reaction networks without the assump
Proving global stability
1. nontrivial stoichiometry classes intersect only repelling faces
of the nonnegative orthant (via a calculation on Γ)
2. the local semiflow is monotone and in fact strongly monotone
in the interior (via examination of Γ and the DSR graph)
3. the partial order satisfies certain additional geometrical
constraints
All this allow us to construct a strict
Liapunov function on the relative
interior of each nontrivial
stoichiometry class. Each such class
contains exactly one equilibrium
which attracts the whole
stoichiometry class.
Murad Banaji
Some results on chemical reaction networks without the assump
Where did the results come from?
Computations or observations on Γ (the constant stoichiometric
matrix) and Dv (the rate derivatives) and/or DSR graphs
GΓ,−Dv T ... We were naturally led to questions such as
◮
Does Γ admit a certain factorisation?
◮
Do submatrices of Γ satisfy certain conditions?
◮
What can we say about ker ΓT and ker Γ?
◮
Is GΓ,−Dv T strongly connected?
◮
What can we say about cycles in GΓ,−Dv T ?
Murad Banaji
Some results on chemical reaction networks without the assump
Concluding remarks
◮
Several results on CRNs are consequences of the weak law
“more substrate ⇒ faster reactions” rather than the
mass-action (MA) law.
◮
However many claims about CRNs with mass action probably
fail when MA is suspended.
◮
Thus some constructions in classical CRNT appear to be
fundamentally “algebraic”, while in others, the algebra hides
more fundamental underlying geometric constraints.
Murad Banaji
Some results on chemical reaction networks without the assump
Concluding remarks
Some guesses:
◮
The class of persistent CRNs with MA is probably much larger
than that of persistent CRNs without MA.
◮
The class of globally stable CRNs with MA is probably much
larger than that of globally stable CRNs with MA.
Murad Banaji
Some results on chemical reaction networks without the assump
Collaborations
Stephen Baigent (UCL) Pete Donnell (Portsmouth) Gheorghe
Craciun (Wisconsin) Casian Pantea (Imperial) David Angeli
(Imperial) Janusz Mierczyński (Wroclaw) Carrie Rutherford
(London South Bank) Andrew Burbanks (Portsmouth)
Murad Banaji
Some results on chemical reaction networks without the assump
Some relevant publications (1)
◮
(with P. Donnell and S. Baigent) P matrix properties,
injectivity and stability in chemical reaction systems, SIAM J
Appl Math, 67(6) (2007), 1523-1547
◮
Monotonicity in chemical reaction systems, Dynamical
Systems 24(1) (2009) 1-30
◮
(with G. Craciun) Graph-theoretic criteria for injectivity and
unique equilibria in general chemical reaction systems, Adv
Appl Math, 44 (2010) 168-184
◮
(with G. Craciun) Graph-theoretic approaches to injectivity
and multiple equilibria in systems of interacting elements,
Comm Math Sci, 7(4) (2009) 867-900
◮
(with D. Angeli) Convergence in strongly monotone systems
with an increasing first integral, SIAM J Math Anal, 42(1)
(2010) 334-353
Murad Banaji
Some results on chemical reaction networks without the assump
Some relevant publications (2)
◮
Graph-theoretic conditions for injectivity of functions on
rectangular domains, J Math Anal Appl, 370 (2010) 302-311
◮
(with C. Rutherford) P-matrices and signed digraphs, Discrete
Math 311 (4) (2011) 289-294
◮
(with J. Mierczyński) Global convergence in systems of
differential equations arising from chemical reaction networks,
[preprint on arXiv at http://arxiv.org/abs/1205.1716]
◮
(with A. Burbanks), A graph-theoretic condition for
irreducibility of a set of cone preserving matrices, [preprint on
arXiv at http://arxiv.org/abs/1112.1653].
Murad Banaji
Some results on chemical reaction networks without the assump