Applied Algebraic Geometry 2013: Translated Chemical ...€¦ · The dynamics is governed by a...

Steady States of Mass Action SystemsTranslated Chemical Reaction Networks

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Applied Algebraic Geometry 2013:Translated Chemical Reaction Networks

Matthew Douglas JohnstonVan Vleck Visiting Assistant Professor

University of Wisconsin-Madison

August 1, 2013

Matthew Douglas Johnston Translated Chemical Reaction Networks

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1 Steady States of Mass Action SystemsBackgroundGeneratorsDeficiency Zero Theorem

2 Translated Chemical Reaction NetworksGeneralized Chemical Reaction NetworksTranslated Chemical Reaction NetworksFeinberg/Shinar Example

3 Future Work

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BackgroundGeneratorsDeficiency Zero Theorem

3 Future Work

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Our basic object of study is an elementary reaction, whichconsists of a set of reactants which turns into a set of products:

2H2 + O2k−→ 2H2O

/A collection of simultaneously occurringelementary reactions is called a chemical reaction network and isdenoted by N = (S, C,R) where:

1 S = {A1,A2, . . . ,Am} is the species set;

2 C = {C1, C2, . . . , Cn} is the complex set; and

3 R = {(Ci , Cj) | Ci → Cj} is the reaction set.

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2H2 + O2k−→ 2H2O

Species/Reactants

/A collection ofsimultaneously occurring elementary reactions is called a chemicalreaction network and is denoted by N = (S, C,R) where:

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2H2 + O2k−→ 2H2O

Reactant Complex/

/A collection of simultaneouslyoccurring elementary reactions is called a chemical reactionnetwork and is denoted by N = (S, C,R) where:

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2H2 + O2k−→ 2H2O

Product Complex/

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2H2 + O2k−→ 2H2O

Reaction Constant/

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2H2 + O2k−→ 2H2O

/A collection of simultaneously occurring elementary reactions iscalled a chemical reaction network and is denoted byN = (S, C,R) where:

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2H2 + O2k−→ 2H2O

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2H2 + O2k−→ 2H2O

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2H2 + O2k−→ 2H2O

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The dynamics is governed by a system of autonomous, polynomialODEs called a mass action system:

x =r∑

ki (y′i − yi ) xyi (1)

We have the following important components:

sum over reactions,

ki is the reaction rate constant,

(y′i − yi ) is the reaction vector,

xyi =∏m

j=1(xj)yij is the mass-action term.

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x =r∑

sum over reactions,

xyi =∏m

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x =r∑

sum over reactions,

xyi =∏m

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x =r∑

sum over reactions,

xyi =∏m

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x =r∑

sum over reactions,

(y′i − yi ) is the reaction vector, and

xyi =∏m

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QUESTION:

How do we characterize the steady states of mass actionsystems?

In general, we have the system of algebraic equations

r∑i=1

ki (y′i − yi ) xyi = 0.

These equations can be difficult—nonlinear, high dimensional,unknown parameters, etc. etc. etc.

Worse still, we must intersect this set with the invariant manifold(S + x0) ∩ Rm

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QUESTION:

r∑i=1

ki (y′i − yi ) xyi = 0.

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QUESTION:

r∑i=1

ki (y′i − yi ) xyi = 0.

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QUESTION:

r∑i=1

ki (y′i − yi ) xyi = 0.

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For example, consider the futile cycle [1]:

S + Ek1�k2

SEk3−→ P + E P + F

k4�k5

PFk6−→ S + F .

This network is governed by the mass action system:

x1 = −k1x1x2 + k2x3 + k6x6

x2 = −k1x1x2 + (k2 + k3)x3

x3 = k1x1x2 − (k2 + k3)x3

x4 = k3x3 − k4x4x5 + k5x6

x5 = −k4x4x5 + (k5 + k6)x6

x6 = k4x4x5 − (k5 + k6)x6

We would like to find an alternative approach...

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S + Ek1�k2

SEk3−→ P + E P + F

k4�k5

PFk6−→ S + F .

x1 = −k1x1x2 + k2x3 + k6x6

x2 = −k1x1x2 + (k2 + k3)x3

x3 = k1x1x2 − (k2 + k3)x3

x4 = k3x3 − k4x4x5 + k5x6

x5 = −k4x4x5 + (k5 + k6)x6

x6 = k4x4x5 − (k5 + k6)x6

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S + Ek1�k2

SEk3−→ P + E P + F

k4�k5

PFk6−→ S + F .

x1 = −k1x1x2 + k2x3 + k6x6 = 0

x2 = −k1x1x2 + (k2 + k3)x3 = 0

x3 = k1x1x2 − (k2 + k3)x3 = 0

x4 = k3x3 − k4x4x5 + k5x6 = 0

x5 = −k4x4x5 + (k5 + k6)x6 = 0

x6 = k4x4x5 − (k5 + k6)x6 = 0.

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S + Ek1�k2

SEk3−→ P + E P + F

k4�k5

PFk6−→ S + F .

x1 = −k1x1x2 + k2x3 + k6x6 = 0

x2 = −k1x1x2 + (k2 + k3)x3 = 0

x3 = k1x1x2 − (k2 + k3)x3 = 0

x4 = k3x3 − k4x4x5 + k5x6 = 0

x5 = −k4x4x5 + (k5 + k6)x6 = 0

x6 = k4x4x5 − (k5 + k6)x6 = 0.

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We can reformulate the steady state conditions to decouple thelinear and nonlinear components of the equations!

Rewrite the system of mass action differential equations as

dt= Y Ia R(x)

where:

Y ∈ Zm×n is the complex matrix (yi along columns)

Ia ∈ {−1, 0, 1}n×r is the connections matrix

R(x) ∈ Rr≥0 is the kinetic vector (entries kix

OBSERVATION: Most of this is linear!

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dt= Y Ia R(x)

where:

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dt= Y Ia R(x)

where:

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dt= Y Ia R(x)

where:

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dt= Y Ia R(x)

where:

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dt= Y Ia R(x)

where:

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For the futile cycle

S + Ek1�k2

SEk3−→ P + E P + F

k4�k5

PFk6−→ S + F

we have

Y Ia R(x)

1 0 0 0 0 11 0 1 0 0 00 1 0 0 0 00 0 1 1 0 00 0 0 1 0 10 0 0 0 1 0

−1 1 0 0 0 01 −1 −1 0 0 00 0 1 0 0 00 0 0 −1 1 00 0 0 1 −1 −10 0 0 0 0 1

k1x1x2k2x3k3x3k4x4x5k5x6k6x6

OBSERVATION: Only R(x) is nonlinear!

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S + Ek1�k2

SEk3−→ P + E P + F

k4�k5

PFk6−→ S + F

we haveY Ia R(x)

1 0 0 0 0 11 0 1 0 0 00 1 0 0 0 00 0 1 1 0 00 0 0 1 0 10 0 0 0 1 0

−1 1 0 0 0 01 −1 −1 0 0 00 0 1 0 0 00 0 0 −1 1 00 0 0 1 −1 −10 0 0 0 0 1

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S + Ek1�k2

SEk3−→ P + E P + F

k4�k5

PFk6−→ S + F

we haveY Ia R(x)

1 0 0 0 0 11 0 1 0 0 00 1 0 0 0 00 0 1 1 0 00 0 0 1 0 10 0 0 0 1 0

−1 1 0 0 0 01 −1 −1 0 0 00 0 1 0 0 00 0 0 −1 1 00 0 0 1 −1 −10 0 0 0 0 1

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Now reconsider the general conditions for steady state:

Y Ia R(x) = 0.

Clearly, we need ker(Y Ia) ∩ Rr≥0 6= ∅.

There are two distinct kinds of generators of ker(Y Ia) ∩ Rr≥0 we

will be interested in:

1 E ∈ ker(Ia) (positive circuits)

2 E 6∈ ker(Ia) (stoichiometric generators)

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Y Ia R(x) = 0.

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Y Ia R(x) = 0.

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Y Ia R(x) = 0.

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Y Ia R(x) = 0.

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Consider again the futile cycle:

S + Ek1�k2

SEk3−→ P + E P + F

k4�k5

PFk6−→ S + F .

This network has the stoichiometric matrix

Y Ia =

− 1 1 0 0 0 1− 1 1 1 0 0 01 − 1 − 1 0 0 00 0 1 − 1 1 00 0 0 − 1 1 10 0 0 1 − 1 − 1

We have that ker(Y Ia) ∩ Rr≥0 = λ1E1 + λ2E2 + λ3E3, λi ≥ 0,

where E1 = [1 1 0 0 0 0]T , E2 = [0 0 0 1 1 0]T , andE3 = [1 0 1 1 0 1]T .

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S + Ek1�k2

SEk3−→ P + E P + F

k4�k5

PFk6−→ S + F .

Y Ia =

− 1 1 0 0 0 1− 1 1 1 0 0 01 − 1 − 1 0 0 00 0 1 − 1 1 00 0 0 − 1 1 10 0 0 1 − 1 − 1

where E1 = [1 1 0 0 0 0]T , E2 = [0 0 0 1 1 0]T , andE3 = [1 0 1 1 0 1]T .

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S + Ek1�k2

SEk3−→ P + E P + F

k4�k5

PFk6−→ S + F .

Y Ia =

− 1 1 0 0 0 1− 1 1 1 0 0 01 − 1 − 1 0 0 00 0 1 − 1 1 00 0 0 − 1 1 10 0 0 1 − 1 − 1

where E1 = [1 1 0 0 0 0]T , E2 = [0 0 0 1 1 0]T , andE3 = [1 0 1 1 0 1]T .

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S + Ek1�k2

SEk3−→ P + E P + F

k4�k5

PFk6−→ S + F .

Y Ia =

− 1 1 0 0 0 1− 1 1 1 0 0 01 − 1 − 1 0 0 00 0 1 − 1 1 00 0 0 − 1 1 10 0 0 1 − 1 − 1

where E1 = [1 1 0 0 0 0]T , E2 = [0 0 0 1 1 0]T , andE3 = [1 0 1 1 0 1]T .

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S + Ek1�k2

SEk3−→ P + E P + F

k4�k5

PFk6−→ S + F .

Y Ia =

− 1 1 0 0 0 1− 1 1 1 0 0 01 − 1 − 1 0 0 00 0 1 − 1 1 00 0 0 − 1 1 10 0 0 1 − 1 − 1

where E1 = [1 1 0 0 0 0]T , E2 = [0 0 0 1 1 0]T , andE3 = [1 0 1 1 0 1]T .

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S + Ek1�k2

SEk3−→ P + E P + F

k4�k5

PFk6−→ S + F .

Y Ia =

− 1 1 0 0 0 1− 1 1 1 0 0 01 − 1 − 1 0 0 00 0 1 − 1 1 00 0 0 − 1 1 10 0 0 1 − 1 − 1

where E1 = [1 1 0 0 0 0]T , E2 = [0 0 0 1 1 0]T , andE3 = [1 0 1 1 0 1]T .

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Now consider the connections matrix

− 1 1 0 0 0 01 − 1 − 1 0 0 00 0 1 0 0 00 0 0 − 1 1 00 0 0 1 − 1 − 10 0 0 0 0 1

Only E1 = [1 1 0 0 0 0]T ,E2 = [0 0 0 1 1 0]T ∈ ker Ia.

However, E3 = [1 0 1 1 0 1]T 6∈ ker(Ia) (i.e. it is a stoichiometricgenerator).

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− 1 1 0 0 0 01 − 1 − 1 0 0 00 0 1 0 0 00 0 0 − 1 1 00 0 0 1 − 1 − 10 0 0 0 0 1

Only E1 = [1 1 0 0 0 0]T ,E2 = [0 0 0 1 1 0]T ∈ ker Ia.

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− 1 1 0 0 0 01 − 1 − 1 0 0 00 0 1 0 0 00 0 0 − 1 1 00 0 0 1 − 1 − 10 0 0 0 0 1

Only E1 = [1 1 0 0 0 0]T ,E2 = [0 0 0 1 1 0]T ∈ ker Ia.

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− 1 1 0 0 0 01 − 1 − 1 0 0 00 0 1 0 0 00 0 0 − 1 1 00 0 0 1 − 1 − 10 0 0 0 0 1

Only E1 = [1 1 0 0 0 0]T ,E2 = [0 0 0 1 1 0]T ∈ ker Ia.

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− 1 1 0 0 0 01 − 1 − 1 0 0 00 0 1 0 0 00 0 0 − 1 1 00 0 0 1 − 1 − 10 0 0 0 0 1

Only E1 = [1 1 0 0 0 0]T ,E2 = [0 0 0 1 1 0]T ∈ ker Ia.

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− 1 1 0 0 0 01 − 1 − 1 0 0 00 0 1 0 0 00 0 0 − 1 1 00 0 0 1 − 1 − 10 0 0 0 0 1

Only E1 = [1 1 0 0 0 0]T ,E2 = [0 0 0 1 1 0]T ∈ ker Ia.

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QUESTION:

Why do we care about the distinction between E ∈ ker(Ia) andE 6∈ ker(Ia) anyway?

To answer this, we introduce a network parameter called thedeficiency:

δ = dim(ker(Y ) ∩ Im(Ia)).

/NOTE: If δ = 0 there are no stoichiometric generators.

Steady states are completely characterized for deficiency zerosystems!

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QUESTION:

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QUESTION:

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QUESTION:

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We do not like E3 = [1 0 1 1 0 1]T 6∈ ker(Ia) but can wechange/eliminate it?

Consider the following intuition:

We can move reaction vectors in complex-space byadding/subtracting species on both sides of a reaction.

This changes the reaction graph but does not change thereaction vectors or generators of ker(Y Ia).

Stoichiometric generators may form a positive circuit/cycle!

But we must retain old source monomials (consider later).

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Let’s restructure the futile cycle!

S + Ek1�k2

SEk3−→ P + E

P + Fk4�k5

PFk6−→ S + F

Our new reaction network is:

S + E + Fk1�k2

SE + Fk3−→ P + E + F

P + E + Fk4�k5

PF + Ek6−→ S + E + F

We have new connections in the reaction graph!

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S + Ek1�k2

SEk3−→ P + E

P + Fk4�k5

PFk6−→ S + F

S + E + Fk1�k2

SE + Fk3−→ P + E + F

P + E + Fk4�k5

PF + Ek6−→ S + E + F

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S + Ek1�k2

SEk3−→ P + E (+F )

P + Fk4�k5

PFk6−→ S + F (+E )

S + E + Fk1�k2

SE + Fk3−→ P + E + F

P + E + Fk4�k5

PF + Ek6−→ S + E + F

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S + Ek1�k2

SEk3−→ P + E (+F )

P + Fk4�k5

PFk6−→ S + F (+E )

S + E + Fk1�k2

SE + Fk3−→ P + E + F

P + E + Fk4�k5

PF + Ek6−→ S + E + F

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S + Ek1�k2

SEk3−→ P + E (+F )

P + Fk4�k5

PFk6−→ S + F (+E )

S + E + Fk1�k2

SE + Fk3−→ P + E + F

P + E + Fk4�k5

PF + Ek6−→ S + E + F

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S + Ek1�k2

SEk3−→ P + E (+F )

P + Fk4�k5

PFk6−→ S + F (+E )

S + E + Fk1�k2

SE + Fk3−→ P + E + F

P + E + Fk4�k5

PF + Ek6−→ S + E + F

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S + Ek1�k2

SEk3−→ P + E (+F )

P + Fk4�k5

PFk6−→ S + F (+E )

S + E + Fk1�k2

SE + Fk3−→ P + E + F

P + E + Fk4�k5

PF + Ek6−→ S + E + F

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The restructured graph is:

S + E · · ·

S + E + Fk1�k2

SE + F

· · · SE

k6 ↑ ↓k3

PF · · ·

PF + Ek5�k4

P + E + F

· · · P + F

The stoichiometric generator E3 = [1 0 1 1 0 1]T is now a cyclicgenerator!

Steady state set can be computed by considering ker(Ak) whereAk := Ia Ik ∈ Rn×n (of restructured graph)

but using monomials ofold network!

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S + E · · ·

S + E + Fk1�k2

SE + F

· · · SE

k6 ↑ ↓k3

PF · · ·

PF + Ek5�k4

P + E + F

· · · P + F

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S + E · · ·

S + E + Fk1�k2

SE + F

· · · SE

k6 ↑ ↓k3

PF · · ·

PF + Ek5�k4

P + E + F

· · · P + F

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S + E · · ·

S + E + Fk1�k2

SE + F

· · · SE

k6 ↑ ↓k3

PF · · ·

PF + Ek5�k4

P + E + F

· · · P + F

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S + E · · · S + E + Fk1�k2

SE + F · · · SE

k6 ↑ ↓k3PF · · · PF + E

k5�k4

P + E + F · · · P + F

Steady state set can be computed by considering ker(Ak) whereAk := Ia Ik ∈ Rn×n (of restructured graph) but using monomials ofold network!

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S + E · · · S + E + Fk1�k2

SE + F · · · SE

k6 ↑ ↓k3PF · · · PF + E

k5�k4

P + E + F · · · P + F

Steady state set can be computed by considering ker(Ak) whereAk := Ia Ik ∈ Rn×n (of restructured graph) but using monomials ofold network!

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Generalized Chemical Reaction NetworksTranslated Chemical Reaction NetworksFeinberg/Shinar Example

3 Future Work

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What we have used is power law formalism—framework forsubstituting monomials not implied by network stoichiometry.

A recent network structure extension to this was recentlypresented by Stefan Muller and Georg Regensburger [2].

Definition

A generalized chemical reaction network (S, C, CK ,R) is achemical reaction network (S, C,R) together with a set of kineticcomplexes CK which are in one-to-one correspondence with theelements of C.

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Definition

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Definition

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IDEA: Two sets of complexes, one in graph, one for dynamics!

For example, consider the generalized network:

A1 +A2

k1�k2

......

7A1 +A3 5A2

This gives rise to the generalized mass action system

=dx2dt

= −dx3dt

= −k1x71x3 + k2x52 .

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A1 +A2

k1�k2

......

7A1 +A3 5A2

=dx2dt

= −dx3dt

= −k1x71x3 + k2x52 .

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A1 +A2

k1�k2

......

7A1 +A3 5A2

=dx2dt

= −dx3dt

= −k1x71x3 + k2x52 .

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Restructured graph is a generalized chemical reaction network!

S + E · · · S + E + Fk1�k2

SE + F · · · SE

k6 ↑ ↓k3PF · · · PF + E

k5�k4

P + E + F · · · P + F

We have the set of stoichiometric complexes

C = {S + E + F , SE + F ,P + E + F ,PF + E}

and the kinetic complexes

CK = {S + E , SE ,P + F ,PF} .

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S + E · · · S + E + Fk1�k2

SE + F · · · SE

k6 ↑ ↓k3PF · · · PF + E

k5�k4

P + E + F · · · P + F

C = {S + E + F , SE + F ,P + E + F ,PF + E}

CK = {S + E , SE ,P + F ,PF} .

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S + E · · · S + E + Fk1�k2

SE + F · · · SE

k6 ↑ ↓k3PF · · · PF + E

k5�k4

P + E + F · · · P + F

C = {S + E + F , SE + F ,P + E + F ,PF + E}

CK = {S + E , SE ,P + F ,PF} .

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GOAL: Characterize mass action steady states of N .

TOOL: Construct generalized chemical reaction network N .

N must satisfy:

1 Has same reaction vectors as N

2 Reactant complexes in N mapped to reactant complexes in Nas kinetic complexes

Call such a network N a translation of N .

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N must satisfy:

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N must satisfy:

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N must satisfy:

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N must satisfy:

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Best case scenario: Reactant complexes in N are mappeduniquely to reactant complexes in N (proper translations).

Previous example was a proper translation!

S + E · · · S + E + Fk1�k2

SE + F · · · SE

k6 ↑ ↓k3PF · · · PF + E

k5�k4

P + E + F · · · P + F

QUESTIONS:

Does this always happened?

If not, can be resolve the conflict?

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S + E · · · S + E + Fk1�k2

SE + F · · · SE

k6 ↑ ↓k3PF · · · PF + E

k5�k4

P + E + F · · · P + F

QUESTIONS:

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S + E · · · S + E + Fk1�k2

SE + F · · · SE

k6 ↑ ↓k3PF · · · PF + E

k5�k4

P + E + F · · · P + F

QUESTIONS:

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S + E · · · S + E + Fk1�k2

SE + F · · · SE

k6 ↑ ↓k3PF · · · PF + E

k5�k4

P + E + F · · · P + F

QUESTIONS:

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S + E · · · S + E + Fk1�k2

SE + F · · · SE

k6 ↑ ↓k3PF · · · PF + E

k5�k4

P + E + F · · · P + F

QUESTIONS:

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S + E · · · S + E + Fk1�k2

SE + F · · · SE

k6 ↑ ↓k3PF · · · PF + E

k5�k4

P + E + F · · · P + F

QUESTIONS:

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S + E · · · S + E + Fk1�k2

SE + F · · · SE

k6 ↑ ↓k3PF · · · PF + E

k5�k4

P + E + F · · · P + F

QUESTIONS:

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S + E · · · S + E + Fk1�k2

SE + F · · · SE

k6 ↑ ↓k3PF · · · PF + E

k5�k4

P + E + F · · · P + F

QUESTIONS:

Does this always happened? If not, can be resolve the conflict?

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Consider the following signal transduction example [1, 3]:

k1�k2

X2k3−→ X3

k4−→ X4

(+X1 + X3 + X5)

X4 + X5k5−→ X6

k6−→ X2 + X7

(+X1 + X3)

X3 + X7k7−→ X8

k8−→ X3 + X5

(+X1 + X2)

X1 + X7k9−→ X9

k10−→ X1 + X5

(+X2 + X3)

This has the stoichiometric generators:

E1 = [0 0 1 1 1 1 1 1 0 0]T

E2 = [0 0 1 1 1 1 0 0 1 1]T .

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k1�k2

X2k3−→ X3

k4−→ X4

(+X1 + X3 + X5)

X4 + X5k5−→ X6

k6−→ X2 + X7

(+X1 + X3)

X3 + X7k7−→ X8

k8−→ X3 + X5

(+X1 + X2)

X1 + X7k9−→ X9

k10−→ X1 + X5

(+X2 + X3)

E1 = [0 0 1 1 1 1 1 1 0 0]T

E2 = [0 0 1 1 1 1 0 0 1 1]T .

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k1�k2

X2k3−→ X3

k4−→ X4

(+X1 + X3 + X5)

X4 + X5k5−→ X6

k6−→ X2 + X7

(+X1 + X3)

X3 + X7k7−→ X8

k8−→ X3 + X5

(+X1 + X2)

X1 + X7k9−→ X9

k10−→ X1 + X5

(+X2 + X3)

E1 = [0 0 1 1 1 1 1 1 0 0]T

E2 = [0 0 1 1 1 1 0 0 1 1]T .

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k1�k2

X2k3−→ X3

k4−→ X4

(+X1 + X3 + X5)

X4 + X5k5−→ X6

k6−→ X2 + X7

(+X1 + X3)

X3 + X7k7−→ X8

k8−→ X3 + X5

(+X1 + X2)

X1 + X7k9−→ X9

k10−→ X1 + X5

(+X2 + X3)

E1 = [0 0 1 1 1 1 1 1 0 0]T

E2 = [0 0 1 1 1 1 0 0 1 1]T .

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k1�k2

X2k3−→ X3

k4−→ X4 (+X1 + X3 + X5)

X4 + X5k5−→ X6

k6−→ X2 + X7 (+X1 + X3)

X3 + X7k7−→ X8

k8−→ X3 + X5 (+X1 + X2)

X1 + X7k9−→ X9

k10−→ X1 + X5 (+X2 + X3).

E1 = [0 0 1 1 1 1 1 1 0 0]T

E2 = [0 0 1 1 1 1 0 0 1 1]T .

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A possible translation is:

2X1 + X3 + X5 � X1 + X2 + X3 + X5 −→ X1 + 2X3 + X5

↗ ↑ ↓X2 + X3 + X9 X1 + X2 + X8 X1 + X3 + X4 + X5

↖ ↑ ↓X1 + X2 + X3 + X7 ←− X1 + X3 + X6

This has the cyclic generators:

E1 = [0 0 1 1 1 1 1 1 0 0]T

E2 = [0 0 1 1 1 1 0 0 1 1]T .

This is sweet!... but there is some bad news...

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2X1 + X3 + X5 � X1 + X2 + X3 + X5 −→ X1 + 2X3 + X5

↗ ↑ ↓X2 + X3 + X9 X1 + X2 + X8 X1 + X3 + X4 + X5

↖ ↑ ↓X1 + X2 + X3 + X7 ←− X1 + X3 + X6

E1 = [0 0 1 1 1 1 1 1 0 0]T

E2 = [0 0 1 1 1 1 0 0 1 1]T .

Future Work

2X1 + X3 + X5 � X1 + X2 + X3 + X5 −→ X1 + 2X3 + X5

↗ ↑ ↓X2 + X3 + X9 X1 + X2 + X8 X1 + X3 + X4 + X5

↖ ↑ ↓X1 + X2 + X3 + X7 ←− X1 + X3 + X6

E1 = [0 0 1 1 1 1 1 1 0 0]T

E2 = [0 0 1 1 1 1 0 0 1 1]T .

Future Work

2X1 + X3 + X5 � X1 + X2 + X3 + X5 −→ X1 + 2X3 + X5

↗ ↑ ↓X2 + X3 + X9 X1 + X2 + X8 X1 + X3 + X4 + X5

↖ ↑ ↓X1 + X2 + X3 + X7 ←− X1 + X3 + X6

E1 = [0 0 1 1 1 1 1 1 0 0]T

E2 = [0 0 1 1 1 1 0 0 1 1]T .

Future Work

2X1 + X3 + X5 � X1 + X2 + X3 + X5 −→ X1 + 2X3 + X5

↗ ↑ ↓X2 + X3 + X9 X1 + X2 + X8 X1 + X3 + X4 + X5

↖ ↑ ↓X1 + X2 + X3 + X7 ←− X1 + X3 + X6

E1 = [0 0 1 1 1 1 1 1 0 0]T

E2 = [0 0 1 1 1 1 0 0 1 1]T .

This is sweet!

... but there is some bad news...

Future Work

2X1 + X3 + X5 � X1 + X2 + X3 + X5 −→ X1 + 2X3 + X5

↗ ↑ ↓X2 + X3 + X9 X1 + X2 + X8 X1 + X3 + X4 + X5

↖ ↑ ↓X1 + X2 + X3 + X7 ←− X1 + X3 + X6

E1 = [0 0 1 1 1 1 1 1 0 0]T

E2 = [0 0 1 1 1 1 0 0 1 1]T .

Future Work

Consider the reaction translations:

X3 + X7k7−→ X8

k8−→ X3 + X5 ( + X1 + X2)

X1 + X7k9−→ X9

k10−→ X1 + X5 ( + X2 + X3)

In the translation we have X1 + X2 + X3 + X7 corresponds toboth X3 + X7 and X1 + X7.

We want to assign the monomials x1x7 and x3x7 to the samenode but can’t!

The assignment of reactant complexes is not unique (impropertranslation).

Future Work

X3 + X7k7−→ X8

k8−→ X3 + X5 ( + X1 + X2)

X1 + X7k9−→ X9

k10−→ X1 + X5 ( + X2 + X3)

Future Work

X3 + X7k7−→ X8

k8−→ X3 + X5 ( + X1 + X2)

X1 + X7k9−→ X9

k10−→ X1 + X5 ( + X2 + X3)

Future Work

X3 + X7k7−→ X8

k8−→ X3 + X5 ( + X1 + X2)

X1 + X7k9−→ X9

k10−→ X1 + X5 ( + X2 + X3)

Future Work

X3 + X7k7−→ X8

k8−→ X3 + X5 ( + X1 + X2)

X1 + X7k9−→ X9

k10−→ X1 + X5 ( + X2 + X3)

Future Work

X3 + X7k7−→ X8

k8−→ X3 + X5 ( + X1 + X2)

X1 + X7k9−→ X9

k10−→ X1 + X5 ( + X2 + X3)

Future Work

QUESTION:

How can we resolve this problem? (If at all...)

Maybe we can relate the monomials x1x7 and x3x7...

Trivial relationship... substitute this into equations to give(state-dependent) rate constants... but at steady state...

The terms K1 and K3 are state-dependent but the ratio K1/K3

depends only on the rate constants (k2k4/k1k3)!

Future Work

QUESTION:

Future Work

QUESTION:

x1x7 =

Trivial relationship...

substitute this into equations to give(state-dependent) rate constants... but at steady state...

Future Work

QUESTION:

k9x1x7 = k9

Trivial relationship... substitute this into equations to give(state-dependent) rate constants...

but at steady state...

Future Work

QUESTION:

k9x1x7 = k9

)x3x7 = k9

Future Work

QUESTION:

k9x1x7 = k9

)x3x7 = k9

Future Work

Reconsider the translated reaction graph:

2X1 + X3 + X5

X1 + X2 + X3 + X5

−→ X1 + 2X3 + X5

X2 + X3 + X9 X1 + X2 + X8 X1 + X3 + X4 + X5

k2k4k1k3

↖ ↑

X1 + X2 + X3 + X7

←− X1 + X3 + X6

We associate X1 + X2 + X3 + X7 with the monomial x3x7 but atthe expense of having to adjust the rate constants.

This network has the same steady state set!

Future Work

2X1 + X3 + X5

X1 + X2 + X3 + X5

−→ X1 + 2X3 + X5

X2 + X3 + X9 X1 + X2 + X8 X1 + X3 + X4 + X5

k2k4k1k3

↖ ↑

X1 + X2 + X3 + X7

←− X1 + X3 + X6

Future Work

2X1 + X3 + X5

X1 + X2 + X3 + X5

−→ X1 + 2X3 + X5

X2 + X3 + X9 X1 + X2 + X8 X1 + X3 + X4 + X5

k2k4k1k3

↖ ↑

X1 + X2 + X3 + X7

←− X1 + X3 + X6

Future Work

2X1 + X3 + X5

k1�k2

X1 + X2 + X3 + X5k3−→ X1 + 2X3 + X5

↗k10 ↑k8 ↓k4X2 + X3 + X9 X1 + X2 + X8 X1 + X3 + X4 + X5

k2k4k1k3

↖ ↑k7 ↓k5

X1 + X2 + X3 + X7k6←− X1 + X3 + X6

Future Work

2X1 + X3 + X5

k1�k2

X1 + X2 + X3 + X5k3−→ X1 + 2X3 + X5

↗k10 ↑k8 ↓k4X2 + X3 + X9 X1 + X2 + X8 X1 + X3 + X4 + X5

k2k4k1k3

) ↖ ↑k7 ↓k5

X1 + X2 + X3 + X7k6←− X1 + X3 + X6

Future Work

2X1 + X3 + X5

k1�k2

X1 + X2 + X3 + X5k3−→ X1 + 2X3 + X5

↗k10 ↑k8 ↓k4X2 + X3 + X9 X1 + X2 + X8 X1 + X3 + X4 + X5

k2k4k1k3

) ↖ ↑k7 ↓k5

X1 + X2 + X3 + X7k6←− X1 + X3 + X6

Future Work

Summary of steps:

Translate the stoichiometric generators into cyclicgenerators (if possible).

If a node is removed, adjust the rate constants (if possible).

Characterize the steady states of this generalized chemicalreaction network (if possible).

Future Work

Summary of steps:

Future Work

Summary of steps:

Future Work

Summary of steps:

Future Work

3 Future Work

Future Work

Future work:

Algorithmize/computerize translation process.

Determine better conditions for when the rate constantsmay be adjusted.

Further develop theory of generalized chemical reactionnetworks.

Future Work

Future work:

Future Work

Future work:

Future Work

Future work:

Future Work

Thanks for coming out!

Special thanks to Anne Shiu, Carsten Conradi, Stefan Muller,Casian Pantea, and others for many suggestions and constructive

comments

Future Work

Selected Bibliography

Mercedes Perez Millan, Alicia Dickenstein, Anne Shiu, and CarstenConradi.

Chemical reaction systems with toric steady states.

Bull. Math. Biol., 2011.

Stefan Muller and Georg Regensburger.

Generalized mass action systems: Complex balancing equilibria andsign vectors of the stoichiometric and kinetic-order subspaces.

SIAM J. Appl. Math., 72(6):1926–1947, 2012.

Guy Shinar and Martin Feinberg.

Structural sources of robustness in biochemical reaction networks.

Science, 327(5971):1389–1391, 2010.

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