Correspondence of regular and generalized mass action systems · (N4 ) 8 >> >> >> >< >> >> >> >: 2 XD +XT Y k 1 ˛ k 2 X !k 3 + 2 % k

BackgroundCorrespondence Process (Translation)

Summary / Future Work

Correspondence of regular and generalized massaction systems

Matthew Douglas JohnstonVan Vleck Visiting Assistant Professor

University of Wisconsin-Madison

Joint Mathematics Meetings (San Antonio, TX)Saturday, January 10, 2014

Matthew Douglas Johnston Correspondence of mass action systems



1 Background

2 Correspondence Process (Translation)

3 Summary / Future Work




Objective:

Determine the dynamical properties / behavior of systems ofinteracting biochemical species.

Do the systems exhibit:

1 stable behavior?

2 oscillatory behavior?

3 switching behavior / hysteresis?

4 extinction?

5 limit cycles / chaos? etc.




Figure: Picture courtesy of American Society of Microbiology.




Figure: Picture courtesy of Roche Applied Sciences.




Systems Biology (2000-):

“Bottom up” approach (engineer network like circuit)

Modularize the network into functional pathways, e.g.

Protein Activation

A + B −→ 2BB −→ A

Enzymatic Futile Cycle

S + E � SE −→ P + EP + F � PF −→ S + F .

Signaling Network

XD � X � XT −→ Xp

Xp + Y � XpY −→ X + Yp

XT + Yp � XTYp −→ Y




Chemical Reaction Network Theory (1972-):

Relate dynamical properties of system to underlyingnetwork structure (esp. weak reversibility)

Often able to determine system behavior independently ofreaction parameters and initial conditions, e.g.

Deficiency Zero Theorem ([1, 2, 3], 1972)

Deficiency One Theorem ([4], 1987)

Global Attractor Conjecture ([5], 2009)

etc. etc. etc.




Consider the enzymatic futile cycle:

S + Ek1

�k2

SEk3−→ P + E P + F

k4

�k5

PFk6−→ S + F .

Corresponding mass action system is:

xS = − k1xSxE + k2xSE + k6xPF

xE = − k1xSxE + k2xSE + k3xSE

xSE = k1xSxE − k2xSE − k3xSE

xP = k3xSE − k4xPxF + k5xPF

xF = −k4xPxF + k5xPF + k6xPF

xPF = k4xPxF − k5xPF − k6xPF

Not weakly reversible (path connectivity is not symmetric)





S + Ek1

�k2

SEk3−→ P + E P + F

k4

�k5

PFk6−→ S + F .













S + Ek1

�k2

SEk3−→ P + E P + F

k4

�k5

PFk6−→ S + F .













S + Ek1

�k2

SEk3−→ P + E P + F

k4

�k5

PFk6−→ S + F .












My approach: Shift/translate reactions in species space to makeweakly reversible.

(N1)

{S + E � SE −→ P + E (+F )

P + F � PF −→ S + F (+E )

The restructured reaction network is:

(N2)

S + E + F � SE + F

↑ ↓

PF + E � P + E + F

N1 and N2 have the same (toric) steady state set.





(N1)

{S + E � SE −→ P + E (+F )

P + F � PF −→ S + F (+E )


(N2)

S + E + F � SE + F

↑ ↓

PF + E � P + E + F






(N1)

{S + E � SE −→ P + E (+F )

P + F � PF −→ S + F (+E )


(N2)

S + E + F � SE + F

↑ ↓

PF + E � P + E + F





Consider following signal transduction network (where X =EnvZ, Y = OmpR, p = phosphate group) [6]:

(N3)

XD � X −→ XT −→ Xp (+XD + XT + Y )

Xp + Y −→ XpY −→ X + Yp (+XD + XT )

XT + Yp −→ XTYp −→ XT + Y (+XD + X )

XD + Yp −→ XDYp −→ XD + Y (+X + XT )

Rate constants and corresponding mass action system (in 9species) omitted.




Consider following signal transduction network (where X =EnvZ, Y = OmpR, p = phosphate group) [6]:

(N3)

XD � X −→ XT −→ Xp (+XD + XT + Y )

Xp + Y −→ XpY −→ X + Yp (+XD + XT )

XT + Yp −→ XTYp −→ XT + Y (+XD + X )

XD + Yp −→ XDYp −→ XD + Y (+X + XT )

Rate constants and corresponding mass action system (in 9species) omitted.




Restructured reaction network is:

(N4)

2XD + XT + Y � XD + X + XT + Y −→ XD + 2XT + Y

↗ ↑ ↓X + XT + XDYp XD + X + XTYp XD + XT + Xp + Y

↖ ↑ ↓XD + X + XT + Yp ←− XD + XT + XpY

Cannot directly correspond steady state set because pathways overlapat a source =⇒ need steady state algebraic relation:

cXD · cYp =

(k2k4

k1k3

)cXT · cYp




Restructured reaction network is:

(N4)

2XD + XT + Y � XD + X + XT + Y −→ XD + 2XT + Y

↗ ↑ ↓X + XT + XDYp XD + X + XTYp XD + XT + Xp + Y

↖ ↑ ↓XD + X + XT + Yp ←− XD + XT + XpY

Cannot directly correspond steady state set because pathways overlapat a source =⇒ need steady state algebraic relation:

cXD · cYp =

(k2k4

k1k3

)cXT · cYp




Final restructured (and reweighted) reaction network:

(N4∗)

2XD + XT + Yk1

�k2

XD + X + XT + Yk3−→ XD + 2XT + Y

↗k10 ↑k8 ↓k4

X + XT + XDYp XD + X + XTYp XD + XT + Xp + Y

k9

(k2k4k1k3

) ↖ ↑k7 ↓k5

XD + X + XT + Ypk6←− XD + XT + XpY

N3 and N4∗ have the same (toric) steady state set.




Final restructured (and reweighted) reaction network:

(N4∗)

2XD + XT + Yk1

�k2

XD + X + XT + Yk3−→ XD + 2XT + Y

↗k10 ↑k8 ↓k4

X + XT + XDYp XD + X + XTYp XD + XT + Xp + Y

k9

(k2k4k1k3

) ↖ ↑k7 ↓k5

XD + X + XT + Ypk6←− XD + XT + XpY

N3 and N4∗ have the same (toric) steady state set.




Summary of Results:

1 Characterize steady states through network translation(Definition 6 & Theorem 5, Johnston [7]).

2 Algorithmize translation process (Section 4, Johnston [8])

Future work:

1 Develop theory of generalized mass action systems.

2 Wider implementation/application.




Thank you!




.Selected Bibliography

M. Feinberg. Complex balancing in general kinetic systems. Arch. Ration. Mech. Anal., 49:187–194, 1972.

F. Horn. Necessary and sufficient conditions for complex balancing in chemical kinetics. Arch. Ration. Mech.Anal., 49:172–186, 1972.

F. Horn and R. Jackson. General mass action kinetics. Arch. Ration. Mech. Anal., 47:187–194, 1972.

M. Feinberg. Chemical reaction network structure and the stability of complex isothermal reactors: I. Thedeficiency zero and deficiency one theorems. Chem. Eng. Sci., 42:2229–2268, 1987.

G. Craciun, A. Dickenstein, A. Shiu, and B. Sturmfels. Toric Dynamical Systems. J. Symbolic Comput.44:1551–1565, 2009.

G. Shinar and M. Feinberg. Structural sources of robustness in biochemical reaction networks. Science,327(5971):1389–1391, 2010.

M.D. Johnston. Translated chemical reaction networks. Bull. Math. Biol., 76(5):1081–1116, 2014.

M.D. Johnston. A Computational Approach to Steady State Correspondence of Regular and GeneralizedMass Action Systems. Submitted.


Documents

Correspondence of regular and generalized mass action systems · (N4 ) 8 >> >> >> >< >> >> >> >: 2 XD +XT Y k 1 ˛ k 2 X !k 3 + 2 % k