Correspondence of regular and generalized mass action systems

Transcription

1 Correspondence of regular and generalized mass action systems Van Vleck Visiting Assistant Professor University of Wisconsin-Madison Joint Mathematics Meetings (San Antonio, TX) Saturday, January 10, 2014

2 1 Background 2 3

3 Objective: Determine the dynamical properties / behavior of systems of interacting biochemical species. Do the systems exhibit: 1 stable behavior? 2 oscillatory behavior? 3 switching behavior / hysteresis? 4 extinction? 5 limit cycles / chaos? etc.

4 Figure: Picture courtesy of American Society of Microbiology.

5 Figure: Picture courtesy of Roche Applied Sciences.

6 Systems Biology (2000-): Bottom up approach (engineer network like circuit) Modularize the network into functional pathways, e.g. Protein Activation A + B 2B B A Enzymatic Futile Cycle S + E SE P + E P + F PF S + F. Signaling Network XD X XT X p X p + Y X p Y X + Y p XT + Y p XTY p Y

7 Chemical Reaction Network Theory (1972-): Relate dynamical properties of system to underlying network structure (esp. weak reversibility) Often able to determine system behavior independently of reaction 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.

8 Consider the enzymatic futile cycle: S + E k 1 SE k 3 P + E P + F k 4 PF k 6 S + F. k 2 k 5 Corresponding mass action system is: ẋ S = k 1 x S x E + k 2 x SE + k 6 x PF ẋ E = k 1 x S x E + k 2 x SE + k 3 x SE ẋ SE = k 1 x S x E k 2 x SE k 3 x SE ẋ P = k 3 x SE k 4 x P x F + k 5 x PF ẋ F = k 4 x P x F + k 5 x PF + k 6 x PF ẋ PF = k 4 x P x F k 5 x PF k 6 x PF

9 Consider the enzymatic futile cycle: S + E k 1 SE k 3 P + E P + F k 4 PF k 6 S + F. k 2 k 5 Corresponding mass action system is: ẋ S = k 1 x S x E + k 2 x SE + k 6 x PF ẋ E = k 1 x S x E + k 2 x SE + k 3 x SE ẋ SE = k 1 x S x E k 2 x SE k 3 x SE ẋ P = k 3 x SE k 4 x P x F + k 5 x PF ẋ F = k 4 x P x F + k 5 x PF + k 6 x PF ẋ PF = k 4 x P x F k 5 x PF k 6 x PF

10 Consider the enzymatic futile cycle: S + E k 1 SE k 3 P + E P + F k 4 PF k 6 S + F. k 2 k 5 Corresponding mass action system is: ẋ S = k 1 x S x E + k 2 x SE + k 6 x PF ẋ E = k 1 x S x E + k 2 x SE + k 3 x SE ẋ SE = k 1 x S x E k 2 x SE k 3 x SE ẋ P = k 3 x SE k 4 x P x F + k 5 x PF ẋ F = k 4 x P x F + k 5 x PF + k 6 x PF ẋ PF = k 4 x P x F k 5 x PF k 6 x PF

11 Consider the enzymatic futile cycle: S + E k 1 SE k 3 P + E P + F k 4 PF k 6 S + F. k 2 k 5 Corresponding mass action system is: ẋ S = k 1 x S x E + k 2 x SE + k 6 x PF ẋ E = k 1 x S x E + k 2 x SE + k 3 x SE ẋ SE = k 1 x S x E k 2 x SE k 3 x SE ẋ P = k 3 x SE k 4 x P x F + k 5 x PF ẋ F = k 4 x P x F + k 5 x PF + k 6 x PF ẋ PF = k 4 x P x F k 5 x PF k 6 x PF Not weakly reversible (path connectivity is not symmetric)

12 My approach: Shift/translate reactions in species space to make weakly 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.

13 My approach: Shift/translate reactions in species space to make weakly 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.

14 My approach: Shift/translate reactions in species space to make weakly 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.

15 Consider following signal transduction network (where X = EnvZ, Y = OmpR, p = phosphate group) [6]: (N3) XD X XT X p (+XD + XT + Y ) X p + Y X p Y X + Y p (+XD + XT ) XT + Y p XTY p XT + Y (+XD + X ) XD + Y p XDY p XD + Y (+X + XT ) Rate constants and corresponding mass action system (in 9 species) omitted.

16 Consider following signal transduction network (where X = EnvZ, Y = OmpR, p = phosphate group) [6]: (N3) XD X XT X p (+XD + XT + Y ) X p + Y X p Y X + Y p (+XD + XT ) XT + Y p XTY p XT + Y (+XD + X ) XD + Y p XDY p XD + Y (+X + XT ) Rate constants and corresponding mass action system (in 9 species) omitted.

17 Restructured reaction network is: (N4) 2XD + XT + Y XD + X + XT + Y XD + 2XT + Y X + XT + XDY p XD + X + XTY p XD + XT + X p + Y XD + X + XT + Y p XD + XT + X p Y

18 Restructured reaction network is: (N4) 2XD + XT + Y XD + X + XT + Y XD + 2XT + Y X + XT + XDY p XD + X + XTY p XD + XT + X p + Y XD + X + XT + Y p XD + XT + X p Y Cannot directly correspond steady state set because pathways overlap at a source = need steady state algebraic relation: ( ) k2 k 4 c XD c Yp = c XT c Yp k 1 k 3

19 Final restructured (and reweighted) reaction network: (N4 ) 2XD + XT + Y k1 k 2 XD + X + XT + Y k3 XD + 2XT + Y k10 k8 k4 X + XT + XDY p XD + X + XTY p XD + XT + X p + Y ( k2 k k 4 9 k 1 k 3 k7 k5 k XD + X + XT + Y 6 p XD + XT + Xp Y N3 and N4 have the same (toric) steady state set.

20 Final restructured (and reweighted) reaction network: (N4 ) 2XD + XT + Y k1 k 2 XD + X + XT + Y k3 XD + 2XT + Y k10 k8 k4 X + XT + XDY p XD + X + XTY p XD + XT + X p + Y ( k2 k k 4 9 k 1 k 3 k7 k5 k XD + X + XT + Y 6 p XD + XT + Xp Y N3 and N4 have the same (toric) steady state set.

21 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.

22 Thank you!

23 . Selected Bibliography M. Feinberg. Complex balancing in general kinetic systems. Arch. Ration. Mech. Anal., 49: , F. Horn. Necessary and sufficient conditions for complex balancing in chemical kinetics. Arch. Ration. Mech. Anal., 49: , F. Horn and R. Jackson. General mass action kinetics. Arch. Ration. Mech. Anal., 47: , M. Feinberg. Chemical reaction network structure and the stability of complex isothermal reactors: I. The deficiency zero and deficiency one theorems. Chem. Eng. Sci., 42: , G. Craciun, A. Dickenstein, A. Shiu, and B. Sturmfels. Toric Dynamical Systems. J. Symbolic Comput. 44: , G. Shinar and M. Feinberg. Structural sources of robustness in biochemical reaction networks. Science, 327(5971): , M.D. Johnston. Translated chemical reaction networks. Bull. Math. Biol., 76(5): , M.D. Johnston. A Computational Approach to Steady State Correspondence of Regular and Generalized Mass Action Systems. Submitted.