Lecture #6

1. Lecture #6 Open Systems

2. Biological systems are ‘open:’Example: ATP production by mitochondria

3. Outline • Key concepts in the analysis of open systems • The reversible reaction in an open environment • The Michaelis-Menten reaction mechanism in an open environment • Lessons learned

4. Key Concepts • Systems boundary: inside vs. outside • Crossing the boundary: I/O • Inside the boundary: • the internal network; • hard to observe directly (non-invasively) • From networks to (dynamic) models • Computing functional states • Steady states  homeostatic states • Dynamic states  transition from one steady state to another

5. Open Systems: key concepts Physical: i.e., cell wall, nuclear membrane Virtual: i.e., the amino acid biosynthetic pathways Hard: volume = constant Soft: volume = fn(time)

6. Start simple THE REVERSIBLE REACTION IN AN OPEN SETTING

7. b1 is a “forcing function” b2 is a function of the internal state The reversible reaction constant m = 2, n = 4, r = 2 Dim(Null) = 4-2=2 Dim(LNull)=2-2=0 The basic equations - b1 v1 b2 Null(S) Sv=0 type I pathway v1 type III pathway v-1

8. The Steady State Flux Values Dynamic mass balances @ stst dx/dt=0 weights that determine a particular steady state b1 v1 b2 Structure of the steady state solution type I type III

9. The Steady State Concentrations type I pathway type III pathway thus, the flux through pathway III is (k-1/k2) times the flux through pathway I

10. The “Distance” from Equilibrium the difference between life and death G: the mass action ratio Keq: the equilibrium constant G/Keq < 1 the reaction proceeds in the forward direction

11. Dynamic Response of an Open System (x10=1, x20=0) k1 =1 k-1=2 k2 =0.1 b1 =0.01 internal external 1/2 equilibrium line x2,ss x1,ss

12. Response of the Pools disequilibrium = = conservation internal change in p1 small external change in p2 small

13. Dynamic Simulation from One Steady State to Another (b1 from 0.01 to 0.02 at t=0) Large change Small change Distance from Eq Inventory Realistic perturbations are in the boundary fluxes Sudden changes in the concentrations typically do NOT occur

14. Lessons • Relative rates of internal vs. exchange fluxes are important • Open systems are in a steady state and respond to external stimuli • Changes from steady state • Changes in boundary fluxes are realistic • Changes in internal concentrations are not • If internal dynamics are ‘fast’ we may not need to characterize them in detail

15. Towards a more realistic situation THE MICHAELIS-MENTEN MECHANISM IN AN OPEN SETTING

16. Michaelis-Menten Mechanism in an Open Setting output input system boundary

17. The Micaelis-Menten reaction The basic equations

18. The stoichiometric matrix mxn = 4x5 and r= 3 Dim(Null(S)) = 5-3=2: two-dimensional stst flux space Dim(L.Null(S)) = 4-3=1 – one conservation variable: e+x

19. The Steady State Solution the steady state flux balances are which sets the concentrations and the detailed flux solution as before, the internal pathway has a flux of (k-1/k2) times that of the through pathway

20. Dynamic Response Shift b1=0.025 to 0.04 @ t=0 Phase portrait Dynamic response Dynamic response

21. Internal Capacity Constraint Steady state fluxes and maximum enzyme (etot) concentration give b1=k2x2ss<k2etot b1 can be set to over come the capacity of the system (see HW 6.4)

22. Long-term adaptive response:increased enzyme synthesis synthesis degradation See chapter 8 for an example

23. Summary • Open systems reach a steady state -- closed systems reach equilibrium • Living systems are open systems that continually exchange mass and energy with the environment • Continual net throughput leads to a homeostatic state that is an energy dissipative state • Time scale separation between internal and exchange fluxes is important • Internal capacities can be exceeded: • Exchange fluxes are bounded: 0 < b1 < b1,max