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Understanding Open Biological Systems: ATP Production and Dynamic Responses

This lecture explores the fundamental concepts of open biological systems using ATP production by mitochondria as a prime example. Key topics include the analysis of open systems, reversible reactions, the Michaelis-Menten reaction mechanism, and dynamic state transitions. Participants will learn about system boundaries, input/output dynamics, steady states, and how living systems maintain homeostasis. By examining the interplay between internal and external fluxes, attendees will gain insights into the importance of relative rates and adaptive responses in maintaining system stability.

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Understanding Open Biological Systems: ATP Production and Dynamic Responses

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  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

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