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Lecture #6 Open Systems

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

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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

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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 Key Concepts

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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)

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THE REVERSIBLE REACTION IN AN OPEN SETTING Start simple

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The reversible reaction The basic equations constant b 1 is a “forcing function” b 2 is a function of the internal state b1b1 v1v1 b2b2 type I pathway v1v1 v -1 type III pathway Null(S) Sv=0 m = 2, n = 4, r = 2 Dim(Null) = 4-2=2 Dim(LNull)=2-2=0 -

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The Steady State Flux Values Dynamic mass balances b1b1 v1v1 b2b2 type Itype III weights that determine a particular steady state @ stst dx/dt=0 Structure of the steady state solution

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Add a slide detailing how x1ss and x2ss are derived

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The Steady State Concentrations type I pathway type III pathway thus, the flux through pathway III is (k -1 /k 2 ) times the flux through pathway I

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

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Dynamic Response of an Open System (x 10 =1, x 20 =0) x 2,ss x 1,ss equilibrium line 1/2 k 1 =1 k -1 =2 k 2 =0.1 b 1 =0.01 external internal

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Response of the Pools disequilibrium = change in p 1 small = change in p 2 small conservation

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Dynamic Simulation from One Steady State to Another (b 1 from 0.01 to 0.02 at t=0) Realistic perturbations are in the boundary fluxes Sudden changes in the concentrations typically do NOT occur

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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

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

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

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The Micaelis-Menten reaction The basic equations

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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

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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 /k 2 ) times that of the through pathway

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Dynamic Response Shift b 1 =0.025 to 0.04 @ t=0 Phase portraitDynamic response

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Internal Capacity Constraint Steady state fluxes and maximum enzyme (e tot ) concentration give b 1 =k 2 x 2ss

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

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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 < b 1 < b 1,max

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