lecture #6 open systems. biological systems are ‘open:’ example: atp production by mitochondria

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Lecture #6 Open Systems

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Page 1: Lecture #6 Open Systems. Biological systems are ‘open:’ Example: ATP production by mitochondria

Lecture #6

Open Systems

Page 2: Lecture #6 Open Systems. Biological systems are ‘open:’ Example: ATP production by mitochondria

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

Page 3: Lecture #6 Open Systems. Biological systems are ‘open:’ Example: ATP production by mitochondria

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

Page 4: Lecture #6 Open Systems. Biological systems are ‘open:’ Example: ATP production by mitochondria

• 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

Page 5: Lecture #6 Open Systems. Biological systems are ‘open:’ Example: ATP production by mitochondria

Open Systems: key concepts

Physical: i.e., cell wall, nuclear membraneVirtual: i.e., the amino acid biosynthetic pathways

Hard: volume = constantSoft: volume = fn(time)

Page 6: Lecture #6 Open Systems. Biological systems are ‘open:’ Example: ATP production by mitochondria

THE REVERSIBLE REACTION IN AN OPEN SETTING

Start simple

Page 7: Lecture #6 Open Systems. Biological systems are ‘open:’ Example: ATP production by mitochondria

The reversible reactionThe basic equations

constant

b1 is a “forcing function”

b2 is a function of the internal

state

b1 v1 b2 type I pathwayv1

v-1

type III pathway

Null(S)Sv=0

m = 2, n = 4, r = 2

Dim(Null) = 4-2=2Dim(LNull)=2-2=0

-

Page 8: Lecture #6 Open Systems. Biological systems are ‘open:’ Example: ATP production by mitochondria

The Steady State Flux Values

Dynamic mass balances

b1 v1 b2

type I type III

weights that determine aparticular

steady state

@ stst dx/dt=0

Structure of the steady state

solution

Page 9: Lecture #6 Open Systems. Biological systems are ‘open:’ Example: ATP production by mitochondria

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

Page 10: Lecture #6 Open Systems. Biological systems are ‘open:’ Example: ATP production by mitochondria

The “Distance” from Equilibriumthe difference between life and death

: the mass action ratioKeq: the equilibrium constant

/Keq < 1 the reaction proceeds in the forward direction

Page 11: Lecture #6 Open Systems. Biological systems are ‘open:’ Example: ATP production by mitochondria

Dynamic Response of an Open System (x10=1, x20=0)

x2,ss

x1,ss

equilibriumline

1/2

k1 =1k-1=2k2 =0.1b1 =0.01

external

internal

Page 12: Lecture #6 Open Systems. Biological systems are ‘open:’ Example: ATP production by mitochondria

Response of the Poolsdisequilibrium

=change in p1 small

=change in p2 small

conservation

Page 13: Lecture #6 Open Systems. Biological systems are ‘open:’ Example: ATP production by mitochondria

Dynamic Simulation from One Steady State to

Another (b1 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

Page 14: Lecture #6 Open Systems. Biological systems are ‘open:’ Example: ATP production by mitochondria

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

Page 15: Lecture #6 Open Systems. Biological systems are ‘open:’ Example: ATP production by mitochondria

THE MICHAELIS-MENTEN MECHANISM IN AN OPEN SETTING

Towards a more realistic situation

Page 16: Lecture #6 Open Systems. Biological systems are ‘open:’ Example: ATP production by mitochondria

Michaelis-Menten Mechanism in an Open Setting

system boundary

input output

Page 17: Lecture #6 Open Systems. Biological systems are ‘open:’ Example: ATP production by mitochondria

The Micaelis-Menten reactionThe basic equations

Page 18: Lecture #6 Open Systems. Biological systems are ‘open:’ Example: ATP production by mitochondria

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

Page 19: Lecture #6 Open Systems. Biological systems are ‘open:’ Example: ATP production by mitochondria

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

Page 20: Lecture #6 Open Systems. Biological systems are ‘open:’ Example: ATP production by mitochondria

Dynamic ResponseShift b1=0.025 to 0.04 @ t=0

Phase portrait Dynamic response

Dynamic response

Page 21: Lecture #6 Open Systems. Biological systems are ‘open:’ Example: ATP production by mitochondria

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)

Page 22: Lecture #6 Open Systems. Biological systems are ‘open:’ Example: ATP production by mitochondria

Long-term adaptive response:increased enzyme synthesis

synthesis degradation

See chapter 8 for an example

Page 23: Lecture #6 Open Systems. Biological systems are ‘open:’ Example: ATP production by mitochondria

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