lecture #6 open systems. biological systems are ‘open:’ example: atp production by mitochondria
TRANSCRIPT
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
• 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
Open Systems: key concepts
Physical: i.e., cell wall, nuclear membraneVirtual: i.e., the amino acid biosynthetic pathways
Hard: volume = constantSoft: volume = fn(time)
THE REVERSIBLE REACTION IN AN OPEN SETTING
Start simple
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
-
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
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
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
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
Response of the Poolsdisequilibrium
=change in p1 small
=change in p2 small
conservation
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
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
THE MICHAELIS-MENTEN MECHANISM IN AN OPEN SETTING
Towards a more realistic situation
Michaelis-Menten Mechanism in an Open Setting
system boundary
input output
The Micaelis-Menten reactionThe basic equations
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
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
Dynamic ResponseShift b1=0.025 to 0.04 @ t=0
Phase portrait Dynamic response
Dynamic response
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)
Long-term adaptive response:increased enzyme synthesis
synthesis degradation
See chapter 8 for an example
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