Transcript the reversible reaction in an open setting

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
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
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)
Start simple
THE REVERSIBLE REACTION IN AN
OPEN SETTING
The reversible reaction
The basic equations
b1 is a
“forcing
function”
b2 is a
function of
the internal
state
constant
m = 2, n = 4, r = 2
Dim(Null) = 4-2=2
Dim(LNull)=2-2=0
-
Null(S)
Sv=0
b1
v1
v1
v-1
b2
type I pathway
type III pathway
The Steady State Flux Values
Dynamic mass
balances
@ stst dx/dt=0
Structure of
the steady state
solution
b1
v1
type I
weights that
determine a
particular
steady state
b2
type III
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 Equilibrium
the difference between life and death
G:
Keq:
the mass action ratio
the equilibrium constant
G/Keq < 1 the reaction proceeds
in the forward direction
Dynamic Response of an Open
System (x10=1, x20=0)
k1 =1
k-1=2
k =0.1
external 2
b1 =0.01
internal
1/2
equilibrium
line
x2,ss
x1,ss
Response of the Pools
disequilibrium
conservation
=
=
change in
p1 small
change in p2 small
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
Towards a more realistic situation
THE MICHAELIS-MENTEN
MECHANISM IN AN OPEN SETTING
Michaelis-Menten Mechanism in
an Open Setting
output
input
system boundary
The Micaelis-Menten reaction
The 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 Response
Shift 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