TCA
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TCA
Gly
G1P
G6P
F6P
F1-6BP
PEP Pyr
Gly3p
13BPG
3PG
2PG
ATP
NADH
Oxa
Cit
ACA
TCA
Gly
G1P
G6P
F6P
F1-6BP
PEP Pyr
Gly3p
13BPG
3PG
2PG
ATP
NADH
Oxa
Cit
ACA
F6P
F1-6BP
Gly3p
13BPG
3PG
ATP
F6P
F1-6BP
Gly3p
13BPG
3PG
ATP
1 1
q
h
q Vx
x
1
1 y
qk y
1
0 xk x
y
x
Control
Autocatalytic
F6P
F1-6BPGly3p
13BPG
3PG
ATP
1 1
q
h
q Vx
x
1
1 y
qk y
1
0 xk x
y
x
Control
Autocatalytic
1 1
q
h
q Vx
x
1
1 y
qk y
1
0 xk x
y
x
Control
Autocatalytic
Minimal metabolism model• x = ultimate product (ATP)• y = intermediate metabolite • Two feedbacks of x:
– Autocatalytic– Control
RHP z and pHard limits
2 20
1ln ln
z z pS j d
z z p
01
1 2 1
1 2 1
22.41
2 2
hk q
z p
z p
nd
NominalVariable Process
Value?
autocatalysis 1
inhibition 1
2 enzyme 1
q
k
1(1 )
0
1 1
q
h
q Vx
x
1
1 y
qk y
y
xAutocatalytic
Control
produced
consumed
yk yy
x
( )yk y
y
1( )k x y
xAutocatalytic
Control
consumed
produced
1( )k x
x
1 1(1 )
1 1 0x
x q qk x ky
y
consumed
1(1 )
0
x
1(1 )
0
1 1
q
h
q Vx
x
1
1 y
qk y
y
xAutocatalytic
( )yk y
y
1(1 )
0
1 1
q
h
q Vx
x
1
1 y
qk y
y
xAutocatalytic
1( )k x
( )yk y
Control
Enzyme complexity,Strong inhibition
Oscillations
Low autocatalysis Inefficiency,High reaction rates metabolic load
Robust Yet Fragile
1(1 )
0
1 1
q
h
q Vx
x
1
1 y
qk y
y
xAutocatalytic
Control
0 5 10 15 200.8
0.85
0.9
0.95
1
1.05
Time (minutes)
[AT
P]
h >>1
h = 1
Time response)
Fourier
Transform
of error
h hS x = F(
0 2 4 6 8 10-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Frequency
Lo
g(|
Sn
/S0|
)
h >>1
h = 1
Spectrum
1log loghS S
Ideal
0 5 10 15 200.8
0.85
0.9
0.95
1
1.05
Time (minutes)
[AT
P]
h >> 1
h = 1
0 2 4 6 8 10-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Frequency
Lo
g(S
n/S
0)
h >>1
h = 1
Spectrum
Time response
Robust
Yet fragile
0ln lnhS j S j
0 2 4 6 8 10-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Frequency
Lo
g(S
n/S
0)
h = 3
h = 0 Robust
Yet fragile
0
ln 0S j d
0 2 4 6 8 10-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Frequency
Lo
g(S
n/S
0)
h = 0 Robust
Yet fragile
0 0
ln ln 0S j d S j d
log )nx d constant F(
log|S |
Tighter regulation
Transients, Oscillations
Biological complexity is dominated by the evolution of
mechanisms to more finely tune this robustness/fragility tradeoff.
This tradeoff is a law.
lnz p
z p
Re p
2 20
1ln
zS j d
z
0
1ln S j d
0
2 20
11
zd
z
0RHP z
Hard limits
Benefits must be “paid for” within bandwidth z
RHP p
11
0
1
1
qky
y
x
“Optimal” controller
“Optimal” enzyme
Necessity, not accident
2 20
1ln ln
z z pS j d
z z p
0 10
ˆ 1.25h
ˆ 1.5h
q=α=1
ˆ 2h
ˆ 1.5h
0 10
ˆ .5h
ˆ 10h
q=α=0
Necessity, not accident
Synthesis challenges
ˆlarge Enzyme complexity,
small Oscillations
large Inefficiency,small metabolic load
Robust Yet Fragile
h
k
q
• Necessity or accident?• Alternative designs
• Other rates and uncertainty
• Computational complexity– Higher order
dynamics– Global, nonlinear– Comparisons with
data• Robustness is key
Hard limits and tradeoffs
On systems and their components• Thermodynamics (Carnot)• Communications (Shannon)• Control (Bode) • Computation (Turing/Gödel)
• Include dynamics and feedback• Extend to networks• New unifications are encouraging
Robust/fragile
is unifyingconcept
log S d
log S d
benefits costs
log S d
log S d
• benefits = attenuation of disturbance• goal: make this as negative as possible
cost = amplificationgoal: make this small
Constraint:
a
-e=d-u
Control
uPlant
d
delay
u
log S d
Bode
a
ES
D
log S d
benefits costs stabilize
a
-e=d-u
Control
uPlant
d
delay
u
log S d
Bode
a
ES
D
log S d
benefits costs stabilize
Negative is good
Disturbance-e=d-u
ControlSensor
ChannelEncode
PlantRemoteSensor
dd
r
ControlChannel SC
u
CC
log S d
log S d
SC
CClog( )a
http://www.glue.umd.edu/~nmartins/
Nuno C Martins and Munther A Dahleh, Feedback Control in the Presence of Noisy Channels: “Bode-Like” Fundamental Limitations of Performance.Nuno C. Martins, Munther A. Dahleh and John C. Doyle Fundamental Limitations of Disturbance Attenuation in the Presence of Side Information(Both in IEEE Transactions on Automatic Control)
Variety of producers
Electric powernetwork
Variety ofconsumers
• Good designs transform/manipulate energy• Subject (and close) to hard limits
Variety ofconsumers
Variety of producers
Energy carriers
• 110 V, 60 Hz AC• (230V, 50 Hz AC)• Gasoline• ATP, glucose, etc• Proton motive force
Standard interface
Constraint that deconstrains
Disturbance-e=d-u
ControlSensor
ChannelEncode
PlantRemoteSensor
dd
r
ControlChannel
log S d
log S d
benefits
feedback
SC
CCstabilizeremotesensing
remote control
log( )a
costs
Robust
log( )a
Fragile
• Robust designs transform/manipulate robustness• Subject (and close) to hard limits• Fragile designs are far away from hard limits and
waste robustness.
Hard limits and tradeoffs
On systems and their components• Thermodynamics (Carnot)• Communications (Shannon)• Control (Bode) • Computation (Turing/Gödel)
• Include dynamics and feedback• Extend to networks• New unifications are encouraging
Robust/fragile
is unifyingconcept
[a system] can have[a property] robust for [a set of perturbations]
Yet be fragile for
Or [a different perturbation]
[a different property]Robust
Fragile
[a system] can have[a property] robust for [a set of perturbations]
Robust
Fragile
• But if robustness/fragility are conserved, what does it mean for a system to be robust or fragile?
• Some fragilities are inevitable in robust complex systems.
• But if robustness/fragility are conserved, what does it mean for a system to be robust or fragile?
Robust
Fragile
• Robust systems systematically manage this tradeoff.• Fragile systems waste robustness.
• Some fragilities are inevitable in robust complex systems.
Emergent
Cat
abol
ism
Genes
Co-factorsFatty acidsSugars
Nucleotides
Amino Acids Proteins
Pre
curs
ors
DNA replication
Trans*
Carriers
Components and materials:Energy, moieties
Systems requirements: functional, efficient,
robust, evolvable
Hard constraints:Thermo (Carnot)Info (Shannon)Control (Bode)Compute (Turing)
Protocols
Constraints
Diverse
Diverse
UniversalControl
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