TCA

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