Complex Network Architecture: A cartoon guide - cmacs

Complex Network Architecture: A cartoon guide

Reactions Flow

Protein level Reactions

Application Error/flow control

Global

John Doyle

Flow RNA level

Relay/MUX

John G Braun Professor

Control and Dynamical Systems E/F control

E/F control

Local Relay/MUX

Reactions

BioEngineering, Electrical Engineering

Flow

Caltech Relay/MUX

DNA level Physical

Outline • Not merely “complexity, networks, abstraction, recursion, modularity,…” • But very specific forms of these. • Formal methods have great potential • Illustrate with case studies and cartoons: Internet versus bacterial biosphere • Implicitly: importance of formal methods, not merely modeling and simulation

Huge range of dynamics • Spatial • Temporal

Global

RNA level

Relay/MUX

Relay/MUX

Physical

Protein level

Flow

Error/flow control

Local

Flow

Reactions

Application

E/F control

Reactions

Bewildering w/out clear grasp of layered architecture

E/F control

Reactions

Relay/MUX

Flow

DNA level

Network Math and Engineering (NetME) Challenges • Predictive modeling, simulation, and analysis of complex systems in technology and nature • Theoretical foundation for design of network architectures • Balance rigor/relevance, integrative/coherent • Model/simulate is critical but limited – Predicting rare but catastrophic events – Design, not merely analysis – Managing complexity and uncertainty

“Architecture” • Most persistent, ubiquitous, and global features of organization • Constrains what is possible for good or bad • Platform that enables (or prevents) innovation, sustainability, etc, • Internet, biology, energy, manufacturing, transportation, water, food, waste, law, etc • Existing architectures are unsustainable • Theoretical foundation is fragmented, incoherent, incomplete

Stochastics in Biology • Arkin, Gillespie, Petzold, Khammash, El-Samad, Munsky,

Paulsson, Vinnicombe, many others… • Noise in the cellular environment ‣ Elowitz, van Oudenaarden, Collins, Swain, Xie, Elston, ...

• Stochastic Monte Carlo Simulation ‣ Kurtz, Gibson, Bruck, Anderson, Rathinam, Cao, Salis, Kaznessis, ...

• Statistical moment computations ‣ Hespanha, Singh, Verghese, Gomez-Uribe, Kimura

• Density function computations ‣McNamara, Sidje, ...

• Stochastic differential equation approximations ‣ van Kampen, Kurtz, Elf, Ehrenberg,...

• Spatial stochastic models and tools ‣ Elf, Iglesias,…

Very incomplete, idiosyncratic list

Other Influences • Internet (Kelly/Low, Willinger, Clark, Wroclawski, Day, Chang, etc etc) • Biology/Medicine (Savageau, G&K, Mattick, Csete, Arkin, Alon, Caporale, de Duve, Exerc Physio, Acute Care, etc etc…) • Architecture (Alexander, Salingeros,…) • Aerospace (many, Maier is a good book) • Philosophy/History (Fox Keller, Jablonka&Lamb) • Physics/ecology (Carlson) • Management (Baldwin,…) • Resilience/Safety/Security Engineering/Economics (Wood, Anderson, Leveson, …)

Biology versus the Internet Similarities

Differences

• • • • • • • •

• • • • •

Evolvable architecture Robust yet fragile Constraints/deconstrain Layering, modularity Hourglass with bowties Feedback Dynamic, stochastic Distributed/decentralized

• Not scale-free, edge-of-chaos, selforganized criticality, etc

Metabolism Materials and energy Autocatalytic feedback Feedback complexity Development and regeneration • >4B years of evolution • How the parts work?


Differences

• • • • • • • •

• • • • •

Evolvable architecture Robust yet fragile Constraints/deconstrain Layering, modularity Hourglass with bowties Feedback Dynamics Distributed/decentralized


Metabolism Materials and energy Autocatalytic feedback Feedback complexity Development and regeneration • >4B years of evolution Focus on bacterial biosphere

Question: Human complexity Robust  Efficient, flexible metabolism  Regeneration & renewal  Rich microbial symbionts and  Immune systems  Complex societies  Advanced technologies

Yet Fragile  Obesity and diabetes  Cancer  Parasites, infection  Inflammation, Auto-Im.  Epidemics, war, …

 Catastrophic failures

Mechanism? Robust  Efficient, flexible metabolism  Regeneration & renewal  Fat accumulation  Insulin resistance

 Inflammation

Fluctuating energy

Yet Fragile  Obesity and diabetes  Cancer  Fat accumulation  Insulin resistance  Inflammation

Static energy

Robust      

Implications/ Generalizations

Efficient, flexible metabolism Rich microbial symbionts and Immune systems Regeneration & renewal Complex societies Advanced technologies

     

Yet Fragile Obesity and diabetes Parasites, infection Inflammation, Auto-Im. Cancer Epidemics, war, … Catastrophic failures

• Fragility = Hijacking, side effects, unintended… of mechanisms evolved for robustness • Complexity is driven by control, robust/fragile tradeoffs • Math: New robust/fragile conservation laws

my computer

Wireless router

web server

Optical router

HTTP TCP IP

MAC Switch

MAC

MAC

Pt to Pt

Pt to Pt

Physical

Non-networked Systems Functions

Local

Local

control

control

Local

Local

Resources

Network requirements Semantic (functionally) local

Diverse, Distributed Functions

Local

Resources Geographically local

Local

Layered solution


Global, universal control Local

Local Resources

Local

Constraints

Universal control

That deconstrain

Diverse Functions Constraints

Universal control Global, universal control

Diverse Resources That deconstrain

my computer

Wireless router

TCP IP

MAC Switch

Physical

my computer

Wireless router

TCP IP

MAC Switch

Differ in • Details • Scope Error/flow control

Global

Relaying/Multiplexing

Error/flow control

Local

Relaying/Multiplexing

Physical

Wireless router

web server

Optical router

TCP IP

MAC Pt to Pt

Physical

Wireless router

Error/flow control

Global

Relay/MUX

web server

Optical router

TCP IP

Error/flow control

MAC

Local

Pt to Pt

Relay/MUX

Physical

my computer

Wireless router

web server

Optical router

HTTP TCP IP

MAC Switch

MAC

MAC

Pt to Pt

Pt to Pt

Physical

Recursive control structure Application Error/flow control

Global Relay/MUX E/F control Relay/MUX

Local

Physical

E/F control Relay/MUX


Global

Local

Local

Physical

Local

Layered solution


Layered control

Local

Local Resources

Local

How many layers are there?

Layered control

As many as you need.

Layered solution


And layers have sublayers

Instructions Logical

Local Resources

Circuit Circuit Circuit

Physical

And layers have sublayers Instructions Logical Circuit Circuit Circuit

Physical

Instructions Logical Circuit Circuit Circuit

Physical

Layered solution


Layered control

Local

Local Resources

Local

Diverse, Distributed Functions Huge range of dynamics • Spatial • Temporal

Local

Instructions Logical

Circuit Circuit Circuit

Physical

Bewildering w/out clear grasp of layered architecture

telephony

telephony

Diverse applications

TCP IP

MAC Switch

MAC

MAC

Pt to Pt

Pt to Pt

Physical

Ancient network architecture: “Bell-heads versus Net-heads”

Layers (Net) Computer

Operating systems

Pathways (Bell) Communications

Phone systems

Cyber-Physical Theories • • • •

Thermodynamics Communications Control Computation

Cyber • • • •

Physical


• • • •


Internet

Bacteria Case studies

Cyber • • • •


Physical • • • •


Promising unifications

Theoretical framework: Constraints that deconstrain

Applications Deconstrained min

Rx c

x

2

Rx c

x arg max L v, p , p v

xs

2

dt

Rx c

arg max Ls v, p v

Resources Deconstrained

• • • • •

Optimization Optimal control Robust control Game theory Network coding

Architecture is not graph topology.

Application

TCP

IP

Architecture facilitates arbitrary graphs.

Physical

Biology versus the Internet Similarities • • • • • • • •



“Central dogma”

DNA

RNA

Protein

Network architecture?

From Pathways

To Layers? Metabolic pathways

Layered Brain (Hawkins)?

Diverse Physiological Functions

Global, universal control Computational Resources

Diverse Physiological Functions

Global, universal control

Computational Resources

Diverse Physiological Functions Actions

errors

Prediction Goals Actions

Catabolism

Precursors

Inside every cell

Nucleotides

Carriers Core metabolic bowtie

Catabolism

Precursors

Core metabolism

Carriers Inside every cell ( 1030)

Nucleotides

Precursors

Catabolism

Carriers

Gly G1P G6P

Catabolism

F6P F1-6BP

Gly3p ATP 13BPG

3PG

2PG NADH

Oxa

PEP

Pyr

ACA

TCA Cit

Gly G1P G6P F6P F1-6BP

Gly3p 13BPG

3PG 2PG

Oxa PEP

Pyr

ACA

TCA Cit

Gly

Precursors

G1P G6P F6P

metabolites

F1-6BP

Gly3p 13BPG

3PG 2PG

Oxa PEP

Pyr

ACA

TCA Cit

Gly G1P G6P

Enzymatically catalyzed reactions

F6P F1-6BP

Gly3p 13BPG

3PG 2PG

Oxa PEP

Pyr

ACA

TCA Cit

Autocatalytic

G1P G6P F6P

Precursors

Gly

F1-6BP

Gly3p

Carriers

ATP 13BPG

3PG 2PG

Oxa PEP

Pyr

ACA

TCA

NADH Cit

Gly

Autocatalytic

G1P G6P

Rest of cell

F6P

consumed F1-6BP

Gly3p

ATP 13BPG

produced 3PG 2PG

Oxa PEP

Pyr

ACA

TCA

NADH Cit

Gly G1P

Reactions G6P

Control?

F6P F1-6BP

Carriers

Gly3p

Proteins

ATP 13BPG

3PG 2PG

Oxa PEP

Pyr

ACA

TCA

NADH Cit

Gly G1P G6P

Control

F6P F1-6BP

Gly3p ATP 13BPG

3PG 2PG

Oxa PEP

Pyr

ACA

TCA

NADH Cit


Gly3p 13BPG

3PG 2PG

Oxa PEP

Pyr

ACA

TCA Cit

If we drew the feedback loops the diagram would be unreadable. Gly G1P G6P F6P F1-6BP

Gly3p

ATP 13BPG

3PG 2PG

Oxa PEP

Pyr

ACA

TCA

NADH Cit

dx dt

Gly G1P

Sv( x) S Mass &

G6P

Reaction

Energy

F6P

Balance F1-6BP

Gly3p ATP

Stoichiometry matrix

13BPG

3PG 2PG

Oxa PEP

Pyr

ACA

TCA

NADH Cit

flux

dx dt

Gly G1P

Sv( x) Mass &

G6P

Reaction

Energy

F6P

Balance

flux

F1-6BP

Gly3p

Regulation of enzyme levels by transcription/translation/degradation

13BPG

3PG 2PG

Oxa PEP

level

Pyr

ACA

TCA Cit

dx dt

Gly G1P

Sv( x) Mass &

G6P

Reaction

Energy

F6P

Balance

flux

F1-6BP

Gly3p

Error/flow ATP

13BPG

Allosteric regulation of enzymes

3PG 2PG

Oxa PEP

Pyr

ACA

TCA

NADH Cit

Mass &

dx dt

Gly

Sv( x)

Energy

Reaction

Balance

flux

G1P G6P

Reaction

F6P

Error/flow

F1-6BP

Gly3p

Level

ATP 13BPG

3PG 2PG

Oxa PEP

Pyr

ACA

TCA

NADH Cit

Gly G1P

Reactions

G6P

Flow/error

F6P F1-6BP

Protein level

Gly3p ATP 13BPG

3PG 2PG

Oxa PEP

Pyr

ACA

TCA

NADH Cit

Gly G1P

Reactions

G6P

Flow/error

F6P

Layered F1-6BP architecture Gly3p

Protein level

ATP 13BPG

3PG 2PG

Oxa PEP

Pyr

ACA

TCA

NADH Cit

Reactions Flow/error Macromolecules

Diverse Functions Universal control Constraints Global, universal control

Diverse Resources That deconstrain

S

reactions

Enz1 reaction3

P

products Reaction rate

Enz2

Enzyme form/activity

Running only the top layers Mature red blood cells live 120 days

“metabolism first” origins of life?

Reactions Flow/error Protein level Reactions Flow/error RNA level Reactions Flow/error DNA level

Protein

Reactions Flow/error Protein level

RNA

Reactions Flow/error RNA level

DNA

Reactions Flow/error DNA level

Reactions Flow/error Protein level Translation Flow/error RNA level Transcription Flow/error DNA level

Diverse Reactions Flow/error

Protein level Conserved core control

Reactions Flow/error

RNA level Reactions Flow/error

DNA

DNA

Diverse Genomes

DNA

Top to bottom • Metabolically costly but fast to cheap but slow • Special enzymes to general polymerases • Allostery to regulated recruitment • Analog to digital • High molecule count to low (noise)

Rich Tradeoffs

Reactions Flow/err or level Protein Reactions Flow/err orlevel RNA Reactions Flow/err orlevel DNA

Fragility example: Viruses

Reactions Flow

Viral proteins


Viruses exploit the universal bowtie/hourglass structure to hijack the cell machinery.

Flow RNA level Reactions

Viral genes

Flow

DNA level


Differences

• • • • • • • •

• • • • •



Metabolism Materials and energy Autocatalytic feedback Feedback complexity Development and regeneration • >4B years of evolution

Control of the Internet Packets source

receiver

control packets

signaling gene expression metabolism lineage source

receiver

Biological pathways

signaling gene expression metabolism lineage source

receiver

control energy materials

More complex feedback

source

receiver

control

energy materials Autocatalytic feedback

signaling gene expression metabolism lineage What theory is relevant to receiver source these more complex feedback systems?

control energy materials


What theory is relevant to these more complex feedback systems?

1

z ln S j signaling d ln 2 2 z 0 genezexpression

source

z

metabolism lineage

control materials

energy

p p receiver


Precursors

Catabolism

Nucleotides

Carriers

Flow/error Protein level

RNA DNA

Precursors

Biosynthesis

Nucleotides

RNA DNA

Precursors

Biosynthesis

Co-factors

RNA Transc.

xRNA RNA level/ Transcription rate

RNAp

Gene

DNA level

Precursors

Catabolism

AA

RNA Transc. Gene

xRNA RNAp

Precursors

Catabolism

AA

transl.

tRNA

Enzymes Ribosome

ncRNA

mRNA

RNA Transc. Gene

xRNA RNAp

“Central dogma”

Protein AA

Transl. Flow RNA RNA Transc. Flow DNA

transl.

Protein Ribosome

RNA Transc.

Gene

mRNA RNAp

Precursors

Catabolism

Autocatalysis everywhere AA

transl.

Proteins

tRNA

All the enzymes are made from (mostly) proteins and (some) RNA.

Ribosome

RNA transc. xRNA RNAp

This is just charging and discharging

G6P

consumption Rest of cell = discharging

F6P F1-6BP

Gly3p

ATP 13BPG

charging

3PG 2PG

PEP

Pyr

ATP supplies energy to all layers

Rest of cell G6P F6P F1-6BP

ATP

Gly3p 13BPG

3PG 2PG

A*P

PEP

Pyr

Flow/error AMP level

Protein level

RNA DNA

RNA DNA ATP

A*P

Flow/error AMP level

Lots of ways to draw this.

Protein level

RNA DNA

cell

Precursors

Catabolism

AA

transl.

Enzymes

tRNA

Layered

RNA transc. xRNA

S

reactions

P

Enz1 reaction3

tRNA ncRNA

AA

trans.

Reaction rate

Enz2

Enzymes


Enzyme level/ Translation rate RNA form/activity

mRNA RNA Transc. Gene

RNAp

xRNA

RNA level/ Transcription rate

Ribosome

reactions

products

reaction3

Control?

Proteins

trans.

Transc.

All products feedback everywhere

ncRNA

Recursive control structure

Reactions Flow


Application Error/flow control

Global

RNA level

Relay/MUX

E/F control

E/F control

Reactions

Relay/MUX

Flow

Local Relay/MUX

Flow

DNA level Physical

Recursive control structure

Reactions Flow

Protein level

Application Huge range of dynamics Error/flow control • Spatial Global • Temporal Relay/MUX

E/F control

E/F control

Local Relay/MUX

Relay/MUX

Physical

Reactions

Flow RNA level Reactions Floww/out Bewildering DNAgrasp level of clear layered architecture

Horizontal gene transfer HGT and Shared Protocols

Bacteria

What is locus of early evolution?

Eukaryotes Animals

Archaea

Fungi

Plants

Algae

Architecture!?!

reactions

S HGT and

Shared Protocols Bacteria

products Reaction P rate

Eukaryotes

Enz1 reaction3 Enz2 Animals Fungi Plants Archaea

AA

trans.


Enzyme level/ Enzymes Translation rate Algae

mRNA Ribosome

tRNA RNA Transc. RNAp

ncRNA xRNA Gene

RNA form/activity

RNA level/ Transcription rate

Horizontal gene transfer HGT and Shared Protocols

Bacteria

• Not a static database • Not only point mutations

Eukaryotes Animals

Archaea

Gene

Fungi

Plants

Algae

DNA level

Controlled, dynamic

fan-in of diverse inputs

Diverse function Universal Control

Diverse components

universal carriers

Bowties: flows within layers

fan-out of diverse outputs

Essential ideas Robust yet fragile

Constraints that deconstrain


Diverse function

Diverse components


Highly robust • Diverse • Evolvable • Deconstrained Robust yet fragile


universal carriers

Universal Control

Highly fragile • Universal • Frozen • Constrained • Hijacking


Diverse function Universal Control

Diverse components

universal carriers

Bowties: flows within layers


Essential ideas Robust yet fragile


What theory is relevant to these more complex feedback systems?

1

z ln S j signaling d ln 2 2 z 0 genezexpression

source

z

metabolism lineage

control materials

energy

p p receiver



Gly3p ATP 13BPG

3PG 2PG

Oxa PEP

Pyr

ACA

TCA

NADH Cit


Gly3p ATP ATP 13BPG

3PG 2PG

Oxa PEP

Pyr

ACA

TCA

NADH Cit

1

Autocatalytic

0

x Control

Vx q 1 1 xh q

F6P F1-6BP

Gly3p ATP 13BPG

3PG

y

1 q 1

ky y

kx x

1

Autocatalytic

0

x Control

Vx q 1 1 xh q

F6P F1-6BP

Gly3p 13BPG 3PG

ATP

y

1 q 1

ky y

kx x

Autocatalytic

1 (1 0

x

)

Control Vx q 1 1 xh q

1 q

y

1

ky y

Autocatalytic x y

q 1

Vx 1

q

x

1 q h

Control

1

ky

1 0

(1

)

Control theory cartoon

x u

S j

output=x

Controller

+

input 1

x Vx q 1 1 xh q

Caution: mixed cartoon

y

0

1 q 1

1

ky y

u

S j

Hard limits

X j U j

1

output=x

ln S j

d

0

C

0

d

ln X j

d

ln U j

d d

Plant

ln S j

X j ln U j

+

u Entropy rates

[ATP]

1.05

Ideal

1

h >>1 0.95

Time response

0.9

S h = F( x) h

0.85

h=1

Fourier Transform

0

5

10

15

20

Time (minutes) 0.8

h >>1

0.6

Log(|Sn/S0|)

of error

0.8

0.4 0.2

Spectrum

h=1

0 -0.2

log Sh

-0.4

-0.6 -0.8

0

2

4

Frequency

log S1 6

8

10

[ATP]

1.05

1

h >> 1 0.95

Time response

0.9

0.85

Yet fragile

h=1 0.8

0

5

10

15

20

Time (minutes) 0.8

h >>1

Robust

Log(Sn/S0)

0.6 0.4 0.2

Spectrum

h=1

0 -0.2 -0.4

ln Sh j

-0.6 -0.8

0

2

4

Frequency

ln S0 j 6

8

10

ln S j

d

0

0

Yet fragile 0.8

h=3

Robust

Log(Sn/S0)

0.6 0.4 0.2

h=0

0 -0.2 -0.4

-0.6 -0.8

0

2

4

Frequency

6

8

10

[a system] can have [a property] robust for [a set of perturbations]

Fragile

Yet be fragile for [a different property]

Robust

Or [a different perturbation]

Robust yet fragile = fragile robustness

S j

Hard limits

X j U j

1

output=x

ln S j

d

0

C

0

d

ln X j

d

ln U j

d d

Plant

ln S j

X j ln U j

+

u Entropy rates

[ATP]

1.05

1

h >> 1 0.95

Time response

0.9

0.85

Yet fragile

h=1 0.8

0

5

10

15

20

Time (minutes) 0.8

h >>1

Robust

Log(Sn/S0)

0.6 0.4 0.2

Spectrum

h=1

0 -0.2 -0.4

ln Sh j

-0.6 -0.8

0

2

4

Frequency

ln S0 j 6

8

10

X j U j

S j 1

output=x

ln S j

d

C

0

0

+ ln S j 0

z z

2

2

d

z ln z

p p

The plant can make this tradeoff worse.

Plant

1

u

X j U j

S j 1

ln S j

output=x d

0

d 2

z ln z

C

0

1

ln S j

z2

+

p p

Plant

0

z

All controllers: Biological cells: = z

k q

p

RHPzero s 2

q

u k s

k

X j U j

S j 1

ln S j

output=x d

0

d 2

z ln z

C

0

1

ln S j

z2

+

p p

Plant

0

z

Small z is bad.

u z

k q

p

RHPzero s 2

q

k s

k

Small z is bad (oscillations and crashes) 1

ln S j 0

z z

2

Small z = • small k and/or • large q

z

k q

2

d

z ln z

p p

Efficiency = • small k and/or • large q

Correctly predicts conditions with “glycolytic oscillations”

S j

Hard limits

X j U j

1

output=x

ln S j

d

0

C

0

d

ln X j

d

ln U j

d d

Plant

ln S j

X j ln U j

+

u Entropy rates

1

ln S j

d

0

output=x

0

Plant

+

Controller

u

1

ln S j

d

0

output=x

0

Channel

Plant

+

Sensor+ Channel

u

Controller

1

ln S j

d

CFB

Hurts

0

output=x

1

ln S j

d

Csensor

Helps

0

Channel

Plant

+

Controller

Sensor+ Channel

u

Reactions Flow/error Protein level Trans* Flow/error *NA level

Anaerobically on w Autocatalytic

x

Glycogen/ glucose q

w

Vx

1 1

Control

q

y

xh

AA Biosyn

a

1 (1 0

1 q 1

ky y

Nuc Biosyn

)

z n

x Prot1

Protx

x

p1

Polym

p

*NAx *NA1

Polym

Control feedback everywhere

motif rpoH

Other operons

DnaK

Lon

See El-Samad, Kurata, et al…

PNAS, PLOS CompBio

motif rpoH

Other operons

DnaK

Lon

Where are these layers?

Protein

RNA DNA


motif Heat

rpoH folded

unfolded

Other operons

DnaK

Lon

degradation

DnaK

Lon

rpoH

Heat mRNA

Other operons DnaK RNAP

Lon

RNAP DnaK

FtsH

Lon

rpoH

Heat mRNA

DnaK


DnaK

ftsH Lon

RNAP DnaK

FtsH

Lon

Where are these layers?

Protein

RNA DNA


RNA

rpoH

Heat

mRNA

DNA DnaK

Protein Other operons

DnaK

DnaK

Lon

ftsH

RNA RNAP

RNAP DnaK

FtsH

Lon

DNA

RNA

rpoH

Heat

mRNA

mRNA activity is actively controlled. Translation was not shown.

RNA RNAP

RNAP

rpoH

Heat mRNA

DnaK


DnaK

ftsH Lon

RNAP DnaK

FtsH

Lon

rates

rpoH

Heat

mRNA

levels

DnaK

DnaK RNAP

DnaK

ftsH

Lon RNAP

DnaK

FtsH

Lon

Gly G1P

Layered control architectures

G6P F6P F1-6BP

Gly3p ATP

Allosteric

13BPG

3PG 2PG

Oxa PEP

Pyr

ACA

TCA

NADH Cit

Trans*

rates

rpoH

The greatest complexity here is primarily in the control of rates

mRNA

levels

DnaK

DnaK RNAP

Lon

DnaK

FtsH

Lon

Gly G1P G6P

That is not always the case.

F6P F1-6BP

Gly3p ATP

Allosteric

13BPG

2PG

Oxa PEP

Pyr

DnaK

ftsH

RNAP

3PG

Heat

ACA

TCA

NADH Cit

Trans*

rpoH

Heat mRNA

DnaK


DnaK

ftsH Lon

RNAP DnaK

FtsH

Lon

motif rpoH

Other operons

DnaK

Lon

All at the DNA layer

Autocatalytic

x

Glycogen/ glucose q

w

Vx

1 1

1 (1 0

q

Control

1 q

y

xh

1

Reaction?

ky y

z

)

Aerobic

Reaction? AA Biosyn

a x Prot1

p1 p? Prot2

*NA1

p2

*NA?

*NA2

p?

*NA?


x

Glycogen/ glucose

w

q 1 1

Vx

1 (1 0

q

xh

Control

y

1 q 1

ky y

)

z

Protein level

Prot1 Prot2

p1 p2


x

Glycogen/ glucose

w

q 1 1

Vx

1 (1 0

q

xh

Control

y

1 q 1

ky y

)

z

AA Biosyn

a

Control?

x

Prot1 Prot2

p1

degrade

p2

Precursors

Catabolism

AA

transl.

tRNA

Enzymes Ribosome

ncRNA

mRNA

RNA Transc. Gene

xRNA RNAp

Precursors

Catabolism

Autocatalysis everywhere AA

transl.

Proteins

tRNA

All the enzymes are made from (mostly) proteins and (some) RNA.

Ribosome

RNA transc. xRNA RNAp

Aerobically on z Autocatalytic

1 (1 0

x

)

Glycogen/ glucose

z

y

w

Aerobic

Reaction? Reaction?

supply

AA Biosyn

a

Control?

p? p?

p?

p?

Autocatalytic

x

Glycogen/ glucose

w

q 1 1

Vx

1 (1 0

q

xh

Control

y

1 q 1

ky y

z

)

Aerobic

Reaction? Reaction? AA Biosyn

a

x

Prot1 Prot2

p1

degrade

p2

reactions

products

reaction3

Control?

Proteins

trans.

Transc.

All products feedback everywhere

ncRNA


x

Glycogen/ glucose q

w

Vx

1 1

1 (1 0

q

Control

y

xh

1 q 1

ky y

)

z

AA Biosyn

a x

degrade

Prot1

p1 Prot2

*NA1

*NA2

p2


Autocatalytic

x

Glycogen/ glucose q

w

Vx

1 1

1 (1 0

q

Control

1 q

y

xh

1

Reaction?

ky y

z

)

Aerobic

Reaction? AA Biosyn

a x Prot1

p1 p? Prot2

*NA1

p2

*NA?

*NA2

p?

*NA?


x

Glycogen/ glucose q

w

Vx

1 1

Control

q

y

xh

AA Biosyn

a

1 (1 0

1 q 1

ky y

Nuc Biosyn

)

z n

x Prot1

Protx

x

p1

Polym

p

*NAx *NA1

Polym
