Genetic, Protein, Metabolic Networks
Which choice is amenable to modeling?
Masanori Arita (National Institute of Genetics)

Biology, the final frontier…
 Erwin Schroedinger “What is Life?”1944
 Ilya Prigogine “Order out of Chaos”
1984
 ??? “Life out of Math”
2024
But, even single-protein phenomena have notbeen solved…
Protein structure folding Temperature-compensation mechanism
 Molecular recognitionSuccess in Biology
 Great success in “DNA”

Genetics
 Neutral theory of evolution (Motoo Kimura) Evolution by duplication (Susumu Ohno)
Informatics
 Dynamic programming P-value by random walk (Karlin & Altschul)

Other molecules need “killer” models.

Is Network the killer approach?
Most natural networks are:Scale-free Node degree exhibits power-law distribution.
 Exponent λ of y = x-λ is between 2 ~ 3.
(Barabasi Science 1999)
Small-world
 Local friends are clustered.
 Diameter of the network is extremely small.
(Watts & Strogatz Nature 1998)

Clustering and path properties
Average path length

Clustering coefficient Average number of steps

 Density of edges among

between all pairs of nodes
 Random network has O(log n) path.

neighboring nodes
 Random network has avery low density.

Albert, R, Barabasi, AL, Reviews of Modern
Physics 74, 47 (2002)

Why power law?
 Power law≒ lognormal distribution≒ multiplicative process
Real world is governed bymultiplication, not by addition.
The central limit theorem:“cumulative distribution approaches to lognormal (i.e.
power law) distribution”(E Fox-Keller Bioessays 2004, Arita J Biochem 2004)

Why small world?
 a little randomness + regular structure= small world

Real world is governed bylocal interaction, not by global (i.e. regular).
Numerical analysis:“random noise make the world drastically smaller”

So the morals are In biology, scale freeness is a cumulative effect of complexinteractions,and small worldness is due to noise…
 We need to look at more “simple” processes.
 Look at details of the target.

Complexity of the metabolism
L-Glu

EC2.3.1.1
acetyl CoA

N-acetyl
L-Glu

CoAATPEC 2.7.2.8
ADP
N-acetyl
L-Glu 5p

NADPH
EC 1.2.1.38
NAD + PiN-acetyl
L -Glu 5sa
N-acetyl
L- GluL- GluEC 2.6.1.69
-keto Glu

EC 2.3.1.35

N-acetyl
L-Orn

L -OrnacetateEC 3.5.1.16

H2 O

Amino acid glutamine isconverted to ornithinein 5 steps.
Glutamine is used twice toprovide 2 nitrogens.

Textbook notation
L-Glu

EC2.3.1.1
acetyl CoA

N-acetyl
L-Glu

CoAATPEC 2.7.2.8

EC 2.3.1.35

ADP
N-acetyl
L-Glu 5p

-keto Glu

NADPH
EC 1.2.1.38
NAD + PiN-acetyl
L -Glu 5sa

EC 2.6.1.69
N-acetyl
L-Orn

L -OrnacetateEC 3.5.1.16

Textbooks often show thered part only.

H2 O

Databases too.

Ideal explanation (for this pathway)
L--GluEC2.3.1.1
acetyl CoA

N--acetyl
L-Glu

CoAATPEC 2.7.2.8
ADP

Red: carbon
Blue: acetate
Green: nitrogenN--acetyl
L-Glu
5p

NADPH
EC 1.2.1.38
NAD + PiN-acetyl
L --Glu 5sa
EC 2.6.1.69
-ketoGlu
EC 2.3.1.35

N-acetyl
L--Orn

L -OrnacetateEC 3.5.1.16

H2 O

Metabolism is neither
small-world nor
scale-free.

Complexity of protein network
 Famous MAPK cascade has 4 types in human.

Textbook information
 MAPK (Mitogen-activated Protein Kinase) has multiple

instances, under different names.
 Databases contain new/old names.
 Ontology is still a big issue.

Mike Ashburner "Biologists would rather share a toothbrush thanshare a gene name"Complexity of the gene network Dependence?

Finding a good target is difficult.
 Single protein/molecule topics

Protein folding Fragmentation in mass spectrometry
 Retention in chromatography
 Catalytic site design …
 Regular and robust mechanism

Biosynthesis of metabolites (tracer study)
 Circadian oscillation
 :Circadian mechanism(with Dr. Yoshihiko Hasegawa) Gene expression of 24 h ± 1 cycle
 Entrained by (sun) light Mechanism is species-dependent
humanflyPhase response curve Same perturbation results in different shift of the rhythm.

PRC shows its relation.

[Refinetti, Circadian physiology,Taylor & Francis, 2005]

The model Deterministic rate equation of limit cycle
light sensitive parameter
periodic signal

freq.
Molecular species

multiplicative term

 The purpose is to analyze:

intensity

XWhitenoise1XX

3

2

Regularity … period varianceclock Entrainability… range to be able to synchronize toexternal stimuli




freq.



Regularity and entrainability
 Regularity

 Entrainability(Arxiv 1305.0623)

Variational method
 Maximize entrainability under constant regularity
Lagrange multiplier

Entrainability

Regularity

Optimal phase-response curve (PRC)
iPRC
input signal

related to light
entrainment
mechanism

Model parametersPeriod of the oscillation

Variance of the period

Noise intensityAmplitude of the key
molecule affected by lightOptimal PRCs Dead zone exists, a time during which light stimuli

neither advance nor delay the clock Dead zone always exists forIntuitive explanation
 To optimize regularity, zero sensitivity

is optimal (dead zone only).
 To achieve entrainability, some
response is necessary (actual PRC).
 Light response is effective when thekey molecule is abundant.

Observation
 Only under the solar input, we observe the dead zone.
 Dead zone appears for a wide parameter range.

 Actual circadian clocks are optimal forsynchronization to daylight !
(Hasegawa & Arita Interface accepted)
 No dead zone in peripheral clocks.
 When the pulse is long, we do not observe true PRCs.

Take home messages
 Do not work on noisy data.
 Choice of the modeling target is important.
 Keep the model simple.

For math details,
Dr. “Yoshihiko Hasegawa” (Google Sites)