Yuichi TOGASHI （冨樫 祐一） Computational Robotics Research Group,

Department of Computational Science, Graduate School of System Informatics,

Kobe University

細胞内反応ネットワークと 「少数性」効果

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Euglena sp.; Illustration by A. Kouprianov (wikipedia)

Modeling of Cells

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Modeling of Biological Cells

Our aim is, of course, better

understanding of biological phenomena.

And also, construction of artificial

systems inspired by biological systems.

An important feature of the biological

systems: Coexistence of high “stability” (robustness) and

moderate “instability” (plasticity, adaptability)

How to attain this?

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A “Textbook” Picture of a Cell

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From Molecular Biology of the Cell (Alberts et al.)

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Biochemical Processes Modeled

as Reaction-Diffusion Systems

We often model biochemical processes

(and even whole cells / organisms)

as Reaction(-Diffusion) Systems

反応拡散系. Most activities of the cell / organism

depend on chemical reactions!

A variety of enzymes are involved.

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PEESSE

1k

1k

]][[1 SEk

][1 ESk

][]][[][ 11 ESkSEkS

The System is Represented by

[S] [E] [ES]

Biochemistry (N→∞)

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Biochemical Processes Modeled

as Reaction-Diffusion Systems

Turing pioneered the field 60 years ago!

A. M. Turing, Phil. Trans. Roy. Soc. B 237, 37 (1952).

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Biochemical Processes Modeled

as Reaction-Diffusion Systems

Modeling of biochemical processes and

even whole cells / organisms as

reaction-diffusion systems for 60 years.

..., quite successful.

So, what shall we do next?

Nothing is left for theoreticians?

Zebrafish Skin

Damaged → Cure

Experiment

&

Simulation

(Turing Pattern) M. Yamaguchi et al.,

PNAS 104, 4790 (2007). （今は阪大生命機能の

近藤さんらの研究）

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Classical Reaction-Diffusion

Models

As Turing did, we often adopt differential equations of concentrations. （反応速度方程式、反応拡散方程式）

This implies, we assume MOLECULES are:

MEMORYLESS ○ Each reaction event is completed immediately.

TINY (Point-like)

○ Freely go through (uniform medium, normal diffusion, ...).

MANY (No fluctuations)

○ Amounts represented by continuous variables.

),,,( 212

2

Nii

ii cccR

x

cD

t

c

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Molecules in the Cell...

1. MEMORYLESS?

Enzymes are macro-molecular machines.

Reaction cycles

coupled with the motion

Sometimes taking

a long time > ms ～ s.

Reaction event cannot be

completed immediately,

i.e.,

with states or memory.

Even more complex dynamics...

Acetyl-CoA Synthase movie by molmovdb

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Molecules in the Cell...

1. MEMORYLESS?

Cycle-time of enzymes

From: H. Gruler, D. Müller-Enoch, Eur. Biophys. J. 19, 217 (1991);

see also M. Schienbein, H. Gruler, Phys. Rev. E 56, 7116 (1997).

Example:

Cytochrome P-450 dependent

mono-oxygenase system.

Turnover time = 1.54 s.

Sync

Without feedback,

still synchronized!

Pro

duct

Rele

ase

→

(Flu

ore

scent)

Time

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Molecules in the Cell...

2. TINY?

Cells are highly

crowded with

macromolecules. NOT Point-like.

Also structural

elements such as

membranes and

cytoskeletons,

particularly in

eukaryotes. Bacterial cytoplasm model

by S. R. McGuffee and A. H. Elcock,

PLoS Comput. Biol. 6, e1000694 (2010).

(consisting of 275g/l macromolecules)

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Molecules in the Cell...

3. MANY?

A variety of chemicals

in a small system

Eukaryotic cell ～10um

Organelle in the cell

or Bacteria ～1um

→ some must be rare! MW10000 × 10000 types

× 1uM each = 100 g/l.

Direct evidence → Sometimes there,

sometimes not! Quantification of protein copy numbers in e-coli.

From Y. Taniguchi et al., Science 329, 533 (2010).

< 1 molecule

per cell

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Molecules in the Cell...

3. MANY?

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From Molecular Biology of the Cell (Alberts et al.)

Small compartments, Narrow paths … 14 20130306

Classical Reaction-Diffusion

Models

Since then, as Turing did, we often adopt reaction-diffusion equations, i.e. partial differential equations of concentrations.

This means, we assume MOLECULES are:

MEMORYLESS ○ Each reaction event is completed immediately.

TINY (Point-like)

○ Freely go through (uniform medium, normal diffusion, ...).

MANY (No fluctuations)

○ Amounts represented by continuous variables.

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Toward Better Modeling

of Biological Cells ...

In biological cells, we cannot ignore: 1. MEMORY, or DYNAMICS

2. SIZE & SHAPE (exclusion volume)

3. SMALL NUMBERS

of molecules.

Today’s main topic: 3. Effects of Small Numbers. Though we are working also on 1 & 2., and

these problems are related to each other.

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Small Numbers between

Single Molecule Biology

N=1

&

Biochemistry

N≫1（ideally ∞）

N = 0,1,2,3,4,...10,...

not always 0 or 1, but

not so large (< 100～1000).

Correlation / Cooperation ?

Synchronization / Coherence ?

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Molecules in the Cell...

3. Effects of SMALL NUMBERS?

Some chemicals are very rare

(on the order of 1 molecule per cell).

What may happen at such an extremely

low concentration?

Strong fluctuations in the numbers.

Molecular discreteness may matter.

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Molecular Discreteness

For such rare chemicals, molecular discreteness may matter.

We consider two types of discreteness:

Discreteness in Numbers 数の離散性 Numbers of molecules must

be integer (0,1,2,...; NOT 0.5).

Spatial discreteness 空間的離散性 Finite distances (space)

between molecules; NOT gradual concentration.

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Molecular Discreteness

But, you may wonder:

Can A SINGLE Molecule

affect the behavior

of the whole system?

→ Yes! Otherwise, DNA can do nothing.

If it works as a catalyst (→ enzyme)

or a template (→ gene). May induce many reaction events.

Not consumed as a substrate.

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Molecular Discreteness

For such rare chemicals, molecular discreteness may matter.

We consider two types of discreteness:

Discreteness in Numbers Numbers of molecules

must be integer (0,1,2,...; NOT 0.5).

Spatial discreteness Finite distances (space)

between molecules; NOT gradual concentration.

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Discreteness in Numbers

Model: simple is the best!

In the examples below,

represented by:

Sometimes autocatalytic:

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A + B → A + C

A + B → A + A

B

A

C

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Discreteness-induced switching

of the network

We adopt a simple catalytic model:

A + B → A + C; A = catalyst (enzyme)

The point is: If the substrate or catalyst

is exhausted (i.e. = 0 molecule),

the reaction halts = Same as nothing!

Network ← Which chemicals are there.

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III I II

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Discreteness-induced switching

of the network

Network ← Which chemicals are there.

III I II

Large system

(continuum limit)

Small system

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Discreteness-induced switching

of the network

Example: a network model:

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Discreteness-induced switching

of the network

Same model, 1/10 system size.

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Molecular Discreteness

For such rare chemicals, molecular discreteness may matter.

We consider two types of discreteness:

Discreteness in Numbers Numbers of molecules

must be integer (0,1,2,...; NOT 0.5).

Spatial discreteness Finite distances (space)

between molecules; NOT gradual concentration.

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Spatial Discreteness

Catalytic reaction model “A makes B”

Then, A & B react.

“AB model’’ by Shnerb et al.

or

Two B’s react. 3 species model by Togashi & Kaneko

N. M. Shnerb et al., PNAS 97, 10322 (2000).

Y. Togashi and K. Kaneko, PRE 70, 020901 (2004);

Physica D 205, 87 (2005).

A B

B B

B

B B

B

B

B

B B B

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Spatial Discreteness

Slow diffusion, Low density of A,

Short lifetime of B

A

A B

B B

B B

B

B

B B

B

B B

B

B

B B

B

B

B

B B B

B B A B

B B

B

B B

B

B

B

B B B

A B

B B

B

B B

B

B

B

B B B

A B

B B

B

B B

B

B

B

B B B

A B

B B

B

B B

B

B

B

B B B

A B

B B

B

B B

B

B

B

B B B

A B

B B

B

B B

B

B

B

B B B A

B B

B

B

B B

B

B

B

B B B

A B

B B

B

B B

B

B

B

B B B

A B

B B

B

B B

B

B

B

B B B

A B

B B

B

B B

B

B

B

B B B

A B

B B

B

B B

B

B

B

B B B

A B

B B

B

B B

B

B

B

B B B

A B

B B

B

B B

B

B

B

B B B

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Spatial Discreteness

Slow diffusion, Low density of A,

Short lifetime of B

A

A B

B B

B B

B

B

B B

B

B B

B

B

B

B

B

B

B

B B B

B B

→ Acceleration by (Co-)Localization! 30 20130306

Localization and Compartments

Fast diffusion or long lifetime of “B”

→ Almost uniform distribution.

→ No acceleration.

A

A B

B

B B

B B

B

B B

B

B

B

B

B B

B

B

B

B

B B

B

B

B

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Localization and Compartments

How about this?

→ Similar effect

A

A B

B B

B B

B

B

B B

B

B B

B

B

B B

B

B

B

B B B

B B

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Localization and compartments

Effects of compartmentalization Indeed, such experiments have been done

by Matsuura et al., using microreactors.

↑ T. Matsuura et al.,

J. Biol. Chem. 286, 22028 (2011).

← H. Urabe et al.,

Biochemistry 49, 1809 (2010).

四方研（阪大情報）松浦さんらの成果

Tetramer formation

→ the rate is strongly nonlinear

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Spatial Discreteness

& the Cell Environment

The crowded environment in the cell may enhanced the effect by “confinement”. The enzyme and product

lie close to each other for a long time.

Processivity of reactions etc.

Also complex structures such as membranes and cytoskeletons. Small compartments, narrow passages, ...

→ Forming a special “reaction field”?

A B

B B

B

B

B B

B B

● ● ● ●

● ●

● ● ● ●

● ●

●

●

●

● ●

●

●

●

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Spatial Discreteness

& the Cell Environment

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From Molecular Biology of the Cell (Alberts et al.)

Compare the reaction behavior! 35 20130306

Summary

In biological cells, we cannot ignore: 1. MEMORY, or DYNAMICS

2. SIZE & SHAPE (exclusion volume)

3. SMALL NUMBERS

of molecules

MINORITIES (Small numbers of molecules)

may rule the system’s behavior through

their discreteness. Discreteness in Numbers (Integerness)

Spatial Discreteness (Finite Spacing)

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Summary

In biological cells, we cannot ignore: 1. MEMORY, or DYNAMICS

2. SIZE & SHAPE (exclusion volume)

3. SMALL NUMBERS

of molecules It is theoretically & computationally

challenging to fully consider them. A royal road may be: multi-scale modeling

for peta-scale-supercomputers.

Our alternative: seeking principles in “over-

simplified” models and simulations.

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Outlook

More computational power → always detailed/precise models?

Why still using simplified models even on supercomputers?

To draw a schematic view for a much larger (longer) scale, or investigate statistics. Easier to understand. Useful for preliminary trials.

Of course, we appreciate efforts to construct fine-tuned multi-scale models. ○ For computer scientists: There is a difference in the

pattern of computation (resource usage), as compared with detailed models. Perhaps we should care.

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Outlook

An important feature of the biological

systems: Coexistence of high “stability” (robustness) and

moderate “instability” (plasticity,

adaptability)

How to attain this? No clear statement yet.

Now we are extending our models and

theories to large molecular networks,

to suggest possible working mechanisms.

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Outlook

The cell has significant features: Molecular machines (e.g. enzymes) work under

strong fluctuations, with energy ～ kBT.

Even though some types of molecules are not

always there (0 molecule), the system survives.

→ Design principles allowing errors. But different from “fault-tolerant” systems

in the current IT industries…

We hope we can learn from it, with a good

combination of theoretical, computational

and experimental studies.

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Acknowledgments

Collaborators J. Gu, S. Ohnaka, K. Fujimoto, K. Higashikubo,

S. Shinkai, Z. Luo (Kobe)

H. Teramoto, C.-B. Li, T. Komatsuzaki (Hokkaido)

K. Kaneko (Tokyo)

in the framework of the Project

“Spying Minority in Biological Phenomena”. - These works were partially supported by JSPS, JST and MEXT-Japan.

Thank you for your attention!

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