細胞内反応ネットワークと 「少数性」効果 · 2017-04-24 · computational robotics...
TRANSCRIPT
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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