The best material model of a cat is another,
or preferably the same, cat. A Rosenbleuth.
Artificial
Biochemistry
Bitonal
Membrane Systems
Luca Cardelli
Microsoft Research
Trento, 2006-05-22..26
www.luca.demon.co.uk/ArtificialBiochemistry.htm
Membranes are Oriented 2D Surfaces
Diffusion (fast)
Lipid
Extracellular
Space (H2O)
Hydrophilic head
Hydrophobic tail
Flip
(rare)
Lipid Bilayer
Self-assembling
Largely impermeable
Asymmetrical (in real cells)
With embedded proteins
Cytosol (H2O)
Embedded
membrane proteins
Channels, Pumps
(selective, directional)
Luca Cardelli
A 2D fluid inside a 3D fluid!
5nm
5nm
~60 atoms
(Not spontaneous)
2006-05-27
2
Systems of Oriented Membranes
phospholipid bilayer
membrane proteins
(consistently oriented)
Each membrane has two faces. A cytosolic
(~inner) face and an exoplasmic (~outer) face.
Nested membranes alternate orientation.
(E.g. cytosolic faces always face each other,
by definition, or by fusion/fission dynamics)
cytosolic face
cytosolic face
This alternation is illustrated by using two
tones: blue (cytosol(2)) and white (exosol(3)).
Bitonal diagrams.
Bitonal diagrams
cytosol
exosol
~exosol
cytosol
Double membranes (e.g. the nuclear
membrane) gives us blue-in-blue components.
~cytosol
(1) A membrane is built from a phospholipid bilayer that is asymmetrical. Moreover, all real membranes are heavily
sprinkled with proteins: “each type of integral membrane protein has a single specific orientation with respect to
the cytosolic and exoplasmic faces of a cellular membrane, and all molecules of any particular integral membrane
protein share this orientation. This absolute asymmetry in protein orientation confers different properties on the
two membrane faces.” MCB p162.
(2) Short for Cytoplasmic Solution. (3) Short for Exoplasmic Region (I am making this one up).
2006-05-27
Luca Cardelli
Membranes are closed non-intersecting
curves, with an orientation(1).
3
Bitonal Structure
Bitonality
Blue and white areas alternate.
Bitonal Invariant
Bitonality and subsystem coloring is preserved
by reactions. I.e., blue and white fluids
E.R.
9%vol
Golgi
6%vol
Nucleus
6%vol
Mitochondria
2%vol, #1700
never mix and never flip color.
Bitonal Duality
Reactions come in complementary-tone versions.
The cell maintains a strong compartment-based
separation between inside fluids and outside fluids
even when incorporating foreign material.
Chloroplasts
Cytosol
54%vol
Exosol
3%vol,
#900
Mitochondria
acquisition
Mitochondria
to Chloroplasts
Pre-Eukarya
to Eukarya
Luca Cardelli
Evolutionary
explanations
of bitonal
structure
Lysosomes
Vacuoles
Transport
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4
Luca Cardelli
Gradual Transformations
of Membrane Systems
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5
Locally Realizable Reactions
Membrane
System
?
What transformations
“make sense”?
Local (Patch) Reactions
Reactions that obviously “make sense”
from a local, molecular viewpoint
Froth
Fizz
(Symmetric by 90o
rotation.)
(Phospholipids thrown in water
self-assemble into empty
vesicles)
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Luca Cardelli
Switch
6
Gradual Change
A global reaction is a pair of membrane systems (before and after),
but we are only interested in gradual changes; e.g.:
Mito
There are three ways to characterize gradual changes:
• Local interactions of membrane patches.
(What really happens at the biochemical level.)
• A specific set of global reactions that are “biologically meaningful”
(e.g. mitosis, endocytosis) and hence presumably gradually implemented.
Luca Cardelli
• The gradual transformation of “small areas” of a membrane system
in ways that do not “mix fluids” on a large scale.
These turn out to be equivalent!
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Those Global Reactions are Local Reactions
Reactions that “make
sense” from a descriptive,
global viewpoint
Mito
(Fission)
Mate
(Fusion)
(dual)
Endo
(Fission)
(dual)
Same
Local
View!
Exo
(Fusion)
2006-05-27
Luca Cardelli
Switch
8
Luca Cardelli
Bitonal Transformations:
Operational View
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9
Bitonal Reactions
We look for reactions that “preserve” the bitonal coloring of a
membrane system. (And hence preserve proper membrane orientation
and “well-being”.)
Luca Cardelli
?
2006-05-27
10
Froth/Fizz Reaction
The spontaneous appearance/disappearance of empty
bubbles (of the correct tonality).
White
expanse
Blue
expanse
F
F
* Phospholipid molecules
automatically assemble
into closed membranes.
Luca Cardelli
N.B. non-empty membranes should not “spontaneously” be created or
deleted: usually only very deliberate processes cause that. However,
spontaneous froth/fizz seems be harmless; it means that empty
membranes are not observable.
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Mito/Mate Reaction
P
P Q
Q
M
P
Q
P
Q
Dual:
P Q
M
Q
Luca Cardelli
P
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Endo/Exo Reaction
Q
P
P
Q
E
P
Q
Dual:
P
Q
E
Q
P
Q
Luca Cardelli
P
2006-05-27
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Peel/Pad Reaction
Q
Q
P
Q
Dual:
Q
P
Q
Luca Cardelli
Q
2006-05-27
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Bud Reaction
P
Q
Bud
P
Q
Obviously a special case of Mito,
but it can be, both biologically and computationally,
considerably simpler (no arbitrary splitting).
Can also be seen as Pad + Exo:
Q
Pad
P
Q
Exo
P
Q
Luca Cardelli
P
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Bad Bubbles
Bubble catastrophe:
Violates bitonality in context.
Also, ill-toned reaction arrow.
Luca Cardelli
Wrong bubbles:
Violates bitonality.
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Flooding
Violates bitonality in context.
Also, ill-toned reaction arrow.
Flooding
P
Q
P
Q
Flooding in context violates bitonality:
Q
P
Q
Luca Cardelli
P
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Ambients
P
Q
In
P Q
Out
P
Open
Preserve bitonality, but violate stability
for subsystem P (i.e. all membranes of P
must be “flipped” inside-out).
P Q
P
P
Q
P
Q
P Q
P
PI Q
In
Out
Open
PI
Q
PI
Luca Cardelli
Violate bitonality
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Summary: At Least Four Good Reactions
F
Endo/Exo
Mito/Mate
Peel/Pad
P
Q
P Q
Q
E
M
P
P
P
Q
Q
Q
Actually, Peel/Pad is
NOT a bitonal reaction
by my definition, but is
the composition of two
such. Good enough.
Luca Cardelli
Froth/Fizz
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Mito/Mate by 3 Endo/Exo
Q
P
Q
Endo
Exo
Q
Q
Q
P
P
P
Exo
Endo
P Q
(dual)
P Q
P Q
Exo
Endo
Luca Cardelli
P
P Q
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Endo/Exo by Mito/Mate and Peel/Pad
P
Q
Q
Mate
Mito
P
P
Q
Q
Endo/Exo from
Mito/Mate only?
No: depth of
nesting is
constant in
Mito/Mate.
Luca Cardelli
P
Pad
Peel
2006-05-27
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Peel/Pad by Froth/Fizz and Endo/Exo
P
Endo
Exo
P
P
Luca Cardelli
P
Froth
Fizz
2006-05-27
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An (Turing) Complete Set of Reactions
Endo
Exo
Froth
Fizz
Others bitonal reactions are Derivable, e.g.:
Luca Cardelli
Mito
Mate
Are all other derivable? YES!
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Luca Cardelli
Ex: Eukaryotic Mitosis
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Luca Cardelli
Bitonal Transformations:
Topological View
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Depth and Tonality
0
2
3
1
3
1
2
Luca Cardelli
The depth of a point is the number of membranes that contain it.
The tonality of a point is white/blue iff its depth is even/odd.
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Def: Bitonal Reactions
A bitonal (resp. layered) reaction is a pair of membrane systems <M,M’> such that the
points that change tone (resp. depth) form a simply-connected region
(a region not separated by membranes). (Layered ⇒ Bitonal)
Froth
changes tone & depth
simply connected
Mito
changes tone & depth
simply connected
changes tone
simply connected
Exo
Simple Deformation
(Layered & Bitonal)
change depth
not connected
Layered
Bitonal
Reaction
Layered
Bitonal
Reaction
Non-Layered
Bitonal
Reaction
Luca Cardelli
Deform
changes tone & depth
simply connected
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Non-Bitonal Reactions
A bitonal (resp. layered) reaction is a pair of membrane systems <M,M’> such that the
points that change tone (resp. depth) form a simply-connected region
(a region not separated by membranes).
In
In
Pad
change tone & depth
not connected
change tone & depth
not connected
change tone & depth
not connected
but obtainable as the composition
of two bitonal reactions (Froth+Endo)
2006-05-27
Luca Cardelli
Wrap
change tone & depth
not connected
28
Bitonal Transformations
● A transformation is a finite sequence of reactions. A bitonal
transformation is a finite sequence of bitonal reactions.
● We want all “legal” transformations to be bitonal transformations (and
hence “gradual” transformations). E.g.: padding:
Non-Layered Non-Bitonal
Reaction
(but obtainable by two bitonal
reactions)
Luca Cardelli
Pad
not simply
connected
has hole
● Some transformations are inherently non-bitonal.
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Characterization
● Soundness and Completeness Theorem
– A transformation of membrane systems:
●is locally realizable
(realizable by a sequence of switch + froth/fizz)
●iff it is bitonal
(changes tone of at most a simply-connected region at a time)
●iff it is fusion/fission-realizable
(realizable by a sequence of endo/exo + froth/fizz)
● Proof Sketch
1. All local reactions are bitonal reaction. E.g., Switch:
- By cases on the external connectivity of A,B,C,D.
B
E
C
A
Switch
D
B
E
C
D
Luca Cardelli
A
2006-05-27
30
2. All bitonal reactions can be obtained by
sequences of local reactions (switch/froth/fizz)
and deformations.
- By analysis of the simply connected regions
that change tonality, and induction on the number
of membranes that cross such a region.
3. Endo/Exo and Mito/Mate are bitonal reactions.
- They can all be locally implemented by Switch, which is bitonal.
4. Any instance of Switch is an instance of either Endo, Exo, Mito, or Mate,
plus deformations.
- By cases on A,B,C,D connectivity around Switch.
●
Therefore, any bitonal transformation can be written as a sequence of local
reactions, and hence as a sequence of Endo/Exo/Froth/Fizz plus deformations.
Conversely, and such sequence is a bitonal transformation.
2006-05-27
Luca Cardelli
5. Mito/Mate can be encoded by Endo/Exo.
31
Luca Cardelli
Bitonal Calculus
A Prototype for Membrane Calculi
2006-05-27
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Bitonal Calculus
Systems
X ::= k | XmX | hXi
This algebra is a minimal “subset” of
more sophisticated process calculi for
membranes that one may devise.
membrane
Axioms
km
is a comm. monoid
F/F:
E/E:
k hki
XmhYi hhXimYi
Facts
(without using
commutativity)
X Xmk Xmhki
hhXimki hhXii
hXimhX’i hhhXiimX’i hhkmhXiimX’i
hhkimXmX’i kmhXmX’i hXmX’i
Define a simple “type
system” that colors
brackets and operators
with alternating tones.
k hji
XmhYi hfXglYi
Alternative axiomatization: take M/M and P/P
as axioms and derive F/F and E/E as theorems:
F/F:
Subject reduction theorem.
Bitonal coloring is preserve by
reductions.
P/P:
E/E:
hki kmhki hhkimki hhkii k
XmhYi hhXiimhYi hhXimYi
2006-05-27
Luca Cardelli
M/M:
33
Atonal Calculus
Systems
X ::= k | XmX | hXi
membrane
Axioms
km
is a comm. monoid
F/F:
I/O:
k hki
XmhYi hXmYi
violates bitonality
Facts
Atonal emulates bitonal (obviously):
XmhYi XmkmhYi XmhkimhYi hXmkimhYi hXimhYi hhXimYi
Bitonal emulates atonal, based on this translation:
Luca Cardelli
k* = k
(XmY)* = X*mY*
hXi* = hhX*ii “double walling”
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Summary
● Bitonal Membrane Systems
– Algebraically capturing the notion that cytosol/exosol do not “usually” mix
during membrane transformations.
– Characterization theorem: membrane reactions are locally implementable
(switch) iff globally implementable (endo/exo) iff topologically gradual
(bitonal).
● Bitonal Calculus
Luca Cardelli
– A minimalist membrane calculus.
– Bitonal can emulate atonal.
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Luca Cardelli
Q?
2006-05-27
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Luca Cardelli
Appendix
More Examples
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Ex: Molting
P
Pad
P
Aged
Exo
P
Luca Cardelli
P
2006-05-27
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Ex: Autophagic Process
Lysosome and target don’t just merge.
Lysosome
Enzymes
E.R.
1
2
3
4
5?
6?
7
Biologically, Mito/Mate
clearly happens. However,
weird sequences of
Endo/Exo are also common.
Luca Cardelli
Target
2006-05-27
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(fake) Ex: Clean Eating
(why Endo/Exo is “healthier” than Mito/Mate)
Q
P
Q
Either:
Or:
Fizz
Pad
P Q
Mate
P
Endo
Q
P Q
Exo
P Q
P Q
P Q
Exo
Luca Cardelli
P
Dirty!!
Clean!
Exo
P Q
2006-05-27
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Luca Cardelli
Appendix
A Note on Locality
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Locality Postulate
Locality Postulate
Luca Cardelli
Interactions should be local to small membrane patches.
E.g., independent of global membrane properties such as
overall curvature that cannot be observed locally.
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Endo/Exo Violates Locality?
“local view” of reaction
P
Q
E
P
Q
membrane curves right
same local view
R P
membrane curves left
Q
E
R
P
Q
(not a direct
instance of
endo/exo)
Luca Cardelli
Oops…
2006-05-27
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Local-view
Endo/Exo Reaction
P Q
e
P Q
P Q
e
P Q
Q
E
might curve
left or right
Dual:
Global View
P
and:
RP Q
M
P
R
Q
P Q
Ah!
Local Endo/Exo
= co-Mito/Mate
Luca Cardelli
Both:
2006-05-27
44
Local-view
Mito/Mate Reaction
P
Q
m
P
Q
P
Q
m
P
Q
P
Q
might curve
together or apart
Dual:
Global View
and:
P
R P
Q
Q
M
E
R P
Q
Ah!
Local Mito/Mate
= co-Endo/Exo
Luca Cardelli
Both:
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Locality is Not Violated
● Hence, even though Endo/Exo and Mito/Mate strictly violate locality,
●
Luca Cardelli
●
locality is indirectly preserved in a bigger system that includes them
both and their duals.
This needs to be somewhat justified (L.Cardelli: “Bitonal Systems”)
after which we can forget about local-view reactions.
Problem: how to formally represent the local-view reactions?
2006-05-27
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Luca Cardelli
Appendix
Molecules as Small
Membranes
2006-05-27
47
Molecules as Small Membranes
Mol
moln = maten+Kn
Moln = molnhki
n(k) = mateIn
k(n) = KIn
n(k) = dripn(moln)
k(n) = G(moln)
n(k)k(m) = n(k).k(m) etc.
Luca Cardelli
Moln m n(k).k(n).σ|σ0hPi } σ|σ0hMoln m Pi
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Luca Cardelli
Appendix
Local Membrane Reactions
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Membrane Systems
Good Systems
Bad Systems
Luca Cardelli
(Closed non-intersecting curves)
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Bitonal Membrane Systems
Good Bitonal Systems
Bad Bitonal Systems
Luca Cardelli
(Alternating oriented curves)
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Local Unoriented Interactions
Switch
Klein
Luca Cardelli
Unravel
2006-05-27
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Switch as a Bitonal Reaction
Switch
Switch
Luca Cardelli
(dual)
2006-05-27
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Global Effects of Switch
Preserves
Bitonality
Switch
Decreases
Cardinality
Mate
Exo
(selfreactions)
Mito
Endo
(In 3D these might
create a torus:
require elaborate
staging)
2006-05-27
Luca Cardelli
Increases
Cardinality
54
Global Effects of Klein
Violates
Proper
Containment
Luca Cardelli
Produces
Unorientable
Curves
2006-05-27
55
Global Effects of Unravel
Pop
Squish
Burst
Luca Cardelli
Decreases
Cardinality
Violates
Bitonality
Open
2006-05-27
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Global Effects of Pinch
Preserves
Bitonality
Pinch
Nothing here
Drip
Pino
Luca Cardelli
Switch
2006-05-27
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Global Effects of Pass
Violates
Bitonality
Pass
Preserves
Cardinality
In
Luca Cardelli
Out
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Global Effects of Coat
Preserves
Bitonality
Coat
Nothing here
Increases
Cardinality
Phago
Luca Cardelli
Bud
N.B. Pass can be obtained by Coat + Unravel, showing that
Pass violates alternation because Unravel violates
alternation.
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Reductions to Switch
Coat by Switch
Switch
Luca Cardelli
Pass by Switch and Unravel
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Unoriented Local Reactions
Undo Reactions
Luca Cardelli
Coat
Pass
Unravel
Switch
(Irreversible) Fusion
sfes
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Oriented Local Reactions
No Initial Bitonality
(Irreversible
Fusion)
Violates
Membrane
Orientation
Violates
Bitonality
Violates
Bitonality
Luca Cardelli
Coat
Pass
Unravel
Switch
+duals Good Initial Bitonality
2006-05-27
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Local Brane Algebra?
● Hence, Switch and Coat are the “good” oriented reactions.
● Moreover, Coat can be obtained by Switch, and Pass can be obtained by
Coat and Unravel.
● Can we build a membrane algebra just out of local operations such as
Luca Cardelli
Switch and Unravel?
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