from sq3.org.uk

Making Membranes in Artificial Chemistries
Tim J. Hutton
Biomedical Informatics Unit, Eastman Dental Institute, University College London,
256 Grays Inn Road, London WC1X 8LD, UK
[email protected]
Abstract
In this paper we review some of the possible ways of making
membrane-like structures in artificial chemistries. Such implementations range from simulations with accurate physics
and high computational costs to abstract models with very
low computational costs. We observe that various properties
of natural membranes, such as self-assembly and self-repair,
are lacking in some of the systems. Additionally, we present a
novel implementation based on a two-dimensional lattice that
has several desirable features and is computationally cheap.
Introduction
Along with the genetic material, membranes form a fundamental part of living cells. By allowing only small
molecules to pass through, semi-permeable membranes protect the contents of the cell while allowing its metabolism to
operate. By dividing space into compartments, membranes
allow the local chemistry to be manipulated by the genetic
material.
Membranes in Earth biology consist mainly of phospholipids. The physical interaction with water molecules causes
these lipids to self-assemble into energy-minimising bilayer
sheets, with the hydrophobic tails of the lipids on the inside
of the sheet and the hydrophilic heads facing the water on
each side (Fig. 1).
These membranes have some very interesting properties.
They spontaneously assemble from lipids in suspension, and
if perturbed will reform into the energy-minimising state.
Small molecules can migrate through the membrane, although this does not happen quickly. Certain proteins will
spontaneously become inserted into the membrane, opening
channels of communication or conferring other properties.
Controllable pores can even be created in the membrane by
inserting a ring-shaped structure of proteins. See for example (Alberts et al., 1994) for more information on the very
rich array of possibilities.
In the next sections we review the various models that
have been made of membranes, and consider how accurately
various properties of the membranes are recreated. These
are: a) self-assembly, b) self-repair, c) semi-permeability,
and d) whether new structures can be inserted into, or attached to, the membrane. It seems likely that some of these
properties are necessary if complex life forms are to evolve,
since rich interactions (Taylor, 2001) and cellular communication (Suzuki et al., 2003) are thought to be required, although a full evaluation of these requirements is beyond the
scope of this paper. An additional consideration for artificial
life experiments is: e) low computational cost.
Accurate physics
If the physics of interacting atoms must be simulated accurately, then a Molecular Dynamics (MD) approach is required. Published work on lipid bilayers includes (Heller
et al., 1993; Feller et al., 1997; Maillet et al., 1999). Figure 1 shows a cross-section through such a simulated lipid
bilayer, with water molecules above and below.
Figure 1: A molecular dynamics simulation of a segment of
lipid bilayer. Adapted from (Feller et al., 1997).
Such simulations can recreate all of the properties of natural membranes, however they are very expensive to compute,
see (Allen et al., 2001) for an insight into this. To date only
small segments of membrane have been simulated, an entire
vesicle (a bilayer forming a spherical shell) would require
more computing power than is currently available.
Simplified physics
One way to avoid having to simulate all the interatomic
forces accurately is to simplify the physics, reducing the
number of particles, and removing some of the more complicated interactions. Ideally such a simplification will retain the properties of the membranes that are thought to be
important, while allowing the simulation to run on standard
computer equipment.
One such system is presented in Chapter 3 of (Ono, 2001).
The system has three particle types in a two-dimensional
space, with simple spring forces between the different particles. The greatest repulsion is between the water particles and the hydrophobic particles, with lesser forces between the other types. By joining the hydrophobic and hydrophilic particles into amphiphilic pairs, simple bilayers
can be shown to form spontaneously (Fig. 2). The computational cost is sufficiently low for small areas to be simulated
at interactive speeds on desktop computers1 . An earlier system that is closely related is (Edwards et al., 1998).
force on each particle must be computed based on the position of tens of other particles. However, the membranes
retain the properties of self-assembly and self-repair, and
semi-permeability is possible since small particles that do
not have large repulsive forces from the other types will be
able to pass through.
A further simplification that can be made is to reduce
the membrane to a simple chain of bonded particles. Such
a membrane is used in (Hutton, 2004) to enclose a selfreplicating molecule (Fig. 3). Because the simulation uses
only local spring forces for volume-exclusion, and uses far
fewer particles, it is computationally cheaper than the previous system. Semi-permeability is achieved by turning off the
volume-exclusion force for certain types of unbonded particles. Self-assembly and self-repair are not easily achieved
in this system, however.
Figure 3: A chain of bonded particles can be used as a
simple implementation of a membrane. Only certain particles (shown as dots) can pass through the membrane, the
other particles (shown as filled circles) cannot. Adapted
from (Hutton, 2004).
Lattice based physics
Figure 2: A screenshot of a two-dimensional simulation
of bilayers. From an initially random configuration, the
amphiphilic particle-pairs have self-assembled into micelles
and bilayers. (Author’s implementation1 .)
For artificial life experiments, however, this implementation is still expensive. The number of interacting particles is
still large, with at least 100 particles required to make a stable vesicle in two-dimensions. Additionally, the repulsion
force on each particle extends over a large area and so the
1 The source code for this implementation and several of the
other systems in this paper is available at:
http://www.eastman.ucl.ac.uk/∼thutton/
Evolution/Squirm3/development/
Simulations that are based in a continuous space are typically more expensive than ones based on a lattice of points.
This is because the coordinates in grid-based systems can
be expressed in terms of integers rather than floating-point
numbers, and it is cheap to identify the neighbours of each
particle. Lattice gas simulations have been around for
some time, with recent work on lipid particles including
(Boghosian et al., 2000; Mayer and Rasmussen, 2000). Although these are not as physically accurate as MD simulations, the membranes typically have the features of selfassembly and self-repair, with semi-permeability being possible too.
By modelling membrane particles as having a directional
force, and allowing many particles to occupy each grid
point, simulations of self-assembling membranes and protocells can be achieved (Ono and Ikegami, 1999; Ono and
Ikegami, 2001; Ono and Ikegami, 2002; Madina et al.,
2003). Figure 4 (left) illustrates how the repulsive forces
are determined, with the greatest repulsion being along the
axis of each oriented particle. When the simulation is run
(Fig. 4, right) the particles self-assemble to form membrane
sheets and then enclosed vesicles or protocells.
Figure 5: A membrane surrounding a catalyst and other particles in an autopoietic artificial chemistry system. Adapted
from (McMullin and Varela, 1997).
the same reasons as the previous system, it is very cheap
computationally. Self-assembly and self-repair are not easily achievable in this system, however. An additional problem is that molecules sometimes become stuck on the lattice,
when their movement becomes restricted by the constraints
of their bonds.
Figure 4: (left) Oriented particles on a hexagonal lattice exert repulsive forces that vary in different directions. (right)
These particles self-assemble into membranes and protocells. Adapted from (Ono and Ikegami, 2002).
Structurally, these membranes differ from natural membranes in that they readily form three-way junctions. This
means that membranes tend to be shared between neighbouring protocells, and the protocells do not appear as separate entities. The membranes do have the properties of selfassembly and self-repair but it is not clear how genetic material might be represented in such a system.
A different approach to implementing membranes on a
lattice is presented in (Varela et al., 1974; McMullin and
Varela, 1997). Semi-permeable membranes can be created
by linking particles together and allowing only certain types
of particles to diffuse through (Fig. 5). In this system the
bonded particles are not permitted to move, while the other
particles move on a random walk on the lattice. Chemical
reactions between the particles cause the membrane to be
maintained, against a constant rate of membrane disintegration. All computations require only local information, and
the total number of particles is small, thus the system is computationally cheap.
A similar system is presented in (Hutton, 2002). In this
system every particle is permitted to move, with the constraints that at most one particle can occupy each grid point,
and that each particle must remain within the Moore neighbourhood of every particle it is bonded-to (Fig. 6). Semipermeability comes for free in this system, since unbonded
particles can move across diagonal links while bonded particles cannot without violating the constraints mentioned. For
Figure 6: A semi-permeable membrane in a simple latticebased system can be made by linking a chain of particles together. Unbonded particles can pass through this membrane
by crossing diagonal links, while bonded particles cannot
since the bond length is limited to the Moore neighbourhood. (Author’s implementation.)
Extended bonds, with bond-crossing forbidden
To conclude this section on lattice-based physics, we present
some new work aimed at solving the problem just mentioned, that molecules in the system presented in (Hutton,
2002) lack flexibility. The motivation for this is that the system has other desirable properties, not least the low computational cost of this form of membrane.
To improve the flexibility of the linked particles, we extend the permissible bond length to a Moore neighbourhood
of radius N. Even with N = 2 (a 5 × 5 neighbourhood), this
prevents most molecules from getting stuck on the lattice. In
the limit, extending the bond length would cause the spatial
representation to approach that of a continuous one. How-
ever, the membranes do not function in the same way as in
the previous system, since molecules can now pass through
them easily.
To solve this problem, we introduce an additional constraint, that bonds cannot cross each other. Thus a particle is forbidden to make a move if that move would cause
two bonds to become crossed, and a bonding reaction will
only proceed if a bond-crossing is not created. While this
means that a larger neighbourhood must be searched before
each move, the computational costs of this system are still
low. Bond-crossing is detected using a simple formula for
whether two line-segments intersect.
Figure 7 shows a molecule inside a semi-permeable membrane in this system. The molecule cannot escape because
that would cause bonds to become crossed. Unbonded particles can pass through easily.
Figure 7: By extending the permissible bond length and forbidding bonds to become crossed, good flexibility can be
achieved in a two-dimensional lattice-based artificial chemistry without sacrificing the semi-permeability property of
the membranes. In the example shown, each bonded particle must remain within the 5 × 5 neighbourhood of every
particle it is bonded to. (Author’s implementation.)
One drawback of such a system from the point of view of
trying to create artificial life is that the interactions between
cells and their environment is minimal. For the evolutionary growth of complexity (McMullin, 2000) to be observed,
not only is information inheritance required, but also interactions between competiting organisms (Taylor, 2001).
Graph based physics
A different approach to simulating molecular collisions is to
represent each molecule as a node, and each potential interaction as an arc (or edge), in a graph. Possible topologies
include triangular planar graphs (Speroni di Fenizio et al.,
2001) and graphs with the ‘small-world’ property (Suzuki,
2004). Movement is not explicitly represented in these
systems, however the physics of diffusion and hydrophobic repulsion can be implemented by a graph-rewriting process (Suzuki, 2004). Edges are deleted at random, creating
the effect of molecules moving apart. Edges are also created at random (these molecules effectively move towards
each other) but only in such as way as to maintain the smallworld property, and not if a water node would be linked to
a hydrophobic node, for example. This process leads to the
self-assembly of membranes that separate the graph into different regions (Fig. 8).
Figure 8: A proto-cell in a graph-based artificial chemistry.
Water-like particles have been driven to the left and the right
of the graph, with lipid node-pairs forming an interface between the water and the contents of the cell (centre). The
molecules inside the cell are therefore more likely to react
with each other, rather than with molecules outside the cell.
Adapted from (Suzuki, 2004).
One-dimensional physics
A further dramatic simplification of the physics of membranes and cells is to reduce the dimensionality further,
to a one-dimensional system. Such a system is presented
in (Ono and Ikegami, 2000), where repulsion between particles together with diffusion between neighbouring sites
causes membranes to be maintained, and the protocells to
reproduce under the right conditions.
The computational cost of such a system is minimal, since
interactions only need to be computed between neighbouring sites on a one-dimensional line. The membranes are selfrepairing and self-assembling, and also could be permeable
to other types of particle.
The visualisation of these graphs tends to be somewhat
difficult, partly because the dimensionality of the space of
interactions is quite high (each node can have many interaction arcs). In (Suzuki, 2004) it is suggested that this highdimensionality is beneficial for artificial life experiments
since many molecules can be involved in each reaction.
The membranes in this system self-assemble and selfrepair, and semi-permeability is possible by allowing occasional arcs to form between the inside and outside of cells.
Computationally (aside from the visualisation), the system
may be inexpensive, although this is not discussed in the paper.
Abstract physics
Acknowledgements
The final class of system we will briefly review are artificial chemistries without an explicit spatial representation.
Membranes in such systems can be represented using syntactical structures such as λ-expressions (Fontana and Buss,
1996), or in P systems (Paun, 2002), or as multisets (Suzuki
and Tanaka, 2000). Operations in these systems consist of
entities within a membrane reacting with each other, or of
membranes merging or dividing.
By defining the rules of the system appropriately, the
membranes can be made semi-permeable (by allowing certain particles to move between inside and outside) and the
spontaneous formation of membranes is easily achieved.
Self-repair does not have an obvious meaning in this context.
For artificial life experiments, one potential drawback
is that the properties of the membranes cannot usually be
altered by the organisms themselves. Whereas biological
membranes, for example, can have various proteins inserted
into them, allowing other structures to be attached, this is
not typically meaningful in abstract membrane systems.
Another abstract chemical system that includes a membrane is Gánti’s chemoton (Gánti, 1997). However, the selfassembly and self-repair properties of the membrane are assumed, rather than being modelled, and so this system is not
relevant to this paper.
The anonymous reviewers were very helpful in improving
this paper.
Conclusions
In this paper we have reviewed a number of chemical simulation systems from the point of view of creating useful
membranes for artificial life experiments. Systems range
from ones that are physically realistic but expensive to simulate, to abstract models with minimal computational costs.
The results are summarised in Table 1. While no conclusions
are drawn as to which kind of system is the most desirable,
it is suggested that systems at either extreme have limited
potential for making artificial lifeforms that exhibit the evolutionary growth of complexity: molecular dynamics simulations that can simulate entire cells may not be feasible for
decades to come, while abstract chemical systems may lack
the very features of rich interactions and flexibility of function that allow creative design solutions to be found (Taylor,
2001; Suzuki et al., 2003).
We have presented a novel chemical simulation that has a
lattice-based physics in which semi-permeable membranes
with good flexibility can be created. The low computational
cost and rich interactions in the system give it good potential for artificial life experiments. Importantly, the model
is completely compatible with existing systems of selfreplicating, information-carrying molecules (Hutton, 2002;
Hutton, 2004), raising the possibility that the simulated cells
will be able to evolve through interactions in a computationally amenable environment.
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-
a)
X
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difficult
X
difficult
difficult
difficult
X
X
potentially
b)
X
X
difficult
X
X
difficult
difficult
X
X
unclear
c)
X
X
X
potentially
X
X
X
potentially
potentially
X
d)
X
X
X
difficult
potentially
X
X
difficult
potentially
difficult
e)
very high
medium
medium
medium
low
low
low
very low
low
very low
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