4.2 Algorithm

4 Internal Algorithm Review

4 Internal Algorithm Review

This section of (Varela et al. 1974) first
defines alphanumeric codes for the distinct chemical
species or ``elements'' in the model. These are the
same codes already introduced above, with the addition
of a separate code for *bonded* links (`BL`);
However, notwithstanding this, the term ``link'',
denoted `L`, is still usually explicitly qualified as
either ``free'', ``singly bonded'' or ``doubly
bonded''. In fact, the code `BL` is used only twice
by Varela et al. in the subsequent discussion and
definition of the algorithm. Since bonded links are
otherwise generally regarded as simply link particles
with bonds (rather than another distinct kind of
element) I will generally dispense with the `BL`
code.

Next, the overall structure of the algorithm is
described as consisting of two separate phases, the
first concerned with movement, the second with the
various reactions. A final comment here is that
``...[t]he rules by which `L` components bond to
form a boundary complete the algorithm''; this is a
somewhat unfortunate phrasing because it prejudges the
issue of whether bonding will actually lead to
*closed* chains (``boundaries'').

The discrete space, with a rectangular co-ordinate
system, or square lattice, is described next. It is
specified that the initial state of the space should
contain one or more `K` particles, but otherwise be
completely filled by `S`. It is unclear to me why
this restriction is imposed; in any case, one would
expect any comprehensive computer implementation to
allow arbitrary initialisation of the particles in the
space (and, indeed, subsequent intervention and
manipulation of it).

Both movement and reactions depend, in part, on the
relative spatial positions of the interacting
particles; so a convention for labeling relative
(``neighborhood'') positions in the space is
introduced next. Figure 1 is a redrawn
version of this scheme (drawn in terms of lattice
*cells* rather than lattice *points* as in the
original paper). Notice, in particular, that the
neighborhood extends *two* positions away from
the central cell, but only in the orthogonal (not
diagonal) directions. This is an essential device to
support `S` particles permeating through chains of
`L`.

**Figure 1:** Numbering of Neighbor Positions

The only comment about behaviour at the edges of the space occurs parenthetically in the discussion of this neighborhood diagram:

...of course, near the array boundaries, not all of the neighbor locations identified in the figure will actually be found...

I interpret this to mean, as earlier indicated, that the edges are to be regarded as hard limits to the space--no interactions or motions can operate beyond these edges. This is in contrast to, say, making the space effectively toroidal, so that cells on the East edge are considered adjacent to the West edge, and similarly for the North and South edges. I suggest that the latter arrangement might actually be conceptually preferable, because the edges otherwise introduce an asymmetry into the space, which may give rise to artifact phenomena.

There follows a qualitative discussion of motion in the model. This is expressed in terms of the particles being ``ranked'' by ``mass'', where ``mass'' here denotes which kinds of particles may displace which others. This is, at best, a relatively weak analog of newtonian mass, and there are no analogs of force, momentum, energy etc. All motion in the space is fundamentally a form of simple diffusion or random walk.

Roughly speaking then, the ranking by increasing mass
is in the order `S`, (free) `L`, `K`; so `K`
particles can displace, or push away, free `L` and
`S` particles, free `L` particles can displace
`S` particles, and `S` particles can only move
into holes. Bonded `L` particles cannot move at all.
`S` may move through a single thickness chain of
bonded `L` particles, but free `L` and `K`
particles cannot.

The algorithm proper fleshes out these motion rules
more precisely. However, I should note here that this
idea of ranking by ``mass'', with the possibility of
``displacement'', does complicate the algorithm very
substantially and yet seems to me to contribute little
if anything to the utility of the model. The rules on
permeability of chains of `L` particles are clearly
crucial to the possibility of autopoietic organisation
being realised-but that is a completely separate issue
from whether certain particles are ranked by ``mass''
or can ``displace'' other particles. If the idea was
to implement varying degrees of *mobility* for the
different kinds of particles, I suggest this could have
been more simply achieved by qualifying the motions by
a probability (of ``attempted'' motion), per particle,
per timestep. In any case, I conjecture that the
relatively complicated model of motion actually
proposed by Varela et al. does not significantly
affect the capacity for the model to support
autopoietic phenomena.

Next comes a qualitative discussion of the production
reaction, specifying that the two `S` particles must
be immediately adjacent to each other and to a `K`
particle. Thus, taking the `K` to be in the centre of
a neighborhood (position 0 of figure 1),
the `S` particles might be in relative positions 2
and 7, or 5 and 4, but *not* 1 and 3, etc. It is
explicitly stipulated here that the reaction rate is
limited to, at most, production of one `L` per `K`
per timestep. If there are multiple pairs of `S`
particles which could react under the action of a given
`K` in one timestep, then one pair is to be selected
at ``random''. Since the production reaction consumes
two `S` particles to produce one `L` particle, it
also results, in effect, in production of an additional
hole in the space--into which particles may
subsequently diffuse.

The final paragraph of this section is concerned with the disintegration reaction and will be quoted in full:

The disintegration of L's is applied as a uniform probability of disintegration per timestep for each L whether bonded or free, which results in a proportionality between failure rate and size of a chain structure. The sharply limited rate of ``repair'', which depends upon random motion of S's through the membrane, random production of new L's and random motion to the repair site, makes the disintegration a very powerful controller of the maximum size for a viable boundary structure. A disintegration probability of less than about .01 per timestep is required in order to achieve any viable structure at all (these must contain roughly ten L units at least to form a closed structure with any space inside).

This introduces the only explicitly specified parameter
of the model (subsequently labeled ), being the
probability of disintegration per `L` particle
(bonded or otherwise) per timestep. The implication
is that this is the only behaviour or interaction in
the model which will be characterised by such a rate
parameter; one must presume, therefore, that the other
dynamics are intended to proceed at the ``maximum''
rate (consistent with any explicitly stated constraints
or enabling conditions). Thus, all particles
*will* move whenever they can; `S`
particles *will* react (under the action of `K`)
whenever they can; `L` particles *will* form
additional bonds whenever they can.

It is clear that the model could be easily generalised
by *introducing* new rate parameters (effectively,
probabilities of the action, per site, per unit time),
analogous to the disintegration rate parameter.
Setting these parameters to 1 would then recover
precisely the model explicitly specified by
Varela et al. (1974). The ``mobility''
probabilities mentioned earlier are specific examples
of this. I suggest, therefore, that a computer
implementation of the model should support such
additional parameters, thus allowing a wider range of
models, and corresponding phenomenology, to be
explored. These potential additional parameters will
be elaborated below in the detailed discussion of the
algorithm proper.

The final point to be made here is that while Varela
et al. stipulate that should be less than 0.01
(which would yield a composite probability of
disintegration for a 10 particle `L` chain of just
under 0.1, or an expected lifetime, for the chain, of
about 10 timesteps), they do not say precisely how
*much* less. Clearly, the phenomenology of the
model will alter as this parameter is made smaller;
thus, it would have been useful to specify what precise
value(s) were used in the particular experiments
reported in the paper.

4.2 Algorithm

4 Internal Algorithm Review

4 Internal Algorithm Review

Copyright © 1997 All Rights Reserved.

Timestamp: Tue Dec 31 18:43:32 GMT 1996