SELF-ORGANIZATION AND
STOCHASTIC RE-CAUSALIZATION
IN SYSTEM DYNAMICS MODELS
Erik Mosekilde and Steen Rasmussen
Physics Laboratory III
and
Torben Smith Sgrensen
Institute of Physical Chemistry
The Technical University of Denmark
2800 Lyngby, Denmark
Abstract
Self-organization denotes a class of instabilities in
which a system spontaneously generates structure, diversity
and/or specialization. From a thermodynamic point of view,
transitions of this kind, which proceed against the general
tendency for relaxation towards an unstructured equilibrium,
can occur in energetically open systems and under far-from-
equilibrium conditions. The exergy required to build up and
maintain a non-equilibrium (so-called dissipative) structure
can here be extracted from the continuous supply of energy
(and/or resources) .
The interest in self-organizing systems originates in
the work on irreversible thermodynamics performed primarily
by the so-called Brussels school. According to this school,
developments in biological, ecological and social systems
which involve qualitative change, diversification or increased
complexity are also to be viewed as self-organizing processes.
This applies for instance to the build-up of genetic information
the appearance of new species in an ecological system, the in-
troduction of new techniques in a social system, the adoption
of new scientific paradigms, and the penetration of new pro-
ducts.
In the present paper we analyse the basic ideas of
self-organization in terms of concepts familiar to System
Dynamics practitioners. Through a series of relatively simple
models it is shown how System Dynamics can be used as an ef-
ficient tool for modeling self-organizing systems. As a parti-
cular example we consider the evolution of cooperative structu-
res (RNA molecules with their associated enzymes) in a prebio-
tic system,
Introduction
Is the American society stable with respect to a so~-
cial or racial revolution? If the unemployment was to increase
128
would there be a kind of threshold value above which the so-
ciety becomes unstable, and a nationwide upheaval can be trig-
gered by almost any outbreak of local unrest.
These questions belong to a type that System Dynamics
practitioners are not so used to consider. Nontheless, they
appear to be quite relevant in discussing the possible future
developments of USA and, more particularly, for assessing the
limits to a contractive economic policy.
On this background, one may ask if human societies
are operating under such conditions that they are unstable
with respect to certain small disturbances. And the answer
to this question is clearly: yes, sometimes, The human popu-
lation, for instance, may be unstable with respect to the
spread of vira or bacteria from laboratories engaged in gene-
tic engineering or production of biological weapons. In 15-
20 years, when less than half of the population has been
vaccinated, the human race may be unstable with respect to
the reappearance of smallpox. This implies that a local out-
break of this disease could explode into a major epidemic
wave which would propagate all around the world. Even more
frightful, the terror balance may some day turn out to be
unstable with respect to a small disturbance in the Middle
. East, or even in the electronic warning systems.
These examples give an impression of what is meant by
instabilities in social systems. All the examples may be con-
sidered as relaxation type transitions, however, which can
occur in isolated systems. The main idea of Prigogine and his
co-workers '1’?) is that human societies are open systems in
129
which large amounts of energy and resources are processed.
These flows of exergy (useful energy) so to speak "lift" the
systems far away from equilibrium and permit “upward transiti-
ons” in which structure is created, and diversity and specia-
lization spontaneously increased. The evolution of social
systems is thus to be considered as a series of instabilities
leading to an increasing level of complexity.
This is presumably a rather unfamiliar viewpoint to
many system dynamicists. It also appears to conflict with
the usual assumptions about the stability of social systems.
With the present paper we would like to show how the notion
of self-organization can be accomodated within System Dyna-
mics, and also to illustrate how System Dynamics can be used
as a tool for modeling unstable transitions. The contention
that social systems are stable to parameter fluctuations is
a practical working hypothesis in many situations. It is not
a scientific fact, however, and by incorporating the ideas of
the Brussels school, system dynamicists can complement their
understanding of evolving social systems.
3), as there will
There may be some semantic problems!
often be when ideas from different disciplines are brought
together. In particular, the word structure as applied by the
Brussels school usually refers to the spatial and/or temporal
variation of a distributed system. Therefore, self-organization
in irreversible thermodynamics does not necessarily imply a
change in the basic equations of motion, or in the logical
structure as represented by the System Dynamics flow-diagram
130 -
However, in the hierachy of successive bifurcations leading
to different dissipative structures in a hydrodynamic system,
for instance, the equations of motions to be used may change
completely from level to level. Then, we have also a change
in the logical structure. If we want to apply the idea to
social systems, the self-organizing transitions must be re-
Presented as activations of various parts of or causal rela-
tions in a flow-diagram. It is important to notice, however,
that this activation occurs through a spontaneous process,
i.e. as a result of certain noise components becoming unstable.
‘An alternative process in which hidden parts of a
flow-diagram are activated when certain conditions are satis~
fied by the system's macroscopic variables has been investi-
gated by Barry Richmond'4) , we have extended this work by
assuming the activation of certain causal relations to be
determined stochastically, and by introducing auxiliary con-
ditions which evaluate the performance of the system and,
for instance, stabilize particularly advantageous connecti-
ons. We would like to refer to such a process as "stochastic
re-causalization". It does not describe the spontaneous tran-
sitions associated with self-organization in detail. Under
certain conditions it may be a reasonable approximation,
however, which allows one to handle systems in which a large
number of transitions are possible.
Self-organizing processes in nature
Self-organization is the thermodynamic term for a
131
class of processes by which a system under far-from-equilibri-
um conditions spontaneously develops spatial structure and/or
breaks into sustained oscillations. Light amplification by
stimulated emission of radiation (Laser-action) is a typical
example of self-organization. When a gas (a glass or a semi-
conductor) is pumped sufficiently above thermal equilibrium
by illumination for instance with ultraviolet light, it may
suddenly break into a qualitatively different mode of beha-
viour in which the emitted spectrum of incoherent light is
replaced by a single line of sometimes extreme coherency.
This transition occurs because, above a certain pumping thres-
hold, particular components of the system's random noise
become unstable and start to build up exponentially until
after many decades of amplification they are finally limited
by non-linear processes.
Another example of a self-organizing process is
the Bénard instability (©) . A few mm thick layer of oil in
a pan is heated uniformely from below. As long as the tempera-
ture difference between the top and the bottom of the layer
is sufficiently small, heat is transported up through the
layer by conduction. If the heating is intensified, however,
at a certain well defined temperature gradient, regular macro-
(7)
scopic convection cells spontaneously appear‘’). ~ transition
has here occurred in which convection currents representing
a high degree of organization have grown out of irregular
molecular motions: or occasional hydrodynamic disturbances.
The more detailed mechanism involved in thé Bénard
instability is not so difficult to understand. It is related
132
to the decrease of the surface tension of a liquid with in-
creasing temperature. Thus if for instance the temperature of
a certain small area of the surface by a random fluctuation
happens to be higher than elsewhere, oil is drawn away from
this area along the surface. To replace it, oil will start to
rise from below. The rising oil is warmer than the surface
oil, however, and the local increase in surface temperature
will therefore be enhanced. This establishes a positive feed
back which in the end leads to the formation of macroscopic
convection cells. A threshold for the temperature gradient
exists because the convection currents must overcome the
friction associated with the finite viscosity of the oil.
There is no reason to dwell too much with this examp-
ie, however, because we all know of self-organizing processes
in our daily live:
i the growth of snow crystals, the develop-
ment of frostwork on our windows at wintertime, the formation
of sand bars along a beach, the generation of characteristic
washboard groves on top of the sand bars, the division of
the earth's atmosphere into belts of opposite wind directions,
and the formation of low-pressure cyclons, just to mention a
few. None of these phenomena can be described on a purely
deterministic basis because they originate in small irregula-
xities which by virtue of an instability have grown into co-
herent macroscopic patterns.
Besides in lasers, self-organizing processes are
technically employed in microwave generators, Gunn diodes,
multivibrators, etc. In our more technically oriented re-
Search, two of us have been engaged with the stuéy of se1s-
133
organizing effects in connection with diffusion of surface ten-
sion active molecules through liquid-liguid interfaces and in-
(8.9) | ana with the formation of
(10,11) |
stabilities of cell membranes
acoustoelectric high fiela domains
The contribution that the Brussels school has made to
the field (besides introducing the term self-organization) is
mainly to establish the common thermodynamic principles for
these far-from-equilibrium processes. At the same time, the
Brussels school has generalized the idea by postulating that
similar processes play a vital role in the evolution of biolo-
gical, ecological and social systems. From the thermodynamic
point of view, the most fundamental features of such systems
are the unidirectional transformation of large amounts of food
or fuel of medium grade energy content into low grade waste
(entropy) coupled with a simultaneous build-up of high-exergy
structures. Through the degradation of resources, the systems
~ in stead of relaxing towards a state of thermal equilibrium
where no life is possible - become capable of building up and
maintaining a high degree of complexity. This is merely a re~
statement of the second law of thermodynamics, and the way such
systems restructure themselves through non-linear interactions
of sociological, economical or psychological nature, can be
seen as an illustration of the concept of dissipative struc-
tures.
In this way, self-organization becomes a paradigm of
evolution through instabilities, a paradigm which complements
the more mechanistic view of classical System Dynamics.
134
It is also possible that self-organization can occur
in a model which unintendedly becomes unstable to computatio-
nal noise. We shall not enlarge on this problem, however, but
rather try to show how self-organizing processes can be hand-
led with System Dynamics.
As easy as lifting a feather
November 7, 1917 is presumably one of the most sig-
nificant dates‘?2) in modern history. This was the day when
Lenin and his bolschevists took power in the Russian Capital
Petrograd. Eight months before, the Zar regime had stopped to
function. On top of centuries of impoverishment and injustice,
the enormous problems associated with the war had strained
the system above a critical threshold. The population had
revolted all over the country, and the soldiers had refused
to fight.
The fall of the Zar regime left a political vacuum.
officially, the country was ruled by a provisional govern-
ment, but the authority of this government was very limited.
In these months, the political scene was dominated by libe-
rals and socialists who hoped for the rise of a decentrali-
zea and democratic society. But the population was impatient:
. “The revolution has already lasted for 6 weeks, and nothing
has changea", wrote dissatisfied peasants to the government.
Workers, soldiers and farmers formed soviets, committies and
councils. Hitherto forbidden political parties started a
hectic campaign to prepare for the election of a constituent
135
assembly, and national minorities raised demands for home-rule
and independence.
Under these conditions, the bolschevists could take
power through a minor operation involving armed workers and
soldiers from the garrisons in Petrograd, and within a few
days a new government had been established with the promises
of "pease" and "bread". "It was as easy as lifting a feather"
Lenin is reported to have said. (The real build-up of a new
power structure was a tremendous task which lasted decades
rather than days, and in'which terror and civil war were sig-
nificant tools).
It is not our purpose in the present paper to discuss
a model of the Russian revolution. Rather, we would like to
use some af the characteristic elements of the revolutionary
process to illustrate unstable transitions in social systems.
These elements are:
(1) the sudden break-down of an apparently stable structure
when strained above a critical level,
(2) the creation of a "vacuum" in which several rivaling,
more or less organized activities explode, and
(3) the rather unexpected appearance of a new dominant po-
wer which suppresses all alternatives.
As we shall show it is quite possible to represent these ele~
ments in a System Dynamics model.
The flow diagram of figure 1 shows a schematic repre~
sentation of the basic elements of an unstable social process.
As long as the indicator of social tension IST is below a
certain threshold value (threshold for social tension THST),
136
the system is in stable equilibrium in a state in which the
whole population of 20 million people is politically indif-
ferent (politically indifferent persons PIP). Indeed there
is always a small generation of political activists, but un-
der normal conditions their number is vanishing small. As
examples of such activists, the model considers left wing
extremists LWE, moderate political activists MPA, and right
wing extremists RWE.
aE,
nomal accession
oF lefe wing
exerenises
we
left ving
ehtrenices
Pup
political
Thaifferent
persons,
—
Figure 1. To illustra~
Ee the Basic ideas of
unstable transitions
in social systems, this
figure shows the flow~
diagram of a schematic
model of social revolu-
tion.
1
\
1
\
\
\
\
AIRE
association time
for right wing
The generation of political activists is represented
by the three rate variables: net generation of left wing ex-
tremists NGLWE, net generation of moderate political activists
NGMPA, and net generation of right wing extremists NGRWE.
Each of these rate variables has a random accession term pro~
portional to the normal accession rate, and to the fraction
of remaining political indifferent persons. There is also
a defection term which is modelled as a telaxation process,
i.e. it is proportional to the number of activists of each
kind divided by the corresponding average association time.
Together these two mechanisms produce a "back-ground noise"
of activists of the order of a few thousands.
If the indicator of social tension IST becomes lar-
ger than the threshold value THST, the system is no longer
stable in its original (ground) state, but positive feed-
back loops are activated which generate an exponentially
rising number of activists in each of the three groups.
There is now.a characteristic incubation period before the
number of activists has grown by the 3-4 orders of magni-
tude necessary for it to become comparable with the total
population. When this occurs, however, the non-linearity
associated with the total number of people being constant
sets in, the exponential growth ceases, and one group of
activists become dominant. Which group that is going to win
depends crucially upon the parameters used to specify the
random accession’ rates, the defection rates, and the rein-
forcing positive loops.
138
The DYNAMO-program for the simple model of a social
revolution is given in the appendix. Figure 2 shows a charac-
teristic simulation result. The indicator of social tension
(which in general could be defined as an endogenous variable)
has here been taken to increase linearly from 0 to 2 during
the first 20 time units, and hereafter to remain constant
The system then becomes unstable at time 10. After an incuba-
tion period of 7-8 time units, the number of activists become
large enough to be distinguished from zero on the DYNAMO-plot.
The number of political indifferent persons then drops dra~
matically, and after a short struggle the moderate political
activists take over the political scene.
Figure 3 shows similar results only with the modifi-
cation that the average association time for left wing extre-
mists has been increased from 2 to 3 time units. This relati-
vely small parameter change is sufficient for the left wing
extremists to become completely dominant.
The above model is an idealized model of a self-or-
ganizing process. With a few modifications it could represent
for instance the growth of modes in a laser above the pumping
threshold. For a distributed system in which the self-organi-
zing process produces a spatial structure, the three level
variables would represent amplitudes or intensities of va-
rious spatially defined eigen-functions. Our model could also
serve as a kind of switching module in a larger System Dyna-
mics model, i.e. as a module which driven by the occurrence
of instabilities would activate various causal relations or
hidden parts of a flow-diagram
139
Figure 2. Simulation results obtained with the schema-
tic model of social revolution. In this run the mode-
rate political activists are seen to win.
Figure 3. Same as figure 2 except that a relatively
Small parameter change now leads to the result that
left wing extremists take power.
140
Self-organization in a mutating two-species system
Self-organization can occur under somewhat different
conditions with respect to the noise amplitude. In the simple
model of a social revolution discussed in the preceding sec-
tion, the background noise of political activists was relati-
vely strong, and as a consequence the outcome of the revolu-
tionary process was mainly determined by the strength of the
positive growth loops. As an example of the opposite situa-
tion we shall now consider a system which is unstable from
the very beginning, but for which the self-organizing tran-
sition is triggered by a relatively infrequent random event.
Under these conditions there is a considerable uncertainty
with respect to when the transition will occur.
Figure 4 shows the flow-diagram for an ecological
system in which a population of species A is in equilibrium
with a resource pool RES. To indicate that we are consider-
ing an open system, there is a continuous in- and out-flow
of resources as modeled through the rate-variables RIF (re-
source in-flow) and ROF (resouce out-flow). There is also a
resouce usage rate RUR which is proportional to the number
of species A NA and to the individual resource consumption
MCA (metabolic constant for species A). The specific resour-
ce availability SRA, i.e. the amount of resources per unit
of population influences the net growth rate for species A
NGRA. In particular, there is a characteristic value of
SRA above which the growth rate turns positive.
However, the system also includes the infrequent but
possible event of a mutation by which an individual (or a
pair) of species A is transformed into species B. Everything
144
=
inert
Species B] maximum x
pecies 8) Growth rate
for species B
Generation
Figure 4. Flow-diagram for a unstable ecological system.
With the self-organizing transition being triggered by a
relatively infrequent mutation, there is a considerable
uncertainty with respect to when the transition occurs.
that has been said about species A also holds for species B,
except that individuals of species B are assumed to have a
better resource utilization. As a result the population of
species B can maintain a positive net growth rate and reduce
the resource pool at a value of the specific resource availa-
bility SRA which is too small for the population of species
A to grow. The system is thus unstable in its initial state.
If a mutation occurs, the system performs a transition from
its original state into a new state in which the population
142
of species B is in equilibrium with the resource pool, while
population A is extinguished.
The occurrence of a mutation can be expressed as (23/14)
MR. KL=CLIP (0,1,NNAB.K,NMF.K*NA.K*DT)
NNAB.K=NOISE()+.5 is here a random number uniformely distri-
buted between 0 and 1. NMP is the normal mutation frequency,
and’ NMF.K*NA.K*DT thus gives the probability that a mutation
will occur in the time-step DT. The CLIP-function compares
the random number with the probability for a mutation to oc-
cur, and if NNAB.K happens to be smaller than the mutation
probability, this is taken to mean that a mutation does occur
as),
Figures 5 and 6 show the simulation results obtained
with two different initiations of DYNAMO's noise-function.
The initial population of species A is 4000, and the normal
mutation frequency is 2-107>. In figure 5, the mutation oc-
curs before time 20 while in figure 6 the mutation does not
occur until about time 45. This illustrates the sensitivity
of an unstable macroscopic system (thousands of individuals
to a small random fluctuation. As shown in the flow-diagram
of figure 4, we have extended the model a little by assuming
“that species B produce a refuse at a rate which is propor-
tional to the population size. The model also assumes that
the produced refuse is eliminated with a certain characteri-
stic time constant ARET, Figures 5 and 6 show how the’ refuse
(or waste) builds up in the two simulation runs.
143
Figure 5. Simulation results obtained with the model of a
mutating two-species system. The self-organizing transition
occurs at time 18
Resgagadanet ae:
a
Figure 6. Same as figure 5 except that the simulation has
een run with another initiation of DYNAMO's noise-function.
The transition now occurs at time 45.
444
The above process is an unstable transition. One species com-
peting another species away from the scene is not per se a
self-organizing processes, however. This requires in general
the emergence of additional species and/or the division of
function (creation of niches), It might be though that spe-
cies B have a higher degree of internal sophistication than
species A, and that the level of complexity for the system
is increased in this way.
It is also possible, however, to make the system per-
form a self-organizing transition which is more similar to the
dramatic change in the mode of behaviour which characterizes
self-organizing processes -in thermodynamic systems ‘1°,
To do
this we only have to add the assumption that the metabolic
biproducts (refuse) produced by species B with a certain de-
lay become poisonous to this population. (The corresponding
causal relation is indicated with four small arrows in figure
4, the total DYNAMO-program can be found in the appendix).
When a mutation occurs, the system now transfers from its ori-
ginal stationary state into a self-sustained strongly non-
linear oscillation (a limit cycle). This is illustrated in
figure 7. When looking at this figure it should be recalled
that the dramatic macroscopic oscillations in the bi-stable
system are the result of the mutation of one single indivi-
dual.
Information crisis in a prebiotic system
The most fascinating of all self-organizing. processes
is the development of life out of the simple organic and inor-
145
ine, HD RESO PEF
Figure 7. Self-sustained oscillations in the macroscopic
two species system upon the mutation of one single indivi-
dual of species A into species B.
ganic compounds of primordial world. Exactly how things hap-
pened when life began some 3 billion years ago is not known
but we have some general conceptions about the kind of proces~
ses which took place, and the conditions under which they oc~
curred. It is very clear that the evolution must have faced
several severe information crises in which new principles of
organization had to be found to protect the information al-
ready developed and to make continuation of the information
build-up possible. One such crisis was solved through the
development of cells, but even before that time the develop-
ment of sufficiently long RNA-molecules had to find a solu-
tion.
Even at the molecular level, the evolution of life
can be interpreted in terms of Darwinian principles of natu-
146
ral selection. It is a trial and error process in which a
wide spectrum of randomly produced structures are tested with
respect to their ability to survive under the given condi-
espne (27-28),
For simple molecules, the "competition" is a
question of rate constants in the formation process and resi-
stance to decomposition.
A basic step in the evolutionary process was the de-
velopment of self-replicating RNA-molecules. (The appearance
of DNA presumably occurred at a somewhat later stage). This
introduced a first order autocatalytic process or, in System
Dynamics terms, a positive feedback in the generation process.
RNA-molecules are threadlike structures build up as chains of
nucleotides. Each nucleotide consists of a sugar, a phosphate
compound and a nitrogen containing organic base. The sugars and
phosphates are linked together to form the "back-bone" of the
molecules, while the genetic information is encoded as a
particular sequence of the four possible, pairwise comple-
mentary bases.
In a replication process, the string of nucleotides
serves as a template along which complementary nucleotides
are assembled according to the base pairing rulés. With cata-
lyzing effects from various inorganic compounds and from
miscellaneous primitive proteins, chemical forces and thermo-
dynamic laws permit the formation of RNA strings with up to
about 100 hucleotides, corresponding may-be to today's trans—
fer RNA-molecules. At this stage, the information build-up
is terminated by inevitable errors in the replication pro-
147
cess. However, the information carried by such RNA-molecules
is not sufficient for them to produce more specific proteins,
and without the enzymatic effects of such proteins, the neces-
sary error suppression in the replication process can not be
achieved.
To overcome this "crisis" and make possible the pro-
duction of RNA-molecules with several thousands of nucleotides,
a new principle of organization had to be developed. Eigen
7,18) nas suggested that this was acomplished through the
formation of cooperative ‘structures (so-called hypercycles)
between different RNA-subsystems. Such a hypercycle would re-
sult if one type of RNA-molecule by chance happened to pro-
duce a protein which could facilitate and stabilize the pro-
duction of another type of RNA-molecules, and if at the same
time the second type of RNA-molecules produced a protein
which could assist the replication of the first RNA-molecules.
In the beginning, such a narrow closed loop would
probably not have occurred. Rather, the simple proteins pro-
duced by a given type of RNA-molecules would be relatively
unspecific and would catalyse the replication of a great
many other molecules. Other types of RNA-molecules might
also start to produce relatively unspecific, slightly enzy-
matic proteins, and at a certain time, a reinforcing possi-
tive loop involving a large number of RNA-subsystems could
be established. The formation of such a hypercycle would
give the involved RNA-systems an advantage over other RNA-
systems with respect to their rate of production (a second
order autocatalytic process), it would stabilize the total
information carried by the cooperating RNA-systems, and it
148
would permit the gradual development - hand in hand - of lon-
ger and longer RNA-molecules and of more and more specific
proteins.
It is such formations of cooperative structures and
the associated build-up of genetic codes that we have started
to investigate by means of System Dynamics. Clearly, we are
here dealing with a self-organizing process in the sense that
more and more complex structures are generated. The process
also involves characteristic spontaneous transitions starting
as random mutations at the level of individual molecules and,
if a positive feed-back is established, proceeding through
amplifications over tens of decades until macroscopic num-
bers of mutants have been produced.
When several RNA-subsystems are considered, the num-
ber of possible couplings quickly becomes very large, and
combinatory problems become significant. At the same time,
D?-problems tend to arise when a system between periods of
relatively slow development several times has to make tran-
sitions involving amplification from 1 to say 102° molecules.
For these reasons we have decided to model the self-organi-
zing process in an approximative manner for which we have in-
troduced the term "stochastic re-causalization".
In a stochastic re-causalization process it is as-
sumed that all RNA-molecules of a given type mutate at one
and the same time, i.e. the very rapid amplification proces-
ses are simulated by CLIP-functions which suddenly change
the properties of a macroscopic number of molecules. In the
flow-diagram this corresponds to random generation (and/or
149
disconnection) of causal links between RNA-subsystems. Each
of these connections originate in the pool of proteins pro-
duced by one type of RNA-molecules and terminates in the re-
plication rate of another type of RNA-molecules. Since ampli-
fication from molecular level only occurs when a hypercycle
is established, only such combinations of causal links which
give rise to closed loops are allowed.
The stochastic generation of causal links is comple-
mented by functions that continuously evaluate the perfor-
mance of the produced structures and stabilize reinforcing
connections. This expresses the principle of natural selec-
tion which in the present context is a result of the compe-
tition between RNA-subsystems for resources.
It is relatively simple to give examples of re-cau-
salization phenomena in social systems. One could think for
instance of a company which "happened" to start a collabora-
tion with a former competitor. Such a collaboration could im-
ply division of markets or of product selections, or one com-
pany could start to produce semi-manufacture for the other.
If this problem was extended to consider the establishment
of mutually beneficial cooperative structures between several
companies, it would resemble our RNA-problem a good deal.
Stochastic re-causalization and formation of RNA-hypercycles
“At the present stage, our model of RNA-hypercycle
formation operates with 3 RNA-subsystems, only. Figure 8
shows the flow-diagram for each of these subsystems. RNA-
strands of type I=1,2,3 are synthesized from mono-nucleotides
150
EFRI
mi efficiency in
cgncentration BAA synehesis
Yeotides y, Figure 8. Plow-
7 diagram for an RNA-
\ subsystem. The model
consists of three
\ az
2 ae Pood of RRA such sectors, a
ee sevee = itn resource sector and
pag NI a module that con-
Brmenesis] 77S trols the stochastic
ro / ~s intersector connec-
vA tions.
f
/
a
/
sk pool of
its proteins of 2]
PROIS Ee xD
protein protein
Eynthoese| econo
sitson
‘ DCP
/ SS Pisse for provely”
Vue. in
Soot Gacions
PP
oe
ion of ond
peptides
through base-pairing. The rate of this process RNAIS depends
upon the amount of RNAI already formed, the concentration of
nucleotides NUC and the efficiency of the replication pro-
cess EFRI. This efficiency again depends upon possible ca-
talyzing effects of simple proteins produced by other RNA-
“molecules. The produced RNA-molecules are subject to decompo-
sition through hydrolysis and other processes, and the rate
of decomposition RNAID is determined as the amount of RNAI
multiplied by a characteristic decay constant DCR.
Proteins (poly-peptides) of type I are synthesized
from mono-peptides (amino acids) by RNAI-molecules at a rate
151
which is determined by the amount of RNAI-molecules, the con-
centration of mono-peptides PEP, and the efficiency of protein
synthesis EFP. Also proteins decompose, and the characteristic
decay constant for this process is DCP
Besides the three RNA-subsystems, the model also in-
cludes resource pools for mononucleotides and amino acids.
We are considering an open system, and these resources are
therefore continuously supplied. Finally, the model includes
a set of equations that specify the random coupling and de-
coupling of RNA-subsystems. The re-causalizations are treated
as Markov processes according to a formalism that we have
previously describea!t3:14) | pigure 9 gives an overview of
some of the 21 possible hypercycles and combinations hereof
which can exist with 3 RNA-subsystems.
Figures 10 and 11 give examples of the obtained si-
mulation results. On each figure we have plotted the quanti-
ties of each of the three types of RNA. The states of the pos-
sible subsystem to subsystem connections are indicated through
the values (1=connected, O=disconnected) of the dummy variab-
les plotted as A,B,C,D,E and F. From the variation of these
dummies one can determine the development in the structure
of the system. On figure 10 for instance, a hypercycle be-
tween subsystems 1 and 2 is established at about time 24. The
amounts of RNAl and RNA2 hereafter increase significantly,
while the amount of RNA3 is reduced. Figure 11 shows how the
formation of a hypercycle including all the RNA-subsystems
can lead to sustained oscillations in the system. The occur-
rence of such limit cycles has previously been established by
Eigen 19) |
152 <
2 2
Figure 9. 11 out of
the 21 possible hyper-
cycles and combina-
er? Breet tions hereof which
can exist in a model
with 3 RNA-subsystems.
2 2 2 In the case of four
RNA-subsystems there
can be about 500
different combina-
tions of hypercycles,
> % * > seach with its charac-
teristic mode of be-
haviour for the sy-
AWL
AOE
It is characteristic for the re-causalization model
that there is an uncertainty both with respect to which kind
of hypercycle that happens to be established, and with re-
spect to the time that this occurs.
Conclusion
We have discussed how instabilities in energetically
open systems can lead to self-organizing processes. Such transi-
tions exemplify the break down of the law of large numbers. Amp-
lification of random noise plays a significant role, and the ave-
153
bate
28)
Petrie
Figure 10. Simulation results obtained with the model of RNA
Fypereyele formation. In this run, a hypercycle involving
RNA-subsystems 1 and 2 is generated at about time 24.
Figure 11. Same as figure 10 only with a different initia-
tion of DYNAMO's noise function. A hypercycle involving all
three RNA-subsystems is here generated, and the system starts
to oscillate violently.
154
xage macroscopic variables are insufficient to determine
system development. We have also shown how self-organizing
transitions in social and biological systems can be modeled
with System Dynamics, This requires methods capable of hand-
ling stochastic processes, but otherwise it does not imply
expansions or generalizations of System Dynamics as a model-
ing technique.
Rather, it is the ideas that we have come to connect
with System Dynamics about the stability of social systems
which have to be complemented. We consider these ideas as a
practical working hypothesis in many real life problems, and
may be also as part of a tactical defence towards econome~
trics. They do not constitute an indisputable truth, however
According to the Brussels school, the evolution of social and
biological systems can be considered as a series of unstable
transitions interrupted by periods of more deterministic
development. As we have tried to illustrate, this can be a
very fruitful viewpoint, and we feel that it can be accomo-
dated in System Dynamics without changing any of the more
fundamental principles.
Self-organizing phenomena are not at all uncommon in natural
sciences, and there is no reason to expect that the introduc-
tion of such processes in System Dynamics should weaken or
undermine its basic. ideas.
155
Appendix
Since the models are small and the DYNAMO equations
at certain points a little unusual we have found it reasonable
to list below the complete programs for the simple model of
social revolution and for the model of self-organization in
a mutating two-species system:
* sine MODEL OF SOCIAL REVOLUTION ¢REVOL4>
PIP. RePIP. J-CDT> CNGLIE, JK+NGMPA, Jit
ee ie CHNGRME, JK)
PIPI=2E7
LE. KSLNE. J+#(DT> (NGLNE. Jk>
LHE=1. 4E3
MPA. KSMPA. J+CDT) (NGMPA, dk
MPRE3. 6E3
RVE. KSRWE. J4+<DT) (NGRWE, Jk
RUE=1, 5E3
NGLWE. KL=¢¢PIP. K/PIPI 4NALWE#CNDISEC) —
+CLNE, KMGRLMEMMIST, K> (PIP. E/PIPID SERENE TEMES
NALNE=0, SE3
ATLUES2
MGRLWE=2, 4
NGMPA. KL=¢CPIP. K/PIP ID 4NFMPRECMDISE C+, 5>>—¢
; SD -CN PA,
+CMPA. KAMGRMPRAMIST, KD ¢PIP. K/PIPI ee ES?
NAMPR=2, 4E3
ATMPAS3
MGRMPR=4. 9
NGRUE. KL=¢¢PIP. K/PIPI¢NARIEW - ;
FCRHE, KANGRRMEWMIST. EDCPIE ORIPE ey RNE: KCATRHED
NARWE=0, SE
ATRUE=1. 9
MGRRNE=1, 9
MIST. KeHAXCO, IST. K=THST)
THstat
ST. K=TABLECISTT, TIME, K, 0, 60,
ISTT=0/22272 oo 2
SPEC DT=, O5/LENGTH=s0/PLTPE!
eT=4/PIP=1.LNESL, NERS
APO RIGO XMOO GX NO ONKMEr Er erozrz
6
=A,
RWESR(O, 2E73
156
dre SELF-ORGANIZATION IN A MUTATING THO-SPECIES SYSTEM <MUTAL)
RES. K=RES, J+¢DT>CRIF. JK-ROF. JK-RUR, JK?
RES=RESI
RESI=6E4
RIF. KL=NRIF
NRIF=1E4.
ROF, KL=RES. K/ARLT
ARLT=10
RUR. KLSMCAANA, K+MCB4NB, i
SRE. K=RES. KCNA KENB. KD
NA, KSNA_J+¢DT > <NGRA. JK-MR, JK>
NR=CNRIF-CRESIARLT) /MCA
NGRA. KL=NA, KeMGRASMRAA, K
MGRAE2. 9
HRAA, K=TABLECMRAAT, SRR. K, 0, 30, 5)
MRABT=~d/—. 82, 8700. 67. 9e'L
NB. K=NB. J+¢DT 2 CNGRE, JKAMR, JK}
NB=O
NGRE. KL=NB. KANGREAMRAB, i
MGRE=2, 9
MRAB. K=TABLECMRABT, SRA, K+RAMB-COREF, K*OREF, K>, 0, 30, 52
MRABT=-10~. PY-. 6/07. 67. 924.
RAMB=3, 0
HR. EL=CLIPCO, 4. MNAB. K, NMFHNA, KADT-CLIPCO, 1, NNBR, K, NMEANE, ADT)
NNAB, K=NOISECD+, 5
NHBA, KeNDISEQ)+, 5
NHF=26-5
REF. K=REF. J+¢DT<RFGR, Jk
REF=0
RPGR. KL=NE. K+SRFG
SRFG=, 8E-4
RFER, KL=REF. K/ARET
RRET=30
DREF. K=SMDOTHCREF, K, SMT)
SNT=10
SPEC DT=. 4/LENGTH=200/PLTPER=4
PLOT NASR, NB=B/RES=R/REFSW
RUN
NOISE 224587
Rut
NOISE 457403
RUN
NDISE 72e465
RUN
OROM ORE OD PROGR MERA POAEr DON ZOMOMGEZe
157
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The present formulation of course requires that DT is
chosen such that NMF.K*NA.K*DT is much smaller than 1.
If the occurrence of a mutation is a relatively infre-
quent event, the normal mutation frequency must be such
that NMF.K*NA.K*LENGTH is of the order of 1.
Since we are dealing with ordinary differential equa-
tions (as opposed to partial differential equations),
only organization in time can be modeled.
159
17.
18
19
Eigen M., W. Gardiner, P, Schuster and R. Winkler-Os-
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Organization" (1978), Naturwissenschaften, vol. 65, p. 7.
160
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