a meta-level evolution theory connecting physics, chemistry, biology and artificial life

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'''The operator theory''' is a meta-level evolution theory presenting a methodology to extend the concept of evolution beyond biology. It was introduced in 1999 by Gerard Jagers op Akkerhuis and Nico van Straalen [1] and elaborated and discussed in the following papers:[2] [3] [4] [5] [6] [7] .
The operator theory can be regarded as to be based on a special kind of metasystem transitions and contributes to various scientific goals, such as the defining of life, of the organism and of death, the predicting of the future of evolution, the integration of existing periodic tables into a single comprehensive periodic table and the generalization of evolutionary theory.


The operator theory thanks its name to the use of the word “operator” as the generic name for all the systems that are known as particles and organisms(for other uses of this word see also operator). The operator theory has its origin in the analysis of hierarchy in ecology. Existing approaches in this field rank operators (physical particles and organisms) together with systems of interacting operator (“interaction systems”, such as populations and ecosystems). The operator theory now suggests a different approach that uses three fundamental directions for complexity increase, one of which is used for ranking the operators. To rank operators according to complexity levels a mechanism is needed for the identification of these levels. Such a mechanism was offered by the first-next, new type of closed interaction and/or closed shape that nature allows. This mechanism was named “first-next possible closure”.

The counting of the number of first-next possible closures that are required for the construction of a given operator allows the creation of a yardstick for structural complexity, the "operator hierarchy". The operator hierarchy ranks all operators, from physical entities, such as quarks, hadrons, atoms and molecules, to organisms, varying in complexity from prokaryotic cells and eukaryotic cells, to multicellulars (such as fungi or plants) and multicellulars with a neural network.


The developing of integrating theory has a long history in science. Already Aristotle created the ’’scala naturae’’. More recent approaches include Darwin’s evolution theory, Einstein’s theory of general relativity and the as yet unfinished grand unified theory in particle physics. The domains of these theories are biology, physics and particle physics, respectively. It would require a general theory of hierarchic order as envisaged by Von Bertalanffy [8] in order to create interdisciplinary connections.

A new way of looking at hierarchic order was developed during a 1993 ecotoxicology study. This study aimed at integrating ecotoxicological theory, for which purpose a strict ranking was sought for the different targets affected by toxicant action. Examples in the literature generally offered a one-dimensional ranking of system complexity including different types of ranking rules and system types. As an alternative, it was proposed to analyze hierarchy according to three dimensions. The use of these three dimensions subsequently allowed the identification and stringent ranking of all operators (particles and organisms).(Jagers1999/><ref name=Jagers2008) The name was chosen to indicate systems that interact with elements in their environment, while keeping their basic organization intact, much in the same way as mathematical operators act on data without being changed.

Hierarchies based on mixed rules

Figure 1. a comparison of the classical ordering (left half of the figure) and the new approach (right). The conventional approach ranks the complexity of system types using mixed hierarchy rules and system types. The different rules and system types are shown in different colours in the right half of the figure. Green=from the individual to its internal complexity. Orange=from the individual to various subsets that, based on specific interactions, can be recognized in the ecosystem. Blue=from lower level operator (abiotic particles and organisms) to higher level operator. The thin blue lines indicate that the concept of “organism” relates to various complexity levels of operators.
Figure 1. a comparison of the classical ordering (left half of the figure) and the new approach (right). The conventional approach ranks the complexity of system types using mixed hierarchy rules and system types. The different rules and system types are shown in different colours in the right half of the figure. Green=from the individual to its internal complexity. Orange=from the individual to various subsets that, based on specific interactions, can be recognized in the ecosystem. Blue=from lower level operator (abiotic particles and organisms) to higher level operator. The thin blue lines indicate that the concept of “organism” relates to various complexity levels of operators.

Levels in the organization of nature are frequently represented as a one-dimensional sequence of increasingly complex system types, for example from atom, to molecule, organelle, cell, tissue, organ, organism, population, community, ecosystem, planet, solar system, galaxy and universe (see Fig. 1).
The literature contains many examples of these sequences, e.g.[9] [10] [11] [12] [13] [14] (see also biological organization and hierarchy of life).

Figure 1 illustrates a way to analyze the use of hierarchy rules and system types in mixed approaches.<ref name=Jagers2008/> For example the step from atom to molecule represents a direct path from one operator to the next. This is not so for the sequence from molecules to organelles, which has not occurred in this order in nature, because nature only had the means to produce organelles after the construction of the cell. Similarly, the steps from cells to tissues, organs, organ systems and organisms (that is: multicellular organisms) have never followed this sequence. Only after the formation of primitive multicellular organisms, did it become possible for evolution to select for internal organization in organisms in the form of tissues, organs and organ systems. Furthermore, the one-dimensional example for hierarchy in Figure 1 ranks atoms, molecules, cells and multicellular organisms, which are all operators, and switches to interaction systems when it continues with populations and ecosystems. This implies a change of rules and a break in the hierarchy. Finally, populations, communities and ecosystems do not represent physical units. Accordingly, organisms cannot integrate first into populations and than into communities and ecosystems. Instead, populations and communities are subsets of one and the same ecosystem, each subset being based on the use of a different type of interaction as the grouping criterion.

Untwining hierarchy by using three dimensions

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As an alternative to mixed hierarchical rankings, the operator theory introduces a methodology for the untwining of dimensions for hierarchy (Figure 2). The untwined dimensions are: (1) from an operator to a system of interacting operators, called an ‘interaction system’, (2) from an operator to its interior, called ‘internal differentiation’, and (3) from a given operator to the next operator of higher complexity.

Examples of organizational levels in interaction systems are populations (interactions based on mating behavior), caste structure in insect colonies (various caste systems have evolved from the combined effects of competition between colonies and specific reproductive genetics), food chains, symbiosis, etc.

Examples of organizational levels based on internal differentiations are complex molecules in bacteria, organelles in protozoa, and the presence of tissues, organs, organ systems and supportive material (bones, cuticle) in multicellular organisms.

Finally, the third dimension ranks the organizational levels that are linked to the subsequent operators, in this way creating the operator hierarchy. The presence of endosymbiontic cells in eukaryotes and of neural networks in higher animals suggests complexity steps involving internal differentiation. However, as these steps simultaneously comply with the demands for the formation of a next operator, these transitions are regarded as to belong to the sequence of operators.

First-next possible closure

Based on the operator dimension of Figure 2 it is possible to strictly rank levels of complexity. Every step results from a circular/recurrent self-organization process. Concepts generally used in system science to indicate circular/recurrent interactions are metasystem transitions and closure[15] (see also ‘closure’ as used in mathematics and in topology). The operator theory advocates that the construction of any next operator must involve a new type of closed process (a functional closure), a new type of closed shape (a structural closure) or a combination of both (structuro-functional closure).<ref name=Jagers2008/> Based on these rules it is possible to recognize a strict ranking by demanding that every subsequent (combination of) new type (s) of closure(s) is realized as the first possibility nature allows. This demand was named the principle of ‘first-next possible closure’
.<ref name=Jagers2008/>

The first-next possible closure of any operator always builds on operators of the highest lower level possible. For this reason, first-next possible closure has to either involve the simplest possible interaction or the simplest possible internal differentiation. Thus, by first-next possible closure, an operator type at one particular level of complexity self-organizes to the operator type at the next level.

The definition of first-next possible closure allowed the identification of the recursively defined set of the operators. This set includes the quarks, the hadrons, the atoms, the molecules, the prokaryotic cells, the eukaryotic cells, the multicellulars and the animals with neural network. The operator hierarchy suggests that animals with neural network possess a new closure level, that of neural hypercyclicity with sensory interface, and refers to this level of organization as ‘memons’. Accordingly, a plant is strictly a multicellular organism, while a human being is both a memon and a multicellular organism. It is convenient to use the concept of the memon, instead of that of an animal, in the following situations: a. when it comes to separating memons from protozoa and b. when classifying future, intelligent, technical organisms, which are not generally considered as animals.

Because it focuses strictly on operators and their organization, the concept of first-next possible closure does not include the role of the environment. Yet, the environment offers driving forces, limiting conditions and scaffolds that are required for the construction of any next type of operator. For example the nuclear fusion in stars produces the carbon, nitrogen and other heavy elements that play a role in the emergence of complex organic molecules and, thereafter, organisms. By focusing on the closures, however, the operator hierarchy works at a level of abstraction where there is no direct need to explain specifically the role of environmental influences.

Higher order logic of the operator hierarchy

In essence, subsequent first-next possible closures define and connect all the operators in the operator hierarchy. Additionally, similarities in closure types of operators at distinct positions along the linear ranking indicate that this can be reorganized according to a higher order ranking.
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Such meta-level similarities allow the recognition of similarly colored "pearls" that allow a restructuring of the operator hierarchy, from a ‘pearl necklace’ to the meandering path that is presented in Fig 3. For example, hypercyclic organization (which must represent a first-next possible closure!) can be recognized not only in quark-gluon interactions, but also in pion exchange between the protons and the neutrons in the atom nucleus, in the set-wise autocatalytic reactions in a cell and in the interactions in neural networks. Likewise, connectivity between similar-type operators (which must represent a first-next possible closure!) was called "the multi state" and can be recognized in hadrons, in molecules and in prokaryotic and eukaryotic multicellulars.

Because the logic of the operator hierarchy ranks operators according to more than one dimension the scheme is very strict. Using the analogy of the pearl necklace again, one may say that it is easier to delete a pearl from a pearl necklace, than to delete a single operator from the scheme in Fig 3, or add one to it.

Applications of the operator hierarchy

Definitions of life, the organism and death

For many years scientists discuss the question of what is life. Up till now this has not resulted in a commonly accepted definition. While some discussions focus on criteria for a definition and on criteria for ‘living[16] [17] other discussions center on whether or not a definition of life should include ecosystems and evolution.[18] Still other discussions focus on the understanding of the first cell and/or autopoiesis as a basis for a definition of life.[19] [20] [21]

With respect to the above discussions, the operator theory now offers a new context. The operator hierarchy organizes the levels of organization of both non-living and living entities, and for this reason can focus on life as a group property of all the operators that share a specific minimum level of organization of matter.<ref name=Jagers2010/> This general context allows one to include different levels of organization of matter in a definition of life (life being more fundamental than living). Accordingly, life can be defined as “matter with the operator structure and an equal or even higher complexity than that of the cellular operator”. Based on this definition of life, it is possible to strictly define the organism as “any operator that agrees with the definition of life”. This definition includes all hierarchical levels of organization of organisms, including future technical organisms.[22] Living can now be defined as ‘the dynamic activity of an organism’.

The operator theory also suggests a definition for death. Back in 1860 it was realized that as long as the organization of an organism stays intact, it can stop living without dying.[23] This principle was demonstrated using dried or frozen bacteria, which are not living because their dynamics are “turned off” and which can be revived upon wetting or thawing. Their non-living state has been called “viable lifelessness”.[24] Because a frozen or dried bacterium is not dead as long as it retains all structure necessary for becoming active again, the operator theory suggests to define death as ‘the moment when irreversible deterioration has caused the loss of the level of closure that is typical for the organism’.<ref name=Jagers2010/>

Restructuring one-dimensional natural hierarchies

As was explained above, one-dimensional rankings of complexity are frequently based on a variety of ranking rules and system types. The operator hierarchy offers an alternative methodology that uses three dimensions for hierarchy. How the use of these three dimensions contributes to analyzing hierarchy is shown in figure 4.
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The central column in figure 4 represents a one-dimensional simplification of the operator hierarchy that runs from quarks to hadrons, atoms, molecules, prokaryotic cells, eukaryotic cells, multicellulars and the memon (representing all animals with the required neural network with interface). The left column captures various levels of complexity of interaction systems. For example, a planet like the moon consists of atoms and molecules and of various aggregates produced from these operators, but no higher level operators. The right column, finally, presents the levels of internal differentiation that may be present in operators. In the prokaryotic operator this may involve organelles and complex molecules, which evolve after the formation of the cell. In the multicellular operator, such as a plant or fungus, various tissues and organs may evolve in a multicellular organism when its size increases. In memons based on organic cells (“animals”), the internal differentiation may include a complex system of organs, tissues, sensors, muscles and, typically, also a neural network. Even though the neural network with interface developed as an internal differentiation, its typical organization is the cause that it qualifies as the operator at one higher level than multicellularity.

A periodic table for periodic tables

Apart from the periodic table of the elements, a whole range of other periodic tables exist. The operator theory now offers a means to link the existing periodic tables<ref name=Jagers2010/>.

The most well-known periodic table is the periodic table of the elements. Mendeleev introduced this tabular display of the chemical elements in 1869.[25] It organizes the reactivity of atoms. In addition, several other periodic tables exist in other disciplines. For example the ‘standard model’ that is used in particle physics categorizes fundamental particles as either force carrying particles (bosons) or matter particles (fermions). The fermions are subsequently divided into leptons or quarks, both of which are partitioned over three groups of increasing mass. Another fundamental periodic table used in particle physics is the ‘eightfold way’. This table is used to organize the many ways by which quarks can combine into hadrons. Hadrons consisting of two quarks are called mesons while those made up of three quarks are called baryons, and a separate table exists for each of these types. The eightfold way was developed by Gell-Mann and Nishijima and received important contributions from Ne'eman and Zweig.[26] Furthermore, two tables can be considered the fundaments of Mendeleev’s periodic table: the ‘nuclide chart’ and the charts showing which sets of electron shells are to be expected in relation to a given number of protons. Finally, and even though it may seem a bit unusual to regard this arrangement as a periodic table, there are good reasons to include the ‘‘tree of life” in this overview of tabular presentations. The only difference with the other tables is that the tree of life also includes descent, a property that has no meaning in the other periodic tables. In all other aspects, the tree of life has similar properties of creating a unique and meaningful overview of all basal types of operators, which enter the scheme as species.

Every periodic table discussed above is central to its proper field of science. But the tables are not connected. Using the operator hierarchy it is possible to connect the separate tables by focusing on the types of systems (frequently operators) in every table. In this way, the operator hierarchy can be used as a ‘periodic table for periodic tables’. As Table 1 shows, there exists a periodic table for almost every complexity level in the operator hierarchy. Interesting gaps exist at the following positions: the quark-gluon hypercycles, the quark confinement, the molecules, the autocatalytic sets, the cellular membranes, the cyclic CALM networks and the sensory interfaces. With the exception of the molecules, which may not have a periodic table because of the almost unlimited number of combinations that can be made from the various atom species, all the gaps involve hypercyclic sets and interfaces. One may now suggest that it is generally impossible to create periodic tables for hypercyclic sets or for interfaces, but this assumption is at least partially contradicted by the nucleotide chart and the classification of potential electron shells. A reason for the absence of tables for hypercyclic sets may be that the number of possible configurations is so large that it is impossible to classify them, in the same way that it is hard to classify molecular configurations. Such ideas, however, need to be worked out in more detail.

Table 1. Using the operator hierarchy for organising the periodic tables that exist for different types of operators, pre-operator hypercyclic sets and interfaces. Bold text indicates system types that are operators.
{| class="wikitable"
! System type in operator hierarchy
! Specific ‘periodic table’
| fundamental particles || standard model
| quark-gluon hypercycle || ??
| quark confinement || ??
| '''hadrons''' || '''eightfold way'''
| nuclear hypercycle || nucleotide chart
| electron shell || types of shells
| '''atoms''' || '''periodic table of the elements'''
| '''molecules''' || '''??'''
| autocatalytic sets || ??
| cellular membrane || ??
| '''cells (prokaryotic)''' || '''tree of life: prokaryotes'''
| '''eukaryote cells''' || '''tree of life: eukaryotes'''
| '''prokaryote multicellulars''' || '''tree of life: prokaryote multicellulars'''
| '''eukaryote multicellulars''' || '''tree of life: eukaryote multicellulars'''
| cyclic CALM networks || ??
| sensory interface (perception/activation) || ??
| '''memons (hardwired)''' || '''tree of life: animals with brains'''
| '''memons (softwired)''' || '''future ‘tree’ of technical memons'''

Predicting future operators

The evolution theory proposed by Darwin deals with reproducing organisms. Consequently, it can only predict organisms. As the operator theory focuses on transitions in system organization, it becomes possible to analyze evolution without reproduction and to predict other system types than organisms.<ref name=Jagers2001/>[27]
Possibly for the first time in scientific history, the meta-level approach of the operator hierarchy allows the prediction of essential construction properties of future operators. The operator theory now predicts that evolution’s next steps will be various technical memons. The next higher operator above the memons based on organic cells (the so-called “hardwired memons”, which includes all animals with brains) is the so called “softwired memon”. This memon type is called “softwired” because the wiring of its neural network is represented as files of computer code indicating the neurons, their connections and the strength of the connections. In the same way as it is for us humans impossible to experience how our thinking is produced by cellular activity, the softwired memons will not be able to experience how their thinking is the result of the processing of database files. The use of the operator theory allows a different kind of predictions compared to extrapolations based on existing trends in technological development. In addition, the operator hierarchy opens up possibilities to look more than one level ahead.

Redefining the Meme concept

Dawkins [28] defines the meme as: '... new replicators, which I called memes to distinguish them from genes, can propagate themselves from brain to brain, from brain to book, from book to brain, from brain to computer, from computer to computer.' Dawkins' memes are not structural analogues of genes, because the patterns of replicable information in brains, books, etc. do not code for the construction of a memon in the same way as the gene codes for the development of an organism. The operator theory indicates<ref name=Jagers2001/> that it may be practical to introduce additional interpretations of the meme concept, for example: 1. Abstract meme (A-meme): abstractions communicated between memons (melodies, stories, and theories), 2. Physical memes (P-meme): physical models of thoughts (written language, artwork, buildings), 3. Functional meme (F-meme): the actual neural network, or part of it, harboring information, 4. Coding meme (C-meme): C-memes code for neural network architecture just as genes code for catalytic molecules. C-memes could also be considered as technical genes. A C-meme would take the form of a code-string coding for the number of neurons in a network, for the connections of these neurons and in certain cases also for the inhibitory or excitatory strengths of the connections. Based on these three parameters, a C-meme could actually code for the entire knowledge in a given neural network (with interface) and would allow the establishment of Lamarckian inheritance. By demanding that a C-meme codes for the neural network structure of a memon, the analogy between gene and meme can be reestablished.

Evolution, from contingency to predictability

Definitions of Darwinian evolution are based on heredity, the production of variable offspring, and selection, and can be measured from changes in the proportion of genes in a population. The result is a branching speciation process, leading to what is known as the ‘tree of life’. The branching pattern of this tree is considered essentially haphazard, or ‘contingent[29] by some, but others emphasize that there are aspects of predictability in it as well. Predictable trends are reflected in similar body plans, such as wings for flying, legs for walking, etc.,[30] because all organisms have to deal with the same environmental interactions in these processes.[31] Both viewpoints having their merits, the operator hierarchy refocuses on a still higher level of abstraction and predictability; the passing of subsequent levels of complexity that can be recognized in separate branches of the tree of life. The operator theory cannot predict which branch of the tree will develop from a prokaryotic to a eukaryotic complexity, but it advocates that regardless the specific branch, all branches have to pass the levels of complexity that are defined by the closure levels of the operator hierarchy, thus from the prokaryotic level via the eukaryotic level, to the eukaryotic multicellular level and memic level<ref name=Jagers2010/>. In marked contrast to the diversification pattern of the Darwinian evolution tree that for a larger part represents an inherently contingent process, the passing of levels of organization is regarded by the operator hierarchy as representing a strictly stepwise and deterministic process. The recognition of complexity levels therefore adds a non-contingent, predictable aspect to evolution.

Man is not the apex of evolution

Darwin’s thinking obtained a lot of interest because it implied that species were not created but had evolved. Accordingly all species on Earth, from man to great apes, dogs, earthworms, mites and protozoa share a common descend from a universal ancestor cell. The extraordinary mental capacity of human beings is sometimes used to suggest that man forms the apex of evolution. The operator hierarchy now implies that even though the human brain represents a highly evolved operator of the type called ‘hardwired memon’, this level is followed by the still higher level of ‘softwired memons’, the technical neural network organisms.<ref name=Jagers2001/>

In other words, the hardwired memon represents just another kind of operator in the sequence of the operator hierarchy. For this reason there seems to be no scientific basis for considering human beings as some kind of final stage or as a crown on the evolutionary process.

Extraterrestrial life

A field of theory where the operator hierarchy also can be applied is that of astrobiology and extraterrestrial life. Assuming that the operator hierarchy correctly describes all the first-next possible closures that are possible in our universe, then the same sequence of operators must develop anywhere in the universe for as long as ambient conditions allow this to happen. The assumption that the organizational states of matter that are represented by the operators have general validity in the whole universe is supported by astrological observations showing that at least fundamental particles, hadrons, atoms and molecules also exist in distant galaxies. Assuming on these grounds, that the entire operator hierarchy is applicable to all locations in the universe, extraterrestrial life will show the same fundamental closures as life on Earth<ref name=Jagers2010/>. Of course, the actual size, shape, physiology, molecular construction, metabolic pathways, information carrier (on earth various forms of RNA, DNA), color, etc. may differ.


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