Structural Representation of Systems



In this document, we explore notations for representing the structure of systems – both at the macro level and at the micro level. Our background is in IT, so our main interest is in representing the structure of IT systems and their interactions with human systems, business systems and other systems. However, we thought it would be useful to explore the problem, of structural representation, across a range of disciplines. We hope that we have acquired adequate knowledge, of these disciplines, to make the examples, taken from them, useful, but we certainly do not claim any deep knowledge of the disciplines. In this document, we take an example on the borderline between chemistry and biology, a mitochondrion organelle. We gradually introduce the concepts, of our approach to structural representation, as we develop this example. However, we first need to introduce some basic ideas.

Structure of Systems

Biological systems, human systems, and business systems can ultimately be defined, as very complex compositions of chemical and physical systems. At the level of chemistry, systems are either quantities of mass (e.g. neutrons, protons and electrons) or quantities of energy. At this level, within the universe, the total energy and the total mass are both conserved, i.e. both remain constant. At the level of nuclear chemistry, where radioactivity becomes relevant, so that there is seepage between mass and energy, we must take a broader view of mass, and recognise it as a form of energy, as recognised by Einstein. With this broader definition of energy, we still have conservation of energy, within the universe (at least if we ignore black holes). Below the nuclear chemistry level, some physicists discuss the relationship, between energy and information, and take, as the fundamental concept, the conservation of information (thereby including black holes). In this document, we will assume that the term ‘system’ encompasses these definitions.

When IT specialists represent the structure, of their systems, they usually make a strong distinction between the systems and the interfaces between these systems. We believe that this creates an artificial dichotomy, which constrains thinking and becomes entrenched in systems representations (e.g. with boxes and lines in diagrams). It also creates problems when different groups of engineers communicate – a simple interface, in a software engineer’s view of a system, expands into a complex hardware system, as seen by a hardware engineer – similarly the interfaces between chemical systems, as seen by chemists, expand into the complex physical systems, as seen by physicists. The fundamental aim, of our approach to structural representation, is the removal of the rigid distinction between systems and interfaces. We see interfaces as being implemented by means of interactions, hosted by shared component systems.

Given the above, we can take the following view of system structuring:

Nothing is controversial in the above definitions, other than the definitions of interactions. Consider an example of two atoms interacting. There is a force between the interacting atoms, which provides the interaction. However, there is also a force-field, which provides the energy for the force. The force-field is a shared component of the two atoms. The concepts of interaction and system are tightly bound together. An interaction must be hosted by a system. Any system can host an interaction – the only systems, which do not, are ones, which are totally isolated, e.g. the universe. An interface definition would simply define the changes made to the two atoms by the interaction.

We want to explore three ways of representing the structure of systems, as compositions of systems from component systems and their interactions:

By taking very small steps, in the development of our example, we can use just lego-brick diagrams. However, we will explore the use of the textual representation, at the end of the example.

Structural Representation of Mitochondrion

Levels of Abstraction

Throughout its seventy year history, the IT industry has seen, as an ideal, a top-down approach to systems design and evolution. Top-down involves starting with an abstract and undetailed view of the design (emphasising requirements), and then gradually working towards a less abstract and more detailed view (constrained by underlying systems). Top-down proved problematic, because it proved difficult to rigorously relate together the different levels of design. We believe that the strong and inflexible distinction between systems and their interfaces has been at the heart of this problem. This is our motivation for replacing the concept of interfaces with the concept of interactions via shared components.

We cannot take a top-down approach to the design of a mitochondrion, because Darwinian evolution evolved the design billions of years ago. However, we can develop a top-down model of the mitochondrion. It might even hint at the steps, which were taken in Darwinian evolution.

We will develop our top-down model, of the mitochondrion organelle, passing through the following levels of abstraction:

In the following sub-sections, we will develop our top-down mitochondrion model, through these levels of abstraction. We will try to end up with one diagram, showing the complete model.

Mitochondrion Organelle’s Role in the Cell

In our example, we consider the mitochondrion organelle only in its energy generation role. It has other roles, but we mostly ignore them. Hence, the starting point, for our example, could be a model of a cell, just from an energy generation and consumption point of view. To represent this, we use either a directed graph or a lego-brick diagram to show the composition of the cell from component systems:
In the diagram, containment of one brick, within another, means that the contained brick (e.g. energy generation) is a component, of the containing brick (i.e. the cell). The dotted lines and the touching of bricks indicate that there is some interaction, between the energy generation and energy consumption systems, but its nature is unknown at present. As we said, in an earlier section, this may be implemented in one of two ways. Either, there might be a component system (e.g. energy transfer), which is totally shared between the energy generation and consumption systems:
or they might both interact with a third system (an extended energy transfer system), with the nature of these interactions unknown:
Both pairs of diagrams could be valid. They could be considered as different views, as to where we set the boundaries, of the energy generation and consumption systems. The first pair sees the two systems (energy generation and consumption), as bigger systems (containing more components), than those seen in the second pair. This difference would only become apparent, when we developed more detailed diagrams.

An interaction results in the transfer of component systems, across the boundary, between the interacting systems (though this may only be shown in more detailed diagrams). We could show that the transfer is, predominantly, though not exclusively, in one direction, as shown in.
This extra decoration, of the diagrams, should only be viewed, as guidance or clarification, and not, as definitive. There may be movement of component systems in both directions. The true nature, of the movement of component systems, should emerge later in more detailed diagrams.

Acceptable alternatives, for the last lego-brick diagram, could be:

We could encounter systems, which combine component systems in ways, which are, so complex, that it is impossible to represent them in lego-brick diagrams. We can normally get around this problem, by developing our representation, in small steps - we do this, in this document. We could, also, get around the problem, by providing textual equivalents, of the diagrams. For instance, a textual equivalent, of the diagram, on the left, above is:

α:cl{ β:eg{ε:int}, γ:et{ε,ζ}, δ:ec{ζ:int} }
cl=cell eg=energy-generation et=energy-transfer ec=energy-consumption int=interaction
The curly brackets and the Greek letters, together, convey all structural information. The ‘names’, cl, eg, et, and ec, just convey some understanding, of the purpose of the systems, which they name, but do not need to convey any structural information. We only really need the Greek letters for the components, which are shared, so can simplify the above to:
cl{ eg{α:int}, et{α,β}, ec{β:int} }
cl=cell eg=energy-generation et=energy-transfer ec=energy-consumption int=interaction
The directed graph, the textual description and the lego-brick diagram should, each, be able to tell us how to draw and/or write the other two. Therefore, for the rest of the mitochondrion example, we will just use lego-brick diagrams. However, we will revisit the textual notation, at the end of the example.

Now, let us take another view of a cell, one that introduces the chemical processes, which are involved in the energy transfer, which was shown in the earlier diagrams.

Molecules act as containers of potential energy. Energy is required to combine atoms, or molecules, in molecules. The energy is released when the molecule is split into its component atoms, or molecules. This storage and release of energy is exploited, by a cell, in its ATP/ADP cycle
An APT molecule contains three component phosphate molecules, while an ADP molecule contains just two. A mitochondrion organelle, in a cell, uses energy to add one phosphate molecule (Pi) to ADP, to create ATP. Other organelles, in the cell, release (and use) the energy by removing one phosphate molecule from ATP, to create ADP.

The cell takes in nutrients and supports the various functions, of the body, which contains the cell. The ADP/ATP cycle transfers energy from mitochondrion organelle to other organelles. The other resources, shown, are detailed, in the next section.

The relationship, between this diagram and earlier diagrams, can be represented by:

Component Structure of the Mitochondrion Organelle

We are, now, in a position to explore the detailed structure of the mitochondrion organelle, as shown in our earlier diagram (and repeated here):

and take it to the next level of detail:

It is very important to realise that this diagram (like our earlier diagrams) does not attempt to represent physical layout. Physically, in three-dimensional Euclidean space, the mitochondrion has two concentric membranes, or skins, its inner-membrane and its outer-membrane. The space, within the inner-membrane, is called the matrix, and the space, between the two membranes, is called the inter-membrane space. Hence, physically the matrix is within the inner-membrane, but it is, not, in our model, a component system of the inner-membrane. The above diagram (and its equivalent directed graph and textual description) represent the component structure, of the mitochondrion organelle – how it is composed, from component systems.

From an energy point of view, the most important component system of the mitochondrion organelle is the ATP synthesis system. It is responsible, for the conversion of ADP molecules into ATP molecules. As we indicated above, this is the mechanism, by which the mitochondrion organelle stores and transmits energy, for use by other organelles.

The electron-transport-chain creates and maintains a proton-gradient (an excess of protons (H+s) and hence an excess of electro-chemical energy) between the inter-membrane space and the matrix. This enables it to continually deliver the electro- chemical energy, which drives ATP synthesis.

The inner-membrane is impermeable to all chemicals other than O2, CO2, and H2O. However, the mitochondrion needs to transfer NADH, NAD+, ADP, ATP, and H+, between its inter-membrane space and its matrix. It achieves this with special component systems, which exist within the inner-membrane – translocators (as well as shuttles and antiports).

The Krebs Cycle receives food (e.g. glucose), from outside the mitochondrion, and generates the resources, which are needed, by the other component systems of the mitochondrion.

In the following subsections we will give detailed representations of these components.

The ATP Synthesis Motors

Remember that we said, in the previous section, that the inner-membrane, of the mitochondrion, is impermeable to protons. The electron transport chain and the ATP-synthesis component systems provide routes through this impermeableness. The first, the electron transport chain (described in the next section), forces protons, through the inner-membrane, from the matrix to the inter-membrane space, so that there are more protons in the latter than the former, thus forming a proton gradient, which gives electro-chemical energy to the protons. The second, the ATP synthesis system (described here), allows the energised protons to flow, in the opposite direction, down this gradient, and uses the electro-chemical energy, which is released, to perform its ATP synthesis role. The proton gradient is thus analogous (with protons, rather than water) to the huge hydro-powered ‘battery’ system, at Elidir Mountain, in Wales. Darwinian evolution invented such batteries (highly efficient and miniaturised) billions of years ago, before Wales and its mountains even existed.

In the diagram, of the previous section, the ATP synthesis system was represented with:
We can now expand ATP synthesis, to give:
The ATP synthesis system is a very complex chemical structure. However, it effectively functions as two motors, which the chemists have simply named, Fo and F1. One of these motors, Fo, is an electric motor, driven by the electro-chemical potential of the protons, as they flow down the proton gradient. This Fo motor is embedded in the inner-membrane, but it has a rotating spindle, which extends into the matrix. It converts the electro-chemical energy, which it releases, from the protons, into the kinetic energy, of the rotating spindle. The other motor, F1, is contained in the matrix and could, potentially, be driven, by the chemical energy stored in ATP, and could also produce kinetic energy, to drive the spindle. In fact, within the ATP synthesis system, this motor is forced, by the spindle of the other motor, to work in reverse mode, operating as a generator of ATP, using kinetic energy to create ATP from ADP and Pi. We have described this pair of motors, in this way, because they can operate in reverse and they do just that when necessary. It seems that, billions of years before Faraday was born, Darwinian-evolution evolved the electro-chemical motors, Fo and F1, and copied them, into most living cells.

The Fo motor was shown in the last diagram with:
Ten protons need to flow, down through Fo, in order to rotate the spindle (referred to above) through 360° and return it to its initial state. Fo passes through ten states, as the ten protons gradually flow through it. In each state, one proton is being released, from Fo into F1, eight protons remain held in Fo, and another proton is being acquired (from the energised proton flow).

The main component, of the Fo system, is the c-ring, a ring of 10 subunits, each containing a chemical (Asp61), which can absorb a proton. The c-ring also has the spindle (mentioned above), attached at its centre and extending into F1. Each proton enters the c-ring and is absorbed into the adjacent subunit. This changes the shape, of the subunit, and that forces a rotation by 36°, of the c-ring, thus giving kinetic energy to the spindle. The proton is released when the ring has rotated a further 324° (9x36°), and thus made its subunit available for a later proton.

We can name the subunits cri with 0≤i≤9. The labels, i, should be treated as numbers modulo 10. When we look at a non-digital clock face, we see numbers modulo 12. If the clock tells you that you fell asleep at 11 o'clock and awoke at 6-o'clock, then you will know that you have been asleep 7 hours. Likewise, with numbers modulo 10, if i is 6, then i-7 should be understood as 9.

Thus, the c-ring has ten states c-ringi with 0≤i≤9:
At this point, the reader might think that this diagram, compared with earlier ones, is confusing versions and instances of a component, or possibly confusing types and instances. To answer this point, we need to be clearer, about what is represented by a lego-brick. Our view is that each lego-brick represents the component structure, of some system, over some period of time. In the above diagram, each version, c-ringi, of the c-ring lego-brick, represents the component structure of the c-ring, over some period of time, ti. The Fo lego-brick represents the component structure, of Fo, over a longer period, which is the sum of the tis. The component structure of Fo is the union, of the component structures, of the c-ringi for 0≤i≤9.

We can see that each subunit, cri, has two states, which we can name as cri,1 (with proton absorbed) and cri,0 (without proton). Then, each of the 10 states of the c-ring is equivalent to states of its subunits, as defined by:
c-ringi = { cri,0, cri-1,1, cri-2,1, cri-3,1, cri-4,1, cri-5,1, cri-6,1, cri-7,1, cri-8,1, cri-9,1 } with 0≤i≤9.

Up to this point, our diagrams and our textual descriptions have given static views of system structure. The assumption has been that the structure of the systems shown (their sets of component systems and the interactions between them) does not change, at least at the level of detail shown. Similarly, if you viewed a city from outer space, its structure would not appear to change, for quite a long time, even though the city’s very detailed structure was continually and rapidly changing. In describing the Fo system, change is important and must be represented. We now need to move, beyond static representations of structure and try to represent changes of structure. Note, that, already, in the above diagram, ke1i-1 and H+i-10 represent the kinetic energy flow and the proton flow, which resulted from the previous input of H+ and ecei-1, while the other components relate to the current input of H+ and ecei. Thus, the diagram did give some indication of dynamics.

We can represent the transitions, between successive states of the c-ring, with:

Within the c-ring, as the proton moves, from inter-membrane space to matrix, it loses its electro-chemical energy (ecei). Some of this energy is used in absorbing the proton H+i, into subunit cri,0, to create cri,1. The rest is transformed into kinetic energy. Some of this kinetic energy (ke1i) is used to rotate the spindle through 36°, thus transmitting energy to F1. Some (ke2i) is used, by the subunits of the c-ring, to push the rest of the subunits round the spindle (reflecting the fact that the subunits are in immediate proximity, to each other).

We said, in the introductory sections, that an interaction between two systems could only take place by means of shared components and the functions they perform. Here, it is the electro-chemical energy ecei, which drives the interaction. The interaction can be represented, textually, with:

ecei( { epfi, crgi, kefi, pfli } ) = { epfi+1, crgi+1, kefi+1, pfli+1 }
ece=electro-chemical-energy epf=energised-proton-flow crg=c-ring kef=kinetic-energy-flow pfl=proton-flow

We will develop this, in more detail, at the end of the mitochondria example.

The Fo system was comparatively simple, in that it just had 10 similar states. The F1 system is slightly more complex. It was represented above with:

F1 has a ring, containing three identical pairs of proteins, the pair of proteins, which chemists refer to as α and β. Each of the αβ pairs (labelled αβi with i=0,1 and 2) cycles through three states (labelled αβij, with j=0,1 and 2 for each i):

In state j=0, an αβi pair has nothing attached; in state j=1 it has ADP and Pi molecules attached; and in state j=2 it has an ATP molecule attached. The transition, between states j=0 and j=1, attaches ADP and Pi molecules; the transition, between states j=1 and j=2, uses kinetic energy to force the ADP and Pi molecules together and, thus, forms an ATP molecule; the transition between states j=2 and j=0 detaches the ATP molecule, as in:

We need to explain the component, ke. Firstly, it drives each αβi pair (0≤i≤2) through the three states αβik (0≤k≤2), described above. Secondly, it ensures that the three αβi pairs (0≤i≤2) are constrained to always be at different states, in their cycles. The rotation of the spindle provides most of the kinetic energy. However, the reconfiguration of each αβi pair, when forcing ADP and Pi molecules together, also has kinetic effect on neighbouring αβi pairs. It may be that the second source suffices to achieve the second aim, mentioned above. However, we believe that this is still a subject for debate, amongst researchers. The transitions between the three states of F1 (labelled F1i, with 0≤i≤2) can be represented by:

For convenience of drawing the diagram, we have shown the three αβi pairs rotating 120° anticlockwise with each transition. They also rotate physically, but it is not our purpose to represent the physical movement.

Given the above descriptions of F1 and Fo, one would expect Fo to require 9 (or some multiple of 3) protons and their associated electro-chemical energy, to drive each 360° rotation, of the spindle. In fact, chemists believe it receives 10. We understand that research, on Fo, is still underway, but it is believed that more than 3 protons (approximately 3.33) are required to provide enough energy, for the ATP synthesis system to generate each ATP molecule (and the translocator requires one more). Furthermore, this quantity may vary slightly, with variations in the energy gradient, between the inter-membrane space and the matrix.

The Electron Transport Chain

Earlier diagrams have represented the Electron Transport Chain with:

The role, of the Electron Transport Chain, is to maintain the proton gradient, needed by the ATP synthesis system, and described, in the previous section. To do this, it must pump protons, through the inner membrane. This requires energy. The energy is obtained, by moving electrons, between molecules, with an oxidation reaction releasing an electron, from a donor molecule, and a reduction reaction acquiring the electron, for an acceptor molecule. This works, because, in the Electron Transport Chain, the donor molecules chosen (by Darwinian evolution) hold their electrons in higher-energy grips than the acceptor molecules. The spare energy, thus released, can be used, to pump protons, up the gradient, from matrix to inter-membrane space.

Electrons are only free for a very short time, so an oxidation reaction is always quickly followed by a reduction reaction, thus combining the two reactions in a redox reaction. The pairings:
NADH→NAD++2e-+H+ and Q+2e-+2H+→QH2
QH2→Q+2e-+2H+ and cyt-c(oxigenated)+e-→cyt-c(reduced)
cyt-c(reduced)→cyt-c(oxigenated)+e- and O2+4e-+4H+→2H2O

provide the redox reactions of the Electron Transport Chain. Electrons move down the chain - NAD → Q → cyt-c → O2 - with each link in the chain having a weaker hold on its electrons than its predecessor. The three chemical systems, which perform the three redox reactions are referred to as Complex I, Complex III, and Complex IV.

There is a subsidiary path, performed by Complex II, which replaces the first pair with:
FADH2→FAD+2e-+2H+ and Q+2e-+2H+→QH2
At first sight, the name Electron Transport Chain may seem peculiar, as the purpose of the system, so named, is to transfer protons, from the matrix to the inter-membrane space. However, these transfers require energy, and this energy is released, by the movements, of electrons, through the Electron Transport Chain.

We could ask, why has Darwinian Evolution evolved four steps, when it could have used one step, involving just NAD and O2. The answer is that the one step would have been too explosive, releasing a large amount of energy, most of which would have been dissipated as heat. Darwinian Evolution evolved four steps, so that the spare energy could be released slowly, in manageable quantities.

The, above, four steps, of the Electron Transport Chain, differ in their details, but are similar in their essentials. The following shows what they have in common (remember that oxidising removes an electron (thus increases charge) and reduction adds it (thus reducing charge)):
e-=electron eng=energy

The states of the energised proton pool and the krebs cycle systems are both dependent on the number of protons they contain. The state labels m and n represent these numbers. Each of the five Complexes increases m and decreases n; the ATP Synthesis system does the opposite. enga is the energy, which is released by the oxidisation. engb is the electro-chemical energy acquired by a proton, when moved from matrix to the inter-membrane space. engb can vary a bit, depending on the number of protons (n), in the matrix, relative to the number (m), in the inter-membrane space.

We can now develop lego-brick diagrams for the four components, of the Electron Transport Chain, as variations of the above diagram:
Each electron, entering the Electron Transport Chain, is either carried, into ComplexI, within NADH, or carried, directly into ComplexII. In either case, it emerges, carried within QH2. QH2 then carries it, into ComplexIII, and it emerges, carried within Cyt-c(reduced). Cyt-c(reduced) carries it, into ComplexIV, and it finally emerges, carried within H2O, water. Hence each electron, entering the Electron Transport Chain, either passes, through the sequence CompexI – ComplexIII – ComplexIV, or through the sequence CompexII – ComplexIII – ComplexIV.

We should not think, of the Complex systems. in the above sequences, as being locked together. The Inner-Membrane contains many copies, of each, of the Complexes. As we have shown above, the electrons are carried, out, of one Complex, and into the next Complex, in its sequence, by NADH, FADH, QH2, Cyt-c(red), or H2O. There may be a number, of QH2s (a pool of QH2s), waiting to be picked up, by a ComplexIII. Eventually, each of them will be accepted by a ComplexIII. Similarly, there will be pools of NADH, FADH, and Cyt-c(red) waiting for the appropriate Complexes.

The four Complexes and their resource pools can be plugged together::
to give the full Electron Transport Chain.

The energised proton pool is the excess of protons and energy, held in the inter-membrane space, relative to those, held in the matrix. The balance, between the electron transport chain and the atp synthesis systems, will keep the number of protons and the amount of energy, held in the pool, quite close to stability. The energy, in the energised proton pool (fed by endd, enge and engk), will have three component parts:

electrical energy – each of the excess protons carries a charge
chemical energy – diffusion energy from the excess protons
heat energy – a small amount of spare energy, so far insufficient to pump a further proton

In an earlier diagram we provided a context for the Electron Transport Chain:
We have yet to relate the diagrams, of this section, to that context. We can do this, by decorating each, of the four Complex diagrams, with the context:

The diagram for Complex1, presented earlier was a static view of the structure of a ComplexI system – it combined all states of the ComplexI. We should now try to represent the dynamic changes in ComplexI structure. Similar representations could be provided for ComplexII, ComplexIII and ComplexIV. A dynamic view, for complexI, is given by:
ComplexI is a very complex chemical. The five transitions, shown above, are performed by chains of reactions within ComplexI. In the introductory sections we did introduce such chains, as one of two interaction mechanisms. The researchers have yet to fully identify the collections of chemicals (components of ComplexI), responsible for the reactions in these chains.


Earlier diagrams (and a later diagram for the Krebs Cycle) show the Translocators with:

We can now expand this to give:
There are two translocators, one responsible for the flow of inorganic phosphate (Pi). to the matrix, and the other for the swapping of ADP and ATP molecules.

The first transporter allows a proton to move down its energy gradient, from the inter-membrane space, to the matrix. The electro-chemical energy, released by this movement, opens a channel, which allows a phosphate molecule, to make the same journey. Remember, that we said that an average of 3.33 protons needed to release their electro-chemical energy, during ATP synthesis. The additional energy, required by the phosphate-transporter, considered here, means that an average of 4.33 protons, in total, need to release their energy, during the formation of each ATP molecule.

The second transporter swaps ATP and ADP molecules. It moves an ADP molecule, from the inter-membrane space, to the matrix, and then moves an ATP molecule, in the opposite direction. The movement of the ADP molecule unlocks a channel, for the ATP molecule, and vice versa. Thus, this transporter cycles through two states, one state where it can transfer an ADP molecule and the other where it can transfer an ATP molecule. The swapping of ADP and ATP result in some weakening of the electro-chemical energy gradient, between inter-membrane space and the matrix.

The Krebs Cycle

The main function, of the Krebs Cycle (or Citric Acid Cycle or TriCarboxylic Acid Cycle), is to take in the “food”, which is received, by the mitochondrion, and to generate the protons and electrons, which are required, by the Electron Transport Chain.

Earlier diagrams have shown how the Krebs Cycle component, of the Mitochondrion system, interacts with other components, particularly the Electron Transport Chain:
There are also interactions, with the Nutrient Input and Other Resource Flow components:
In this section we will consider the eight components (reactionsij 1≤i≤8) of the Krebs Cycle system, which participate in the interactions, referred to above:
The eight components are here labelled reactionsij, with 1≤i≤8. There are multiple copies of each component, hence the label j. Remember that, earlier, we noted that the number of protons, in the Krebs Cycle system, was significant, because it determined the strength, of the proton gradient, in the ATP Synthesis and Electron Transport Chain components - hence, the label n in Krebs Cyclen.

The eight components, of the Krebs Cycle, can be represented with the following diagrams:
The top six components provide the main path of Krebs Cycle system. They show why the system is called a cycle – each component has an interaction, on its right, which matches with the interaction, on the left, of the next component – the interaction represents one component releasing a chemical and the next one picking it up, possibly after some delay. The seventh and eighth components feed the main path.

The interactions, at the top of the first five components (and the eighth), match those shown earlier, for the Electron Transport Chain and Other Resource Flow systems. The interactions, at the top of the seventh component, are the mirror image of those, on the third, so they represent a possible connection, between these components.

The other interactions shown above, except H2O and HCO3- are all internal to the Krebs Cycle. The H2O interactions represent the release of water molecules. Water can flow freely within and beyond the mitochondrion, so the water molecules may be picked up by systems, inside or outside the mitochondrion. The HCO3- interaction represents reaction7j picking up a bicarbonate molecule. This molecule will have previously been created, by a reaction, between CO2 and H2O.

We have shown just eight reactions, because these reactions interact, in obvious ways, with each other and with the other components, of the mitochondrion, However, in most descriptions of the Krebs Cycle, two of the reactions, reaction1j and reaction 5j, are broken down into subsidiary reactions, as shown below:
Thus the Krebs Cycle, as usually represented, has ten components, in its main loop, plus the two which feed the loop. The chemists have given names to these twelve components:

reaction1Aj = aconitase-dehydration reaction1Bj = aconitase-hydration
reaction1Cj = isosatrate-dehydrogenase1 reaction1Dj = isosatrate-dehydrogenase2
reaction2j = a-ketoglutarate-dehydrogenase reaction3j = succinyl-coa-synthetase
reaction4j = succinate-dehydrogenase reaction5Aj = fumarase
reaction5Bj = malate-dehydrogenase reaction6j = citrate-synthase

reaction7j = pyruvate-carboxylase reaction8j = pyruvate-dehydrogenase

or, sometimes, slight variants of these names.

We could take these components down to the next level of detail. We will do that, here, for the first component, reaction1Aj:


The Aconitase molecule acts, as catalyst, for this reaction and the next one, reaction1Bj. The His-101 component, of Aconitase, takes a hydroxyl group (HO), from the Citrate molecule, combines it with a proton, and thus forms an H2O molecule. The Ser-642 component, of the Aconitase molecule, takes another proton from Citrate. These two actions result in cis-Aconitase. The next reaction, reaction1Bj, reverses the above two actions, with His-101 and Ser-642 reversing each other’s actions. Before reaction1Bj, the cis-Aconitase is reoriented relative to the Aconitase molecule, so that the positions of the hydroxyl group and the proton are swapped, thus forming D-Isocitrate.

Textual Representation

When we introduced our directed graph and lego-brick notations:
we also introduced our textual notation:
cl{ eg{α:int}, et{α,β}, ec{β:int} }
cl=cell eg=energy-generation et=energy-transfer ec=energy-consumption int=some-interaction
The curly brackets and the Greek letters, together, convey all the structural information, which was conveyed by the diagrams. The ‘names’, cl, eg, et, and ec, just convey some understanding, of the purpose, of the systems, which are represented by the lego-bricks, but do not need to convey any structural information. We only really need the Greek letters, for the components, which are shared, such as the two interactions, a, ß, shown above.

For some systems, it could prove impossible to represent their structures, using directed-graph or lego-brick diagrams. The textual notation has the merit that it can represent these structures. It would also make our examples more amenable to simulation. However, it has the considerable disadvantage, of being harder to read and understand. That was our reason for dispensing with it, in the previous sections. We now briefly explore its use.

A slightly more complicated example is given by:
cl{ cy{mo{ α:at, β:ad, γ:or, δ }, oo{ α, β, γ, ε }, δ:ni, ε:bf } }
cl=cell cy=cytoplasm mo=mitochondrion oo=any-other-organelle int=interaction
ni=nutrient at=atp-flow ad=adp&p-flow or=other-resource bf=bodily-function

Note that the directed graph, the lego-brick diagram, and the textual description should, each, tell us how to draw and/or write the other two.

The textual representation can represent interactions as functions:
ecei( { epfi{ αi+1:epfi+1, βi:H+i, γi:ecei }, crgi{ δi:cri0ii, ζi:ke2i}, { εk:crk1k, ηk:int}, βk:H+kk} }i-9≤k≤i-1 },
            θi-1:kefi-1{ kefi-2, keli-1 }, λi-10:pfli-10{ pfli-11, H+i-10 } } )
= { αi+1{ epfi+2, βi+1:H+i+1, γi+1:ecei+1 }, crgi+1{ cri+1,0i+1i+1, ζi+1:ke2i+2}, k}i-8≤k≤i },
       kefi{ θi-1, ke1i }, pfli-9{ λi-10, H+i-9 } }
ece=electro-chemical-energy epf=energised-proton-flow crg=c-ring kef=kinetic-energy-flow pfl=proton-flow
H+=proton cr=c-ring-subunit ke=kinetic-energy

Here, we have named the function, ecei, after the component, ecei, which supports it.

Functions, representing interactions, can be composed together:
cx10( cx10{ α:ox, β:pp, γ:rc }, δ:NADH )
= cx11{ α{ δ }, β, γ }
cx11( cx11{ α:ox{ NADH{ ε:NAD+, ζ:H+, η:2e-, θ:engc } }, β:pp, γ:rc }, κ:4H+ )
= cx12{ α{ ε, ζ, η, θ }, β{ η, θ, κ }, γ }
cx12( cx12{ α:ox{ NAD+, H+, η:2e-, θ:engc }, β:pp{ η, θ, κ:4H+ }, γ:rc }, λ:2H+, μ:Q )
= cx13{ α{ }, β{ κ, ζ:4engd, η, π:engc-4engd }, γ{ η, π, λ, μ } }
cx13( cx13{ α:ox, β:pp{ 4H+, 4engd, η:2e-, π:engc-4engd }, γ:rc{ η, π, λ:2H+, μ:Q } } )
= cx14{ α, β{ }, γ{ τ:QH2{ λ, μ } } }
cx14( cx14{ α:ox, β:pp, γ:rc{ τ:QH2 } } )
= cx10{ α, β, γ{ } }
ece=electro-chemical-energy epf=energised-proton-flow crg=c-ring kef=kinetic-energy-flow pfl=proton-flow
H+=proton cr=c-ring-subunit ke=kinetic-energy
Combining the five functions in one composite function gives us:

cx14( cx13( cx12( cx11( cx10( cx10{ α:ox, β:pp, γ:rc }, NADH{ δ:NAD+, ε:H+, ζ:2e-, engc{ η:4engd, θ:engc-4engd } } ), κ:4H+ ), λ:2H+, μ:Q ) ) )
= { cx10{ α, β, γ }, δ, ε, κ, η, QH2{ λ, μ } }
or if the composite function cx14( cx13( cx12( cx11( cx10( ) ) ) ) is treated as one function:
cx1( α:cx10, NADH{ β:NAD+, γ:H+, 2e-, engc{ δ:4engd, ε:engc-4engd } }, ζ:4H+, η:2H+, θ:Q )
= { α, β, γ, ζ, δ, QH2{ η, θ, ε } }
and, as we would expect, that is equivalent to our original diagram, the one which combined the states:

Consolidated Diagrams

We can put the lego-brick diagrams, of the previous sections, together in one diagram. At A4 size this is:

Additional Observations

When people talk of a system in the real world, they are usually imprecise. For instance, when we talk of the system, London, we may mean the ancient City of London, with its corporation and Lord Mayor, we may mean Greater London, with its more recently established mayor and councillors, we may mean London as a financial entity, which is of concern to the City, we may be thinking of London as the capital of Great Britain (for now), we may include the people, who live and work in London and the infrastructure, which feeds into London. These are best thought of as different, though closely related, systems, with different, but overlapping, sets of component systems.

We should add a further observation, at this point. We need to recognise that there is an infinite number of systems, as any composition of a set of systems defines an additional system, even if its component systems are remote, totally unrelated, and never interact (i.e. never affect each other’s behaviour, directly or indirectly). Hence, we must concentrate our attention on “interesting” systems, each of which is composed of component systems, which do interact, at some points in time.

Given our emphasis on shared component systems, we could, in theory, represent the structure of all systems, with a gigantic directed graph. The nodes, in the graph, would represent systems. A line, in the graph, would indicate that the system, which is represented at the bottom of the line, is a component of the system, which is represented at the top. The conservation rules, introduced above, mean that every system, considered as a collection of components, exists over all time, in some form, though it may be quiescent (not interacting) most of the time - the collection of atoms and energy, in Caesar’s last breath, though dispersed and transformed, still exists. Thus, our directed graph must remain static, representing the structure of all possible systems (including uninteresting ones), over all time. This leaves us with an important question – how is change represented?

In the real world, we see gradual change in systems. How does this gradual change relate to the static graph view, introduced above? The systems, which are currently active (engaged in interactions), could be represented, by an active subgraph, of our full directed graph. Most of the active graph will be in the lower, more detailed, levels of the full graph, as there reside the rapidly changing systems. The systems in the higher levels change (as sets of direct component systems) very slowly. This reflects reality - if you viewed a city or country very closely, you would see its detailed structure changing continually and rapidly, but if you viewed it from outer space, its visible, top-level, component structure would not appear to change.

The active subgraph gradually moves within the full graph. What triggers this movement? Proximity gives the clue – for elementary chemical systems, it is physical proximity, which triggers an interaction (i.e. a chemical reaction). When close enough, the physical attraction of the protons of one chemical system, for the electrons of the other, is large enough to force a reconfiguration, of the electrons, and a binding, of the chemical systems. We need to generalise this concept of proximity. We believe that systems can only interact (in other words can only change each other), by means of shared component systems. This means that the interaction may be implemented in one of two ways. Either there might be a component system, which is totally shared between the two systems, or the two systems might both interact with a third system. These two forms of interaction provide our generalised understanding of proximity. Note that this is a recursive definition of an interaction – an interaction can only be activated, if all its component interactions can be activated, and this definition could nest down through many levels. Hence, it is primitive interactions, low in the directed graph, which implement those higher up. We referred to one primitive interaction above – the attraction between a proton, of one chemical system (an atom), and an electron, of the other – this attraction was a shared component between the two chemical systems.

In “Origin of the Species”, Darwin identified two orthogonal drivers for evolution -

Darwin gave a very full definition, of the latter, but lacked the modern science needed to deal with the former. Together, these drivers controlled the gradual evolution, of our active graph, as it moved across the full directed graph, with the first driver generating change and the second driver pruning redundant changes. The active graph has gradually increased in complexity, as new systems and new mechanisms, for combining systems, have evolved. The first Darwinian driver was given a good start - at the birth of the universe (with chemistry based on a fortuitous choice of physical constants) and at the later creation of our planet (with volatile climate and core, and a conveniently placed star). Change was thus built in at the beginning, so Darwinian evolution prospered. The electromagnetic motor, found in every mitochondrial organelle, is just one example of the inventive capability of Darwinian evolution. Humans have, only recently, begun to reinvent mechanisms, which emerged billions of years ago from Darwinian evolution.

Our Mitochondrial Model and the Viable System Model (VSM)

The previous sections have only considered a mitochondrion, when it is behaving normally. We need to add other components and systems to deal with evolutionary change and abnormal behaviour. During the Second World War and afterwards Stafford Beer developed his Viable System Model. His model is now used widely, by people who are developing IT systems and Business Systems. Supporters of the model have developed numerous variations, of this model, to cope with a wide range of applications. However, the view, which underlies these models can be captured, in the following lego-brick diagram:

Beer stated that a viable system is any system, which is organised in such a way, as to meet the demands, of surviving in the changing environment. The Viable System Model (VSM) defined the set of components, which should be embedded in any system, which is viable. Key to this model are the components-1 to 5, shown in the above diagram:

A component of a viable system may, itself, be a viable system. This is shown in the diagram, with the VSM within a component-1. However, it is not entirely clear, how the components 4 and 5, of the contained, component, viable system, would relate to the higher-level components 4 and 5, of the containing, viable system.

It is believed that mitochondria originally existed, as independent bugs. At some point in time, billions of years ago, a mitochondrion was enveloped, as food, by a cell. The cell started using the energy generated, by the mitochondrion, so creating a symbiotic relationship, between cell and mitochondrion. The mitochondrion later started using capabilities, provided by the cell, and discarded its own version of these capabilities, thus creating the mutual dependency between cell and mitochondrion. This relationship was so successful that Darwinian evolution, eventually, made it the standard, for all cells. Because of this evolution, mitochondria ceased to be viable systems, but became vital components of the, still viable, cells.

It would be interesting to have a Viable System Model, for the natural systems that have evolved in chemistry and biology. If we applied this model to a cell, the mitochondrion would be an important component-1, within the cell. There are various chemical signals that are generated, by the mitochondrion, when exceptional circumstances are detected, e.g. shortage of oxygen, or increased demand for energy (because of exercise). Chemical signals are returned, telling the mitochondrion what changes are required, e.g. closing or reversing of ATP Synthesis, or creation of additional mitochondria. The component-2s would be responsible, for these two-way signals. The components 3 to 5 would be sited, mainly, in the nucleus, of the cell, with components 4 and 5 being concerned, with aging and Darwinian evolution. Currently, the details of this Viable Natural System Model are beyond our capabilities.


A Mitochondrion

ATP Synthesis

Electron Transport System


Krebs Cycle

Cell Membrane