Strange attractors

Attractors are abstract mathematical concepts that when plotted in 3-D graphics look rather pretty (right) and in biology represent structured patterns of behaviour that characterise all manner of life processes from heart rhythms to gene expression. They are vital for the understanding of disease, healing, inheritance and evolution.

Watch the video below to see some typical behaviour in this animation of the famous Lorenz Attractor.

We just saw several remarkable characteristics:

  • Rapid convergence to the attractor no matter what the start point
  • No deviation from the attractor once it is reached
  • The existence of two distinct semi-stable sub-states
  • Movement between the sub-states is rapid, with no intermediate states
  • Behaviour is pseudo-rhythmic or rhythmic with random variations
  • Precise behaviour is unpredictable but overall behaviour is highly stable
  • A self-organising character with spontaneous pattern generation

All these phenomena are instrumental in keeping order in biological systems where stable outcomes are paramount despite persistent noise and perturbations.

Heart rate and temperature must return to normal after physical exertion, vitamin and hormone concentrations must be regulated despite irregular supply and non-matching demand and molecular order must be preserved for functional gene expression and cell division.

Cymatic attractors: In the Amazing Resonance Experiment below, a different mechanism produces behaviour with nevertheless similar characteristics. Several discrete states are available and as before, there are no intermediate states. However. here the transition is not random but can be controlled by specific frequencies. The movement of each individual grain of sand is locally random but the sum of all this randomness is a well defined pattern. Disturbing the pattern by dragging a finger through it will have no long term effect; the whole thing will simply re-organise.

Different varieties of attractors are available:

  • Single point – static attractor
  • Limit cycle – periodic attractor
  • Chaotic – Strange attractor
  • Toroidal – quasi-periodic attractor

Single point attractor: Behaviour will always converge to a single point no matter what the starting point. One example might be the healing of a wound which, no matter what its shape or extent will converge towards a single body shape if possible. The attractor is a dynamic system which has a single goal: the completion of the phenotype.

Another example might be the maintenance of body temperature, where again, a single point goal is to be achieved whether one has just run a marathon in hot weather or swum through an icy lake. The goal is the same; to maintain a constant body temperature. Here though the body has the option of varying the goal in the case of a fever, say, where a new goal is set by some other process and a higher temperature is maintained for a prescribed length of time.

Limit cycle attractors: Here, a process, whether it be regulation of metabolite levels, neural network activity or organised molecular activity (gene expression) will tend towards a cycle of activity, i.e. a stable dynamic state as opposed to a static fixed state.

One example is a cell cycle, of growth, regulation and division where everything must happen like clockwork and any deviation from the correct procedure must be met by either immediate correction or termination of the whole process. This works at least in part by means of feedback systems and checkpoints to validate the whole process. We have here a closed loop control system.

Another example might be maintenance of diurnal rhythm where the same cyclic pattern must be attained no matter what slight deviations have been made along the way.

Another image of the Lorentz attractor showing two clear pseudo-cyclic sub-states. A dot placed at random anywhere in this diagram will quickly gravitate towards one of these loops at random.

In other cases, randomly placed elements within a loop will cycle within that loop and remain there until some disturbance pushes them within range of another cycle.

Attractor ‘wells’: In this image of the famous Julia Set the black dots (barely visible) at the centre of the grey areas represent the goals of fixed point attractors and the grey areas themselves are the attractor wells or catchment areas of those sub-states. Any process caught in these areas will inevitably make its way to the fixed point at the centre.

So we can have a situation where biological processes are happily running and remain stable to local ‘noise’ but may sometimes be disturbed so much that a phase-change occurs and they are now exhibiting different, yet still stable behaviour, whilst obeying the same set of organisational rules.

We have discovered a way for the same physical substances conforming to the same physical laws to demonstrate markedly different modes of existence and execute very distinct biological functions.

Think of the importance of this with regards to say, gene expression (protein construction), where the end point of cellular activity must be well defined yet at the same time independent of the precise physical composition of the cell. If the cell contains the same atomic elements from one moment to the next then how can its function vary so much and why is it so organised, so apparently goal-oriented? From whence does the difference originate?

Another way to visualise the concept of attractor wells is seen in the diagram below where we have cell (blue) in its usual functioning state and another (red) in a deviant attractor state (cancer, say) . The normal attractor state is in a deep and hence stable ‘well’ but the red cell has somehow found itself in an alternative state which fortunately here is an inherently unstable state – depicted here as a shallow well. (Diagram from Naarala et al)

In the normal course of cellular life, the contents are subjected to many mechanical and energetic disturbances meaning that unstable states will likely not survive very long and that the cell will make its way back to the main attractor by use of other unstable states as ‘stepping stones’.

A ‘3D’ visualisation of the state-space below highlights the idea that normal ‘resting’ states are likely to be low-energy states requiring minimal upkeep whereas deviant states are high energy states requiring additional input and adding to their hopefully already unstable nature. So there is a natural tendency then to move towards a healthy state and a tendency to stay there as this is now a state of both least energy and greatest stability. (Diagram from Uthamacumaran)

Note that the diagrams show attractor behaviour in terms of paths in 3D or 2-D space but this is just an abstraction for illustrative purposes. In reality, the spatial dimensions shown are representative of various parameters of the mechanics of molecular activity: temperature, the concentration of Calcium ions, the rate of production of a particular enzyme for example.

The illustrations are two dimensional depictions of three dimensional processes but in reality the number of dimensions is much higher with one estimate being at around 100,000.

Cancer: As suggested by the previous diagram, cancer is now thought by some to be a result of altered gene expression brought on by a sudden change in attractor state at the sub-cellular level.
Such a state change can be triggered by a relatively weak stimulus below the level that would normally be expected to cause physical damage.

The view of carcinogenesis advanced here sees such mutations as the consequence of an underlying and more fundamental epigenetic process that leads the system into a specific domain of the state space associated with malignancy, via a series of randomly adopted variant attractors” – Baverstock

The cell has wandered (as in the diagram) away from the main attractor state and into an alternative mode of functioning where growth and reproduction are unrestrained.

Stem cell differentiation. A fertilised egg will repeatedly divide into identical cells which only after many such divisions do begin to differentiate into nerve or muscle cells etc.
How does this happen then? If each cell contains the same physical substance and the same information, how do the differences arise?

The answer must be that somehow all the information required to make a complete organism is present in the cellular attractor of the cell. It is this attractor that is responsible for phenotype and the the progression of this attractor that organises development of the organism from egg to foetus and from toddler to adulthood.

Many disease states and healing patterns display attractor-like qualities:

  • Sudden onset of many stereotypical symptoms (state-change)
  • Onset sometimes with no apparent cause
  • Stability of state, whether disease or health, to large perturbations
  • Persistent disease state long after the cause has disappeared
  • Miracle cures from spa water, acupuncture, supplementation, horse-paste, etc.
  • Spontaneous remission with no apparent cause
  • Unstable oscillatory behaviour – bipolar disorder for example
  • Synchrony with seasons (although attractors are only pseudo-periodic they can be entrained to resonate at certain frequencies – Strogatz)
  • Hormesis (non-linear dose-response curve)
  • State-dependent response to stimuli (different people will respond differently to the same medication)
  • Allergic reactions

Evolution is by random mutation and incremental changes in phenotype according to Darwin. However, what is observed in the fossil record is a process of punctuated equilibrium.

What we actually see is that once a species has been established, it will remain stable for many millennia with only minor variations. Major changes are not the accumulation of minor changes but are seen as sudden changes in the phenotype upon which further minor variations will be superimposed.

Variations in dairy cattle for example arise through selective breeding and will revert to ‘breed average’ if not maintained carefully. “Our varieties certainly do occasionally revert in some of their characters to ancestral forms.” – Charles Darwin Chapter 1 of “On the Origin of Species”. This is not consistent with Darwinian evolution but perfectly in tune with attractor behaviour.

Evolution then has the character of an unfolding attractor that gains physical expression via organic life forms. The long term stability with minor variations is indicative of a stable attractor subject to minor perturbations. The sudden emergence of a new species is typical of an attractor phase-change.

Since according to evolutionary theory it is the phenotype that is selected for and since the phenotype is the expression of a multi-dimensional attractor, we must now consider attractors themselves as inheritable entities: “Variant attractors are a source of evolutionarily selectable variation in addition to genetic variation” – Baverstock

So all the information required to make a new life is somehow collected together in the gametes and passed on to the next generation. This is similar to Darwin’s pangenesis hypothesis but instead of the information being stored in physical particles (gemmules) it is stored in the behaviour or configuration of a multi-dimensional attractor.

Exosomes are extracellular vesicles generated by all cells and they carry nucleic acids, proteins, lipids, and metabolites. They are mediators of near and long-distance intercellular communication in health and disease and affect various aspects of cell biology.” – NIH

So small extracellular particles are carrying biologically meaningful information and if biology is organised around attractor states then the exosomes must be carrying information that constructs or modifies the attractors driving cellular processes. But of all biology is organised by attractors then the information within the exosomes is itself an active physical realisation of an attractor.

This does not resemble the transference of digital (genetic) information so much as the merging of an operating system update.

The same process then happens with both exosomes and inheritance. Keith Baverstock gives an analogy of two manufacturing companies that merge into one. All the staff with similar but slightly different skills and abilities continue to perform their usual functions with slight adaptations and the factory keeps running with a very similar but not quite identical product. So rather than encode digital instructions for a new product and pass them from factory to factory, just move some staff around.

Other forms distant cellular communication exist without any apparent exchange of physical substance. Various solutions have been proposed including acoustic vibrations and bio-photons. This maybe so but the information itself needs to be of a biologically meaningful nature and that now means that an attractor state is somehow transmitted.

Self-sustaining attractor states are the basic units of biological organisation and are identified with biological communication and organisation at all scales from cellular to organism wide. They are instrumental in gene-expression, inheritance, evolution, embryonic development and maintenance of form and function. Dysfunction leads to characteristic patterns observable in disease states, regulatory disorders and cancer.

Cognisance of these patterns is essential for the appreciation of non-linear dose responses (hormesis) and indeed the relationship between any external stimulus and the resulting effect on the body.


The gene: An appraisal – Keith Baverstock

Epigenetic Regulation of the Mammalian Cell – Keith Baverstock, Mauno Rönkkö

Electromagnetic Fields, Genomic Instability and Cancer: A Systems Biological View – Jonne Naarala, Mikko Kolehmainen, Jukka Juutilainen

A Review of Dynamical Systems Approaches for the Detection of Chaotic Attractors in Cancer Networks – Abicumaran Uthamacumaran

Determining Relative Dynamic Stability of Cell States Using Boolean Network Model – Joo, Zhou, Huang, Cho

Sync: The emerging science of Spontaneous Order – Stephen Strogatz

The biology, function, and biomedical applications of exosomes – Raghu Kalluri, Valerie S. LeBleu

Attractors – Wikipedia

Transgenerational epigenetic inheritance – Wikipedia
(Lists self-sustaining loops as one form of epigenetic inheritance)

An attractor looking a bit like a bird’s skull

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