May 25, 2013

Reflections on Chaos in Biological Evolution

As mentioned in my last post, this month is the 50th anniversary of the landmark paper by Ed Lorenz that inaugurated the scientific study of chaos (e.g. chaos theory). In this post, we will explore the potential contributions of chaos theory and dynamical systems (Figure 1) to the study of biological evolution from (hopefully) an intuitive perspective.

Figure 1. When people think of chaos theory, they generally think of something called a strange attractor. Two examples of these can be seen above. Figures are from [1] and [2].

Figure 2. LEFT: image from the Ian Malcolm novelty t-shirt. RIGHT: the butterfly effect as an attractor in phase space.

Coincidentally, this year is also the 20th anniversary of the movie "Jurassic Park" (based on the novel by Michael Crichton). The mathematician in the lost group (Ian Malcolm) was obsessed with chaos theory (Figure 2). The novel [3] features more extensive commentary by the fictional Dr. Malcolm, so go there if you want more insight than the movie offers.

The slant on chaos in "Jurassic Park" (or Crichtonian chaos) was on the unpredictablity of living systems: as Ian Malcolm said, "life finds a way". Unfortunately, the "Jurassic Park" argument from chaos was done using a populist, moralistic frame -- as in "ultimately, genetic engineering and otherwise screwing with natural processes is bad". This is unfortunate, since the chaotic and dynamical systems perspective can provide much insight into natural variation and evolution.

Figure 3. Here is a slide demonstrating what a fractal tree looks like (with a philosophical lesson attached). Notice its self-similar nature. Code for generating fractals can be found on OpenProcessing.

Stephen J. Gould wrote much about the role of chance in evolution [4], but did so in a qualitative fashion. While chaos is typically a memoryless phenomenon (and so does not account for historical contingency per se) [5], chaos does provide a tractable mechanism for chance events to unfold in a coherent fashion. Thus, chaos theory can bridge the gap between measurement and theory, which draws from the wide arc of natural history.

Besides the potential use of strange attractors embedded in phase space to characterize fitness landscape activity, there are interesting parallels between biological classification (e.g. phylogenies and gene genealogies) and fractal tree structures (Figure 3). There are also possible uses for Feigenbaum's logistic maps to characterize processes such as speciation and intraspecific population structure. In the second part of this post, I will focus on a number of examples that present an argument for why chaos is relevant to Darwinian evolution (Figure 4).

Figure 4. In [2], Motter and Campbell mention that chaos theory has helped replaced (for the most part) a mechanistic world-view. Here is what a non-chaotic world may look like (COURTESY: Co-axial World ad, Omega watches). For what its worth, the ad has a quasi-creationist theme, which hints at the critical underlying role of chaos in evolution.

The first of these is a set of papers by Robertson (an approach I will call Robertsonian chaos) that focus on the role of feedback on fitness [6, 7]. According to Robertsonian chaos, feedback acts upon the fitness and observed diversity of natural populations as a consequence of natural selection. In some cases, this feedback can actually decrease fitness. The important thing to remember is that the scenario of feedback leading to chaos is distinct from the ratchet mechanism for natural selection I covered in a previous Synthetic Daisies post.

As a chaotic (e.g. non-directional) mechanism, feedback can also act as a source of instability, and produce opportunities for evolutionary change. In [6], Robertson argues that due to its exponential effect, feedback dominates the dynamics of an evolutionary system. Using an electrical circuit as a metaphor, feedback provided by the action of natural selection acts as an amplifier (Figure 5).

Robertson's proposal for the role of feedback in evolution is fundamentally different from teleological, goal-oriented mechanisms. Instead, it enables "open-ended" evolution. This is also different from other mechanisms such as transgenerational epigenetic inheritance or facilitated variation [8] in that feedback is explicitly tied to outcome (e.g. evolutionary dynamics).

On the other hand, feedback does not seem to be a fundamental property of natural selection. That is, natural selection results in differential reproduction, but does not do so in a deterministic manner. One thing is certain: while feedback may act as an amplifier of natural selection, its mode of action is subtle [9].

Figure 5. Annotated version of Figure 1 from [6], which shows how the feedback mechanism works for a population embedded in a fitness landscape.

The second perspective is a book by Arun Holden called "Chaos" [10], which focuses on applications of chaos to physiology and biology. M. Holden contributes a chapter called "What is the use of chaos?", which lays out the functional roles of chaos in biological contexts.

The case is made for five potential functions: 1) diversity generation, 2) diversity preservation, 3) system maintenance, 4) the interaction of population dynamics and genetic structure, and 5) the dissipation of disturbances. Of these five candidates, two are particularly intriguing.

One of these is maintenance, which is referred to in the chapter as "disentrainment". Disentrainment is the an intentional lack of coordination among system components. For example, disentrainment in a social system among individual actors would impede cooperation or perhaps even competition (Figure 6). In neural systems, neurons are often entrained in small populations. If entrainment (see Figure 6 for visual aid) is too extensive and includes too many neural populations simultaneously, electrical activity can cascade out of control [11]. If entrainment is not extensive enough, there is no coherence to the system.

Thus, evolutionary order on the edge of chaos in the form of self-regulating population structure and diversity prevents evolving systems from responding to natural selection and other population processes with only an all-or-nothing [12] response. While such a response is occasionally useful (e.g. biochemical ultrasensitivity), chaos provides options to an evolutionary systems (at least the highly evolvable ones).

The other is the dissipation of disturbances, which may be compared to robustness in evolving biological systems. The dissipation of disturbances involves a mechanism which keeps an evolving system from experiencing dangerous disruptions. This is particularly intriguing in light of the important role disturbances play in evolving ecosystems.

The role of chaos in dissipating disturbances seems to be counter-intuitive at first glance. However, a naturally chaotic system (e.g. a system that can visit a wider range of states in the course of its normal behavior) can recover from disturbances much more efficiently than systems that are more conservative. In conjunction with natural diversity, chaotic behaviors such as transience and itinerance [13] can actually save an evolutionary system from itself.

Figure 7. A more historical view of the "routes to chaos".......

There are likely other examples I have missed. I also have not discussed the differences between deterministic and stochastic chaos [14], which is important in the analysis of biological systems and is deserving of its own discussion. Hopefully, this post serves as inspiration for future work and discussion (see Figure 7). Long live chaos!


[1] Arbesman, S.   The Fiftieth Anniversary of Chaos. Social Dimension blog, May 17 (2013).

[2] Motter, A.E. and Campbell, D.K.   Chaos at Fifty. Physics Today, May, 27 (2013).

[3] Crichton, M.   Jurassic Park. A.A.Knopf (1990).

[4] Gould, S.J.   The Structure of Evolutionary Theory. Harvard University Press (2000).

[5] Kautz, R.   Chaos: the science of predictable random motion. Oxford University Press, Oxford, UK (2011).

[6] Robertson, D.S. and Grant, M.C.   Feedback and Chaos in Darwinian Evolution: Part I. Theoretical Considerations. Complexity, 10-14 (1996).

[7] Robertston, D.S.   Feedback theory and Darwinian evolution. Journal of Theoretical Biology, 152, 469 (1991).

[8] Pigliucci, M. and Muller, G.B.   Evolution: the extended synthesis. MIT Press, Cambridge, MA (2010).

[9] To frame this in terms of speculative mathematical modeling, another possible source of nonlinearity is the interaction (e.g. mathematical convolution) between the fitness value and selection coefficient functions over time sampled at every generation). As both of these functions are semi-independent, their interaction could produce nonlinear effects on a population's subsequent fitness.

See this Synthetic Daisies post from April for more information on the convolution of stationary noisy signals for more information.

[10] Holden, A.V.   Chaos. Princeton University Press, Princeton, NJ (1986).

For an introduction to the field (with relevance to chance and historical contingency), please see the following books:

[a] Ruelle, D.   Chance and Chaos. Princeton University Press, Princeton, NJ (1991).

[b] Smith, L.A.   Chaos: a very short introduction. Oxford University Press, Oxford, UK (2007).

[11] This is the mechanism behind epileptic seizures that originate in the medial temporal lobe of the brain. For more information, please see:

Litt, B., Esteller, R., Echauz, J., D'Alessandro, M., Shor, R., Henry, T., Pennell, P., Epstein, C., Bakay, R., Dichter, M., and Vachtsevanos, G.   Epileptic seizures may begin hours in advance of clinical onset: a report of five patients. Neuron, 30(1), 51-64 (2001).

[12] All-or-nothing responses resemble step-functions. On the other hand, critical responses (or first-order phase transitions, which rely on thresholds that resemble step functions) may be useful for evolving systems as well.

[13] For various perspectives on these concepts, please see:

[a] Zimmermann, M.   Transient Behavior. ZhurnalyWiki, 1999.

[b] Tsuda, I.   Chaotic itinerancy. Scholarpedia, 8(1), 4459 (2013).

[14] Casdagli, M.   Chaos and Deterministic versus Stochastic nonlinear modeling. Santa Fe Institute Working Paper, 1991-07-029.


  1. Happy 50th to Chaos indeed.
    The geometries underlying "edge of chaos" and optimal "routes to chaos,"inspired the Asynsis principle as a theory of design in nature.
    The conjectures that have ensued have much in common with your Robertson & Holden references. With the Constructal law team, I believe that we have begun to reveal the geometric signatures of the thermodynamics & feedback-led, analogical, optimal evolution of persistent complexity in the face of entropy.
    More 4 Less. A new optimal design & construction paradigm and effective Complexity ToE.
    Asynsis Principle-Constructal law Team @ ARUP Foresight & Innovation, London Fri May 3 2013 via

  2. Origin of life and its evolution are the result of action of laws of hierarchical thermodynamics.

    Criterion of evolution

    The approval about the reduction of the entropy of living systems as a result of biological evolution is incorrect. The criterion of evolution of living system is the change (during evolution) of the specific free energy (Gibbs function, G) of this living system. The evolution of living system takes place against the background of flows of energy (e.g., light, energy of physical fields) from the environment. It increases its specific free energy. At the same time, the specific free energy of this living system is decreased as a result of spontaneous processes in this system.
    Thus, the total change in the specific free energy of a living system is composed of two parts: 1. The change of free energy due to the inflow of external energy (G1> 0) and 2. The change of free energy due to spontaneous transformations in the system (G2 < 0) . The evolving system constantly adapts to a changing environment. The principle of substance stability contributes to this adaptation.
    Thermodynamics of evolution obeys the generalized equation of Gibbs (that is the generalized equation of the first and second laws of thermodynamics)*. Biological evolution and the processes of origin of life are well described by the hierarchical thermodynamics, established on the firm foundation of theory of JW Gibbs. Our theory created without the notion on dissipative structures of I. Prigogine and negentropy of L. Boltzmann and E. Schrodinger.
    “Thermodynamics serves as a basis for optimal solutions of the tasks of physiology, which are solved by organisms in the characteristic process of life: evolution, development, homeostasis, and adaptation. It is stated that the quasi-equilibrium thermodynamics of quasi-closed complex systems serves as an impetus of evolution, functions, and activities of all levels of biological systems’ organization. This fact predetermines the use of Gibbs’ methods and leads to a hierarchical thermodynamics in all spheres of physiology. The interaction of structurally related levels and sub-levels of biological systems is determined by the thermodynamic principle of substance stability. Thus, life is accompanied by a thermodynamic optimization of physiological functions of biological systems. Living matter, while functioning and evolving, seeks the minimum of specific Gibbs free energy of structure formation at all levels. The spontaneous search of this minimum takes place with participation of not only spontaneous, but also non-spontaneous processes, initiated by the surrounding environment.”
    Works of the author:

    Georgi Gladyshev
    Professor of Physical Chemistry

    *) The generalized equation of Gibbs (See: )

    P.S. Lastly, it is important to take into account, from the viewpoint of hierarchical thermodynamics, that anti-aging diets and many drugs can be used for the prophylaxis and treatment of cardiovascular diseases, cancer, and for numerous other illnesses.