Skip to main content

Elastic Multi-scale Mechanisms: Computation and Biological Evolution


Explanations based on low-level interacting elements are valuable and powerful since they contribute to identify the key mechanisms of biological functions. However, many dynamic systems based on low-level interacting elements with unambiguous, finite, and complete information of initial states generate future states that cannot be predicted, implying an increase of complexity and open-ended evolution. Such systems are like Turing machines, that overlap with dynamical systems that cannot halt. We argue that organisms find halting conditions by distorting these mechanisms, creating conditions for a constant creativity that drives evolution. We introduce a modulus of elasticity to measure the changes in these mechanisms in response to changes in the computed environment. We test this concept in a population of predators and predated cells with chemotactic mechanisms and demonstrate how the selection of a given mechanism depends on the entire population. We finally explore this concept in different frameworks and postulate that the identification of predictive mechanisms is only successful with small elasticity modulus.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4


  1. This adaptation explains why the rolling stones have survived so many time despite drug and general health abuse.

  2. This relation is valid since we relate two distances. No additional mathematical structures are here involved.

  3. Both conditions are dangerous, since dynamical systems could be used to describe systems that are oee. This can be wrong if the existence of the gap between trajectories describing the environment is not recognized.

  4. Otherwise we must assume for instance more than one adaptive response associated to different models




  • Adams A, Zenil H, Davies PCW, Walker SI (2017) Formal definitions of unbounded evolution and innovation reveal universal mechanisms for open-ended evolution in dynamical systems. Sci Rep 7:997

    Article  PubMed  PubMed Central  Google Scholar 

  • Ahn AC, Tewari M, Poon C-S, Phillips RS (2006). The limits of reductionism in medicine: could systems biology offer an alternative? PLoS Med 3:e208

    Article  PubMed  PubMed Central  Google Scholar 

  • Alon U (2006) An Introduction to systems biology: design principles of biological circuits. Chapman & Hall/CRC Mathematical & Computational Biology, Boca Raton

    Google Scholar 

  • Arditi R, Ginzburg LR (1989) Coupling in predator-prey dynamics: ratio-dependence. J Theor Biol 139:311–326

    Article  Google Scholar 

  • Barabási A-L (2009) Scale-free networks: a decade and beyond. Science 325:412–413

    Article  PubMed  Google Scholar 

  • Bitbol M, Luisi PL (2004) Autopoiesis with or without cognition: defining life at its edge. J R Soc Interface 1:99–107

    CAS  Article  PubMed  PubMed Central  Google Scholar 

  • Buescu J, Graça DS, Zhong N (2011) Computability and dynamical systems. In: Dynamics, games and science I. Springer, Berlin, Heidelberg, pp 169–181

  • Carroll JW (2016). Laws of nature. In: Zalta EN (ed) The Stanford encyclopedia of philosophy. Metaphysics Research Lab, Stanford University, Stanford

    Google Scholar 

  • Castelvecchi D (2015) Paradox at the heart of mathematics makes physics problem unanswerable. Nature 528:207

    Article  Google Scholar 

  • Chang H, Levchenko A (2013) Adaptive molecular networks controlling chemotactic migration: dynamic inputs and selection of the network architecture. Philos Trans R Soc Lond B 368:20130117

    Article  Google Scholar 

  • Cubitt TS, Perez-Garcia D, Wolf MM (2015) Undecidability of the spectral gap. Nature 528:207–211

    CAS  Article  PubMed  Google Scholar 

  • Danchin A (2009) Bacteria as computers making computers. FEMS Microbiol Rev 33:3–26

    CAS  Article  PubMed  Google Scholar 

  • Descartes R. Treatise on man

  • Ellis GFR (2012) Top-down causation and emergence: some comments on mechanisms. Interface Focus 2:126–140

    Article  PubMed  Google Scholar 

  • England JL (2012) Statistical physics of self-replication. ArXiv12091179 Cond-Mat Physicsphysics Q-Bio

  • England JL (2015) Dissipative adaptation in driven self-assembly. Nat Nanotechnol 10:919–923

    CAS  Article  PubMed  Google Scholar 

  • Goldenfeld N, Woese C (2011) Life is physics: evolution as a collective phenomenon far from equilibrium. Annu Rev Condens Matter Phys 2:375–399

    CAS  Article  Google Scholar 

  • Hernández-Orozco S, Hernández-Quiroz F, Zenil H (2016). Undecidability and irreducibility conditions for open-ended evolution and emergence. ArXiv E-Prints arXiv:1606.01810

  • Khanin R, Wit E (2006) How scale-free are biological networks. J Comput Biol 13:810–818

    CAS  Article  PubMed  Google Scholar 

  • Landau LD (2004) Theory of elasticity. Butterworth-Heinemann Ltd, Oxford

    Google Scholar 

  • Marzen SE, DeDeo S (2017). The evolution of lossy compression. J R Soc Interface.

    PubMed  PubMed Central  Google Scholar 

  • Maturana HR, Varela FJ (2004) el Arbol del conocimiento: las bases biológicas del entendimiento humano. Lumen, Buenos Aires

    Google Scholar 

  • Ochoa JGD (2014) Relative constraints and evolution. Int J Mod Phys C 25:1450030

    Article  Google Scholar 

  • Peper A (2009) Intermittent adaptation. A theory of drug tolerance, dependence and addiction. Pharmacopsychiatry 42(Suppl 1):S129–S143

    Article  PubMed  Google Scholar 

  • Rathgeber S (2002). Practical rheology. In: 33. IFF-Ferienkurs 2002, Institut Für Festkörperforschung: soft matter, complex materials on mesoscopic scales. Schriften des Forschungszentrum Jülich, Jülich, p C9.2

  • Rosenberg A (1997) Reductionism redux: computing the embryo. Biol Philos 12:445–470

    Article  Google Scholar 

  • Tabery J, Piotrowska M, Darden L (2005) Molecular biology. Stanford encyclopedia of philosophy

  • Thienpont B, Steinbacher J, Zhao H, D’Anna F, Kuchnio A, Ploumakis A, Ghesquière B, Van Dyck L, Boeckx B, Schoonjans L et al (2016) Tumour hypoxia causes DNA hypermethylation by reducing TET activity. Nature 537:63–68

    CAS  Article  PubMed  PubMed Central  Google Scholar 

  • Tononi G, Sporns O, Edelman GM (1999) Measures of degeneracy and redundancy in biological networks. Proc Natl Acad Sci USA 96:3257–3262

    CAS  Article  PubMed  PubMed Central  Google Scholar 

  • Turing A (1936) On computable numbers, with an application to the entscheidungsproblem. Proc Lond Math Soc 42:230–265

    Google Scholar 

  • Wang CJ, Bergmann A, Lin B, Kim K, Levchenko A (2012a) Diverse sensitivity thresholds in dynamic signaling responses by social amoebae. Sci Signal 5:ra17–ra17

    PubMed  Google Scholar 

  • Wang Y, Xu M, Wang Z, Tao M, Zhu J, Wang L, Li R, Berceli SA, Wu R (2012b) How to cluster gene expression dynamics in response to environmental signals. Brief Bioinform 13:162–174

    Article  PubMed  Google Scholar 

Download references


I want to acknowledge the constructive input from two anonymous referees who helped me to refine and set this work on a solid basis. Without this collaborative work with the reviewers it hasn’t been possible to give the final form of this manuscript. I am also grateful to Elena Ramírez for decisive discussions that provided the fundamental elements exposed in this manuscript.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Juan G. Diaz Ochoa.

Appendix: Model Equations

Appendix: Model Equations

Structure: Molecular Network

The implemented equations for a chemotactic response were adapted from Wang et al. (2012a). Adaptive response relies on a ‘negative feedback adaptive’-based response (iFF) which is described by the following equations

$$\begin{gathered} \frac{{{\text{d}}x(t)}}{{{\text{d}}t}}={\text{K}}1{\text{cb}} \cdot y(t) \cdot \frac{{\left( {1 - x(t)} \right)}}{{\left( {\left( {1 - x(t)} \right)+{\text{K2cb}}} \right)}} - {\text{Fb}} \cdot {\text{K2Fbb}} \cdot \frac{{x(t)}}{{x(t)+{\text{K1Fbb}}}} \hfill \\ \frac{{{\text{d}}y(t)}}{{{\text{d}}t}}={\text{K}}2{\text{ac}} \cdot z(t) \cdot \frac{{\left( {1 - x(t)} \right)}}{{\left( {\left( {1 - x(t)} \right)+{\text{K}}2{\text{cb}}} \right)}} - {\text{Fb}} \cdot {\text{K2Fbb}} \cdot \frac{{x(t)}}{{x(t)+{\text{K}}1{\text{Fbb}}}} \hfill \\ \frac{{{\text{d}}z(t)}}{{{\text{d}}t}}= - z\left( t \right)+{\text{Force}} \hfill \\ \frac{{{\text{d}}r(t)}}{{{\text{d}}t}}=x\left( t \right) - r(t) \hfill \\ \end{gathered}$$

We assume that \(r(t)\) is the response function, \(x\left( t \right)\) is an activating pathway, \(y(t)\) is a delayed inhibitory pathway, \({\text{Force}}\) the external stimuli in the network, defined as a stimulus that growths proportional to the time

$${\text{Force}}\,(t)={\text{Ampl}} \times 0.0007 \times t$$

with \({\text{Ampl}}=0.5\); and \(x\left( t \right)\), \(y\left( t \right)\), and \(z\left( t \right)\) the internal concentrations in the network. For this case, we use the following parameters, extracted from the supplementary material from Wang et al. (2012a).

Parameter adaptive response Value
K1cb 0.01
K2cb 0.2
K1Fbb 0.1
K2Fbb 0.01
K1ac 1
K2ac 1
K1bc 2
K2bc 0.1

If organisms have a non-adaptive response then

$$S=~ - 1$$

When the organism switchs to a non-adaptive response, a negative integral feedback (NFB) is implemented. In this case the following network of interactions were defined

$$\begin{gathered} \frac{{{\text{d}}x(t)}}{{{\text{d}}t}}={\text{K1cb}} \cdot y(t) \cdot \frac{{\left( {1 - x(t)} \right)}}{{\left( {\left( {1 - x(t)} \right)+{\text{K2cb}}} \right)}} - {\text{Fb}} \cdot {\text{K2Fbb}} \cdot \frac{{x(t)}}{{x(t)+{\text{K1Fbb}}}} \hfill \\ \frac{{{\text{d}}y(t)}}{{{\text{d}}t}}={\text{K2ac}} \cdot z(t) \cdot \frac{{\left( {1 - y(t)} \right)}}{{\left( {\left( {1 - y(t)} \right)+{\text{K1ac}}} \right)}} - {\text{K2bc}} \cdot \frac{{x(t) \cdot y(t)}}{{y(t)+{\text{K1bc}}}} \hfill \\ \frac{{{\text{d}}z(t)}}{{{\text{d}}t}}= - z\left( t \right)+{\text{Force}} \hfill \\ \frac{{{\text{d}}r(t)}}{{{\text{d}}t}}=z\left( t \right) - r(t) \hfill \\ \end{gathered}$$

with the following parameters, extracted from Chang and Lexchenko (2013).

Parameter non-adaptive response Value
K1cb 0.0007
K2cb 0.0769
K1Fbb 0.1
K2Fbb 0.1473
K1ac 4.7339
K2ac 0.4695
K1bc 0.0069
K2bc 0.0790

If organisms have a non-adaptive response then


Population Dynamics

The set of equations above describe the response of the organism to external conditions depending on the changes in the concentration of the intracellular molecules.

Now, we introduce the population dynamics, which depends on the consumption of resources as well as the balance between the population of consumers \(C(t)\) and predated cells with chemotaxis \(P(t)\) (owning an adaptive/non-adaptive response). We describe this population using the Lotka–Volterra equations:

$$\begin{aligned} {\text{Ingest}}C(t) &= v \cdot P(t) \cdot C(t) \hfill \\ {\text{Growth}}P(t) &= rG \cdot P~(t) \cdot \left( {1 - \frac{{P(t)}}{K}} \right)~ \hfill \\ {\text{Mort}}C(t) &= rM \cdot C(t) \hfill \\ ~\frac{{{\text{d}}P(t)}}{{{\text{d}}t}} &= {\text{Growth}}P(t) - {\text{Ingest}}C(t) \hfill \\ \frac{{{\text{d}}C(t)}}{{{\text{d}}t}} &= {\text{Ingest}}C(t) \cdot AE - {\text{Mort}}C(t), \hfill \\ \end{aligned}$$

where the parameter \(v\) represents the velocity of propagation of the prey, which is a parameter that depends on the kind of response of the organisms that belong to \(P(t)\). In our experiment, we assume that organisms with non-adaptive response have a slow propagation velocity; we consider therefore the following parameters.

Non-adaptive response (sustained response) Adaptive response (fast relaxation)
\(v=0.01\) \(v=0.05\)

For organisms with distorted mechanisms we impose the following conditions

$$r{\text{II}} = \left\{ {\begin{array}{*{20}l} {0.01\quad {\text{if }}r\left( t \right) < 0.3{\text{ and }}r\left( t \right)\,{\text{is non-adaptive}}} \\ {0.05\quad ~{\text{if }}r\left( t \right)> 0.3{\text{ and }}r\left( t \right)\,{\text{is adaptive}}} \\ \end{array} } \right.$$

For all the experiments, we considered the following initial conditions: x = 0.0, y = 0.0, z = 0.0, r = 0.0, P = 1, C = 0. For the population dynamics, we use normalized values.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Diaz Ochoa, J.G. Elastic Multi-scale Mechanisms: Computation and Biological Evolution. J Mol Evol 86, 47–57 (2018).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Systems biology
  • Elastic mechanisms
  • Computational theory
  • Turing machines
  • Evolution
  • Open-ended evolution