Elastic Multi-scale Mechanisms: Computation and Biological Evolution

Abstract

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.

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

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

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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

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

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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

$$S=~1$$

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:

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

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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

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

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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.