Self-organization and information in biosystems: a case study

Abstract

Eigen’s original molecular evolution equations are extended in two ways. (1) By an additional nonlinear autocatalytic term leading to new stability features, their dependence on the relative size of fitness parameters and on initial conditions is discussed in detail. (2) By adding noise terms that represent the spontaneous generation of molecules by mutations of substrate molecules, these terms are taken care of by both Langevin and Fokker–Planck equations. The steady-state solution of the latter provides us with a potential landscape giving a bird’s eye view on all stable states (attractors). Two different types of evolutionary processes are suggested: (a) in a fixed attractor landscape and (b) caused by a changed landscape caused by changed fitness parameters. This may be related to Gould’s concept of punctuated equilibria. External signals in the form of additional molecules may generate a new initial state within a specific basin of attraction. The corresponding attractor is then reached by self-organization. This approach allows me to define pragmatic information as signals causing a specific reaction of the receiver and to use equations equivalent to (1) as model of (human) pattern recognition as substantiated by the synergetic computer.

Appendix

Derivation of Eq. (1) and equivalence to Eigen’s selection equation

Equation (1) here denoted as (18)

$$\frac{{{\text{d}}n_{k} }}{{{\text{d}}t}} = \lambda_{k} n_{k} + Cn_{k}^{2} - Dn_{k} \sum\limits_{l = 1}^{N} {n_{l} }$$

(18)

is derived from the kinetics of replication. I start from equations similar to, but not identical with those used for replication in a flow reactor. In particular, we are dealing with N species and two substrates 1 and 2 and I use for the number densities or concentrations: n _k for species k, A for substrate 1, and B for substrate 2. The generation rate of species k is of the form:

$$\frac{{{\text{d}}n_{k} }}{{{\text{d}}t}} = \mathop {\mu_{k} An_{k} }\limits_{\text{I}} + \mathop {\nu Bn_{k}^{2} }\limits_{\text{II}} - \mathop {\kappa_{k} n_{k} }\limits_{\text{III}} ,$$

(19)

where the three terms describe autocatalytic production (I), autocatalytic production of second order (II), and the k-dependent decay (III). The generation rates of the two substrates are

$$\frac{{{\text{d}}A}}{{{\text{d}}t}} = \gamma \left( {A_{0} - A} \right) - \sum\limits_{l = 1}^{N} {\mu_{l} } An_{l} \;{\text{and}}$$

(20)

$$\frac{{{\text{d}}B}}{{{\text{d}}t}} = \varGamma \left( {B_{0} - B} \right) - \sum\limits_{l = 1}^{N} \nu Bn_{l} .$$

(21)

I assume $\gamma$ and $\varGamma$ being large, so that in adiabatic approximation

$$\frac{{{\text{d}}A}}{{{\text{d}}t}} = 0\;\;{\text{and}}\;\;\frac{{{\text{d}}B}}{{{\text{d}}t}} = 0$$

(22)

(20) and (22) lead to

$$A \approx A_{0} \left( {1 - \frac{1}{\gamma }\sum {\mu_{l} n_{l} } } \right),$$

(23)

where ${{\mu_{l} } \mathord{\left/ {\vphantom {{\mu_{l} } \gamma }} \right. \kern-0pt} \gamma }$ is assumed to be finite. Equations (4) and (5) lead to

$$B \approx B_{0} ,$$

(24)

under the assumption ${\nu \mathord{\left/ {\vphantom {\nu \varGamma }} \right. \kern-0pt} \varGamma }$ being negligibly small. Next, I assume $\mu_{k} = \mu$ for all k. Inserting Eqs. (23) and (24) into (19) leads to Eq. (18) with the obvious abbreviations:

$$\mu A_{0} - 2\kappa_{k} = \lambda_{k} ,\;\;D = {{\mu^{2} A_{0} } \mathord{\left/ {\vphantom {{\mu^{2} A_{0} } \gamma }} \right. \kern-0pt} \gamma },\;\;C = B_{0} \nu .$$

(25)

To retrieve Eigen’s original equation (Eigen 1969), I put C = 0 and use the substitution $n_{k} \to \lambda_{k} n_{k}$ for all k. This yields

$$\lambda_{k} \frac{{{\text{d}}n_{k} }}{{{\text{d}}t}} = \lambda_{k}^{2} n_{k} - D\lambda_{k} n_{k} \sum {\lambda_{l} n_{l} }$$

(26)

or, after division by $\lambda_{k}$

$$\frac{{{\text{d}}n_{k} }}{{{\text{d}}t}} = \lambda_{k} n_{k} - Dn_{k} \sum {\lambda_{l} n_{l} } .$$

(27)

Finally, I obtain in relative concentrations $x_{k} = {{n_{k} } \big/{\sum\nolimits_{l = 1}^{N} {n_{l} } }}$ with D = 1 (Schuster 2011):

$$\frac{{{\text{d}}x_{k} }}{{{\text{d}}t}} = \lambda_{k} x_{k} - x_{k} \sum\limits_{l = 1}^{N} {\lambda_{l} x_{l} } .$$

(28)