# Increase in Complexity and Information through Molecular Evolution

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## Abstract

**:**

## 1. Introduction

## 2. A Minimal Model for Competition, Cooperation, and Mutation

## 3. Competition and Cooperation

## 4. Stochastic Kinetics in the Competition-Cooperation System

## 5. Competition, Mutation and Quasispecies

- (i)
- A single-peak fitness landscape is applied that assumes equal fitness for all mutants and a higher fitness for the master sequence (Section 6).
- (ii)
- A uniform mutation rate per site and replication event, p, is assumed. In other words the frequency of mutation is assumed to be independent of the nature and the position of the mutated nucleotide. The mutation matrix is largely simplified by the uniform error rate assumption$${Q}_{ij}(p)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}Q\phantom{\rule{0.166667em}{0ex}}{\epsilon}^{{d}_{\mathrm{H}}({\mathrm{X}}_{i},{\mathrm{X}}_{j})}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{with}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}Q={(1-p)}^{\nu}={Q}_{ii}\phantom{\rule{0.166667em}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}i=1,\cdots ,n\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{and}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\epsilon =\frac{p}{1-p}\phantom{\rule{0.166667em}{0ex}}.$$
- (iii)
- Mutational backflow in the kinetic differential Equation (10) is neglected. We rewrite Equation (10b) for $j=m$ and partition in two contributions coming from correct copying of the template ${\mathrm{X}}_{m}$ and from incorrect copying of all other ${\mathrm{X}}_{i}$ with $i\ne m$, and neglect the second term:$$\frac{\mathrm{d}{x}_{m}}{\mathrm{dt}}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}a\left({Q}_{mm}\phantom{\rule{0.166667em}{0ex}}{f}_{m}\phantom{\rule{0.166667em}{0ex}}{x}_{m}\phantom{\rule{0.166667em}{0ex}}+\sum _{i=1,i\ne m}^{n}{Q}_{mi}\phantom{\rule{0.166667em}{0ex}}{f}_{i}\phantom{\rule{0.166667em}{0ex}}{x}_{i}\right)\phantom{\rule{4pt}{0ex}}-\phantom{\rule{4pt}{0ex}}r\phantom{\rule{0.166667em}{0ex}}{x}_{m}\phantom{\rule{4pt}{0ex}}\approx \phantom{\rule{4pt}{0ex}}{x}_{m}({Q}_{mm}{f}_{m}a\phantom{\rule{0.166667em}{0ex}}-r)\phantom{\rule{4pt}{0ex}}.$$

## 6. Sequence Space, Fitness Landscape, and Population Dynamics

## 7. Cooperation and Mutation

## 8. Information, Transitions and the Competition-Cooperation-Mutation Model

## 9. Discussion

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**

**Three processes in evolution.**The sketch presents the three basic processes in biological evolution that are considered in the minimal model presented here and their interplay: (i) selection operating on fitness differences; (ii) mutualistic catalysis operating on replication and leading to cooperation between competitors; and (iii) variation introducing changes into phenotypic properties. On the three Cartesian coordinate axes the pure processes, selection, cooperation and variation are plotted. Typical representatives of mathematical models for the pure processes are (i) the selection equation for asexual reproduction (see, for example, [33]), the hypercycle equation [28], and the random drift equation of neutral evolution [34]. They give rise to optimization, mutualistic coupling, and neutral evolution. In the minimal model the three coordinate axes are quantitatively scaled by parameters. Selection is determined by the differences in the fitness parameters of pairs of species ${\mathrm{X}}_{i}$ and ${\mathrm{X}}_{j}$, $\Delta {f}_{ij}={f}_{i}-{f}_{j}$, a measure for cooperation is the catalytic parameter ${h}_{ij}$, which measures the catalytic enhancement of the template-induced production of ${\mathrm{X}}_{i}$ by the molecular species ${\mathrm{X}}_{j}$, and the frequency of mutation that is determined by a mean mutation rate parameter per site and generation denoted by p. Combination of two pure processes gives rise to different evolutionary phenomena: Combination of (i) and (ii) enables the occurrence of transitions from competition to cooperation in the sense of major transitions. In Section 2 we show the existence of a threshold in the required resources. Cooperation can occur only if the resources exceed this threshold value. The combination of (i) and (iii) yields Darwinian optimization based on mutation and selection [30,31]. Populations converge to a unique stationary mutant distribution called quasispecies provided the replication is sufficiently accurate: $p<{p}_{\mathrm{cr}}\approx \Delta f\phantom{\rule{-0.166667em}{0ex}}/\phantom{\rule{-0.166667em}{0ex}}({\overline{f}}_{-m}\xb7\nu )$ where $\Delta f={f}_{m}-{\overline{f}}_{-m}$ is the fitness difference between the fittest genotype and the average fitness of all except the best where the genotype giving rise to the fittest phenotype is denoted by ${\mathrm{X}}_{m}$ that is denoted by ${\overline{f}}_{-m}$. The chains length of the replicated polynucleotides is given by ν. The third combination involving (ii) and (iii) is less well known and involves the role of mutation in replicating collectives with mutualistic interactions. One unexpected but straightforward to interpret phenomenon is the role of mutation in stochastic processes. Without mutation a molecular species goes irreversibly extinct when the corresponding variable becomes zero. Mutation, however, may bring the species back (Section 7). Real systems, of course, reside in the positive orthant $\Delta f>0$, $h>0$, and $p>0$. Idealized cases are situated in the coordinate planes: $\mathcal{A}$ with $p=0$, $\mathcal{B}$ with $h=0$, and $\mathcal{C}$ with $\Delta f=0$.

**Figure 2.**

**A mechanism for correct replication and mutation as parallel reaction channels.**Mutation is represented as replication error in the sense that the nucleotide sequence of the copy differs from that of the original. The initiation of the process is sketched here as the attachment of building blocks (A) and template (${\mathrm{X}}_{i}$) to the replicating enzyme (blue). The rate parameter, ${w}_{ji}={Q}_{ji}\xb7{f}_{i}$, contains two factors: (i) the frequency ${Q}_{ji}$ at which the mutant ${\mathrm{X}}_{j}$ is obtained as an error copy of the template ${\mathrm{X}}_{i}$; and (ii) the rate parameter ${f}_{i}$ for the replication of ${\mathrm{X}}_{i}$ being a measure for fitness, which is a product of the rate constant of reaction (1b) and the concentration of the resources, ${f}_{i}={k}_{i}a$. Since a copy has to be error-free or incorrect, we have a conservation relation ${\sum}_{j=1}^{n}{Q}_{ji}=1$. For many purposes the elements of the mutation matrix are approximated by the assumption of a uniform error rate p as expressed in Equation (3). The replication process is completed by the release of template and copy from the enzyme. The polymerase chain reaction (PCR) with a DNA polymerase of the bacterium Thermus aquaticus (Taq) may serve as an example of an in vitro copying reaction [43].

**Figure 3.**

**Sequence of phases in a stochastic trajectory.**A stochastic trajectory simulating competition and cooperation of two species in a flow reactor is shown in the plot above. The stochastic process is assumed to start with an empty reactor except seeds for the two autocatalysts and can be partitioned into four phases: (I) fast raise in the concentration of A; (II) a random phase that decides into which final state—${S}_{0}$, ${S}_{1}^{(1)}$, ${S}_{1}^{(2)}$ or ${S}_{2}$—the trajectory progresses; (III) the approach towards the final state; and (IV) fluctuations around the values of the quasi-stationary state. Parameter values: ${k}_{1}=0.099$ [M${}^{-1}$t${}^{-1}$], ${k}_{2}=0.101$ [M${}^{-1}$t${}^{-1}$], ${l}_{1}=0.0050$ [M${}^{-2}$t${}^{-1}$], ${l}_{2}=0.0045$ [M${}^{-2}$t${}^{-1}$], ${a}_{0}=200$, $r=4.0$ [V t${}^{-1}$], pseudorandom number generator: ExtendedCA, Mathematica, seeds $s=631$. Initial conditions: $A(0)=0$, ${X}_{1}(0)={X}_{2}(0)=1$. Color code: $\mathrm{A}(t)$ black, ${\mathrm{X}}_{1}(t)$ red, and ${\mathrm{X}}_{2}(t)$ green.

**Figure 4.**

**Oscillating stochastic trajectories of the competition-cooperation system with**$\mathit{n}\ge \mathbf{4}$. The figures show single stochastic trajectories computed by means of the Gillespie algorithm [58] for systems with $n=4$ (

**top**) and $n=5$ (

**bottom**) in the flow reactor. Fluctuating oscillations increase in amplitude until one subspecies is wiped out and then the entire system in driven into extinction. Choice of parameters, upper plot: ${a}_{0}=400$, $r=0.5\phantom{\rule{0.166667em}{0ex}}$ [V t${}^{-1}$], ${k}_{1}=0.090,\phantom{\rule{0.166667em}{0ex}}{k}_{2}=0.097,\phantom{\rule{0.166667em}{0ex}}{k}_{3}=0.103,\phantom{\rule{0.166667em}{0ex}}$ ${k}_{4}=0.110\phantom{\rule{0.166667em}{0ex}}$ [M${}^{-1}$t${}^{-1}$], ${l}_{1}={l}_{2}={l}_{3}={l}_{4}=0.01\phantom{\rule{0.166667em}{0ex}}$ [M${}^{-2}$t${}^{-1}$]; lower plot: ${a}_{0}=4000$, $r=0.5\phantom{\rule{0.166667em}{0ex}}$ [V t${}^{-1}$], ${k}_{1}=0.0090,\phantom{\rule{0.166667em}{0ex}}{k}_{2}=0.0095,\phantom{\rule{0.166667em}{0ex}}{k}_{3}=0.0100,\phantom{\rule{0.166667em}{0ex}}{k}_{4}=0.105,\phantom{\rule{0.166667em}{0ex}}{k}_{5}=0.0110\phantom{\rule{0.166667em}{0ex}}$ [M${}^{-1}$t${}^{-1}$], ${l}_{1}={l}_{2}={l}_{3}={l}_{4}={l}_{5}=0.001\phantom{\rule{0.166667em}{0ex}}$ [M${}^{-2}$t${}^{-1}$]. Initial conditions, upper plot: $A(0)=0$, ${X}_{1}(0)={X}_{2}(0)={X}_{3}(0)={X}_{4}(0)=5$ and lower plot: $A(0)=0$, ${X}_{1}(0)={X}_{2}(0)={X}_{3}(0)={X}_{4}(0)={X}_{5}(0)=5$. Color code: A black, ${X}_{1}$ red, ${X}_{2}$ green, ${X}_{3}$ yellow, ${X}_{4}$ blue, and ${X}_{5}$ cyan.

**Figure 5.**

**A sketch of quasispecies evolution.**The quasispecies ${\overline{\mathsf{{\rm Y}}}}_{1}$ around ${\mathrm{X}}_{m}^{(1)}$ covers only a tiny part of sequence space. It is replaced by ${\overline{\mathsf{{\rm Y}}}}_{2}$ when a new and more efficient sequence ${\mathrm{X}}_{m}^{(2)}$ is formed by mutation that builds up its mutant cloud, ${\overline{\mathsf{{\rm Y}}}}_{2}$ with ${\mathrm{X}}_{m}^{(2)}$ as master sequence. Later on ${\overline{\mathsf{{\rm Y}}}}_{2}$ in turn may be replaced by ${\overline{\mathsf{{\rm Y}}}}_{3}$ with ${\mathrm{X}}_{m}^{(3)}$, etc. The selective value of the master sequence in a quasi-stationary population, ${W}_{mm}={Q}_{mm}{f}_{m}\approx {(1-p)}^{\nu}{f}_{m}$, is plotted as a function of time along a typical evolutionary trajectory [72,73]. The trajectory approaches stepwise an optimal value ${W}_{\mathrm{opt}}$. Since sequence space is inexhaustible the optimization process goes on until the population has come close to an optimum under the current environmental conditions. Because of the enormous size of sequence space, however, the evolutionary process is unlikely to reach the global optimum. The pink zones indicate transitions from one local quasispecies to another. The light blue zones indicate the loss in fitness caused by the build-up of a mutant cloud. The selective value ${W}_{mm}^{(i)}$ is used here as an approximation for the eigenvalue ${\lambda}_{0}^{(i)}$: ${W}_{mm}^{(i)}\approx {\lambda}_{0}^{(i)}$.

**Figure 6.**

**A sketch of evolution on a rugged landscape.**Realistic fitness landscapes exhibit three features: (i) high dimensionality; (ii) ruggedness; and (iii) a high degree of neutrality. The figure sketches evolutionary paths in a rugged landscape, which are understood as adaptive, i.e., non-descending walks on a fitness landscape. Populations at replication-mutation equilibrium cover a certain (very small) part of sequence space and can bridge narrow clefts. Wider valleys are unsurmountable obstacles for one-dimensional adaptive walks. The enlarged insert shows how such a valley may be circumvent along a neutral path in another dimension. A sufficiently high degree of neutrality is a necessary prerequisite for efficient adaptive walks on rugged landscapes.

**Figure 7.**

**Oscillating trajectories of the cooperation-mutation system leading to extinction.**The figures show single stochastic trajectories computed by means of the Gillespie algorithm [58] for mechanism (1) in the flow reactor with ${k}_{1}={k}_{2}={k}_{3}={k}_{4}=({k}_{5}=)\phantom{\rule{0.166667em}{0ex}}0$. The number of subspecies is four in the upper plot and five in the lower plot ($n=4,5$). The somewhat irregular oscillations grow in both cases until one species dies out and then the hypercycle is extinguished subspecies by subspecies until only compound A remains. In the four-membered system the stochastic oscillations show a kind of beat. The mutation matrix $\mathrm{Q}$ for the four membered system is taken from the uniform error rate model (3) for chain length $\nu =2$. The mutation matrix $\mathrm{Q}$ in the case $n=5$ is built on a pentagram and is symmetric, ${Q}_{ij}={Q}_{ji}$ and has the elements ${Q}_{11}={Q}_{22}=\cdots ={Q}_{55}={(1-p)}^{2}$, ${Q}_{12}={Q}_{23}={Q}_{34}={Q}_{45}={Q}_{51}=p(1-p)$ and ${Q}_{13}={Q}_{24}={Q}_{35}={Q}_{25}={Q}_{14}={p}^{2}$. In both cases the subspecies are equivalent with respect to mutations. Choice of parameters, upper plot: ${a}_{0}=200$, $r=0.5\phantom{\rule{0.166667em}{0ex}}$ [V${}^{-1}$t${}^{-1}$], ${l}_{1}={l}_{2}={l}_{3}={l}_{4}=0.1\phantom{\rule{0.166667em}{0ex}}$ [M${}^{-2}$t${}^{-1}$] and $p=0.001$; lower plot: ${a}_{0}=400$, $r=0.5\phantom{\rule{0.166667em}{0ex}}$ [V${}^{-1}$t${}^{-1}$], ${l}_{1}={l}_{2}={l}_{3}={l}_{4}={l}_{5}=0.01\phantom{\rule{0.166667em}{0ex}}$ [M${}^{-2}$t${}^{-1}$] and $p=0.002$. Pseudorandom number generator: Extended CA (Mathematica 10), seed: $s=089$ (upper plot) and $s=919$ (lower plot). Initial conditions: $A(0)=0$, ${X}_{1}(0)={X}_{2}(0)={X}_{3}(0)={X}_{4}(0)=4$ (upper plot) and $A(0)=0$, ${X}_{1}(0)={X}_{2}(0)={X}_{3}(0)={X}_{4}(0)={X}_{5}(0)=5$ (lower plot). Color code: A black, ${X}_{1}$ red, ${X}_{2}$ yellow, ${X}_{3}$ green, ${X}_{4}$ blue, (and ${X}_{5}$ cyan).

**Figure 8.**

**Mutations preventing extinction.**The figures show an image section of a single stochastic trajectory computed by means of the Gillespie algorithm [58] for mechanism (1) with ${k}_{1}={k}_{2}={k}_{3}={k}_{4}=0$. The lower plot is an enlargement of the upper plot. The subspecies ${\mathrm{X}}_{2}$ (yellow) dies out at $t=1648.5$. Consequently the copy numbers of ${\mathrm{X}}_{1}$ (red) go down and for some while ${X}_{1}(t)$ fluctuates between 1 and 2 copies. Eventually ${\mathrm{X}}_{2}$ (yellow) is created by mutation at $t=1657.2$ and the copy number ${X}_{2}(t)$ increases rapidly because ${\mathrm{X}}_{3}$ is present in large numbers $({X}_{3}(t)$, green). The episode, however, is not over yet since ${\mathrm{X}}_{1}$ dies out at $t=1659.6$ and consequently ${\mathrm{X}}_{4}$ (blue) goes down in numbers and almost dies out. The subspecies ${\mathrm{X}}_{1}$ comes back by mutation at $t=1661.1$ but dies out again by accident immediately afterwards at $t=1661.2$. Finally ${\mathrm{X}}_{1}$ comes back again at $t=1662.0$ by another mutation and ${X}_{1}(t)$ increases fast because ${X}_{2}(t)$ is already high, and with the consecutive increase of ${X}_{1}(t)$ followed by ${X}_{4}(t)$ the stochastic oscillations are restored. Choice of parameters: ${a}_{0}=200$, $r=0.5\phantom{\rule{0.166667em}{0ex}}$ [V${}^{-1}$t${}^{-1}$], ${l}_{1}={l}_{2}={l}_{3}={l}_{4}=0.1\phantom{\rule{0.166667em}{0ex}}$ [M${}^{-2}$t${}^{-1}$], $p=0.01$. Pseudorandom number generator: Extended CA (Mathematica 10), seed: $s=521$. Initial conditions: $A(0)=0$, ${X}_{1}(0)={X}_{2}(0)={X}_{3}(0)={X}_{4}(0)=2$. Color code: A black, ${X}_{1}$ red, ${X}_{2}$ yellow, ${X}_{3}$ green, and ${X}_{4}$ blue.

**Figure 9.**

**The information entropy of populations on rugged model landscapes.**The information entropy H is plotted as a function of the mutation rate p for six (upper plot) and seven (lower plot) landscapes with rugged scatter of different amplitude d. The upper plot was calculated with $s=919$ and shows the information entropy $H(p)=-{\sum}_{i=1}^{{2}^{\nu}}{\overline{\mathsf{\xi}}}_{i}(p){log}_{2}{\overline{\mathsf{\xi}}}_{i}(p)$ as a function of the mutation rate p for a typical strong quasispecies, where the master sequence is the same in the entire range of mutation rates $0\le p<{p}_{\mathrm{cr}}$. For the lower plot seeds $s=637$ were chosen and in this case we observe four different master sequences: ${\mathrm{X}}_{0}$ for $0\le p<0.00065$, ${\mathrm{X}}_{1003}$ for $0.00065<p<0.00177$, ${\mathrm{X}}_{923}$ for $0.00177<p<0.00276$, and ${\mathrm{X}}_{247}$ for $0.00276<p<{p}_{\mathrm{cr}}$. The four regions are separated by transitions between quasispecies indicated by dashed red lines. The individual curves reach the maximal value, ${H}_{max}=\nu $ [bits] for binary and $2\phantom{\rule{0.166667em}{0ex}}\nu $ [bits] for 4 letter sequences, near the error threshold. Parameter choice: $\nu =10$, ${f}_{0}=1.0$ and ${f}_{1}=1.1$ [M${}^{-1}$t${}^{-1}$], color code: $d=0$ black, $d=0.5$ turquoise, $d=0.7$ blue, $d=0.8$ yellow, $d=0.9$ green, $d=0.95$ violet, and $d=1.0$ red. Binary sequences are used and they are characterized by their decadic equivalents: “0” ≡ 0000000000, “1” ≡0000000001, “2” ≡ 0000000010, ⋯ , “1023” ≡ 111111111. Pseudorandom number generator: ExtendedCA, Mathematica, Wolfram.

**Figure 10.**

**The paradigm of structural biology.**RNA sequences are considered as information carriers or genotypes. Evaluation of genotypes in order to obtain the semantic information is performed in two steps: RNA sequences (red spheres), are folded into secondary structures (blue spheres corresponding to individual shapes in short-hand notation: “·” stands for an unpaired nucleotide, parentheses “( )” stand for a base pairs), which are considered as phenotypes. Parameter values entering population dynamics result from evaluation of structures with respect to fitness in reproduction. Fitness values are commonly non-negative. The two evaluations may be seen as consecutive discrete mappings from sequence space $\mathcal{Q}$ into shape space $\mathcal{S}$ and from shape space $\mathcal{S}$ into non-negative real numbers ${\mathbb{R}}_{+}$. Both mappings are context dependent, in particular structures and functions depend on environmental conditions. Redrawn from [32] (p. 71).

**Table 1.**

**Asymptotically stable stationary states of Equation (4) with**$\mathit{n}\mathbf{=}\mathbf{2}$ [9]

**.**The three stationary states are ordered with respect to increasing $$