The error threshold in evolution

In the preceding section, we discuss the dynamics of evolution of information sets and related phenotypes using the approach of Eigen. The analysis also results in an important constraint to

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the copying fidelity of information sets. It relates to the amount of information that the set can maintain and communicate. There exists an intuitively logical restriction to the error rate if we want to maintain and progress a stretch of information of a given size. If the error rate is too high, the direction of evolution towards superior copies of the information set disappears. Making too many errors causes the message to lose its coherence. The math shows that the limit weakly depends on the superiority function defined as the excess reproduction rate of the copy of information with the largest selection advantage. It is present as a logarithm. However, it shows strong dependence on the error rate in reproduction. The size of the DNA that shows directed evolution in Darwinian selection proves inversely proportional to the error rate.

For ease of discussion, we assume that the selection advantage of the most superior copy exceeds that of the next best set by a factor of about 3 (or more precisely a factor e, being the base of the natural logarithm). In that case, a copying fidelity of .99 allows maintaining a message of maximum 100 symbols. If we add two nines to the copying fidelity, i.e. assume it equal .9999, a message of 10,000 symbols just survives. Thus maintaining the genetic information of E. coli, with a genome of 4. 10⁶ symbols, requires an error rate below 2.5. 10^-7 . The human genome of 3. 10⁹ DNA bases, requires an extremely high copying fidelity.

This observation results in an important feature of the development of dissipative structures that employ a large set of information. Computer experiments on maintaining a given copy of information under a selective pressure result in the following picture. If the length of a message is beyond the copying fidelity limit, the message quickly disintegrates and the information

“melts” away. This is also the case if the target message initially is the only one present. If the copying fidelity exceeds the limit, a very different picture emerges; the mixture evolves quickly to the sequence with the highest competitiveness and its closely related companions in the quasi-species. The optimal quasi-species emerges as dominant even if initially not present at all. The speed of evolution increases with decreasing value of the copying fidelity unto the copying fidelity limit. If the copying fidelity gets lower than the limit, the optimal quasi-species does not emerge or, as said, even melts away if initially dominant.

The process of evolution under a selective pressure shows to be very effective. If we return to the example (Note 7.1) of a six-digit code and a copying fidelity of 5/6, a stringent selective pressure results in a code close the optimal one of 6 sixes after 15-20 attempts on the average. If no selective pressure exists, it takes on the average more than 40,000 attempts before we come close to the optimal sequence. The “learning by doing” behavior clearly pays out.

Analyzing the strategies that evolved in biology leads to the following picture. In nature, the copying fidelity is close to the limit required to maintain the genome given its size. Thus RNA-based viruses (phages), genome size of 1000-10,000 bases, allow an error rate of close to .001 to .0001. The prokaryotic bacteria, typical genome size of 5.10⁶, allow an error rate of the order of .0000001. We can formulate this observation in a different way: The copying fidelity of the information set of an organism determines the genome size it can maintain. The trend in evolution to higher genome sizes requires the invention of increasingly sophisticated copying mechanisms. One can speculate that, when reaching the genome size of the higher animals and the ancestors of humankind, the further expansion of the information set on which competitiveness rests requires another strategy. The capabilities of the brain, allowing exogenic evolution, complement the potential of DNA based evolution. Part of the information storage and communication divorces from the DNA molecules and exogenous information, not bound to the physical DNA, appears on the evolutionary stage. Still later, information storage and communication increasingly divorces from the genetic code as mechanism such as language, writing, science and education, academic and industrial research, take their role as information storage and processing mechanisms complementing the role of DNA and the brain. Firms are examples of information processing organizations and the rules of information transmission subject to error (or experimentation) we discuss in this section, apply to these institutions.

85 7.4. Models of Darwinian systems: The Hypercycle.

This section analyzes more complex systems that replicate and are subject to selection through competition. These more complex systems exhibit features that appear in structures such as organisms, markets and industries.

Evolution on earth results in complex chemical machineries in which a separation develops between information and function. Complex information sets, RNA or DNA molecules, are responsible for storing and communicating information. Protein structures result from the translation of this information into a complex biological molecular machinery. These functional structures compete for scarce resources and the feedback leads to evolution of the information sets. This leads through a process of mutation and selection to increasingly competitive structures. As this discussion reveals, the information set and the functional structure interact to form a closed cycle.

We want to reemphasize the reasons for such separation of information and function. These are manifold but one of the most basic ones is that the coding and reproduction and competition functions pose very different requirements. It is unlikely that one molecular species optimally combines these requirements. Alternatively phrased, separation of these functionalities leads to new sources of competitive advantage and it inevitably appears in the competition for scarce resources.

Of course, separating the functions in different entities leads to coordination problems, e.g. their production and interaction needs to be fine-tuned. In biology, the separation of functions proves necessary when the structures become very complex and need large chunks of information for their instructed creation. In functional biological systems, this leads to types of organization that we can model by “Hypercycles”. We will discuss some of the interesting properties of these systems. The work of Eigen presents a detailed discussion.

Fig. 7.2 provides a schematic representation of a simple Hypercycle. The Hypercycle consists of a number of information setsI_i. These entities carry only a part of the total information set of the Hypercycle. The size of these information sets is below the error threshold resulting from the copying fidelity limit. This guarantees their conservation against error copies, of course with the exception of copies that increase the competiveness of the structure. The information sets have a self-reproduction capability indicated by the open circles in Fig. 7.2. The functional structure resulting from the information set directly preceding an information set catalyzes this

Fig. 7.2. A schematic representation of a Hypercycle.

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reproduction process. An important feature results from closure of the Hypercycle, i.e. the product of the last information set in the system catalyses the synthesis of the first one. If the cycle is not closed the structures in the Hypercycle do not cooperate rather they compete and concerted action fails to result.

Hypercycles show at least the following properties:

1. The overall Hypercycle exhibits autocatalytic growth. The elements of the Hypercycle grow in a concerted fashion if sources of free energy, or economic value in our extended concept, in the environment allow this. This is one of the requirements for sustained evolution. Different Hypercycles engage in competition for scarce resources.

They exhibit Darwinian competition and selection.

2. Hypercycles show non-linear kinetics leading to strong selection behavior. A Hypercycle, once established, resists substitution by other emerging cycles. Their substitution requires a substantial selection advantage of the challenger.

3. Its strong selection behavior allows it to evolve quickly and to exploit small differences in selective advantage. It is very effective in improvement through learning by doing once established as a closed loop.

4. The cyclic arrangement allows the system to use more information than consistent with stability in the light of the fidelity of the copying mechanism used (see Section 7.3). The hypercyclic cooperation allows escaping the fidelity limit.

5. The system selects against so-called parasitic branches, i.e. branches that attach to the Hypercycle and replicate with it, but do not contribute to its competitiveness. In addition, parts of the cycle that cease to be functional and do no longer contribute to the competitiveness of the overall cycle, automatically disappear if this results in increased competitiveness.

6. There is an advantage for the system to escape into a closed compartment. In this way, it can evolve and use pieces of information to which its competitors have no access. This also results in protecting itself against pieces of information that evolve elsewhere and pollute the cycle. We recognize a known feature of organisms that have cell walls and membranes. This also is a feature of human organizations, such as firms. Firms generally have defined well-policed interfaces with the environment. In such organizations, restrictions exist to the exchange of materials, resources and information with the environment.

7. Individual Hypercycles do not cooperate but compete. They may link, however, resulting in larger functional entities in which two or several Hypercycles cooperate. Their cooperative rather than competitive behavior critically depends upon the strength of coupling between the Hypercycles. This mimics processes of fusion and alliance in industries. It also resembles critical stages in biological evolution that depend on the merger of separate organisms into one functional biological entity.

Hypercycles, although rather complex from the mathematical point of view, present a highly schematized and simplified picture of the reality of markets in which industries compete. Still, these mathematical abstractions show a rich variety of interesting features.