1. Evolution of Replicator Systems Bratus A. S. 1,2 , Yakushkina T
Evolution of Replicator Systems Bratus A.S.1,2 , Yakushkina T., Drozhzhin S. (1) Lomonosov Moscow State University, (2) Russian University of Transport
MOSCOW 06 November 2018

2. Darwinian Evolution
Variation. Among individuals within any population,
there is variation in morphology, physiology, and behavior.
Heredity. Offspring resemble their parents more than they resemble unrelated individuals.
Selection. Some forms are more successful at surviving and reproducing than other forms in a given environment.
 Charles Darwin (1809–1882)

3. Basic evolutionary concepts: replication and replicator
Replication - is the process of making a replica (a copy) of something.
DNA replication is the process by which a double-stranded DNA molecule is copied to produce two identical DNA molecules
Which objects can we call replicators?
What are the as units of selection?
How can we formalize the influence of the Darwinian triad?

4. Different approaches to define replicator
self-replicating object with hereditary variation1;
any entity that causes certain environments to copy it2;
any entity in the universe of which copies are made. A germ-line replicator, as distinct from a deadend replicator, is the potential ancestor of an indefinitely long line of descendant replicators.
1) A.Markov, Evolution --- АСТ, Corpus, 2014 (in Russian).
2) Deutsch D. The fabric of reality. --- New York: Allen Lane, 1997.
3) Dawkins R. The selfish gene. --- New York: Oxford University Press, 1976.

5. General model of Darwinian evolution
For a correct mathematical formalization of the evolutionary process, one has to define:
a replicator as information unit
a low that defines the replication frequency
mutation frequencies and other crucial parameters

6. Population dynamics: Fisher’s model
Consider a population with a finite number of different species,
defined by 𝑁 𝑖 𝑡 𝑖=1 𝑛 :
𝑟 𝑖 – the fitness of the i-th species
𝑓(𝑡) – the mean fitness
𝑑 𝑁 𝑖 (𝑡) 𝑑𝑡 = 𝑟 𝑖 𝑁 𝑖 𝑡 , 𝑁 𝑖 0 = 𝑁 𝑖 0
𝑢 𝑖 𝑡 = 𝑁 𝑖 (𝑡) 𝑘=1 𝑛 𝑁 𝑘 (𝑡) , 𝑖=1 𝑛 𝑢 𝑖 𝑡 =1
𝑑 𝑢 𝑖 𝑑𝑡 = 𝑢 𝑖 𝑟 𝑖 −𝑓 𝑡 , 𝑓 𝑡 = 𝑖=1 𝑛 𝑟 𝑖 𝑢 𝑖 (𝑡)
Sir Ronald Aylmer Fisher  (1890 -1962)

7. Fisher’s fundamental theorem of natural selection
The rate of increase in (mean) fitness of any organism at any time is equal to its genetic variance in fitness at that time.
𝑑𝑓(𝑡) 𝑑𝑡 = 𝑖=1 𝑛 𝑟 𝑖 𝑢 𝑖 = 𝑖=1 𝑛 𝑟 𝑖 2 𝑢 𝑖 − 𝑖=1 𝑛 𝑟 𝑖 𝑢 𝑖 2 ≥0
𝑖=1 𝑛 𝑢 𝑖 (𝑡) =1, 𝑢 1 ≥0

8. Fitness landscape
Poelwijk, Frank J., et al. "Empirical fitness landscapes reveal accessible evolutionary paths." Nature  (2007): 383.
Pesce, Diego, Niles Lehman, and J. Arjan GM de Visser. Phil. Trans. R. Soc. B (2016):

9. Fitness landscape
Adaptive fitness landscape for replicator systems: to maximize or not to maximize. Bratus A.S., Semenov Y.S., Novozhilov A.S. Math. Model. Nat. Phenom. (2018, in press)

10. Fitness landscape
A schematic example of the hypercycle system’s landscape. (a) The unique internal equilibrium coincides with the global fitness maximum and globally asymptotically stable. (b) The internal equilibrium is no longer a global maximum, the solutions are attracted to the limit cycle.
Adaptive fitness landscape for replicator systems: to maximize or not to maximize. Bratus A.S., Semenov Y.S., Novozhilov A.S. Math. Model. Nat. Phenom. (2018, in press)

11. Particular Cases of the Replicator Equation

12. Replicator Equation

13. Principle of Competitive Exclusion

14. Hypercycle Replication

15. The quasi-species theory

16. Error threshold phenomena

20. Proposed approach:
The specific time of the evolutionary adaptation of the system parameters (evolutionary time) is much slower than the time of internal evolutionary process
This assumption leads to the fact that evolutionary changes of the system parameters happen in a steady-state of the corresponding dynamical system
Maximization of the mean fitness in a steady-state is performed by varying the system parameters depending on evolutionary time under the condition of bounded values of these parameters
In other words, we suppose that Fishers’ fundamental theorem of natural selection is valid in evolutionary time scale for steady-states

21. Problem statement Replicator system: 𝑢 𝑖 = 𝑢 𝑖 𝑨𝒖 𝑖 − 𝑚 𝒖 , 𝑖 = 1,… 𝑛.
𝑢 𝑖 = 𝑢 𝑖 𝑨𝒖 𝑖 − 𝑚 𝒖 , 𝑖 = 1,… 𝑛.
where 𝑨= 𝑎 𝑖𝑗 ,
𝒖 𝑡 ∈ 𝑆 𝑛 = 𝒖 ∈ ℝ 𝑛 , 𝒖≥0, 𝑖=1 𝑛 𝑢 𝑖 =1
𝑚 𝑢 = 𝑖=1 𝑛 𝑎 𝑖𝑗 𝑢 𝑖 𝑢 𝑗 =(𝑨𝒖, 𝒖)
Steady-state:
𝑨𝒖= 𝑚 𝒖 ⋅𝟏, 𝟏= 1,…, 1 , 𝒖∈ 𝑆 𝑛
For permanent systems:
lim 𝑡→∞ 1 𝑡 0 𝑡 𝑚 𝑠 𝑑𝑠 = 𝑚 (𝒖)

22. Mathematical programming problem
In evolutionary time scale 𝜏:
𝒖 𝜏 ∈ 𝑆 𝑛 ,𝜏≥0
𝑨 𝜏 𝑢 𝜏 = 𝑚 𝜏 ⋅𝟏
𝑖,𝑗 =1 𝑛 𝑎 𝑖𝑗 ≤ 𝑄 2 , ∀𝜏≥0
where 𝑄 is a constant parameter to control the limit of the resources.
Maximization process:
𝑚 𝜏 = 𝑨 𝜏 𝑢 𝜏 , 𝑢 𝜏 →𝑚𝑎𝑥
There exist a unique maximum of the mean fitness m τ .

23. Fitness variation and linear programming problem
The evolutionary time change 𝜏→𝜏+ℎ
Maximization process:
𝛿 𝑚 =𝑚(𝜏) 𝑨 −𝟏 𝜏 𝑨 1 (𝜏) 𝑢 𝜏 , 𝟏 →𝑚𝑎𝑥
where
𝟏=(1,…,1)
𝑖,𝑗=1 𝑛 𝑎 𝑖𝑗 𝜏 𝑎 𝑖𝑗 ′(𝜏) ≤0
𝑎 𝑖𝑗 ′ 𝜏 ≤𝜀

24. The result of the iteration algorithm of evolutionary process for the hypercycle system
𝑢 𝑖 = 𝑢 𝑖 𝑢 𝑖−1 − 𝑚 𝒖 , 𝑖=1,…,𝑛
𝑢 0 = 𝑢 𝑛 , 𝒖∈ 𝑆 𝑛
𝑚 𝒖 = 𝑖=1 𝑛 𝑢 𝑖 𝑢 𝑖−1
For the replicator system, the matrix takes the form:
𝑨= ⋯ ⋮ … ⋱ ⋮ ⋯ 1 0

25. The result of the iteration algorithm of evolutionary process: numerical calculations
Bratus A., Drozhzhin S. Yakushkina T.,  On the Evolution of Hypercycles // Mathematical Biosciences,2018 (in press). 

26. Adaptation of the fitness landscape in the hypercycle model

27. The result of the iteration algorithm of evolutionary process: system with a parasite
𝑢 𝑖 = 𝑢 𝑖 𝑢 𝑖−1 − 𝑚 𝒖 , 𝑖=1,…,5
𝑢 6 = 𝑢 6 (1.7 𝑢 5 − 𝑚 (𝒖))
𝑚 𝒖 = 𝑖=1 𝑛 𝑢 𝑖 𝑢 𝑖− 𝑢 5 𝑢 6

28. The result of the iteration algorithm of evolutionary process: stabilization effect

29. The result of the iteration algorithm of evolutionary process: penalty function
𝐹 𝒖 = 𝑚 𝒖 − 𝑖=1 𝑛 𝑢 𝑖 𝑢 𝑖
𝑖=1 𝑛 𝑢 𝑖 =1
ln 𝑖=1 𝑛 𝑢 𝑖 𝑢 𝑖 = 𝑖=1 𝑛 𝑢 𝑖 𝑙𝑛 𝑢 𝑖

30. Conclusion
One of the central assumptions made for this analysis is that evolutionary time of the replicator system is much slower than the internal system dynamics time
We proposed an algorithm for the process of the fitness landscape adaptation using the example of the hypercycle system under the condition on the fitness matrix coefficients
We showed that this process of evolutionary adaptation provides persistence against parasite in the hypercycle system
We showed the existence of the phase transition in adaptation process, which is similar to the error threshold in the Eigen model

31. Research group:

32. Спасибо за внимание!