Thursday, May 2, 2013

A Mathematical Model for Evolution

One of the fundamental criticisms of the theory of evolution has been that it lacks a mathematical basis. From its inception evolution has used qualitative concepts such as “random mutations” and “natural selection” to describe how genes can spontaneously mutate and cause the organism to be more adaptive to its environment, thus giving it an advantage in the number and “fitness” of its offspring. Now that may be changing.

A team of scientists from the Chinese Academy of Sciences and the State University of New York at Stony Brook, led by Professor Wang, has published a paper that examines evolutionary dynamics from a mathematical perspective. (1,2) The study uses mathematical formulas to describe a new theory of evolution in which two forces are at play: an underlying emergent 3-D “fitness” landscape and an evolutionary force called “curl flux” which causes individuals and species to move through the fitness landscape in a spiraling manner. The hypothesis envisions endless co-evolution between individuals within species or between two different species by movement through the fitness landscape via curl flux. The curl flux can be created by interactions between individuals of different species which can result in continuous and endless co-evolution, a version of the Red Queen Hypothesis.

The Red Queen Hypothesis states that a species will continually evolve in order to protect itself from a parasite that is continually evolving. It was first proposed by evolutionary biologist Leigh Van Valen in 1973 and it gets its name from the Red Queen in Lewis Carroll’s Through The Looking Glass, who remarks that in her kingdom it is necessary to run faster and faster just to stay in place. As each species evolves to acquire an advantage over the other, a curl flux interaction with the fitness landscape produces endless evolution. A high degree of genetic variation, much of which is not immediately advantageous, results as a result of this interaction between the Ying and Yang of the fitness landscape and the curl flux. This can also explain the evolution of sexual reproduction, which increases the degree of genetic variability. Whenever conditions change or the parasitic species evolves a new capability, a gene that up to now was not important in the gene pool of the defending organism could now contribute to its survival.

The hypothesis is proposed via mathematical equations which describe the constant interaction between the fitness landscape and the curl flux force which drives evolution. In this hypothesis there is no guarantee for the survival of the fittest, just for a constant evolution, a running in place just to survive. Nor is it possible for evolution to ever go backward due to the mathematical non-zero nature of the flux. The authors draw an analogy between this process and the quantum particle and wave duality.

As an interesting aside, the counterpart to the Red Queen Hypothesis is the Red King Hypothesis which refers to an altruistic relationship between two individuals of different species. The partner who is evolving the slowest benefits because it does not have to invest much in improving the relationship. A study (3,4) now shows that this strategy of slow evolution only works when there are two parties involved. Once there are three or more, it becomes advantageous to evolve rapidly.

It remains to be seen if this attempt at a mathematical hypothesis for evolution will withstand criticism, but it is heartening that evolution is finally being considered with a mathematical rigor not previously seen.

1.     American Institute of Physics (AIP) (2012, August 8). Physics and math shed new light on biology by mapping the landscape of evolution. ScienceDaily. Retrieved May 1, 2013, from­ /releases/2012/08/120808132454.htm
2.     Feng Zhang, Li Xu, Kun Zhang, Erkang Wang, and Jin Wang. The potential and flux landscape theory of evolution. The Journal of Chemical Physics, 2012 DOI: 10.1063/1.4734305
3.     Max-Planck-Gesellschaft (2012, September 24). Red king or red queen: In relationships based on mutuality, number of individuals involved can determine rate at which species evolve. ScienceDaily. Retrieved May 1, 2013, from­ /releases/2012/09/120924080259.htm
4.     C. S. Gokhale, A. Traulsen. Mutualism and evolutionary multiplayer games: revisiting the Red King. Proceedings of the Royal Society B: Biological Sciences, 2012; DOI: 10.1098/rspb.2012.1697