BEACON Researchers at Work: Mathematical modeling of evolution

This week’s BEACON Researchers at Work post is by MSU graduate student Masoud Mirmomeni.

Dog licking kitten on head

Photo from http://www.cuteheaven.com

I bet that the very first time you saw this kind of picture, an unconscious “ahh!” came out. Well, it worked on me. But now, I am seeing these kind of pictures under another perspective. 

Nature is full of sophisticated phenomena. Although some of these phenomena are known, partly known or remain mysterious, nature is beautiful. But it is all this complexity that makes nature beautiful, and on the other hand hard to analyze. This complexity is a result of many completely or partly unknown chemical, physical, biological, or sociological processes such as evolutionary processes. Evolutionary processes increase diversity at every level of biological organisms. Among interesting behaviors in living organisms, cooperative behaviors, especially altruistic behaviors, are more difficult to explain.

In evolutionary biology, a behavior is considered as an altruistic behavior when its organism benefits other organisms, at a cost to itself. One way to measure costs and benefits of a behavior is to look at the reproductive fitness, or expected number of offspring. Many species in nature behave altruistically. For example, dogs sometimes adopt orphaned kittens, squirrels, or even ducks! In many bird species, a breeding pair sometimes receives help in raising its young from other birds called “helpers.” These birds usually protect the nest from predators and help the breeding pair to feed the fledglings. In the presence of predators, vervet monkeys give alarm calls to warn their fellow monkeys – even if they put themselves in danger by attracting attention of predators to themselves. Even microorganisms can display altruism. For example, the slime mold Dictyostelium discoideum shows altruistic behavior when food is scarce. When food is plentiful, they live in soil as single cells, feasting on bacteria. However when starved, they form a multicellular fruiting body with a ball of spores at the tip. Around one fifth of them will die and become the stalk that lifts the spores above the ground as the result of this altruistic behavior, but the chance of dispersing to more favorable environments increase.

(a) William Donald Hamilton; (b) George Price

Evolution of altruistic behavior has been addressed by many researchers (e.g. R.A. Fisher, J.B.S. Haldane, W.D. Hamilton, G. Price, and D.C. Queller). Around 50 years ago, William Donald Hamilton proposed a theory to explain the evolution of altruism among relatives based on the idea of inclusive fitness. This theory is called Hamilton’s rule (HR), which is usually interpreted as specifying the conditions under which the indirect fitness of altruists sufficiently compensates the immediate self-sacrifice of altruists. Simultaneously, George Price presented the more thorough mathematical treatment given to this theory and developed the Price Equation. In his theory, Price shows how a trait evolves over time, depending on the trait’s fitness and the fidelity with which the characteristic is transmitted to the next generation. 

The Price equation was a simple, general, and profound insight into the nature of selection, and it was a new mathematical formulation for evolutionary change in the population. The novelty of the Price Equation is that it does not change the fundamental simplicity of evolutionary change; however by making a few minor rearrangements and changes in notation, the equation provides an easier and more natural way to reason about complex problems. It provides us a way to understand the effects that gene transmission and natural selection have on the proportion of genes within each new generation of a population. The Price Equation shows that the change in the fitness mean is proportional to the fitness variance. The Price Equation, given in equation 1, proposes a model that unifies all types of selection (chemical, sociological, genetic, and every other kind of selection):

wavg × Δz_avg = COV(wi , zi) + E(wi×Δz_i),

where w is fitness and z is a quantitative character, which can be anything such as complexity of species. This covariance equation shows that what matters in kin selection is not common ancestry, but statistical associations between the genotypes of donor and recipient. The Price Equation adds considerable insight into many evolutionary biology problems by partitioning selection into meaningful components. It’s an important theorem because it associates entities from two populations, called the ancestral and the descendent populations.

Photo of Masoud MirmomeniAs a Ph.D. student in Chris Adami’s lab, I am working on the evolution of altruistic behavior under William Punch and Chris Adami’s supervision. My main focus is to extend the Price Equation and Queller’s formula from their linear form to a generalized form based on information theory. By generalizing these theories to a nonlinear form. First we will have a better model for this nonlinear process. Then, we can address recent arguments about the validity of the Price Equation (refer to M. Van Veelen’s papers in 2005 and 2011).

Currently, I am trying to address the following questions:

“If the Price Equation is not valid, what would be the proper probabilistic model/equation that can properly describe the selection problem in general?”

What is the best way to extend the Price Equation to hold for all populations?”

Screen shot of Avida populationI am using AVIDA to test the validity of the Price Equation. Because there is no assumption regarding the trait in the Price Equation, I chose the genomic complexity of AVIDIANs in the population as the trait (zi). The genomic complexity is defined based on information theory and Shannon entropy.

For more information about Masoud’s work, you can contact him at mirmomen at msu dot edu.

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