Evolution speed models

Pure attempt to quantify evolution speed

Speed of evolution and purpose of evolution are linked terms.
They can be defined loosely.

Evolution and environment pressure

Terminology

Environmental factor (will be often referred to simply as “factor”):

Environmental factor is any external or internal pressure on a species behavior or characteristic that will determine survival advantage and therefore direct the evolutionary process. Example of external pressure: cold temperature. Species learn to make warm clothes or have thick fur – or they will die. Example of an internal pressure: gender – male vs. female gender directly dictates certain behaviors respectively that will affect species survival.

Force (description of an environmental factor):

Force is a relative measure of the effect of an environmental factor on population evolution. This measurement can be done in many different (imperfect) ways, depending on what is most suitable for the present purpose.

To measure the force of a very strong (forceful) factor, one might look at the number of populations it took to kill off the species after applying this factor in the opposite direction to what is advantageous to survive. For example, if thick fur is needed to survive the cold, it would be measurement of the number of populations it took to kill off the species not adapted to the cold by exposing them to cold. To measure the force of a weaker (less forceful factor) one might measure how many populations it takes to develop the necessary trait after applying the factor (how many populations it took to develop thick fur or learn to make warm clothes). Later we will develop an all-encompassing model that covers both strong and weak factors.

Population dynamics model

For an all-encompassing way to measure force (which includes both of the above listed examples), let us begin by looking at the wave-form change in population the results from applying the factor of a specific force (Figure 1). After application of an environmental factor to the population (ex: cold) at time 0, there will first be a dip in the population since all those with thin fur will die. Over time, the remaining thick-furred creatures will reproduce and there will be a rise in the population. This is true of any factor – it will always lead to this type of fluctuation in a population because it will be advantageous to one subset while a disadvantage to another.

Figure 1. Population dynamics under factors of various forces.

For strong factors the dip will be more abrupt and closer to 0 (blue line). The following rise will restore the population with thick-fur coats. If the factor is an extreme of strength and kills off the whole population, the dip will be permanent (red line). For weaker factors, the dip will be more gradual, farther away from 0 and less deep since those with existing thick fur will gradually replace the population (purple line). The rise will too be more gradual and closer to zero than for a strong factor.

While Figure 1 demonstrates a time vs. population representation of the influence three factors with 3 different values of force, imagine now that the factor force is represented on the Z axis, and the population curves over time will take the shape of an irregular wavy plane rather than a line. Thus, the red, the blue and the purple may be viewed as three cross-sections of this plane at various points on the z-axis representing factor force.

Take it a step further, realizing that population dynamics are determined by not factors in isolation, but by factors combined with each other at different proportions. Imagine a multidimensional space, with all the possible combinations of all possible factors. One may always take a slice of this multi-dimensional space to study the effect of a certain factor while holding others relatively constant, meaning this is a slice of a combination where factor X has 100% influence, while all the others have close to 0 influence – this scenario is represented for 3 types of factors in Figure 1). One may also study combinations of factors in various proportions ( 70% to 30% to 0 to 0 etc). In these studies, the overall model (this multidimentional space) must always hold in order that an accurate conclusion is made and is not limited by potential.

Take it another step further, realizing that the ‘speed of evolution’ may be defined in lots of different ways. If by evolution we mean population size, then we will measure it by population size vs. time. If we mean sophistication and intelligence, then we will measure average intelligence or the total number of intelligent individuals. Bacteria, for example, are very evolved population-wise, but are completely unevolved in other ways. Therefore, imagine all the different types of “speeds of evolution” coexisting in this multidimensional space as well. The “speed of evolution” which will dictate survivability may differ for various species, so that for bacteria – intelligence is not required for survival, but for humans as we know them today – it may be.

This is difficult to imagine in picture-form, but for theoretical purposes this model exactly describes, in an all-encompassing way and with 0 limitations, the type of multidimensional space that determines population dynamics. Therefore, the only way to formulate correct conclusions with no room for mistake, is by ensuring that these conclusions fit such a model.

Major theorems

Let us consider that the multidimensional space delineated above is describing 1) speed of evolution (evolution of a characteristic vs. time) and 2) environmental factors (internal and external).

Let us call the subspace of all combinations of speeds of evolution of various characteristics ‘subspace S’, where For an all-encompassing way to measure force (which includes both of the above listed examples), let us begin by looking at the wave-form change in population the results from applying the factor of a specific force (Figure 1). After application of an environmental factor to the population (ex: cold) at time 0, there will first be a dip in the population since all those with thin fur will die. Over time, the remaining thick-furred creatures will reproduce and there will be a rise in the population.This is true of ANY factor – it will always lead to this type of fluctuation in a population because it will be advantageous to one subset while a disadvantage to another.

Take it another step further, realizing that the ‘speed of evolution’ may be defined in lots of different ways. If by evolution we mean population size, then we will measure it by population size vs. time. If we mean sophistication and intelligence, then we will measure average intelligence or the total number of intelligent individuals. Bacteria, for example, are very evolved population-wise, but are completely uninvolved in other ways. Therefore, imagine all the different types of “speeds of evolution” coexisting in this multidimensional space as well. The “speed of evolution” which will dictate survivability is may differ for various species, so that for bacteria – intelligence is not required for survival, but for humans as we know them today – it may be.

S = the number of combinations/dimensions in the subspace.

Let us call the subspace of all combinations of environmental factors (including all possible distribution of weights amongst factors) ‘subspace F’, where F = the number of combinations/dimensions in the subspace.

Therefore, in the entire multidimensional space describing population dynamics there are S dimensions describing speeds of evolution and F dimensions describing acting environmental factors. This entire contains S+F dimensions.

Theorem 1. The differential of S-dimensional space onto F-dimensional space is maximum in a direction of a subspace in which the variation is a more uniform part of F. (shortened, this theorem says: S is maximum where F is uniform).

Biological application: evolution is fastest when all individuals in population experience pressure of the environment in the same direction.

Theorem 2: For a factor of strong force, the differential of S towards evolution when this factor is applied is equal to the differential of S towards degradation when the factor is applied in an opposite direction.

Biological application: If sexual reproduction involving each partner mating with 1 unique partner is a strong factor contributing to evolution progress, then deviating from this model is as strong of a factor in contributing to degradation.

This theorem is universal and can be applied to any concept, not just biological evolution of a biological species. One may either use any simple experiments to prove its correctness, or one may apply it to more complicated situation in order draw a great multitude of useful conclusions and corollaries – and this may be extended all the way to predicting with mathematical precision how society structure will affect human evolution. Below we will list a few very simple examples and corollaries to demonstrate. Then, in further sections we go into detailed set of experiments – of various complexity (from fruit flies to social experiments). Finally, we will try to extend this theorem to draw conclusions and corollaries applied to different fields.

Examples

The Mountain Example.

One example of theorem is an application to a non-biologic concept is evolution of mountain discussed earlier. Here evolution can be described as its stability in all directions. Since stability is directly dependent on its support at each side of the mountain, different sides of the population ‘mountain’ constitute this population. The wind is an environmental factor that affects evolution of the population ‘mountain’ by molding the mountain, or in other words, but applying force to individuals constituting this population. Evolution towards stability will be maximum when the environmental factor wind is informally applied to all members of the population – when it blows equally from all directions. If the environmental factor ‘wind’ is not applied uniformly to all the population members, the mountain will eventually collapse.

An application of Theorem 2 will be the following. If we block the wind from one direction, the destabilizing force on the mountain will be as effective at leading to collapse as the stabilizing force was effective at maintaining stability when the wind in that direction existed.

The wind must be distributed equally to all individuals (all sides of the mountain) in order for the mountain evolve in stability. If the wind force = 0 in all directions (in this case the force is also equal in all directions and is equal to 0), assuming no other factors affect its evolution, it will also remain stable. If any other factors are contributing to its evolution of stability, these factors too will have to be applied equally in all directions.

The Dog Example

Suppose that we are discussing evolution of a population of wild dogs living in snowy mountains. The only type of evolution that exists for this population is fur color. For our dogs living in the mountains, the best way to avoid being eaten by an eagle would be to mask the white fur against the snow. Therefore, white fur color is an evolutionarily advantageous trait that must be mixed into the population. The eagle predator who eats these dogs is a strong external environmental factor. Subspace S contains S dimensions, where S = 2. Since white fur helps mask the dog against the snow so that the eagle will not see them, subspace S is defined by the number of individuals in the population who have white fur. For the purpose of analyzing evolution, a unit of time is a generation, so we will measure time in generations passed, rather than conventional time units. Such evolution may be graphed in a 2-dimensional graph, where the number of individuals with white fur is on the y-axis and the number of generations passed is on x-axis. For simplicity of analysis we will keep the population size i constant.

Subspace F contains F dimensions where F = 1 and is measured as the ability of an eagle to spot the dog in the white mountain. Therefore, evolution dynamics of this population is described by a space containing subspaces S and F, which has S+F = 2 + 1 = 3 dimensions.

If evolution goes on in its natural way, the eagle will keep eating the dark furred dogs that it sees, while the white furred dogs will remain. As a result, over time (over generations), an evolutionarily advantageous trait – white fur- will be mixed into the entire population.

Now, what would happen if the environmental factor (white eagle) is not applied homogeneously to the entire population? We can take an extreme example of this non-homogeneity by dressing all white dogs in the population in white coats for several generations. This will completely stall evolution towards white fur. As a result, the dog population will never evolve to adapt to the environmental factor. The dramatic effects of this can be seen if we simply take out all the white dogs from each generation. Over time (generations), the population will become extinct (eaten by the eagle). Here, application of a forceful external factor in one direction (homogeneously) is as forceful in aiding Evolution Progress as application of the same factor in the opposite direction (non-homogeneously) in leading to extinction.
The above analysis concludes that white fur is a very strong internal factor affecting dog population survival. We have already determined in Part I that sexual reproduction involving 2 unique partners is a forceful factor affecting evolution progress. Theorem 1, applied here, dictates that both partners must experience the environmental factor equally (it must be distributed homogeneously). What would happen if in each generation we collect all dogs of gender A and make them wear white coats? In this instance, the population will never evolve towards white fur, just as if we selectively picked out white-fur dogs. The eagle will keep eating all the black-furred males, while persistence of black females will continually result in generations with black males. Eventually a point may arise when no males remain. Since sexual reproduction is a very strong factor affecting evolution over time, diminishing the benefits of sexual reproduction by interfering with the maximally rapid gene mixing it provides, will prevent Evolution Progress and will eventually lead to extinction. ← need a simulation of this and also I don’t like the way all this is worded and need help!

Evolution as reproductive mechanism

Terminology

Population

Population is an entity that consists of a collection of objects which acquire characteristics as a result of forces, both internal and external, exerted on these objects. For main applications of the theories described below, the most common application of population is to biology, where a population refers to a set of individuals of the same species acquiring new traits as a result of mutations, passing of the mutation to succeeding generations, and selection of individuals with advantageous mutations.

Evolution

Evolution is the acquirement (by the population) of a characteristic that serves as an adaptation to the environment. We may now define the population (above) as an entity consisting of objects which undergo evolution as a result of forces that act on the population.

Generation Test

If we place all traits on a spectrum from detrimental, to least advantageous to most advantageous, this spectrum may be measured in the number of generations that acquired a mutation. A detrimental mutation in generation n passed to generation n+1 will result in the death of generation n+1, so it will stand the test of 0 generations. A transiently advantageous trait (thick fur in cold weather) acquired by generation n may be passed to generation n+1 and then generation n+2, but by generation n+3 weather will warm up and thick furred animals will die of heat. Thus, the thick fur mutation only stands the test of 3 generations and is not evolutionary advantageous. On the farthest right of the spectrum will be the evolutionary advantageous traits got passed to succeeding generations and were therefore mixed into the entire population. Figure 1 is a graphical representation of this principle.

Evolutionary advantageous trait

A trait is evolutionary advantageous to the population when it serves the purpose of adaptation of the population to the environment for each succeeding generation. For a transiently advantageous trait to become an evolutionary advantageous trait, it must be consistently passed on to succeeding generations (it stands the Generation Test). An example of the differences is as follows: Storage of fat into adipocytes was adapted in an environment where food is scarce. This ability is so powerful that it was transiently advantageous, was acquired by the population and remained throughout generations. Now even if the environment changes where food is not scarce anymore, this trait has no negative impact on the species and is well equipped for a future food shortage, making it an evolutionary advantageous trait. Contrast this to unchanging fur thickness which only has its worth in a cold climate and it quickly depreciates once weather gets warmer.

Figure 1. Generation Test required for selection of evolutionary advantageous traits.

Evolution progress

Evolution progress (E) is the acquirement by the entire population of an evolutionarily advantageous trait. Progress can only proceed as a result of acquirement of new evolutionarily advantageous mutations by individuals followed by mixing of these mutations into the entire population. For simplicity, in the following analysis we will equate a mutation with a trait. Evolutionary progress of a population containing a constant number of individuals (i) can be measured as the number of advantageous mutations acquired by the population per generation. Since a mutation must stand the test of generations in order to be considered advantageous, we can calculate evolutionary progress at a segment of time by observing the number of advantageous mutations M acquired by the population of fixed size i after N generations:

Evolution Progress Equation:

E = M/N

Keeping i fixed, and assuming that each mutation in the set M is advantageous, E is maximal when M is maximal. Figure 2 is a graphical representation of evolutionary progress.

Figure 2. Evolution Progress and evolutionarily advantageous trait mixing. Generation test washes out detrimental mutations, while advantageous mutations are mixed into the population.

Theoretical principles of evolutionary advantageous mutation pool mixing

In the following discussion we will describe reproductive characteristics that allow for maximal evolutionary progress.

Let m = the number of mutations received by one organism.

Let x = number of parents in generation n passing mutations to each offspring in generation n+1.

Let N = number of generations

Let i = number of individuals in the population (remains fixed)

The following assumptions will be made to make the calculations more straight-forward:

Assumption 1. Each parent of generation n has one mutation.

Assumption 2. Generation count begins after the parent generation containing mutations.

Assumption 3. During each ‘reproductive period’ all members of the population mate with a unique partner/set of partners determined by the number x.

What values of x allow for best evolutionary progress? In other words, what values allow for advantageous (generation-tested) traits to be mixed into the entire population at maximum speed? Intuitively, it can be immediately suppose that this value must be such that:

Traits are mixed into the population fast enough that advantageous traits are received by the entire gene pool (maximize m)
Traits are mixed into the population slow enough that the evolutionarily advantageous nature of these traits is confirmed by the ‘Generation Test’ – as it is passed to a maximum number of generations (maximize N)

Let us then derive a formula that describes the relationship between m, x and N.

Suppose that x = 1 (Representing asexuality or cloning as a model for reproduction). Then after N = 2 generations, m for each individual will be equal to 1 (since each offspring receives only 1 mutation from 1 parent). After N = 3 generations, m is still 1 and so on. There is no mixing and m remains 1 regardless of the number of generations and regardless of the number of individuals in the population (Figure 3).

Figure 3. Lack of complete evolutionary progress when x = 1.

Suppose that x = 2 (this is the sexual reproduction model involving 2 individuals). In the first generation generation (N=1), each individual will receive 2 of the original i mutations (one from each parent), so m=2. In the second generation each individual will receive 4 mutations (each from one parent containing 2 mutations) and so on.

Suppose that x = 3 (A theoretical model involving anything over 2 individuals required to mate). In the first generation, m = 3 (one from each of the three parents). In the second m = 9 and so on.

Therefore, the relationship between x, N and m can be described by the formula:

From this we derive

Supposing that each individual in the parent generation had one mutation, after complete mixing of these mutations into the gene pool through generations, each individual in the population will acquire all mutations of the original population (m = i). This leads to what we will call the “Parent Quantity Equation”, which helps understand the effect of the number of genders involved in reproduction on the rate of evolutionarily advantageous trait acquirement:

Parent Quantity Equation:

Supposing that an advantageous mutation is one that remains in the population through as many generations as possible, a mutation is most likely advantageous when N is large. So for a fixed ‘i’:

When 1 < x < 2, N approaches infinity (this means that the population will never acquire all mutations of the original parent generation, as there is no mixing).

When x = 2, N is large and, by definition, mutations acquired by the population are evolutionarily advantageous.

When x > 2, N begins to decrease and to approach 0, reducing the number of generations that test each mutation.

These results may be represented in the graph below.

Figure 4. Graphical representation of the Parent Quantity Equation. Generation Test N represents the number of generations to complete gene mixing of i advantageous mutations into the population. The graph is created with i=1000, but conclusions will hold true for any i.

Note, that while the graph below demonstrates all possible values for x, for our current purpose we are considering the simplest physiologically relevant situations where x is a whole number.

Conclusion 1: x = 2 is the most efficient way to acquire evolutionary advantageous mutations by the population. This translates into sexual reproduction involving two genders.

Evidence-based

The following is a selection of experimental proof of concept that x=2 is the most efficient form of reproduction. These and other examples will be later revisited when we discuss different forms of mating that may resemble asexuality in its affect on trait mixing.

Sex increases the efficacy of natural selection in experimental yeast populations.

Transitions to Asexuality Result in Excess Amino Acid Substitutions

Further applications

For the following analysis let us consider the fact that in real life Assumption 3 does not always hold.

Assumption 3. During each ‘reproductive period’ all members of the population mate with a unique partner/set of partners determined by the number x.

From hereon we can continue our analysis with x = 2, since the Parent Quantity Equation already determined that this is the most evolutionary advantageous number of genders. Graphical representation of such a process, with Assumption 3 in place is demonstrated in Figure 5.

Figure 5. Maximally efficient Evolutionary Progress

How does violation of Assumption 3 affect evolutionary progress? In other words, how does violation of this assumption affect the rate of evolutionary advantageous trait acquirement by the population of size i?

Suppose that all members of the population do not mate with a unique partner. This can occur in two ways, which for analytical purposes we will consider separately. In the first instance, some individuals will mate with more than one partner. In the second some of the individuals will not mate at all.

Instance 1: some members mate with more than one partner. To determine how this affects the rate of population mixing and the number of generations going through Generation Test we will go back to our original analysis of generation n containing i individuals (generation size kept constant). Again, suppose that each individual has unique mutations and each has a unique advantageous mutation.

To determine the direction in which such mating affects gene mixing rate we will analyze the extreme situation where individual of gender A in generation n will mate with all individuals of gender B in the population and each succeeding generation will engage in the same type of mating. With x = 2, each offspring in the second generation will acquire 2 new mutations. However, of these two new mutations will be the same for all offspring and will have come from a single individual of gender A. This is technically equivalent to all members of gender B mating with clones of gender A. Therefore, third generation will acquire 3 mutations from the original generation, rather than 4. Genetic mixing will be incomplete. Parent Quantity Equation does not hold, and gene mixing rate is dramatically reduced (Figure 6).

Figure 6. Multiple partner mating slows down evolutionary progress.

Going back to the Evolution Progress equasion, at fixed i and fixed x:

E = M/N

Multiple partner mating reduces E by reducing M and therefore puts a break on natural evolution.

Instance 2: some members do not mate. In this instance, the gene pool of mutation is reduced (m is always less than i and can never equal i), resulting in the same detrimental effect on E as multiple partner mating. Complete mixing never occurs.

Conclusion 2: Sexual reproduction involving two genders, where each individual mates with one unique partner, is most advantageous for Evolution Progress.

In other words, both multiple partner mating (as would be observed with polygamy) and artificial isolation of mating groups (as would be observed with eugenics) – as far as genetic mixing is concerned – approach asexual reproduction in their evolutionary disadvantageous nature and thus slow down evolution of a species.

Speed of evolution determinant

imagine a model that does not model specific details, but models results. Three scenarios.

Model 1) evolution with asexual reproduction, without parallel inheritance with speed X1. All organisms are under environmental pressure Y in the same direction.

Model 2) Evolution with asexual (no parallel inheritance) reproduction. But in this case half the population experiences pressure in direction +Y and the other half in director -Y. Speed of evolution in direction +Y is X2. Let X2<X1 or 0<X2<X1/2

X2>X1 can be by any order of magnitude.

Let us try to correlate these models with our understanding of the speed of evolution to see which correlates better. In case of asexual reproduction, the chance of favorable mutation is proportion to the number of tries, which is equal to the number of organisms. Therefore in scenario 2, speed of evolution for the half that is evolving in direction Y is cut by half and speed of evolution of the entire population is equal to 0.

As such, for bacteria, who reproduce so quickly that the changes in environment are too slow to be significant, sexual reproduction is not as advantageous than for organisms for which environmental changes are more palpable due to slower reproductive times.

The most equivalent model that is not contradictory to real observation dictates that in scenario 2 evolution of entire population will be 0, while evolution of the half that evolves in direction Y is slower than in scenario 2.

Model 3) Evolution is speeded up by factor A, in the direction of evolutionary pressure (it can encompass evolution with parallel inheritance or not). Then the speed of evolution will be speeded up K times. Speed of evolution will be KX1.

Model 4). Includes scenario 2 with factor A. Speed of evolution X1/2K

The speeds of evolution will be ranked in the following way:

Model 3>Model 1> Model 2>Model 4

Model 5) Factor A works only if the entire population evolves in the same direction.

When we consider evolution with parallel form of inheritance, than the instance with factor A will be slower when half the population evolves in one direction, while the other half evolves in the opposite.

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Now we will create a model of sexual reproduction and will check how it coincides with the above. The property of sexual reproduction is such that it gives an advantage for evolution.

There is a space of all environmental factors that produce pressure on evolution. In this case, if all the organisms are in one spot of this space, the n parallel inheritance will have no effect of evolution. If all organisms are in different spots statically, then sexual reproduction will slow evolution. If all organisms are moving within the environment space, then sexual reproduction will speed up evolution.