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A Simple Model of Unbounded Evolutionary Versatility as a Largest-Scale Trend in Organismal Evolution
The idea that there are large-scale trends in the evolution of biological organisms, such as increasing complexity, is highly controversial. But Peter Turney presents a simple computational model showing that local adaptation to a dynamic, randomly changing environment results in a global trend towards increasing evolutionary versatility, which implies an accelerating evolutionary pace, and that this trend can continue without bound if there is sufficient ongoing change in the environment.
Abstract
The idea that there are any large-scale trends in the evolution
of biological organisms is highly controversial. It is commonly
believed, for example, that there is a large-scale trend in evolution
towards increasing complexity, but empirical and theoretical arguments
undermine this belief. Natural selection results in organisms that
are well adapted to their local environments, but it is not clear
how local adaptation can produce a global trend. In this paper,
I present a simple computational model, in which local adaptation
to a randomly changing environment results in a global trend towards
increasing evolutionary versatility. In this model, for evolutionary
versatility to increase without bound, the environment must be highly
dynamic. The model also shows that unbounded evolutionary versatility
implies an accelerating evolutionary pace. I believe that unbounded
increase in evolutionary versatility is a large-scale trend in evolution.
I discuss some of the testable predictions about organismal evolution
that are suggested by the model.
1. Introduction
Ruse argues that almost all evolutionary theorists (before, after,
and including Darwin) believe that there is progress in evolution
[26]. Progress implies that there is a large-scale trend and that
the trend is good [4, 5]. For example, it is commonly believed by
the layperson that there is a large-scale trend in evolution towards
increasing intelligence, and that this trend is good. Several scientists
have suggested that we should focus on the (scientific) question
of whether there are any large-scale trends, without regard to the
(non-scientific) question of whether such trends are good [4, 5,
21, 22]. McShea presents an excellent survey of eight serious candidates
(“live hypotheses”) for large-scale trends in evolution:
entropy, energy intensiveness, evolutionary versatility, developmental
depth, structural depth, adaptedness, size, and complexity [21].
Complexity appears to be the most popular candidate.
The standard objection to large-scale trends in evolution is that
natural selection is a local process that results in organisms that
are well adapted to their local environments, and there is no way
for this local mechanism to yield a global trend. On the other hand,
it does seem that complexity (for example) has increased steadily
since life on earth began. This seems to suggest that natural selection
favours increasing complexity. However, many evolutionary theorists
deny that there is any driving force, such as natural selection,
behind any of the apparent large-scale trends in evolution.
Gould has presented the most extensive arguments against a driving
force [16, 17]. Gould admits that there may be large-scale trends
in evolution, but he argues that any such trends are, in essence,
statistical artifacts. For example, if we consider the evolution
of life since the first appearance of prokaryotes, the mean level
of complexity would necessarily increase with time, because any
organism with significantly less complexity than a prokaryote would
not be able to live [16, 17]. According to Gould, the apparent trend
towards increasing com- plexity is due to random variation in complexity
plus the existence of a minimum level of complexity required to
sustain life; there is no selective pressure that drives life towards
increasing complexity. I discuss Gould’s arguments in more
detail in Section 2.
Of the eight live hypotheses for large-scale trends in evolution,
this paper focuses on evolutionary versatility. I believe that there
is indeed a selective advantage to increasing evolutionary versatility.
Evolutionary versatility is the number of independent dimensions
along which variation can occur in evolution [21, 25, 34, 35, 36,
37]. It is possible that increasing evolutionary versatility may
be the driving force behind other apparent evolutionary trends,
such as increasing complexity. I discuss the concept of evolutionary
versatility in Section 3.
In Section 4, I introduce a simple computational model of unbounded
evolutionary versatility. As far as I know, this is the first computational
model of an evolutionary mechanism for one of the eight live hypotheses
for large-scale trends in evolution. In this model, the population
evolves in a series of eras. During each era, the fitness landscape
is constant, but it randomly changes from one era to the next era.
The model shows that there is a long-term trend towards increasing
evolutionary versatility, in spite of the random drift of the fitness
landscape. In fact, when the fitness landscape is constant, evolutionary
versatility is bounded. In this model, unbounded evolutionary versatility
requires a dynamic fitness landscape. The point of this model is
to show that it is possible, in principle, for natural selection
to drive evolution towards globally increasing evolutionary versatility,
without bound, even though natural selection is a purely local process.
I discuss some related work in Section 5. My simple computational
model of evolutionary versatility is related to Bedau and Seymour’s
model of the adaptation of mutation rates [8]. The primary focus
of Bedau and Seymour was the adaptation of mutation rates, but the
primary focus of this paper is evolutionary versatility. Bedau and
Seymour’s model does not address evolutionary versatility.
The core of this paper is the experimental evaluation of the model,
in Section 6. In the first two experiments, I show that there are
parameter settings for which evolutionary versatility can increase
indefinitely. In the third experiment, I show that evolutionary
versatility is bounded when the fitness landscape is static. In
the remaining experiments, I examine a wide range of settings for
the parameters in the model. These experiments show that the behaviour
of the model is primarily determined by the parameters that control
the amount of change in the fitness landscape.
In Section 7, I discuss the implications of the model. One of the
most interesting implications of the model is that increasing evolutionary
versatility implies an accelerating evolutionary pace. This leads
to testable predictions about organismal evolution.
I discuss limitations and future work in Section 8 and I conclude
in Section 9.
2. Arguments Against Large-Scale Trends in Evolution
Natural selection produces organisms that are well adapted to their
local environments. The major objection to large-scale trends in
evolution is that there is no way for local adaptation to cause
a large-scale trend [16, 17, 22]. For example, although the environments
of primates may favour increasing complexity, the environments of
most parasites favour streamlining and simplification [17]. There
is no generally accepted theoretical explanation of how natural
selection could cause a large-scale trend. Van Valen’s Red
Queen hypothesis [33] attempts to explain how natural selection
could cause a trend towards increasing complexity (based on coevolution),
although Van Valen’s hypothesis has been criticized [19, 23].
In this paper, I propose an explanation of how natural selection
could cause a trend towards increasing evolutionary versatility.
Attractive features of my proposal are that it can easily be simulated
on a computer and that it leads to testable (in principle) predictions.
If there were some constant property, shared by all environments,
then it would be easy to see how there could be a large-scale trend,
due to long-term adaptation to this constant property. However,
the computational model in Section 4 shows that there can be a large-scale
trend even when the fitness landscape changes completely randomly
over time.
Aside from theoretical difficulties with large-scale trends, there
is the question of whether there is empirical evidence for any large-scale
trend. McShea’s survey of candidates for large-scale trends
does not address the issue of evidence for the candidates [21],
but, in another paper, he finds that there is no solid evidence
for a trend in most kinds of complexity [20].
Gould argues that, even if there were empirical evidence for a
large-scale trend, that does not imply that there is a driving force
behind the trend [16, 17]. Gould argues that evolution is performing
a random walk in complexity space, but there is a constraint on
the minimum level of complexity. When the complexity of an organism
drops below a certain level (e.g., the level of prokaryotes), it
can no longer live. Gould’s metaphor is that evolution is a
drunkard’s random walk, but with a wall in the way (i.e., a
bounded diffusion process). This wall of minimum complexity causes
random drift towards higher complexity. This random drift does not
involve any active selection for complexity; there is no push or
drive towards increased complexity.
In summary, (1) it is not clear how local selection can produce
a global trend and (2) observation of a global trend does not imply
that there is a driving force behind the trend. However, (1) my
model shows one way in which local selection can produce a global
trend and (2) the model makes testable (in principle) predictions.
Evolutionary versatility is the number of independent dimensions
along which variation can occur in evolution [21, 25, 34, 35, 36,
37]. A species with high evolutionary versatility has a wide range
of ways in which it can adapt to its environment. Vermeij has argued
that there should be selection for increased evolutionary versatility,
because it can lead to organisms that are more efficient and better
at exploiting their environments [34, 35, 36, 37].
An important point is that evolutionary versatility requires not
merely many dimensions along which variation can occur, but also
that the dimensions should be independent. Pleiot- ropy is the condition
in which a single gene affects two or more distinct traits that
appear to be unrelated. When N traits, which appear to vary on N
dimensions, are linked by pleiotropy, there is effectively only
one dimension along which variation can occur. Several authors have
suggested that it would be beneficial for the genotype-phenotype
map to be modular, since increasing modularity implies increasing
independence of traits [2, 27, 30, 38]. McShea points out the close
connection between evolutionary versatility and modularity [21].
Evolutionary versatility seems to be connected to several of the
seven other “live hypotheses” [21]. Increasing evolutionary
versatility implies increasing complexity, since the organisms must
have some new physical structures to support each new dimension
of variation. The dimensions are supposed to be independent, so
the new physical structures must also be (at least partially) independent.
The increasing accumulation of many independent new physical structures
implies increasingly complex organisms. Among the other live hypotheses,
developmental depth, structural depth, adaptedness, and perhaps
energy intensiveness may be connected to evolutionary versatility
[21].
Evolutionary versatility also seems to be related to evolvability
[1, 2, 12, 13, 32, 38]. Evolvability is the capacity to evolve [12,
13]. An increasing number of independent dimensions along which
variation can occur in evolution implies an increasing capacity
to evolve, so it would seem that any increase in evolutionary versatility
must also be an increase in evolvability. On the other hand, some
properties that increase evolvability may decrease evolutionary
versatility. For example, selection can be expected to favour a
constraint that produces symmetrical left-right development [12,
13]. For humans, if a sixth finger were a useful mutation, then
it would likely be best if the new fingers appeared simultaneously
on both hands, instead of requiring two separate mutations, one
for the left hand and another for the right hand. In general, selection
should favour any constraint that produces adaptive covariation
[24]. Such constraints increase evolvability, but they appear to
decrease evolutionary versatility [21].
Increasing evolutionary versatility suggests an increasing number
of independent dimensions, but adaptive covariation suggests a decreasing
number of independent dimensions. Vermeij reconciles these forces
by proposing that increasing evolutionary versatility adds more
dimensions, which are then integrated by adaptive covariation, so
that new dimensions are added and integrated in an ongoing cycle
[21, 37].
Evolutionary versatility also appears to be related to the Baldwin
effect [3, 18, 31]. The Baldwin effect is based on phenotypic plasticity,
the ability of an organism (the phenotype) to adapt to its local
environment, during its lifetime. Examples of phenotypic plasticity
include the ability of humans to tan on exposure to sunlight and
the ability of many animals to learn from experience. Phenotypic
plasticity can facilitate evolution by enabling an organism to benefit
from (or at least survive) a partially successful mutation, which
otherwise (in the absence of phenotypic plasticity) might be detrimental.
This gives evolution the opportunity to complete the partially successful
mutation in future generations.
Evolution is not really free to vary along a given dimension if
all variation along that dimension leads to death without children.
Thus phenotypic plasticity increases the effective number of dimensions
along which variation can occur in evolution. The Baldwin effect
can therefore be seen as a mechanism for increasing evolutionary
versatility.
The following simple model of unbounded evolutionary versatility
has three important features: (1) The fitness function is based
on a shifting target, to demonstrate that a large-scale trend is
possible, even when the optimal phenotype varies with time. In fact,
in this model, the target must shift, if the model is to display
unbounded evolutionary versatility. (2) The length of the genome
can change. There is no upper limit on the possible length of the
genome. This is necessary, because if the length were bounded, then
there would be a finite number of possible genotypes, and thus there
would be a bound on the evolutionary versatility. (3) The mutation
rate is encoded in the genome, so that the mutation rate can adapt
to the environment. This allows the model to address the claim that
mutation becomes increasingly harmful as the length of the genome
increases. Some authors have argued that natural selection should
tend to drive mutation rates to zero [42]. Of course, if the mutation
rate goes to zero, this sets a bound on evolutionary versatility.
Table 1 shows the parameters of the model and their baseline values.
In the experiments that follow, I manipulate these parameters to
determine their effects on the behaviour of the model. The meaning
of the parameters in Table 1 should become clear as I describe the
model.
Table
1
Figure 1 is a pseudo-code description of the model of evolutionary
versatility. In this model, a genome is a string of bits. The model
is a steady-state genetic algorithm (as opposed to a generational
genetic algorithm), in which children are born one-at-a-time [28,
29, 39, 40]. (In a generational genetic algorithm, the whole population
is updated simultaneously, resulting in a sequence of distinct generations.)
Parents are selected using tournament selection [9, 10]. In tournament
selection, the population is randomly sampled and the two fittest
individuals in the sample are chosen to be parents (see lines 15
to 17 in Figure 1). The selective pressure can be controlled by
varying the size of the sample (TOURNAMENT_SIZE). A new child is
created by applying single-point crossover to the parents encoded
in the child’s genome (lines 22 to 28). Mutation can flip a
bit (from 0 to 1 or from 1 to 0) in the genome or it can add or
delete a bit, making the bit string longer or shorter.
The initial section of a genome (the first MUTATION_CODE_LENGTH
bits) encodes the mutation rate for that genome. The remainder of
the genome (which may be null) encodes the phenotype. The phenotype
is a bit string, created from the genome by simply copying the bits
from the genotype, beginning with the MUTATION_CODE_LENGTH plus
one bit of the genotype and continuing to the end of the genotype.
If the length of the genome is exactly MUTATION_CODE_LENGTH (as
it is when the simulation first starts running), then the phenotype
is the null string.
The fitness of the phenotype is determined by comparing it to a
target. The target is a random string of bits. The fitness of the
phenotype is the number of matching bits between the phenotype and
the target (lines 31 to 33). If the phenotype is null, the fitness
is zero. The length of the target grows, so that the target is always
at least as long as the longest phenotype in the population (lines
29 to 30). When a new child is born, if it is fitter than the least
fit individual in the population, then it replaces the least fit
individual (lines 34 to 36).
The target is held constant for an interval of time called an era.
At the end of an era, the target is randomly changed. Each time
the target changes, it is necessary to re-evaluate the fitness of
every individual (lines 37 to 40). Instead of dividing a run into
a series of eras, the model could have been designed to have a small,
continuous change of the target for each new child that is born.
(This is a special case of the current model, where TARGET_CHANGE_RATE
is small and ERA_LENGTH is one.) The main motivation for dividing
the run into a series of eras is to increase the computational efficiency
of the model, since it is computationally expensive to re-evaluate
the fitness of every individual each time a new child is born. (Actually,
it would only really be necessary to re-evaluate TOURNAMENT_SIZE
individuals each time a new child is born.) It could also be argued
that organismal evolution is characterized by periods of stasis
followed by rapid change (e.g., punctuated equilibria), so this
feature of the model makes it more realistic.
Figure
1: A pseudo-code description of the model of evolutionary
versatility. The model is a steady-state genetic algorithm with
crossover and mutation. Mutation can flip a bit in the genome or
increase or decrease the genome length by one bit. The mutation
rate is encoded in the genome. Parents are chosen by tournament
selection.
Recall that evolutionary versatility is the number of independent
dimensions along which variation can occur in evolution. In this
model, the evolutionary versatility of a genome is the length of
the genome minus MUTATION_CODE_LENGTH. This is the length of the
part of the genome that encodes the phenotype. The first MUTATION_CODE_LENGTH
bits are not independent and they do not directly affect the phenotype,
so I shall ignore them when counting the number of independent dimensions
along which variation can occur. Each remaining bit in the genome
is an independent dimension along which variation can occur. The
dimensions are independent because the fitness of the organism is
defined as the number of matches between the phenotype and the target;
that is, the fitness is the sum of the fitnesses for each dimension.
Fitness on one dimension (a match on one bit) has no impact on fitness
on another dimension (a match on another bit).
Note that increasing evolutionary versatility (i.e., increasing
genome length) does not necessarily imply increasing fitness, because
(1) the additional bits do not necessarily match the target and
(2) a mutation rate that enables evolutionary versatility (genome
length) to increase also makes the genome vulnerable to disruptive
(fitness reducing) mutations. However, the design of the model implies
that increasing genome length will tend to be correlated with increasing
fitness.
5. Related Models
The most closely related work is the model of Bedau and Seymour
[8]. In Bedau and Seymour’s model, mutation rates are allowed
to adapt to the demands of the environment. They find that mutation
rates adapt to an optimal level, which depends on the evolutionary
demands of the environment for novelty. My model is similar, in
that mutation rates are also allowed to adapt. Other work with adaptive
mutation rates includes [6, 11, 14, 41]. Bedau and Seymour’s
model and my model are distinct from this other work in that we
share an interest in the relationship between the adaptive mutation
rates and the evolutionary demands of the environment for novelty.
The main difference between this paper and previous work is the
different objective. None of the previous papers were concerned
with large-scale trends in evolution. As far as I know, this is
the first model to show how it is possible for evolutionary versatility
to increase without bound.
6. Results of Experiments with the Model
This section presents eight experiments with the model of evolutionary
versatility. The first experiment examines the behaviour of the
model with the baseline parameter settings. The second experiment
runs the model for ten million births, but is otherwise the same
as the baseline case. This experiment gives a lower resolution view
of the behaviour of the model, but over a much longer time scale.
These two experiments support the claim that the model can display
unbounded evolutionary versatility, given suitable parameter settings.
The third experiment uses the baseline parameter settings, except
that the target is held constant. With a constant target, the mutation
rate eventually goes to zero and the population becomes static.
The results show that, in this model, unbounded evolutionary versatility
requires a dynamically varying target. The remaining experiments
vary the parameters of the model, one at a time. These experiments
show that the model is most sensitive to the parameters that determine
the pace of change in the target. In comparison, the parameters
that do not affect the target have relatively little influence on
the large-scale behaviour of the model.
Figure 2 shows the results with the baseline parameter settings
(see Table 1). Since the model is stochastic, each run is different
(assuming the random number seed is different), but the general
behaviour is the same for all runs (assuming the parameters are
the same). In this experiment, I ran the model 100 times and averaged
the results across the 100 runs.
For this experiment, the length of an era is 100 children. At the
start of each new era, the fitness drops. However, the overall trend
is towards increasing fitness (see the first plot in Figure 2).
Although the probability that a mutation will increase the genome
length is equal to the probability that a mutation will decrease
the genome length, there is a steady trend towards increasing genome
length (the second plot in Figure 2). The mutation rate decreases
steadily (third plot). Although the length of an era is fixed, in
each era, the increase in fitness since the start of the era is
greater than the corresponding increase for the previous era (fourth
plot). This shows that the pace of evolution is accelerating.
Figure
2: Experiment 1: Baseline parameter values. These four
plots show the fitness, genome length, mutation rate, and fitness
increase as functions of the number of children that have been born.
The target for the fitness function changes each time one hundred
children are born. The fitness increase is the increase in fitness
since the most recent change in the target. All values are averages
over the whole population, for one hundred separate runs of the
baseline configuration (2,000 individuals times 100 runs yields
200,000 samples per value).
Evolutionary versatility is given by the genome length (minus MUTATION_CODE_LENGTH).
The steady growth in the genome length (in the second plot in Figure
2) shows that evolutionary versatility is increasing, at least over
the relatively short time span of this experiment.
6.2 Experiment 2: Longer Run Length
The steady decrease in the mutation rate in the first experiment
suggests that the mutation rate might go to zero. If the mutation
rate is zero, then the fitness can no longer increase without bound.
The fitness would vary randomly up and down as the target changed
each era, but the fitness would always be less than the genome length,
which would become a constant value.
In the second experiment, I ran the model until 10,000,000 children
were born (RUN_LENGTH = 10,000,000), in order to see whether the
trends in Figure 2 would continue over a longer time scale. I ran
the model 10 times and averaged the results across the 10 runs.
In the first experiment, the population averages (for fitness, genome
length, mutation rate, and fitness increase) were calculated each
time a new child was born. In the second experiment, to increase
the speed of the model, the population averages were only calculated
each time 10,000 children were born. Figure 3 shows the results
for the second experiment.
Figure 3 shows that the trends in Figure 2 continue, in spite of
the much longer time scale. The only exception is the mutation rate,
which quickly falls from its initial value of 0.5 to hover between
0.03 and 0.05. There is no indication that the mutation rate will
go to zero. However, since the model is stochastic, there is always
a very small (but non-zero) probability that the mutation rate could
go to zero.
In Figure 3, the fitness increase is calculated as the average
fitness of the population at the end of an era minus the average
fitness of the population at the start of the same era. The fitness
increase is calculated each era and then the average fitness increase
is calculated for each 10,000 births. Since there are 100 births
in an era, there are 100 eras in each sample of 10,000 births, so
each value in the plot of the fitness increase is the average of
100 eras and 10 runs. The values in the other three plots (fitness,
genome length, and mutation rate) are averages over 10 runs.
Figure
3: Experiment 2: Longer run length. These four plots show the
fitness, genome length, mutation rate, and fitness increase as functions
of the number of children that have been born. As in the first experiment,
the target for the fitness function changes each time one hundred
children are born. All values are averages over the whole population,
for ten separate runs of the baseline configuration. The values
are calculated once for each ten thousand children that are born.
6.3 Experiment 3: Static Target
This experiment investigated the behaviour of the model when the
target was static. As in the second experiment, the population averages
for fitness, genome length, and mutation rate were calculated once
every 10,000 births. I ran the model 10 times and averaged the results
across the 10 runs. I used the baseline parameter settings, except
for RUN_LENGTH, ERA_LENGTH, and TARGET_CHANGE_RATE. I set both RUN_LENGTH
and ERA_LENGTH to 10,000,000 and I set TARGET_CHANGE_RATE to zero.
Figure 4 shows the results of the runs. In all 10 runs, the mutation
rate was zero, for every member of the population, long before 10,000,000
children were born. The longest run lasted for 349,000 births, the
shortest run lasted for 37,100 births, and the average run lasted
for 165,300 births. In comparison, in the second experiment, all
10 runs ran for 10,000,000 children, with no sign that the mutation
rate would ever reach zero. These experiments support the claim
that (in this model) unbounded evolutionary versatility requires
a dynamic target. The following two experiments investigate the
amount of change in the target that is needed to ensure unbounded
evolutionary versatility.
6.4 Experiment 4: Varying Rate of Change of Target
In the fourth experiment, the rate of change of the target was
varied from 0.0 to 0.2. The RUN_LENGTH was constant at 1,000,000.
The remaining parameters were set to their baseline values. Figure
5 shows the behaviour of the model, averaged over ten separate runs.
The time of the birth of the last novel child (i.e., the time at
which the mutation rate becomes zero for every member of the population)
was around the birth of the 100,000th child when TARGET_CHANGE_RATE
was 0.0, but it quickly rose to around the 1,000,000th child as
TARGET_CHANGE_RATE approached 0.1 (see the first plot in Figure
5). It could not go past 1,000,000, because RUN_LENGTH was 1,000,000.
I conjecture that there is a threshold for TARGET_CHANGE_RATE at
approximately 0.1, where the average time of birth of the last novel
child approaches infinity as RUN_LENGTH approaches infinity.
When TARGET_CHANGE_RATE was 0.08, 40% of the ten runs made it all
the way to the 1,000,000th birth with a mutation rate above zero.
When the TARGET_CHANGE_RATE was 0.1, this went to 90% (second plot
in Figure 5). The average fitness of the population at the time
of the birth of the 1,000,000th child (the final fitness) rose steadily
as TARGET_CHANGE_RATE increased from 0.0 to 0.1 (third plot). Above
0.1, it could not rise significantly, because of the limit set by
RUN_LENGTH. I conjecture that it would rise to infin- ity as RUN_LENGTH
rises to infinity. The average mutation rate of the population at
the time of the birth of the 1,000,000th child (the final mutation
rate) increased steadily as TARGET_CHANGE_RATE increased, even past
the 0.1 threshold.
Figure
4: Experiment 3: Static target. These three plots show the fitness,
genome length, and mutation rate as functions of the number of children
that have been born. Since the target is static, the fitness increase
is undefined. All values are averages over the whole population,
for ten separate runs of the model. The values are calculated once
for each ten thousand children that are born.
This experiment suggests that a relatively high amount of change
is required to ensure that evolutionary versatility will increase
without bound. When the target is changed once every hundred children
(ERA_LENGTH = 100), the target must change by at least 10% (TARGET_CHANGE_RATE
= 0.1). If there is less environmental change than this, the mutation
rate eventually drops to zero.
Figure
5: Experiment 4: Varying rate of change of target. In this experiment,
TARGET_CHANGE_RATE varies from 0.0 (its value in Experiment 3) to
0.2 (its value in Experiments 1 and 2). The RUN_LENGTH is 1,000,000.
When TARGET_CHANGE_RATE is about 0.1, there is a qualitative change
in the behaviour of the model. This threshold appears to separate
bounded evolutionary versatility (as in Experiment 3; left of the
vertical dotted line) from unbounded evolutionary versatility (as
in Experiment 2; right of the vertical dotted line). All values
in the plots are based on ten separate runs of the model.
6.5 Experiment 5: Varying Length of Era
In the fifth experiment, the length of an era was varied from 100
to 1000. The RUN_LENGTH was constant at 1,000,000. The remaining
parameters were set to their baseline values. Figure 6 shows the
behaviour of the model, averaged over ten separate runs. Like Experiment
4, this experiment supports the hypothesis that a relatively high
amount of change is required to ensure that evolutionary versatility
will increase without bound. When the target changes by 20% each
era (TARGET_CHANGE_RATE = 0.2), the length of an era cannot be more
than 200 children (ERA_LENGTH = 200), if the mutation rate is to
stay above zero.
6.6 Experiment 6: Varying Tournament Size
In the sixth experiment, the tournament size was varied from 100
to 1000. The baseline value for TOURNAMENT_SIZE was 400. Larger
tournaments mean that there is more competition to become a parent,
so there is higher selective pressure. The RUN_LENGTH was constant
at 1,000,000 and the remaining parameters were set to their baseline
values. Figure 7 shows the behaviour of the model, averaged over
ten separate runs. The results suggest that the model will display
unbounded evolutionary versatility as long as TOURNAMENT_SIZE is
more than about 200. Compared to ERA_LENGTH and TARGET_CHANGE_RATE,
the behaviour of the model is relatively robust with respect to
TOURNAMENT_SIZE. The model displays unbounded evolutionary versatility
for a relatively wide range of values of TOURNAMENT_SIZE.
6.7 Experiment 7: Varying Population Size
In the seventh experiment, the size of the population was varied
from 1000 to 3000. The baseline value of POPULATION_SIZE was 2000.
The RUN_LENGTH was constant at 1,000,000 and the remaining parameters
were set to their baseline values. Figure 8 shows the behaviour
of the model, averaged over ten separate runs.
Figure
6: Experiment 5: Varying length of era. In this experiment,
ERA_LENGTH varies from 100 (its value in Experiments 1 and 2) to
1000 (its length in Experiment 3 was 10,000,000). The RUN_LENGTH
is 1,000,000. When ERA_LENGTH is about 200, there is a qualitative
change in the behaviour of the model. This threshold appears to
separate unbounded evolutionary versatility (as in Experiment 2;
left of the vertical dotted line) from bounded evolutionary versatility
(as in Experiment 3; right of the vertical dotted line). All values
in the plots are based on ten separate runs of the model.
When the population is small, the model is more susceptible to
random variations. With a large population, the model will tend
to behave the same way, every time it runs. Figure 8 suggests that
the model becomes unstable when the population size is less than
about 2000, although there is no sharp boundary at 2000. This is
unlike Experiment 4, where there is a sharp boundary when TARGET_CHANGE_RATE
is 0.1, and Experiment 5, where there is a sharp boundary when ERA_LENGTH
is 200.
Figure
7: Experiment 6: Varying tournament size. In this experiment,
TOURNAMENT_SIZE varies from 100 to 1000. The RUN_LENGTH is 1,000,000.
Larger tournaments mean greater selective pressure. The results
suggest that there is unbounded evolutionary versatility as long
as TOURNAMENT_SIZE is greater than about 200 (see the first two
plots). In the third plot, the final fitness (the average fitness
of the population at the time of the birth of the 1,000,000th child)
continues to rise even when TOURNAMENT_SIZE is greater than 200
and 100% of the runs reach the 1,000,000th child with a non-zero
mutation rate (see the second plot). This suggests that there is
an advantage to higher selective pressure, beyond what is needed
to obtain unbounded evolutionary versatility.
Figure
8: Experiment 7: Varying population size. In this experiment,
POPULATION_SIZE is varied from 1000 to 3000. The baseline value
of POPULATION_SIZE is 2000. These plots suggest that, in most runs,
we will have unbounded evolutionary versatility, even when the population
size is only 1000 individuals. However, it appears that the model
becomes less stable when the population size is below about 2000.
With smaller populations, there is more risk that the mutation rate
could fall to zero by random chance.
6.8 Experiment 8: Varying Number of Bits for Encoding Mutation
Rate
In the final experiment, the number of bits in the genome used
to encode the mutation rate was varied from 5 to 15. The baseline
value of MUTATION_CODE_LENGTH was 10. The RUN_LENGTH was constant
at 1,000,000 and the remaining parameters were set to their baseline
values. Figure 9 shows the behaviour of the model, averaged over
ten separate runs.
Figure
9: Experiment 8: Varying number of bits for encoding mutation
rate. In this experiment, the number of bits in the genome used
to encode the mutation rate is varied from 5 to 15. The model displays
unbounded evolutionary versatility when the number of bits was more
than about 8. When MUTATION_CODE_LENGTH is less than 8, it seems
that quantization effects make the model unstable. When the encoding
is too short, the ideal mutation rate may lie between zero and the
smallest value that can be encoded, so the genetic algorithm is
forced to set the mutation rate to zero, even though this is less
than ideal.
The results suggest that the model will display unbounded evolutionary
versatility when the MUTATION_CODE_LENGTH is greater than about
8. If the code length is less than 8 bits, the model becomes susceptible
to quantization errors. For example, 6 bits can only encode values.
If the ideal mutation rate is between 0 and , then the genome may
be forced to set the mutation rate to zero, although a non-zero
value (but less than 0.015873) would be better.
7. Implications of the Model
I do not claim that the model shows that there is a large-scale
trend towards increasing evolutionary versatility in organismal
evolution; I claim that the model supports the idea that, under
certain conditions, it is possible for evolutionary versatility
to increase without bound. In this model, there is active selection
for increased evolutionary versatility; there is a selective force
that drives the increase; it is not merely a statistical artifact,
due to a bounded diffusion process. The model shows how a purely
local selection process can yield a global trend.
The model also shows that the environment must be highly dynamic
(the target for the fitness function must change significantly and
repeatedly) to sustain increasing evolutionary versatility. If the
environment is not sufficiently dynamic, the disruptive effects
of mutation will outweigh the beneficial effects, and selection
will drive the mutation rates to zero. When the mutation rate is
zero throughout the population, the genome length can no longer
increase, so the evolutionary versatility is bounded by the length
of the longest genome in the population.
I believe that, in fact, there is a large-scale trend towards increasing
evolutionary versatility in organismal evolution. Although the model
does not (and cannot) prove this belief, the model suggests a way
to test the belief, because the model predicts that, where there
is increasing evolutionary versatility, there should be an accelerating
pace of evolution (see the fourth plots in Figures 2 and 3). Therefore,
I predict that we will find evidence for an accelerating pace in
the evolution of biological organisms.
It is difficult to objectively verify the claim that the pace of
evolution is accelerating. The natural measure of the pace of evolution
is the historical frequency of innovations, but the analysis is
complicated by several factors. One confounding factor is that the
record of the recent past is superior to the record of the distant
past, which may give the illusion that there are more innovations
in the recent past than the distant past. Another confounding factor
is population growth. We may expect more innovations in recent history
simply because there are more innovators. A third factor is difficulty
of counting innovations. There is a need for an objective threshold
on the importance of the innovations, so that the vast number of
trivial innovations can be ignored.
I suggest some tests that avoid these objections. I predict that
the fossil record will show a decreasing recovery time from major
catastrophes, such as mass extinction events, ice ages, meteorite
impacts, and volcanic eruptions. Also, I predict a decrease in the
average lifetimes of species, as they are out-competed by more recent
species at an accelerating rate [15]. These two tests do not involve
counting the frequency of innovations, which makes them relatively
objective.
8. Limitations and Future Work
There are several limitations to this work. One limitation is that
we cannot run the model to infinity, so we cannot prove empirically
that evolutionary versatility will grow to infinity. I conjecture
that, with the baseline settings of the parameters (Table 1), the
expected (i.e., mean, average) evolutionary versatility of the model
will rise to infinity as RUN_LENGTH rises to infinity. This conjecture
can be supported by empirical evidence (Figure 3), but it can only
be proven by theoretical argument. I have not yet developed this
theoretical argument.
Another limitation of the model is its abstractness. A more sophisticated
model would include (1) a non-trivial genotype-phenotype mapping,
(2) an internal, implicit fitness function, instead of the current,
external, explicit fitness function, (3) a genotype-phenotype mapping
and fitness function that allow varying degrees of dependence and
independence among the dimensions (i.e., traits, characteristics)
along which variation can occur in evolution, (4) the possibility
of covariation, (5) coevolution, (6) multiple species, (7) predator-prey
relationships, and so on. However, the point of this exercise was
to make the model as abstract as possible, in order to identify
the minimum elements that are needed to display unbounded evolutionary
versatility. The abstractness of the model was intended to make
it more clear and susceptible to analysis.
There might seem to be some conflict between this model and the
“no free lunch” theorems [43]. Informally, the “no
free lunch” theorems show that there is no universal optimization
algorithm that is optimal for all fitness landscapes. For example,
one “no free lunch” theorem (Theorem 1 in [43]) shows
that, for any two optimization algorithms and , the average fitness
obtained by equals the average fitness obtained by , when the average
is calculated over all possible fitness landscapes, sampled with
uniform probability. If my model can reach infinite fitness levels,
for some fitness landscapes, does this violate a “no free lunch”
theorem, since then the average fitness must also be infinite? There
is no problem here, because the “no free lunch” theorems
are concerned with the fitness after a finite number of iterations,
not with the fitness after an infinite number of iterations (in
my case, an infinite number of children).
The model that is presented here is not intended to be a new, superior
form of optimization algorithm. The intent of the model is to show
that it is possible, under certain conditions, for evolutionary
versatility to increase without bound. Furthermore, the model is
intended to show that local selection (in this case, local to a
certain period of time) can drive a global trend (global across
all periods of time) towards increasing evolutionary versatility.
The model is not universal; it will only display unbounded increase
in evolutionary versatility for certain parameter settings and for
certain fitness landscapes. The fitness landscape is defined by
the parameters TARGET_CHANGE_RATE and ERA_LENGTH and by the general
design of the model (Figure 1). Experiments 1 and 2 show that the
model appears to display unbounded evolutionary versatility for
the baseline fitness landscape (the fitness landscape that is defined
by the parameter settings in Table 1), but experiments 4 and 5 show
that there are neighbouring fitness landscapes for which evolutionary
versatility is bounded. The experiments here have only explored
a few of the infinitely many possible fitness landscapes. Of the
fitness landscapes that were explored here, only a few appeared
to display unbounded evolutionary versatility.
9. Conclusions
This paper introduces a simple model of unbounded evolutionary
versatility. The model is primarily intended to address the claim
that natural selection cannot produce a large-scale trend, because
it is a purely local process. The model shows that local selection
can produce a global trend towards increasing evolutionary versatility.
The model suggests that this trend can continue without bound, if
there is sufficient ongoing change in the environment.
For evolutionary versatility to increase without bound, it must
be possible for the lengths of genomes to increase. If there is
a bound on the length of the genomes, then there must be a bound
on the evolutionary versatility. A model of unbounded evolutionary
versatility must therefore allow mutations that occasionally change
the length of a genome. It seems possible that, once genomes reach
a certain length, the benefit that might be obtained from greater
length would be countered by the damage that mutation can do to
the useful genes that have been found so far. At this point, evolutionary
versatility might stop increasing.
To address this issue, the model allows the mutation rate to adapt.
The experiments show that, indeed, if there is little change in
the environment, then the damage of mutation is greater than the
benefit of mutation, so the mutation rate goes to zero and evolutionary
versatility stops increasing. However, if there is sufficient change
in the environment, it appears that the mutation rate reaches a
stable non-zero value and evolutionary versatility continues to
increase indefinitely.
Perhaps the most interesting observation is that the fitness increase
during an era grows over time (see the fourth plots in Figures 2
and 3). That is, increasing evolutionary versatility leads to an
accelerating pace of evolution. One of the most interesting questions
about this model is whether it plausible as a highly abstract model
of the evolution of life on earth. One test of its plausibility
is to look for signs that the pace of organismal evolution is accelerating.
For example, does the fossil record show a decreasing recovery time
from major catastrophes? Is there a decrease in the average lifetimes
of species?
If there is evidence that the pace of evolution is accelerating,
evolutionary versatility may be better able to account for this
than the other seven live hypotheses [21]. It is not clear how any
of the other hypotheses could be used to explain the acceleration,
although it seems to be a natural consequence of increasing evolutionary
versatility.
Acknowledgments
Thanks to the reviewers for their very helpful comments on an earlier
version of this paper. Thanks to Dan McShea for many constructive
criticisms and general encouragement.
References
1. Aboitiz, F. (1991). Lineage selection and the capacity to evolve.
Medical Hypotheses, 36, 155-156.
2. Altenberg, L. (1994). The evolution of evolvability in genetic
programming. In: Advances in Genetic Programming, K. E. Kinnear
Jr., (ed.). MIT Press.
3. Anderson, R.W. (1995). Learning and evolution: A quantitative
genetics approach. Jour-nal of Theoretical Biology, 175, 89-101.
4. Ayala F.J. (1974). The concept of biological progress. In Studies
in the Philosophy of Biology, ed. F.J. Ayala, T. Dobzhansky,19:339-55.
New York: Macmillan.
5. Ayala F.J. (1988). Can “progress” be defined as a
biological concept? In Evolutionary Progress, ed. M Nitecki, pp.
75-96. Chicago: University of Chicago Press.
6. Bäch, T. (1992). Self-adaptation in genetic algorithms.
In F.J. Varela and P. Bourgine (eds.), Towards a Practice of Autonomous
Systems. MIT Press, pp. 263-271.
7. Baldwin, J.M. (1896). A new factor in evolution. American Naturalist,
30, 441-451.
8. Bedau, M.A., and Seymour, R. (1995). Adaptation of mutation
rates in a simple model of evolution. Complexity International,
2.
9. Blickle, T., and Thiele, L. (1995). A Comparison of Selection
Schemes used in Genetic Algorithms. Technical Report No. 11. Gloriastrasse
35, CH-8092 Zurich: Swiss Federal Institute of Technology (ETH)
Zurich, Computer Engineering and Communications Networks Lab (TIK).
10. Blickle, T., and Thiele, L. (1995). A mathematical analysis
of tournament selection, In Proceedings of the Sixth International
Conference on Genetic Algorithms, ICGA-95, L.J. Eshelman, Ed., Morgan
Kaufmann, San Mateo, CA, pp. 9-16.
11. Davis, L. (1989). Adapting operator probabilities in genetic
search. Proceedings of the Third International Conference on Genetic
Algorithms, ICGA-89, Morgan Kaufmann, San Mateo, CA, pp. 61-69.
12. Dawkins, R. (1989). The evolution of evolvability. In: Artificial
Life, C. Langton, (ed.). Addison-Wesley.
13. Dawkins, R. (1996). Climbing Mount Improbable. New York: W.W.
Norton and Co.
14. Fogel, D.B., Fogel, L.J., and Atmar, J.W. (1991). Meta-evolutionary
programming. In R.R. Chen (Ed.), Proceedings of the Twenty-fifth
Asilomar Conference on Signals, Systems, and Computers, pp. 540-545,
California: Maple Press.
15. Gilinsky, N.L., and Good, I.J. (1991). Probabilities of origination,
persistence, and extinction of families of marine invertebrate life.
Paleobiology, 17, 145-166.
16. Gould S.J. (1988). Trends as changes in variance: A new slant
on progress and directionality in evolution. Journal of Paleontology,
62, 319-29.
17. Gould S.J. (1997). Full House: The Spread of Excellence from
Plato to Darwin. New York: Harmony.
18. Hinton, G.E., and Nowlan, S.J. (1987). How learning can guide
evolution. Complex Systems, 1, 495-502.
19. Lewin, R. (1985). Red Queen runs into trouble? Science, 227,
399-400.
20. McShea, D.W. (1996). Metazoan complexity and evolution: Is
there a trend? Evolution, 50, 477-492.
21. McShea, D.W. (1998). Possible largest-scale trends in organismal
evolution: Eight “Live Hypotheses”. Annual Review of Ecology
and Systematics, 29, 293-318.
22. Nitecki, M.H. (1988). Evolutionary Progress. Edited collection.
Chicago: University of Chicago Press.
23. Raup, D.M. (1975). Taxonomic survivorship curves and Van Valen's
Law. Paleobiology, 1, 82-86.
24. Riedl, R. (1977). A systems-analytical approach to macro-evolutionary
phenomena. Quarterly Review of Biology, 52, 351-370.
25. Riedl, R. (1978). Order in Living Organisms: A Systems Analysis
of Evolution. Translated by R.P.S. Jefferies. Translation of Die
Ordnung Des Lebendigen. New York: Wiley.
26. Ruse, M. (1996). Monad to Man: The Concept of Progress in Evolutionary
Biology. Massachusetts: Harvard University Press.
27. Simon, H.A. (1962). The architecture of complexity. Proceedings
of the American Philosophical Society, 106, 467-482.
28. Syswerda, G. (1989). Uniform crossover in genetic algorithms.
Proceedings of the Third International Conference on Genetic Algorithms
(ICGA-89), pp. 2-9. California: Morgan Kaufmann.
29. Syswerda, G. (1991). A study of reproduction in generational
and steady-state genetic algorithms. Foundations of Genetic Algorithms.
G. Rawlins, editor. Morgan Kaufmann. pp. 94-101.
30. Turney, P.D. (1989). The architecture of complexity: A new
blueprint. Synthese, 79, 515-542.
31. Turney, P.D. (1996). How to shift bias: Lessons from the Baldwin
effect. Evolutionary Computation, 4, 271-295.
32. Turney, P.D. (1999). Increasing evolvability considered as
a large-scale trend in evolution. In A. Wu, ed., Proceedings of
1999 Genetic and Evolutionary Computation Conference Workshop Program
(GECCO-99 Workshop on Evolvability), pp. 43-46.
33. Van Valen, L. (1973). A new evolutionary law. Evolutionary
Theory, 1, 1-30.
34. Vermeij, G.J. (1970). Adaptive versatility and skeleton construction.
American Naturalist, 104, 253-260.
35. Vermeij, G.J. (1971). Gastropod evolution and morphological
diversity in relation to shell geometry. Journal of Zoology, 163,
15-23.
36. Vermeij, G.J. (1973). Biological versatility and earth history.
Proceedings of the National Academy of Sciences of the United States
of America, 70, 1936-1938.
37. Vermeij, G.J. (1974). Adaptation, versatility, and evolution.
Systematic Zoology, 22, 466-477.
38. Wagner, G.P. and Altenberg, L. (1996). Complex adaptations
and the evolution of evolvability. Evolution, 50, 967-976.
39. Whitley, D., and Kauth, J. (1988). GENITOR: A different genetic
algorithm. Proceedings of the Rocky Mountain Conference on Artificial
Intelligence, Denver, CO. pp. 118- 130.
40. Whitley, D. (1989). The GENITOR algorithm and selective pressure.
Proceedings of the Third International Conference on Genetic Algorithms
(ICGA-89), pp. 116-121. California: Morgan Kaufmann.
41. Whitley, D., Dominic, S., Das, R., and Anderson, C.W. (1993).
Genetic reinforcement learning for neurocontrol problems. Machine
Learning, 13, 259-284.
42. Williams, G.C. (1966). Adaptation and Natural Selection. New
Jersey: Princeton University Press.
43. Wolpert, D.H., and Macready, W.G. (1997). No free lunch theorems
for optimization. IEEE Transactions on Evolutionary Computation,
1, 67-82.
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