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356
CHEMISTRY & BIODIVERSITY – Vol. 6 (2009)
Simulations in Evolution. II. Relative Fitness and the Propagation of Mutants
by Bernard Testa* a ) and Andrzej J. Bojarski b )
a
) Department of Pharmacy, University of Lausanne, Hospital Centre, BH04-CHUV, Rue du Bugnon 41,
CH-1011 Lausanne (phone: þ 41 21 314 4237; fax þ 41 21 314 4992; e-mail: [email protected])
b
) Department of Medicinal Chemistry, Institute of Pharmacology of the Polish Academy of Sciences,
12 Smetna St., PL-31343 Krakow
In Neo-Darwinism, variation and natural selection are the two evolutionary mechanisms which
propel biological evolution. Our previous article presented a histogram model [1] consisting in
populations of individuals whose number changed under the influence of variation and/or fitness, the
total population remaining constant. Individuals are classified into bins, and the content of each bin is
calculated generation after generation by an Excel spreadsheet. Here, we apply the histogram model to a
stable population with fitness F1 ¼ 1.00 in which one or two fitter mutants emerge. In a first scenario, a
single mutant emerged in the population whose fitness was greater than 1.00. The simulations ended
when the original population was reduced to a single individual. The histogram model was validated by
excellent agreement between its predictions and those of a classical continuous function (Eqn. 1) which
predicts the number of generations needed for a favorable mutation to spread throughout a population.
But in contrast to Eqn. 1, our histogram model is adaptable to more complex scenarios, as demonstrated
here. In the second and third scenarios, the original population was present at time zero together with two
mutants which differed from the original population by two higher and distinct fitness values. In the
fourth scenario, the large original population was present at time zero together with one fitter mutant.
After a number of generations, when the mutant offspring had multiplied, a second mutant was
introduced whose fitness was even greater. The histogram model also allows Shannon entropy (SE) to be
monitored continuously as the information content of the total population decreases or increases. The
results of these simulations illustrate, in a graphically didactic manner, the influence of natural selection,
operating through relative fitness, in the emergence and dominance of a fitter mutant.
1. Introduction. – In a previous article [1], we used histograms to simulate the
evolving distribution of a given phenotypic character in a model population. The initial
distribution of the phenotype was taken to be Gaussian, and it evolved generation after
generation under the influence of the two factors driving evolution, namely variation
and selection.
The first influencing factor was variation, namely random changes in the
distribution of the character which enlarge the variability from which natural selection
can choose. This was modeled by assuming that, during each cycle, each bin lost a fixed
percentage of its content to its two adjacent bins. The resulting histograms became
broader and flatter while retaining their bilateral symmetry. The second influencing
factor was natural selection, which constrains this variability by favoring the survival
and fertility (i.e., fitness) of the better adapted organisms. This was modeled by
assuming a differential fertility factor which increased linearily from the lower bins to
2009 Verlag Helvetica Chimica Acta AG, Zrich
CHEMISTRY & BIODIVERSITY – Vol. 6 (2009)
357
the higher bins. This fertility factor was applied as a multiplication factor during each
cycle. The resulting histograms became skewed to the right.
These two influencing factors were applied separately or in combination. A simple
Excel sheet was used to calculate and plot the histograms, as the model population
changed generations after generations.
Population geneticists have proposed formulae which model the influence of
natural selection on the expansion of a mutation in a population or species. One such
formula allows one to calculate the number of generations ( ¼ Time) needed for a
favorable mutation to spread throughout a population [2 – 4]. This formula (Eqn. 1) is:
Time ¼ 2/s ln(2N)
(1)
where s is the selection coefficient (the difference in fitness between the mutants and
the non-mutants), N is the number of individuals in the population, and ln is the natural
logarithm. With s ¼ 0.01 and N ¼ 10,000, it thus takes 1981 generations for the mutation
to be fixed, i.e., for the mutants to have replaced the original population.
Given this simple formula and the flexibility of our histogram model [1], we
undertook a study aimed at comparing the results of simulations with the predictions of
Eqn. 1. The two influencing factors were those of Eqn. 1, namely the size of the
population and the selection coefficient. Furthermore, the versatility of the histogram
model allows its application to more complex situations – some of which are simulated
here – such as the simultaneous or staggered appearance, and propagation of two
favorable mutations in the population. One confirmatory conclusion to emerge from
the plots is that even minute differences in fitness allow favorable mutants to propagate
and replace an original population within relatively short time periods. Above all, the
graphical display of genetic changes in a population has considerable didactic value,
allowing the influence of variation and/or selection to be visualized and better grasped.
2. Layout of the Study. – A large population of individuals with a given genotype
was assumed. It was also assumed that this genotype was identical in all individuals of
the original population and allowed them to be classified into a single bin in the
histogram (Bin 1). The fitness of these individuals was set at F1 ¼ 1.00 (i.e., 1.00
offspring per individual). In a first scenario, the simulations began when a single
mutant emerged in the population whose fitness was greater than 1.00; this mutant was
classified into a distinct bin (Bin 2) and had a fitness F2 > 1.00. The simulations ended
when the original population was down to a single individual.
In this scenario and the two outlined below, the available resources were assumed to
be constant, and the total population was kept constant generation after generation.
This was achieved by a simple normalization procedure at each generation. Except
when stated differently, there was no rounding off to the nearest unit at each
generation, allowing a closer analogy with Eqn. 1.
In the second and third scenarios, the large original population was present at time
zero together with two mutants which were assumed to differ from the original
population by two higher and distinct fitness values (F3 > F2 > 1.00). These mutants
were classified into Bin 3 and Bin 2, respectively.
358
CHEMISTRY & BIODIVERSITY – Vol. 6 (2009)
In the forth scenario, the large original population was present at time zero together
with one fitter mutant. After a number of generations, when the mutant offspring had
multiplied, a second mutant was introduced whose fitness was even greater.
In the four scenarios, the Shannon entropy (SE) was calculated for each histogram,
and the increase or decrease in SE was monitored over the specified number of
generations. Briefly, the SE is a measure of the relative information content of a data
set [5]. Thus, if, for datasets 1 and 2, one finds that SE(2) < SE(1), this means that data
set 2 contains more information than data set 1. Shannons theory is originally one of
digital communication [6], but it is finding more and more applications in chemistry
and biology [7] [8]. SE was calculated from these histograms using Eqn. 2:
SE ¼ X
pi log2 pi
(2)
i
where pi is the probability in each bin, obtained from the count in each bin (ci ):
c
pi ¼ P i
ci
(3)
i
All calculations (number of individuals in each bin, corresponding SE values) were
carried out automatically in ad-hoc programmed Excel spreadsheets.
3. First Scenario: The Propagation of a More Fertile Mutant. – Here, original
populations of 4999, 9999, 19999, 49999, or 99999 were present at time zero together
with 1 mutant, making a total of 5000, 10000, 20000, 50000, or 100000 individuals. The
fitness of the non-mutants was set at F1 ¼ 1.00. Four fitness factors were explored for the
mutant and its offspring, namely F2 ¼ 1.05, 1.10, 1.15, or 1.20. The simulations were
carried out until the original population had almost vanished, i.e., was down to 1
individual.
The results for a total number of 10000 individuals are presented in Fig. 1. As
expected, the number of generations decreased from 377 to 101 as F2 increased from
1.05 to 1.20. The larger values of F might seem excessive. However, some evolutionary
changes and even speciation events are known to be extremely rapid [9 – 13], and our
aim here was to cover a broad range of scenarios.
The variations in SE are comparable for all values of F2 and are shown in Fig. 1, a – d,
below the respective populations plot. As seen, the information content is maximal
when one population predominates overwhelmingly, and is minimal (i.e., SE ¼ 1.00)
when the two populations are equal in number.
A comparison between the results of the histogram model and those of Eqn. 1 is
provided in the Table. The most conspicuous result of this comparison is that the
differences between the two models are globally small or very small. A discrete trend is
visible such that the predictions of Eqn. 1 are very slightly larger than those of the
histogram model for a small selection factor (s ¼ 0.05), the difference decreasing as the
selection factor increases (s ¼ 0.10). The predictions of Eqn. 1 then become marginally
smaller than those of the histogram model for the larger values of the selection factor
(s ¼ 0.15 and 0.20). Taken globally, this comparison brings evidence that both models
yield comparable if not similar results.
Fig. 1. The propagation of a favorable mutation with a fitness of 1.05 (a), 1.10 (b), 1.15 (c), and 1.20 (d) in a total population of 10000 individuals. The fitness
of the original population was 1.00. At time zero (Generation 0), the original population was 9999 and the number of mutants was 1. The changes in Shannon
entropy (SE) are shown underneath the corresponding population plots.
CHEMISTRY & BIODIVERSITY – Vol. 6 (2009)
359
CHEMISTRY & BIODIVERSITY – Vol. 6 (2009)
Fig. 1 (cont.)
360
CHEMISTRY & BIODIVERSITY – Vol. 6 (2009)
361
Table. A Comparison between the Histogram Model and the Outcomes of Eqn. 1
Initial populations
Bin A
Bin A:
4999
9999
19999
49999
99999
Bin A:
4999
9999
19999
49999
99999
Bin A:
4999
9999
19999
49999
99999
Bin A:
4999
9999
19999
49999
99999
Number of generations for mutation to be fixed
Bin B
F ¼ 1.00 // Bin B:
1
1
1
1
1
F ¼ 1.00 // Bin B:
1
1
1
1
1
F ¼ 1.00 // Bin B:
1
1
1
1
1
F ¼ 1.00 // Bin B:
1
1
1
1
1
Histogram model a )
F ¼ 1.05 // s ¼ 0.05
349
377
406
444
472
F ¼ 1.10 // s ¼ 0.10
179
193
208
227
242
F ¼ 1.15 // s ¼ 0.15
122
132
142
155
165
F ¼ 1.20 // s ¼ 0.20
93
101
109
119
126
Eqn. 1 b )
Difference
368
396
424
460
488
19
19
18
16
16
184
198
212
230
244
5
5
4
3
2
123
132
141
154
163
1
0
1
1
2
92
99
106
115
122
1
2
3
4
4
a
) The histogram model was initiated with 1 mutant and ended with 1 non-mutant. b ) Eqn. 1 begins with
0% and ends with 100% mutants.
A possible distortion of results caused by the rounding off procedure was
investigated in separate simulations. Here, the total numbers of individuals at
Generation zero were as above, but the number of initial mutants was 30, and their
fitness values were as above. At each generation, the number of individuals in Bins 1
and 2 were either not rounded off, or rounded off to the nearest unit. The simulations
were carried out until the original population was down to 30 individuals. The numbers
of generations determined under these various conditions and with or without rounding
off were practically identical (difference 2).
4. Second and Third Scenarios: The Simultaneous Propagation of Two Competing
Mutants. – In these scenarios, an original population of 9998 individuals was present at
time zero together with two competing mutants. These mutants had distinct and larger
fitness factors than the original population, and they were classified into Bin 2 and
Bin 3. The variables here was the fitness of the two mutants and their offspring.
In the second scenario, the difference between F2 and F3 was taken to be gradually
smaller. Fig. 2 depicts the plots obtained by keeping the fitness of the mutants in Bin 3
constant (F3 ¼ 1.10), and varying the fitness of the mutants in Bin 2 (F2 ) from 1.05 to
362
CHEMISTRY & BIODIVERSITY – Vol. 6 (2009)
1.099. Thus, the selection factors differentiating the two mutations were 0.05, 0.020,
0.008, and 0.001. The resulting population plots show how the fitter mutation markedly
dominated over the lesser fit one even when the difference in fitness was as small as
0.001. The corresponding variations in SE are shown underneath the corresponding
population plots, confirming that information was minimal when the three bins were
equally populated, leveled off while the two mutant populations competed once the
original population had vanished, and slowly tended to zero again when the fitter
mutation took the overhand.
In the third scenario, the fitness factors F2 and F3 were both set at different values,
with the ratio (F3 F1)/(F2 F1) remaining constant. Thus, F2 and F3 equalled,
respectively, 1.045 and 1.05, 1.09 and 1.10, 1.18 and 1.20, and 1.27 and 1.30. As seen in
Fig. 3, the practical outcome of these variations was simply to shorten the time needed
for the mutants to overtake the original population, then for the fitter mutants to
replace the lesser fit ones. Thus, Fig. 3, a, appears as the expanded first part of Fig. 3, b –
d, since, for example, the situation reached after 193 generations in Fig. 3, a, took only
36 generations in Fig. 3, d. The same is obviously true for the SE plots.
5. Fourth Scenario: The Staggered Propagation of Two Competing Mutants. – This
scenario explored a more complex situation, at the same time demonstrating the
illustrative and didactic value of the histogram model. Here, the original population of
9999 individuals (F1 ¼ 1.00) was present at time zero together with one fitter mutant
(F2 ¼ 1.10). After 99 generations, the original population was down to 4439 individuals,
and the first mutants had reached 5561 individuals. At the next generation, a new
mutant was introduced (Bin 3) whose fitness was F3 ¼ 1.20, the total population was
normalized to 10,000 individuals, and the simulation was continued.
The complex variation in the three populations is seen in Fig. 4, a, where the period
before Generation 99 was identical to that in the corresponding part of Fig. 1, b. From
Generation 100 on, however, the fitter mutant in Bin 3 grew rapidly and forced the
lesser-fit mutants in Bin 2 to start declining after 146 generations. After 253
generations, the fitter mutants made up > 99% of the total number of individuals.
The resulting variations in SE are shown in Fig. 4, b; they reached a maximal value
(SE ¼ 1.00), when two populations were in equal numbers and one population was
almost nil, and minimal values when one population predominated vastly.
6. Conclusions. – The first objective of this study was to check the consistency of the
histogram model with the classical Eqn. 1. As shown in the Table, the discrepancies
between the two outcomes are minute or negligible. In our opinion, they result from a
fundamental difference between the two approaches, in the sense that Eqn. 1 is a
continuous model, whether the histogram model is based on a discontinuous (i.e., stepby-step) iteration. The close similarity between the classical Eqn. 1 and the histogram
model thus validates the latter.
The second and major objective of the work was to simulate the influence of natural
selection in more complex scenarios, and to illustrate these in a graphically didactic
manner. Highly simplified scenarios were chosen, namely a stable population (F ¼ 1.00)
challenged by the emergence of one or two fitter mutations. The first scenario (one
fitter mutation) is the one modeled by Eqn. 1 and exemplified in Fig. 1, a – d. In more
Fig. 2. The propagation of two favorable mutations appearing simultaneously in a population of fitness F ¼ 1.00. One mutation had a fitness F2 of 1.050 (a),
1.080 (b), 1.092 (c), and 1.099 (d), the other a constant fitness F3 ¼ 1.10. The total population was 10000 individuals. At time zero (Generation 0), the original
population was 9998 and the number of mutants was 1 þ 1. The changes in SE are shown underneath the corresponding population changes.
CHEMISTRY & BIODIVERSITY – Vol. 6 (2009)
363
CHEMISTRY & BIODIVERSITY – Vol. 6 (2009)
Fig. 2 (cont.)
364
Fig. 3. The propagation of two favorable mutations appearing simultaneously in a population of fitness F ¼ 1.00. The respective fitness factors of the two
mutations (F2 and F3 ) were set at different values, the ratio (F3 – F1)/(F2 – F1) remaining constant. Thus, F2 and F3 equalled, respectively, 1.045 and 1.05 (a),
1.09 and 1.10 (b), 1.18 and 1.20 (c), and 1.27 and 1.30 (d). The total population was 10,000 individuals. At time zero (Generation 0), the original population
was 9998, and the number of mutants was 1 þ 1. The changes in SE are shown underneath the corresponding population changes.
CHEMISTRY & BIODIVERSITY – Vol. 6 (2009)
365
CHEMISTRY & BIODIVERSITY – Vol. 6 (2009)
Fig. 3 (cont.)
366
CHEMISTRY & BIODIVERSITY – Vol. 6 (2009)
367
Fig. 4. The propagation of two favorable mutations appearing in a staggered fashion in a population of
fitness F ¼ 1.00 (a). The total population was 10,000 individuals. At time zero (Generation 0), the
original population was 9999, and the number of mutants in Bin 2 was 1 (F2 ¼ 1.10). At Generation 100, a
second mutant (1 individual; F3 ¼ 1.20) was introduced in the simulation. The corresponding changes in
SE are presented in b.
complex scenarios, two fitter mutations were introduced, either simultaneously (Figs. 2
and 3), or in a staggered manner (Fig. 4). In these scenarios, the appearance of a
mutation was part of the settings and not a monitored outcome. In other words, the
focus of this article was exclusively on the influence of natural selection independently
of variation understood as the results of the many processes generating new genotypes
and phenotypes. Future simulations should explore the influence of random variations.
REFERENCES
[1] B. Testa, A. J. Bojarski, Chem. Biodiversity 2007, 4, 2458.
[2] M. Kimura, T. Ohta, Genetics 1969, 61, 763.
[3] S. B. Carroll, Endless Forms Most Beautiful, W. W. Norton & Co., New York, 2006.
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CHEMISTRY & BIODIVERSITY – Vol. 6 (2009)
[4] S. B. Carroll, The Making of the Fittest, Quercus, London, 2008.
[5] C. Shannon, Bell Syst. Tech. J. 1948, 27, 379 and 623; http://cm.bell-labs.com/ms/what/shannonday/
paper.html.
[6] S. W. Golomb, Science (Washington, DC) 2001, 292, 455.
[7] A. J. Bojarski, M. Nowak, B. Testa, Cell. Mol. Life Sci. 2003, 60, 2526 – 2531.
[8] A. J. Bojarski, M. Nowak, B. Testa, Chem. Biodiversity 2006, 3, 245.
[9] J. Maynard Smith, E. Szathmáry, The Origins of Life, Oxford University Press, Oxford, 2000.
[10] E. Mayr, What Evolution Is, Orion Books, London, 2002.
[11] C. de Duve, Life Evolving, Oxford University Press, Oxford, 2002.
[12] R. Dawkins, The Ancestors Tale, Weidenfeld & Nicolson, London, 2004.
[13] B. S. Guttman, Evolution, Oneworld Publications, Oxford, 2005.
Received October 10, 2008
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