The aim of this interactive class is to demonstrate how replicants that can inherit mutations can undergo evolution by natural selection if the inheritable characteristics can affect their reproductive success.
Each student was provided with a deck of square cards showing an empty \(8 \times 8\) grid. This deck of cards was available to the students for the whole duration of the class.
The experiment consists of repeating a procedure called a generation several times. Each generation consists of the following three steps:
Each student was provided with a template card, on which some cells of the grid were crossed with an X, forming a pattern. For example, a pattern may look like this:
However, the template card was provided face down, so that the students could not see the pattern.
During the memorization phase, the students were allowed to look at the pattern on their template card for \(10\) seconds and asked to try to memorize it. At the end of the \(10\) seconds, the students had to turn the template card face down again.
As soon as they had turned the template face down, they had to copy the observed pattern onto the empty cards they were provided (without being able to look at the template again). They were asked to try to create as many correct copies of the template as possible over \(30\) seconds. They were told that translations or rotations of the pattern on the grid count as correct replicas. For example, the following three cards display the same pattern.
The templates were collected into a pile called the “generation record”, which was put aside. Instead, the replicas were all put into a bag. The number of replicas largely exceeded the number of students. Each student was then randomly assigned a new template card randomly drawn from the bag. The remaining cards in the bag were discarded.
Once each student has a new template, a new generation can start. The process is repeated until a desired number of generations is reached.
For each generation, the pile of template cards was collected (the “generation record”). Therefore, we had one pile of \(N\) cards per generation (where \(N\) is the number of students in the class). For each pile, the number of copies of each type of pattern was counted. Translated or rotated patterns count as the same pattern (see Fig 3).
The obtained counts were organized into a table.
The class was composed of \(24\) students (\(N=24\)). At generation 0, the “population” of template cards that had been distributed to the students was composed of the following three patterns (8 copies each):
We named the patterns with the letters A, B and C.
The results of the experiment are summarized in the following table.
| Generation | A | B | C |
|---|---|---|---|
| 0 | 8 | 8 | 8 |
| 1 | 16 | 1 | 2 |
| 2 | 21 | 1 | 0 |
We can also visualize the number of A, B and C over generations through a line plot as follows.
import matplotlib.pyplot as plt
plt.rcParams['figure.figsize'] = [5, 2.5]
plt.plot([8, 16, 21], label='A')
plt.plot([8, 1, 1], label='B')
plt.plot([8, 2, 0], label='C')
plt.xticks(list(range(3)))
plt.xlabel('generation')
plt.ylabel('number of individuals')
plt.grid()
plt.legend()
plt.show()
👀 The students were asked to comment on the results shown in the table. They made two observations:
Observation 1:
The number of A cards increased each generation, while the number of B and C types decreased very quickly.
Observation 2:
The total number of A, B and C cards in generations \(1\) and \(2\) is no longer \(24\), even though in each generation one card was assigned to each student (so \(24\) in total).
💡 Students were asked to explain the observations.
Interpretation of Observation 1 (Hypothesis 1):
The students noted that reproducing pattern A was easier than reproducing patterns B or C. More copies of A were generated in each generation and, therefore, the bag contained more copies of A than the other types of patterns. As a consequence, students were more likely to receive an A card in the next generation, when cards were randomly drawn from the bag.
Interpretation of Observation 2 (Hypothesis 2):
One student suggested that, due to copying errors, some cards in generations \(1\) and \(2\) may not correspond to any of the three original patterns, A, B and C. That could explain why the count of A+B+C was no longer \(24\).
🔍 We then tested the students’ explanations.
Testing Hypothesis 1
By looking at cards that went into the bag at the end of generation 0, which had been kept aside, we could test Hypothesis 1. The majority of the cards were of type A, confirming that more copies of A were produced, compared to B and C cards, although the templates were 8 per type and they all had the same amount of replication time. Hypothesis 1 was confirmed.
Testing Hypothesis 2
By inspecting the piles of cards for generations \(1\) and \(2\) (the “generation records”), we found some cards that didn’t correspond to any of the three original patterns A, B and C. These cards were termed mutants. The mutants were \(5\) in generation 1 and \(2\) in generation 2 (see Mutants, below). When taking the mutants into account, the total number of cards per generation was indeed \(24\). Hypothesis 2 was confirmed.
Thus, we updated the table to account for the mutants. This way, all the cards were counted at each generation. For simplicity, we categorized all the mutants under a single class named X.
| Generation | A | B | C | X |
|---|---|---|---|---|
| 0 | 8 | 8 | 8 | 0 |
| 1 | 16 | 1 | 2 | 5 |
| 2 | 21 | 1 | 0 | 2 |
We concluded that the frequency of each type of pattern can change across generations due to chance (after all, the cards composing each new generation are randomly sampled from the bag) but also partly due to another logical principle. The principle is the following: if a pattern is more successful at being replicated than the others, its frequency in the population is more likely to increase. Now that we understand these two principles acting on the frequencies of the competing replicants, we are ready to name them according to the scientific literature:
The overall process that the population of \(N\) replicants underwent, where the frequencies of the different types change over generations, is called EVOLUTION.
We discussed that evolution is not completely predictable because chance is involved in several parts of the process. However, some scenarios may be more/less predictable than others. As an exercise, try to think about the role that the population size \(N\) plays. For example, compare the scenarios where \(N\) is very small/very large. Which one will evolve in a more predictable way?
The balance between genetic drift and natural selection (see Conclusions to check the terminology) can also depend on the difference in reproductive success (evolutionary biologists use the term fitness) between the different types of replicants. Compare the scenarios where the types are very similar/very different in reproductive success. Which one will evolve in a more predictable way?
The experiment we performed with the replicards is a demonstration of how evolution by natural selection works. However, it has two important limitations: 1. We only run a couple of generations, but it would be interesting to follow the evolution of the replicards through many generations. 2. We only experimented with a given initial condition (patterns A, B and C in equal frequencies), but it would be interesting to test patterns that are more different or more similar to each other in terms of their reproductive success.
To that end, we can use computer simulations to get very quick empirical answers to our questions.
We can use the programming language Python to run computer simulations of the replicards’ evolution, by importing the public Python library for the replicards (the source code is here). The code provides the following types of custom Python classes
Organism : objects of this class represent the individuals composing the population (like the replicards); they can reproduce (i.e., create a copy of themselves).Population : this is a handy class that can be used to generate reports of the evolution of a population; it can visually summarize the results of a simulation with a plot.as well as the create function
create : a function that can be used to generate the initial population. It requires to specify three parameters: (1) the type of organism, i.e., whatever name that identifies that type; (2) the (expected) reproductive success of that type of organism; (3) the number of organisms of that type that must be created.We import the needed code from the replicards library, as well as the random library as follows
import random
from replicards import Population, createWe can now set the initial condition (the population at “Generation 0”) for our evolutionary simulation.
First, we set how many “organisms” per type are present at the beginning. Here we simulate the in-class experiment by setting three patterns, 8 copies each, for a total population size of \(N=24\).
n_a = 8 # Initial number of A
n_b = 8 # Initial number of B
n_c = 8 # Initial number of CNext, we can decide how easy it will be to replicate each type of pattern, by setting the expected (average) number of copies each pattern get at each generation.
w_a = 12 # Expected number of offspring for A per generation
w_b = 8 # Expected number of offspring for B per generation
w_c = 4 # Expected number of offspring for C per generationWe are now ready to run a computer simulation.
# Initialize population
pop = Population(create('A', w_a, n_a) +
create('B', w_b, n_b) +
create('C', w_c, n_c))
# Population size
N = pop.size
print("The population size is N =", N)
# Keep evolving until fixation is reached
while not pop.fixation:
# Reproduction based on fitness
pop_offspring = []
for org in pop.organisms:
pop_offspring += org.reproduce()
parents_and_offspring = pop.organisms + pop_offspring
# Maintain population size constant at N, the environment's carrying capacity
new_pop = random.sample(parents_and_offspring, N)
# Update the population
pop.update(new_pop)
# Plot results of the evolutionary simulation
pop.plot_evolution()The population size is N = 24
The graph displays how many individuals of each type are present in each generation, while the total population size remains constant. The simulation stopped when one of the types took over the population completely.
If you are running this as a Jupyter Notebook*, you can go back to the Settings and change the parameters that specify the initial conditions. In particular, if you have thought about the Homework questions, it is time for you to put your thoughts to the test! For example, to get the answer to Question 1, you can change the population size to make it very big or very small, and run the simulation several times to see if results become more consistent or more unpredictable. Similarly, for Question 2 you can play with the w_a, w_b and w_c parameters, which control the reproductive success (the “fitness”) of each type. You can make it more unequal (for example, 12, 2, 1) or more similar to each other (for example, 12, 11, 10).
*The Jupyter Notebook can be downloaded here.
When running the same simulation (with the same initial population) several times, we observed different outcomes. Why is that?
In the code, the w_a, w_b and w_c parameters define the average number of offspring per generation. However, the exact number of offspring is randomly assigned (if you are statistically minded, the precise way this is done is by sampling from a Poisson distribution). Moreover, at each generation, \(N\) individuals are randomly sampled (to keep the population size constant). These random processes make the evolution to some extent unpredictable. Thus, they simulate the phenomenon that we called genetic drift (see Conclusions to check the terminology).
The computer simulations can help us demonstrate empirically the balance between random chance and natural selection in evolving populations (see the paragraph “Run your own experiments”).
However, a limitation of our computer simulations is that the replication occurs without errors. No mutations are introduced, and therefore, no new types. On the contrary, the experiment performed by the students with the replicards included accidental mutations, allowing for the appearance of mutants.
What is a mutation?
We reasoned that not all modifications qualify as mutations. For example, a card got bent during the replication process. However, when being used as a template, the copies of that card (the “offspring”) were not bent because the students were only copying the pattern of X marks, not every possible physical feature. Evolution acts on mutations, which are not any kind of modifications but rather the modification of inheritable traits. In the case of the replicards, the inheritable traits are the positions of the X marks, which are a form of digital information. Actual living beings don’t carry grids of X marks, but rather a sequence of nucleotides making a long molecule called DNA, which is also a form of digital information. Modifying other parts of the organism would constitute a modification but not a mutation. For example, if I lose a finger in an accident, my children will still be born with 10 fingers, not 9. Instead, if the DNA sequence in my “germline” cells gets mutated in some parts (genes) related to finger development (for example, HoxA and HoxD genes), my offspring could have a number of fingers different from 10, even if I have 10 fingers (as in the case of people with polydactyly).
What do you think could happen if we were to continue the replicards protocol for a very very large number of generations?
A student suggested that eventually the population would be taken over by a replicard containing a single X. This pattern could be produced if, at the end of the 30 seconds of the replication phase, a student had just started making a new copy of the template. The prematurely interrupted replication process would leave the card filled with only one X mark. If this card is lucky enough to be sampled from the bag, it could end up dominating the population due to its high reproductive success (high fitness). This high success would be simply due to the fact that many more copies of it could be made in 30 seconds, compared to other more complicated patterns.
By inspecting the patterns found in the mutants (see the Mutants paragraph), we can infer their phylogenetic tree, which is a convenient way to represent how the types are related, analogous to family trees. In the tree below, we can see that all the mutants were descendants of pattern C. This is not too surprising because we reasoned that pattern C was the hardest to replicate and so the easiest to get wrong when trying to copy it. The mutants C\(\alpha\), C\(\beta\) and C\(\gamma\) appeared at generation 1. We can also see that patterns C\(\alpha\) and C\(\gamma\) went extinct at generation 2.
The replicards were \(7 \times 7\) square grids (of size 7 cm \(\times\) 7 cm), printed on thick paper. The X marks on the cells were drawn by the students using pencils.
We have made a PDF file containing blank square grids and grids with the three different patterns to be printed if needed.
The code was written in Python and is available at https://github.com/eliamascolo/replicards.
Replicards package: conda or pip
We would like to thank the students of classes 3°A liceo scientifico and 1°Cl liceo classico (academic year 2024-2025) of the Istituto Salesiano S. Ambrogio in Milan and their science teacher Simone Assanelli for crash-testing our Replicards game. The results in this notebook were produced by these students.