## Saturday, February 27, 2016

### Felsenstein: Wright, Fisher, and the Weasel

Joe Felsenstein has posted a fine explanation of how Richard Dawkins’s monkey/Shakespeare model of cumulative selection, better known as the Weasel program, relates to the classic Wright-Fisher model in population genetics. The relation is approximate, because a finite set of offspring is sampled in the monkey/Shakespeare model, while an infinite set of offspring is sampled in the Wright-Fisher model. Joe raised the question of just how well certain results derived for a special case of Wright-Fisher apply to a Weasel-ish evolutionary process. And it happens that I had on hand some code that was easily adapted to address the question. I’m sharing it here, without explanation, and inviting readers to join the discussion at The Skeptical Zone.

### A nice illustration of how nice Numeric Python array processing is…

… not that I’m going to explain it.

import numpy as np
import numpy.random as rand
import matplotlib.pyplot as plt

LENGTH=28
ALPHABET_SIZE=27
N_OFFSPRING=100

class Evolve(object):
"""
Model somewhat like Wright-Fisher, with Dawkins's Weasel as a special case.

See Felsenstein (2016), "Wright, Fisher, and the Weasel," The Skeptical
Zone, http://theskepticalzone.com/wp/wright-fisher-and-the-weasel/.
"""
def __init__(self, alphabet_size=ALPHABET_SIZE, n_offspring=N_OFFSPRING,
mutate_rate=1.0/LENGTH, selection=0.0):
"""
Initialize evolutionary run.
"""
self.alphabet_size = alphabet_size
self.mutate_rate = mutate_rate
self.selection = float(selection)
self.probability = np.empty(n_offspring, dtype=float)
self.offspring = np.empty(n_offspring, dtype=int)
self.n_match = rand.binomial(LENGTH, 1.0 / alphabet_size)
self.history = [self.n_match]

def generation(self):
"""
Extend evolutionary run by one generation.

Probability of selecting an offspring is proportional to its fitness
(1+selection)**n_match, where n_match is the number of characters
matching the target.
"""
n = len(self.offspring)
loss_rate = self.mutate_rate
gain_rate = self.mutate_rate / (self.alphabet_size - 1.0)
self.offspring[:] = self.n_match
self.offspring += rand.binomial(LENGTH - self.n_match, gain_rate, n)
self.offspring -= rand.binomial(self.n_match, loss_rate, n)
np.power(self.selection + 1.0, self.offspring, self.probability)
self.probability /= np.sum(self.probability)
self.n_match = rand.choice(self.offspring, size=1, p=self.probability)
self.history.append(self.n_match)

def extend(self, n_generations=10000):
"""
Extend evolutionary run by n_generations generations.
"""
for _ in xrange(n_generations):
self.generation()

def get_history(self, begin=0, end=None):
"""
Return list of numbers of matching sites for specified generations.
"""
return self.history[begin:end]

def report(self, begin=1000):
"""
Report on some quantities addressed by Felsenstein (2016).
"""
u = self.mutate_rate
s = self.selection
q = u / (1 + s * (1 - u))
p = u * (1 + s) / (self.alphabet_size - 1 + u * s)
h = self.history[begin:]
print '*'*72
print 'Sentence length L               :', LENGTH
print 'Alphabet size A                 :', self.alphabet_size
print 'Number of offspring             :', len(self.offspring)
print 'Mutation rate u                 :', u
print 'Selection parameter s           :', s
print 'p = u * (1 + s) / (A-1 + u * s) :', p
print 'q = u / (1 + s * (1 - u))       :', q
print 'Assume equilibrium at generation:', begin
print 'Theoretical prob. of match at site p/(p+q)   :', p/(p+q)
print 'Expected number of matching sites Lp/(p+q)   :', LENGTH*p/(p+q)
print 'Mean of observed numbers of matching sites   :', np.mean(h)
print 'Std. dev. of observed nums. of matching sites:', np.std(h)
print '*'*72
plt.plot(self.history)
m = np.mean(h)
plt.plot((0, len(self.history)-1), (m, m), 'k-')
m = LENGTH * p/(p+q)
plt.plot((0, len(self.history)-1), (m, m), 'k--')

def example():
"""
Plot numbers of matching sites for runs varying in just one parameter.
"""
n_gens = 100000
results = []
for s in [0, 2, 5, 15]:
e = Evolve(n_offspring=1000, selection=s, alphabet_size=27)
e.extend(n_gens)
print 'selection (s):', s
e.report(begin=5000)
results.append(e)
plt.title('Weasel-ish Runs with Selection Parameter s in {0, 2, 5, 15}')
plt.ylabel('Matching Characters')
plt.xlabel('Generations')
plt.show()
return results

results=example()
# plt.close()


### Clunky, but effective

Here’s the textual output corresponding to the four plots in the figure above, going from bottom to top (blue, green, red, cyan). The idea of this output is to make sure that I’m on the same page with other folks working on the problem.

selection (s): 0
************************************************************************
Sentence length L               : 28
Alphabet size A                 : 27
Number of offspring             : 1000
Mutation rate u                 : 0.0357142857143
Selection parameter s           : 0.0
p = u * (1 + s) / (A-1 + u * s) : 0.00137362637363
q = u / (1 + s * (1 - u))       : 0.0357142857143
Assume equilibrium at generation: 5000
Theoretical prob. of match at site p/(p+q)   : 0.037037037037
Expected number of matching sites Lp/(p+q)   : 1.03703703704
Mean of observed numbers of matching sites   : 1.02184187535
Std. dev. of observed nums. of matching sites: 0.97972454909
************************************************************************
selection (s): 2
************************************************************************
Sentence length L               : 28
Alphabet size A                 : 27
Number of offspring             : 1000
Mutation rate u                 : 0.0357142857143
Selection parameter s           : 2.0
p = u * (1 + s) / (A-1 + u * s) : 0.0041095890411
q = u / (1 + s * (1 - u))       : 0.0121951219512
Assume equilibrium at generation: 5000
Theoretical prob. of match at site p/(p+q)   : 0.252049180328
Expected number of matching sites Lp/(p+q)   : 7.05737704918
Mean of observed numbers of matching sites   : 7.03611540931
Std. dev. of observed nums. of matching sites: 2.26819646257
************************************************************************
selection (s): 5
************************************************************************
Sentence length L               : 28
Alphabet size A                 : 27
Number of offspring             : 1000
Mutation rate u                 : 0.0357142857143
Selection parameter s           : 5.0
p = u * (1 + s) / (A-1 + u * s) : 0.00818553888131
q = u / (1 + s * (1 - u))       : 0.00613496932515
Assume equilibrium at generation: 5000
Theoretical prob. of match at site p/(p+q)   : 0.571595558153
Expected number of matching sites Lp/(p+q)   : 16.0046756283
Mean of observed numbers of matching sites   : 15.9726634456
Std. dev. of observed nums. of matching sites: 2.5666185402
************************************************************************
selection (s): 15
************************************************************************
Sentence length L               : 28
Alphabet size A                 : 27
Number of offspring             : 1000
Mutation rate u                 : 0.0357142857143
Selection parameter s           : 15.0
p = u * (1 + s) / (A-1 + u * s) : 0.021534320323
q = u / (1 + s * (1 - u))       : 0.00230946882217
Assume equilibrium at generation: 5000
Theoretical prob. of match at site p/(p+q)   : 0.903141702516
Expected number of matching sites Lp/(p+q)   : 25.2879676704
Mean of observed numbers of matching sites   : 25.1902401027
Std. dev. of observed nums. of matching sites: 1.54315605955
************************************************************************