variant_demo.ipynb

variant_demo.md

Some setup first...

# ye olde imports
import math, random
from statistics import mean, geometric_mean

Imagine a virus which is, somehow, a single protein with only 3 "amino acids" (actually just uppercase letters here, oh well)

Currently, only one type is circulating, which is "XYZ".

EXISTING_TYPE = "XYZ"

But there's a possible variant, the Super Nasty Variant, "BAD", which escapes antibodies and is hard to even create good antibodies against (meaning even a vaccine targeted specifically at it has trouble):

SUPER_NASTY_TYPE = "BAD"

Now we'll run a comically oversimplified simulation of evolution: starting from a population of 100 copies of the existing virus (XYZ)...

POPULATION_SIZE = 100

def make_starting_population(): return [EXISTING_TYPE] * POPULATION_SIZE

Each generation, every virus makes a copy of itself (to form a population twice as big), possibly mutating in the process.

Then, each virus may die (in this case, be neutralized by antibodies or whatever); the odds of survival each generation is the virus's fitness.

Note that the steady-state here happens for viruses with fitness of 0.5 (they make a copy of themselves each generation, but on average the original or the copy dies).

def evolve(starting_population, fitness):
    population = starting_population

    new_population = [virus for virus in population] 
    
    
    # each virus makes a copy of itself, maybe mutating...
    for virus in population:
        new_population.append(maybe_mutate(virus))

    # but then some maybe die, based on their fitness:
    population = []
    
    for virus in new_population:
        if fitness(virus) > random.random():
            # virus survived
            population.append(virus)
            
    if not population:
        population = [EXISTING_TYPE] # goofy hack so that the virus never completely dies out

    return population

MAX_GENERATIONS = 5000 # timeout to keep things sensible
def run_experiment(fitness_fn, quiet=True):
    population = make_starting_population()

    if not quiet: print("Generation\tPop. size\tAvg. fitness\tMax. fitness")
    for generation in range(MAX_GENERATIONS):
        if generation % 100 == 0:
            fitnesses = [fitness_fn(virus) for virus in population]
            avg_fitness = geometric_mean(fitnesses)
            max_fitness = max(fitnesses)
            if not quiet: print("{:5}\t\t{:8}\t{:5.4f}\t\t{:5.4f}".format(generation, len(population), avg_fitness, max_fitness))

        population = evolve(population, fitness_fn)
        if SUPER_NASTY_TYPE in population:
            if not quiet: print("Super Nasty Variant evolved in {} generations!".format(generation))
            return generation
        
    return MAX_GENERATIONS # didn't evolve at all, time ran out

def run_many_experiments(fitness_fn, num_experiments):
    results = []
    for experiment in range(num_experiments):
        generations_to_nasty = run_experiment(fitness_fn, quiet=True)
        results.append(generations_to_nasty)
        if generations_to_nasty != MAX_GENERATIONS:
            print("Experiment #{}: Super Nasty Variant evolved in {} generations".format(experiment, generations_to_nasty))
        else:
            print("Experiment #{}: Super Nasty Variant never evolved".format(experiment))
    print("On average, the variant showed up at generation {}".format(mean(filter(lambda r: r is not None, results))))

In this model, "mutations" are just random amino acid substitutions:

ERROR_RATE = 1e-3 # 1/1000 chance per amino acid to replace it with a random one

def random_letter():
    return chr(ord("A")+random.randint(0, 26))

def maybe_mutate(virus):
    # random substitutions
    newvirus = virus
    
    for i in range(len(newvirus)):
        if random.random() < ERROR_RATE:
            # mutate this position
            newvirus = newvirus[:i] + random_letter() + newvirus[i+1:]
    return newvirus

If each of the individual 3 substitutions (X->B, Y->A, Z->D) required to get from the existing type to the Super Nasty Type confers a small advantage on its own...

SINGLE_SUBSTITUTION_FITNESS_GAIN = 1.05 # 5% advantage per substitution

BASE_FITNESS = 0.50001 # just a biiit above steady state...

def fitness_normal(virus):
    num_mutations = 0
    if virus[0] == "B": num_mutations += 1
    if virus[1] == "A": num_mutations += 1
    if virus[2] == "D": num_mutations += 1

    return BASE_FITNESS * pow(SINGLE_SUBSTITUTION_FITNESS_GAIN, num_mutations)

then within a few thousand generations, the Super Nasty Variant usually shows up:

run_many_experiments(fitness_normal, num_experiments=50)

Experiment #0: Super Nasty Variant evolved in 1241 generations
Experiment #1: Super Nasty Variant evolved in 442 generations
Experiment #2: Super Nasty Variant evolved in 2245 generations
Experiment #3: Super Nasty Variant evolved in 432 generations
Experiment #4: Super Nasty Variant evolved in 324 generations
Experiment #5: Super Nasty Variant evolved in 2128 generations
Experiment #6: Super Nasty Variant never evolved
Experiment #7: Super Nasty Variant never evolved
Experiment #8: Super Nasty Variant evolved in 437 generations
Experiment #9: Super Nasty Variant evolved in 435 generations
Experiment #10: Super Nasty Variant evolved in 2282 generations
Experiment #11: Super Nasty Variant evolved in 1179 generations
Experiment #12: Super Nasty Variant evolved in 2802 generations
Experiment #13: Super Nasty Variant never evolved
Experiment #14: Super Nasty Variant evolved in 1586 generations
Experiment #15: Super Nasty Variant evolved in 180 generations
Experiment #16: Super Nasty Variant evolved in 1589 generations
Experiment #17: Super Nasty Variant evolved in 447 generations
Experiment #18: Super Nasty Variant evolved in 1154 generations
Experiment #19: Super Nasty Variant evolved in 2522 generations
Experiment #20: Super Nasty Variant evolved in 2864 generations
Experiment #21: Super Nasty Variant evolved in 479 generations
Experiment #22: Super Nasty Variant evolved in 2289 generations
Experiment #23: Super Nasty Variant evolved in 3444 generations
Experiment #24: Super Nasty Variant evolved in 1355 generations
Experiment #25: Super Nasty Variant evolved in 270 generations
Experiment #26: Super Nasty Variant evolved in 766 generations
Experiment #27: Super Nasty Variant evolved in 2741 generations
Experiment #28: Super Nasty Variant evolved in 548 generations
Experiment #29: Super Nasty Variant evolved in 1571 generations
Experiment #30: Super Nasty Variant evolved in 607 generations
Experiment #31: Super Nasty Variant evolved in 2030 generations
Experiment #32: Super Nasty Variant evolved in 1845 generations
Experiment #33: Super Nasty Variant evolved in 820 generations
Experiment #34: Super Nasty Variant evolved in 264 generations
Experiment #35: Super Nasty Variant evolved in 1435 generations
Experiment #36: Super Nasty Variant never evolved
Experiment #37: Super Nasty Variant evolved in 4584 generations
Experiment #38: Super Nasty Variant evolved in 960 generations
Experiment #39: Super Nasty Variant evolved in 878 generations
Experiment #40: Super Nasty Variant evolved in 306 generations
Experiment #41: Super Nasty Variant evolved in 988 generations
Experiment #42: Super Nasty Variant evolved in 352 generations
Experiment #43: Super Nasty Variant evolved in 486 generations
Experiment #44: Super Nasty Variant evolved in 892 generations
Experiment #45: Super Nasty Variant evolved in 658 generations
Experiment #46: Super Nasty Variant evolved in 4847 generations
Experiment #47: Super Nasty Variant evolved in 4615 generations
Experiment #48: Super Nasty Variant never evolved
Experiment #49: Super Nasty Variant evolved in 2146 generations
On average, the variant showed up at generation 1829.3

But if at least 2 mutations must occur together to get a fitness gain...

def fitness_new(virus):
    num_mutations = 0
    if virus[0] == "B": num_mutations += 1
    if virus[1] == "A": num_mutations += 1
    if virus[2] == "D": num_mutations += 1

    # now, without at least 2 mutations, it's no better than no mutations:
    if num_mutations < 2: num_mutations = 0
        
    return BASE_FITNESS * pow(SINGLE_SUBSTITUTION_FITNESS_GAIN, num_mutations)

... it's much rarer (and slower) for the variant to evolve in the first place, even though it's just as bad when it does show up.

run_many_experiments(fitness_new, num_experiments=50)

Experiment #0: Super Nasty Variant never evolved
Experiment #1: Super Nasty Variant evolved in 4135 generations
Experiment #2: Super Nasty Variant never evolved
Experiment #3: Super Nasty Variant never evolved
Experiment #4: Super Nasty Variant never evolved
Experiment #5: Super Nasty Variant never evolved
Experiment #6: Super Nasty Variant never evolved
Experiment #7: Super Nasty Variant evolved in 1162 generations
Experiment #8: Super Nasty Variant never evolved
Experiment #9: Super Nasty Variant never evolved
Experiment #10: Super Nasty Variant never evolved
Experiment #11: Super Nasty Variant never evolved
Experiment #12: Super Nasty Variant never evolved
Experiment #13: Super Nasty Variant never evolved
Experiment #14: Super Nasty Variant never evolved
Experiment #15: Super Nasty Variant evolved in 2569 generations
Experiment #16: Super Nasty Variant never evolved
Experiment #17: Super Nasty Variant evolved in 2209 generations
Experiment #18: Super Nasty Variant never evolved
Experiment #19: Super Nasty Variant evolved in 4961 generations
Experiment #20: Super Nasty Variant never evolved
Experiment #21: Super Nasty Variant never evolved
Experiment #22: Super Nasty Variant evolved in 4375 generations
Experiment #23: Super Nasty Variant never evolved
Experiment #24: Super Nasty Variant never evolved
Experiment #25: Super Nasty Variant evolved in 1390 generations
Experiment #26: Super Nasty Variant never evolved
Experiment #27: Super Nasty Variant never evolved
Experiment #28: Super Nasty Variant never evolved
Experiment #29: Super Nasty Variant never evolved
Experiment #30: Super Nasty Variant never evolved
Experiment #31: Super Nasty Variant evolved in 1398 generations
Experiment #32: Super Nasty Variant never evolved
Experiment #33: Super Nasty Variant never evolved
Experiment #34: Super Nasty Variant evolved in 2539 generations
Experiment #35: Super Nasty Variant never evolved
Experiment #36: Super Nasty Variant never evolved
Experiment #37: Super Nasty Variant never evolved
Experiment #38: Super Nasty Variant never evolved
Experiment #39: Super Nasty Variant never evolved
Experiment #40: Super Nasty Variant evolved in 2509 generations
Experiment #41: Super Nasty Variant never evolved
Experiment #42: Super Nasty Variant evolved in 4194 generations
Experiment #43: Super Nasty Variant never evolved
Experiment #44: Super Nasty Variant never evolved
Experiment #45: Super Nasty Variant never evolved
Experiment #46: Super Nasty Variant never evolved
Experiment #47: Super Nasty Variant never evolved
Experiment #48: Super Nasty Variant never evolved
Experiment #49: Super Nasty Variant never evolved
On average, the variant showed up at generation 4528.82