In [1]:
import numpy as np
import scipy.optimize
import pandas as pd

import matplotlib.pyplot as plt
import seaborn as sns
sns.set()


## Read in data and plot¶

In [2]:
# load in the data, you will need to change the path to the file
data["ratio"] = data["wildtype / mL"] / data["delta / mL"]

Out[2]:
group number time (hr) time (min) time since start (min) dilution factor µL plated number of wildtype colonies number of delta colonies wildtype / mL delta / mL ratio
0 1 10 30 0 10 100 87 106 8700 10600 0.820755
1 2 10 30 0 10 100 98 110 9800 11000 0.890909
2 2 11 45 75 10 100 46 42 4600 4200 1.095238
3 2 12 30 120 10 100 88 107 8800 10700 0.822430
4 2 13 30 180 10 100 37 21 3700 2100 1.761905
In [3]:
# plot
times = data["time since start (min)"]
N_wildtype = data["wildtype / mL"]
N_delta = data["delta / mL"]

plt.plot(times, N_wildtype, 'r.')
plt.plot(times, N_delta, 'b.')
plt.xlabel("time (min)")
plt.ylabel("cells / mL")
plt.legend(["wildtype", "∆lac"]);


Great! This looks like what we expected! Wildtype is definitely taking over. Now let's consider the ratio of wildtype over the delta strain over time. If these two strains are both growing expontially, then we expect their ratio to be exponential as well.

In [4]:
plt.plot(times, data["ratio"], 'r.')
plt.xlabel("time (min)")
plt.ylabel("wt / delta");


## Computing a selection coefficient¶

As we saw by eye, the wildtype strain takes over. Fitting an expoential to the ratio will elucidate the selection coefficient. The ratio of two exponentially growing strains will look like this over time, where $r_1$ and $r_2$ are the corresponding growth rate:

$$\frac{N_1(t)}{N_2(t)} = \frac{N_1(0) e^{r_1t}}{N_2(0) e^{r_2t}} = \frac{N_1(0)}{N_2(0)} e^{(r_1-r_2)t}$$

Let's determine the best fit for data above to get at this difference in growth date, or the selection coefficient. Let's first define an exponetial fucnction, the function we wish to fit.

In [5]:
def exp_curve(times, A, b):

return A * np.exp(b*times)


Now, we will use scipy's optimize function to get the best fit. The inputs to the function are:

• The function we are fitting, exp_curve() in this case
• The x-values of our data
• The y-values of our data
• The lower bounds and upper bounds on our parameters (A and b in order)
In [6]:
popt_ratio, _ = scipy.optimize.curve_fit(exp_curve,
data["time since start (min)"],
data["ratio"],
bounds=([0.01, 1/1000], [2, 1/10]))


Let's see what parameters we got!

In [7]:
print("Fit for A:", np.round(popt_ratio[0],5))
print("Fit for b:", np.round(popt_ratio[1],5), "per min")

Fit for A: 0.60192
Fit for b: 0.00293 per min


And let's plot the fit.

In [8]:
# plot data
plt.plot(data["time since start (min)"], data["ratio"], 'r.')

# time values for making a smooth curve
time_range = np.linspace(0, 700, 200)

# plot fits
plt.plot(time_range, exp_curve(time_range, popt_ratio[0], popt_ratio[1]), color="red")
plt.xlabel("time (min)")
plt.ylabel("wt / detla");