Module III - Day 4

Python Made Easy: Science and Math Edition

Sep-Dec 2025 batch, Vikrant Patil

Date: 28th Feb 2026

Live note are here https://vikrant.dev/python-made-easy-science-math/students-module3-day4.html

Please login to https://traininghub.vikrant.dev and create a notebook with name module3-day4.ipynb

Mathematical Modeling

Growth of a Yeast Culture

Suppose we start with some culture of yeast bacterias in milk/medium. So we want to observer how much yeast bomass is present at given time frame. So we take observans each hour.

Time in hours (n)	Yeast Biomass (Pn)	Change (Pn+1 - Pn)
0	9.6	8.7
1	18.3	10.7
2	29.0	18.2
3	47.2	23.9
4	71.1	48.0
5	119.1	55.5
6	174.6	82.7
7	257.3

delta(P_n) = P_n+1 - P_n = k*P_n(capacity - P_n)

After puting the values from table we can solve the equation for capacity of 665 to get value of k. it comes out to be 0.00082

P_n+1 - P_n = 0.00082*P_n(665 - P_n)

Now we can write python program to actually find out values.

from matplotlib import pyplot as plt
%matplotlib inline


def model(previous, k, capacity):
    return previous + k*previous * (capacity - previous)

def yeast_mass(n, k=0.00082, capacity=665):
    yeast = [9.6]
    for i in range(n):
        yeast.append(model(yeast[-1], k, capacity))
    return yeast

def saveplot(massdata):
    fig, ax = plt.subplots(figsize=(7,7))
    ax.plot(massdata, "xb") # your may change
    ax.set_xlabel("Time") # x label change
    ax.set_ylabel("Yeast Biomass") # 
    ax.set_title("Yeast Biomass Estimation")
    ax.legend()
    ax.grid(True, alpha=0.3)
    fig.tight_layout()
    fig.savefig("yeast.png", dpi=150)

massdata = yeast_mass(20)
saveplot(massdata)

/tmp/ipykernel_5422/3371913838.py:20: UserWarning: No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.
  ax.legend()

plt.plot(massdata)

a, b = 2, 3

A = (5, 8)

x, y = A

import numpy as np

np.linspace(0.,10, 11)

array([ 0.,  1.,  2.,  3.,  4.,  5.,  6.,  7.,  8.,  9., 10.])