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Mathematical Models of Mitosis and Meiosis

1. Mitosis Models

Mitosis is the process where a single cell divides to form two genetically identical daughter cells. The models focus on cell cycle dynamics and chromosome segregation.

Cell Cycle Model (ODE-Based)

The concentration of cyclins can be modeled by the differential equation:

\[ \frac{dC}{dt} = k_1 G_1 - k_2 C \]

where:

• \( C \) is the cyclin concentration
• \( k_1 \) is the production rate
• \( k_2 \) is the degradation rate

Stochastic Model (Markov Process)

Cell cycle phase transitions can be modeled probabilistically:

\[ P(G_1 \to S) = 1 - e^{-\lambda t} \]

2. Meiosis Models

Meiosis involves two division cycles, leading to four non-identical haploid gametes. Models describe recombination and chromosome segregation.

Meiotic Division Model

Population dynamics during meiosis can be modeled as:

\[ \frac{dM_1}{dt} = r G - d M_1, \quad \frac{dM_2}{dt} = f(M_1) - d M_2 \]

Recombination Models

Crossing-over follows a Poisson distribution:

\[ P(k) = \frac{\lambda^k e^{-\lambda}}{k!} \]

Interference between crossovers can be modeled using a Gamma distribution:

\[ P(d) = \frac{\beta^\alpha d^{\alpha-1} e^{-\beta d}}{\Gamma(\alpha)} \]

Chromosome Segregation Errors

The probability of correct chromosome segregation is:

\[ P_{\text{correct}} = 1 - P_{\text{nondisjunction}} \]

where nondisjunction errors follow:

\[ P_{\text{nondisjunction}} = \frac{e^{-\mu t}}{1 + e^{-\mu t}} \]

Conclusion

Mathematical models provide insights into the regulation of mitosis and meiosis, helping in genetic research and understanding of diseases like cancer.

1. Mathematical Models of Mitosis

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1. Mitosis Model (Cell Cycle Regulation)

This model simulates the concentration of cyclins over time using Ordinary Differential Equations (ODEs).

Implementation Using SciPy

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import solve_ivp

# Define the ODE model for cyclin concentration
def mitosis_ode(t, C, k1, k2):
    dCdt = k1 - k2 * C  # Rate of cyclin accumulation and degradation
    return [dCdt]

# Parameters
k1 = 0.02  # Cyclin production rate
k2 = 0.01  # Cyclin degradation rate
C0 = [0]   # Initial cyclin concentration
t_span = (0, 100)  # Simulation time
t_eval = np.linspace(*t_span, 500)

# Solve the ODE
solution = solve_ivp(mitosis_ode, t_span, C0, args=(k1, k2), t_eval=t_eval)

# Plot the result
plt.plot(solution.t, solution.y[0], label="Cyclin Concentration")
plt.xlabel("Time")
plt.ylabel("Cyclin Level")
plt.title("Cyclin Regulation in Mitosis")
plt.legend()
plt.grid()
plt.show()

🔹 Explanation:

• Models cyclin accumulation and degradation.
• Uses an ODE solver to compute concentration over time.

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2. Stochastic Mitosis Model (Markov Chain)

This model simulates cell cycle transitions (G1 → S → G2 → M) probabilistically.

Implementation Using Markov Chains

import numpy as np

# Transition probabilities between cell cycle phases
transition_matrix = {
    "G1": {"G1": 0.2, "S": 0.8},
    "S": {"S": 0.3, "G2": 0.7},
    "G2": {"G2": 0.4, "M": 0.6},
    "M": {"M": 0.2, "G1": 0.8},  # Cycle restarts
}

# Simulate the cell cycle
def simulate_mitosis(steps=20):
    state = "G1"
    history = [state]

    for _ in range(steps):
        next_state = np.random.choice(
            list(transition_matrix[state].keys()),
            p=list(transition_matrix[state].values())
        )
        history.append(next_state)
        state = next_state

    return history

# Run the simulation
cell_cycle_history = simulate_mitosis(30)
print("Cell cycle sequence:", " → ".join(cell_cycle_history))

🔹 Explanation:

• Uses Markov Chains to simulate stochastic transitions.
• Each phase transitions probabilistically to the next.

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3. Meiosis Model (Recombination & Chromosome Segregation)

This model simulates random crossover events in meiosis using a Poisson distribution.

Implementation of Recombination Model

import numpy as np
import matplotlib.pyplot as plt

# Simulate crossover events using Poisson distribution
lambda_crossover = 2  # Average number of crossovers per chromosome pair
num_chromosomes = 1000  # Simulating 1000 meiosis events

crossovers = np.random.poisson(lambda_crossover, num_chromosomes)

# Plot histogram of crossover events
plt.hist(crossovers, bins=10, edgecolor="black", alpha=0.7)
plt.xlabel("Number of Crossovers")
plt.ylabel("Frequency")
plt.title("Distribution of Crossover Events in Meiosis")
plt.show()

🔹 Explanation:

• Uses a Poisson distribution to model random crossovers.
• Visualizes the distribution of crossover events.

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4. Chromosome Segregation Error Model

This model simulates nondisjunction errors (e.g., incorrect chromosome separation leading to trisomy).

Implementation Using Probability Model

import numpy as np

# Probability of correct segregation
def segregation_error(n, mu=0.02):
    return 1 - (1 / (1 + np.exp(-mu * n)))  # Sigmoid function

# Simulating 50 meiosis events
meiosis_events = np.arange(1, 51)
errors = [segregation_error(n) for n in meiosis_events]

# Plot the error probability
plt.plot(meiosis_events, errors, marker="o", linestyle="-", color="red")
plt.xlabel("Meiosis Event")
plt.ylabel("Nondisjunction Probability")
plt.title("Probability of Chromosome Segregation Errors in Meiosis")
plt.grid()
plt.show()

🔹 Explanation:

• Uses a sigmoid function to model error probability.
• Higher values of $n$ (number of meiosis events) increase the probability of errors.

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Conclusion

These Python models simulate key biological processes:

1. ODE Model for mitosis (cyclin regulation).
2. Markov Chain Model for stochastic cell cycle transitions.
3. Poisson Model for meiotic crossover events.
4. Probability Model for chromosome mis-segregation errors.

This work is licensed under a Creative Commons Attribution 4.0 International License.