Simulateing Aging using a simple model

State Variables

• ( F(t) ): Functional chaperone concentration
  
• ( U(t) ): Non-functional chaperone concentration
  
• ( D(t) ): Degraded (dead) protein concentration

Parameters

• \(N\): Number of proteins produced

• $k_{} $: Chaperone synthesis rate per cell

• $ f_1 $: Fraction of synthesized chaperones that are functional

• $ f_2 $: Refolding rate constant (U → F)

• $ f_3 $: Misfolding rate constant (F → U)

• $ k_d $: Degradation rate of both F and U

System of ODEs

$ \[\begin{aligned} \frac{dF}{dt} &= f_1 \cdot k_{\text{syn}} \cdot N + f_2 \cdot F \cdot U - f_3 \cdot F - k_d \cdot F \\ \frac{dU}{dt} &= (1 - f_1) \cdot k_{\text{syn}} \cdot N + f_3 \cdot F - f_2 \cdot F \cdot U - k_d \cdot U \\ \frac{dD}{dt} &= k_d \cdot F + k_d \cdot U \end{aligned}\]

$

from ipywidgets import interactive, FloatSlider, IntSlider, VBox, Label, HBox, Layout, interactive_output
import matplotlib.pyplot as plt
import numpy as np
from scipy.integrate import odeint
from IPython.display import display
#  ODE system
def chaperone_dynamics(y, t, N, k_syn, f1, f2, f3, kd):
    F, U, D = y
    dFdt = f1 * k_syn*N + f2 * F * U - kd * F - f3 * F
    dUdt = (1 - f1) * k_syn*N + f3 * F - f2 * F * U - kd * U
    dDdt = kd * F + kd * U
    return [dFdt, dUdt, dDdt]

# Plotting function
def plot_chaperone_dynamics(N, k_syn, f1, f2, f3, kd, t_max):
    F0 = f1 * N
    U0 = (1 - f1) * N
    y0 = [F0, U0, 0.0]
    t = np.linspace(0, t_max, 50000)
    solution = odeint(chaperone_dynamics, y0, t, args=(N, k_syn, f1, f2, f3, kd))
    F, U, D = solution.T
    plt.figure(figsize=(8, 5))
    plt.plot(t, F, label='Functional (F)')
    plt.plot(t, U, label='Non-functional (U)')
    plt.xlabel('Time (s)')
    plt.ylabel('Chaperone Level')
    plt.title('Chaperone Dynamics (Interactive)')
    plt.grid()
    plt.legend()
    plt.show()

def param_box(slider, text):
    return VBox([slider, Label(text)], layout=Layout(width='350px'))

N_slider = param_box(
    IntSlider(min=1, max=100, step=1, value=1, description='N'),
    "Number of proteins synthesis at once"
)

k_syn_slider = param_box(
    FloatSlider(min=0.01, max=1, step=0.01, value=0.1, description='k_syn'),
    "Rate of chaperone synthesis."
)

f1_slider = param_box(
    FloatSlider(min=0.01, max=1.0, step=0.01, value=0.1, description='f1'),
    "Frac. of synthesized that are functional."
)

f2_slider = param_box(
    FloatSlider(
        min=0.0, max=0.005, step=0.0001, value=0.0005, 
        description='f2', 
        readout=True,
        readout_format='.4f',
    ),
    "Refolding eff. (U → F)."
)

f3_slider = param_box(
    FloatSlider(
        min=0.0, max=0.01, step=0.0001, value=0.001, 
        description='f3',
        readout=True,
        readout_format='.4f',
    ),
    "Misfolding rate (F → U)."
)

kd_slider = param_box(
    FloatSlider(
        min=0.0, max=0.01, step=0.0001, value=0.005, 
        description='kd',
        readout=True,
        readout_format='.4f',
    ),
    "Degradation rate for F and U."
)

tmax_slider = param_box(
    IntSlider(min=0, max=1e4, step=100, value=2000, description='t_max'),
    "Max value of time to simulate."
)


controls = HBox([
    VBox([N_slider, k_syn_slider, f1_slider],layout=Layout(border='1px solid gray', padding='10px')),
    VBox([f2_slider, f3_slider, kd_slider],layout=Layout(border='1px solid gray', padding='10px')),
    VBox([tmax_slider],layout=Layout(border='1px solid gray', padding='10px'))
])



widget = interactive_output(
    plot_chaperone_dynamics,
    {
        'N': N_slider.children[0],
        'k_syn': k_syn_slider.children[0],
        'f1': f1_slider.children[0],
        'f2': f2_slider.children[0],
        'f3': f3_slider.children[0],
        'kd': kd_slider.children[0],
        't_max': tmax_slider.children[0]
    }
)

display(controls, widget)