MICHAELIS_MENTEN

Overview

The MICHAELIS_MENTEN function numerically solves the Michaelis-Menten system of ordinary differential equations (ODEs) for enzyme kinetics, modeling how substrate concentration changes over time during an enzyme-catalyzed reaction. This implementation uses SciPy’s solve_ivp to integrate the system of differential equations.

Michaelis-Menten kinetics, named after Leonor Michaelis and Maud Menten, is the foundational model in enzyme kinetics. The model describes how an enzyme (E) binds to a substrate (S) to form an enzyme-substrate complex, which then releases a product (P):

E + S \underset{k_{-1}}{\stackrel{k_{+1}}{\rightleftharpoons}} ES \xrightarrow{k_{cat}} E + P

The rate of substrate consumption follows the Michaelis-Menten equation:

v = \frac{dP}{dt} = \frac{k_{cat} \cdot E \cdot S}{K_m + S}

where k_{cat} is the catalytic rate constant (turnover number), representing the maximum number of substrate molecules converted to product per enzyme molecule per unit time. The Michaelis constant K_m equals the substrate concentration at which the reaction rate is half of V_{max}, providing a measure of enzyme-substrate affinity.

This function integrates the ODE system where substrate concentration decreases according to the reaction rate (dS/dt = -v) while enzyme concentration remains constant (dE/dt = 0), reflecting the catalytic nature of enzymes. The function supports multiple numerical integration methods including explicit Runge-Kutta (RK45, RK23, DOP853) for non-stiff problems and implicit methods (Radau, BDF, LSODA) for stiff systems. For more details on the integration methods, see the SciPy solve_ivp documentation.

Michaelis-Menten kinetics has broad applications beyond biochemistry, including pharmacokinetics, blood alcohol clearance, and ecological models such as the Monod equation for microbial growth.

This example function is provided as-is without any representation of accuracy.

Excel Usage

=MICHAELIS_MENTEN(s_initial, e_initial, k_cat, k_m, t_start, t_end, timesteps, mm_method)

s_initial (float, required): Initial substrate concentration.
e_initial (float, required): Initial enzyme concentration.
k_cat (float, required): Catalytic rate constant (turnover number).
k_m (float, required): Michaelis constant.
t_start (float, required): Start time for integration.
t_end (float, required): End time for integration.
timesteps (int, optional, default: 10): Number of timesteps to solve for.
mm_method (str, optional, default: “RK45”): Integration method for ODE solver.

Returns (list[list]): 2D list with header [t, S, E], or error message string.

Examples

Example 1: Demo case 1

Inputs:

s_initial	e_initial	k_cat	k_m	t_start	t_end	timesteps
10	1	2	1	0	5	10

Excel formula:

=MICHAELIS_MENTEN(10, 1, 2, 1, 0, 5, 10)

Expected output:

t	S	E
0	10	1
0.5556	8.995	1
1.111	8.001	1
1.667	7.02	1
2.222	6.057	1
2.778	5.115	1
3.333	4.201	1
3.889	3.324	1
4.444	2.498	1
5	1.746	1

Example 2: Demo case 2

Inputs:

s_initial	e_initial	k_cat	k_m	t_start	t_end	timesteps	mm_method
5	0.5	1.5	0.5	0	2	10	RK23

Excel formula:

=MICHAELIS_MENTEN(5, 0.5, 1.5, 0.5, 0, 2, 10, "RK23")

Expected output:

t	S	E
0	5	0.5
0.2222	4.849	0.5
0.4444	4.698	0.5
0.6667	4.547	0.5
0.8889	4.398	0.5
1.111	4.248	0.5
1.333	4.099	0.5
1.556	3.951	0.5
1.778	3.803	0.5
2	3.656	0.5

Example 3: Demo case 3

Inputs:

s_initial	e_initial	k_cat	k_m	t_start	t_end	timesteps	mm_method
8	2	3	2	0	3	10	RK45

Excel formula:

=MICHAELIS_MENTEN(8, 2, 3, 2, 0, 3, 10, "RK45")

Expected output:

t	S	E
0	8	2
0.3333	6.435	2
0.6667	4.957	2
1	3.598	2
1.333	2.404	2
1.667	1.436	2
2	0.7457	2
2.333	0.3366	2
2.667	0.1368	2
3	0.05251	2

Example 4: Demo case 4

Inputs:

s_initial	e_initial	k_cat	k_m	t_start	t_end	timesteps
2	0.2	1	0.2	0	0.5	10

Excel formula:

=MICHAELIS_MENTEN(2, 0.2, 1, 0.2, 0, 0.5, 10)

Expected output:

t	S	E
0	2	0.2
0.05556	1.99	0.2
0.1111	1.98	0.2
0.1667	1.97	0.2
0.2222	1.96	0.2
0.2778	1.95	0.2
0.3333	1.939	0.2
0.3889	1.929	0.2
0.4444	1.919	0.2
0.5	1.909	0.2

Python Code

from scipy.integrate import solve_ivp as scipy_solve_ivp
import numpy as np

def michaelis_menten(s_initial, e_initial, k_cat, k_m, t_start, t_end, timesteps=10, mm_method='RK45'):
    """
    Numerically solves the Michaelis-Menten system of ordinary differential equations for enzyme kinetics using scipy.integrate.solve_ivp.

    See: https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.solve_ivp.html

    This example function is provided as-is without any representation of accuracy.

    Args:
        s_initial (float): Initial substrate concentration.
        e_initial (float): Initial enzyme concentration.
        k_cat (float): Catalytic rate constant (turnover number).
        k_m (float): Michaelis constant.
        t_start (float): Start time for integration.
        t_end (float): End time for integration.
        timesteps (int, optional): Number of timesteps to solve for. Default is 10.
        mm_method (str, optional): Integration method for ODE solver. Valid options: RK45, RK23, DOP853, Radau, BDF, LSODA. Default is 'RK45'.

    Returns:
        list[list]: 2D list with header [t, S, E], or error message string.
    """
    # Validate inputs
    try:
        s0 = float(s_initial)
        e0 = float(e_initial)
        kcat = float(k_cat)
        km = float(k_m)
        t0 = float(t_start)
        t1 = float(t_end)
        ntp = int(timesteps)
    except Exception:
        return "Invalid input: all initial values, parameters, and timesteps must be numbers."
    if t1 <= t0:
        return "Invalid input: t_end must be greater than t_start."
    if ntp <= 0:
        return "Invalid input: timesteps must be a positive integer."
    if s0 < 0 or e0 < 0 or kcat < 0 or km < 0:
        return "Invalid input: concentrations and rate constants must be non-negative."
    allowed_methods = ['RK45', 'RK23', 'DOP853', 'Radau', 'BDF', 'LSODA']
    if mm_method not in allowed_methods:
        return f"Invalid input: mm_method must be one of {allowed_methods}."

    # Create time array for evaluation
    t_eval = np.linspace(t0, t1, ntp)

    # Michaelis-Menten ODE system: dS/dt = -v, dE/dt = 0
    def mm_ode(t, y):
        S, E = y
        v = (kcat * E * S) / (km + S) if (km + S) != 0 else 0.0
        dSdt = -v
        dEdt = 0.0
        return [dSdt, dEdt]

    try:
        sol = scipy_solve_ivp(
            mm_ode,
            [t0, t1],
            [s0, e0],
            method=mm_method,
            dense_output=False,
            t_eval=t_eval,
            rtol=1e-6,
            atol=1e-8
        )
    except Exception as e:
        return f"ODE integration error: {e}"

    if not sol.success:
        return f"ODE integration failed: {sol.message}"

    # Prepare output: header row then data rows
    result = [["t", "S", "E"]]
    for i in range(len(sol.t)):
        t_val = float(sol.t[i])
        s_val = float(sol.y[0][i])
        e_val = float(sol.y[1][i])
        # Check for nan/inf values
        if not (np.isfinite(t_val) and np.isfinite(s_val) and np.isfinite(e_val)):
            return "Invalid output: nan or inf detected."
        result.append([t_val, s_val, e_val])
    return result

Overview

Excel Usage

Examples

Python Code

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