Julia: Michaelis-Menten Kinetics – Alicia’s Side Projects

Basic Michaelis-Menten enzyme kinetics.

Chemical and Rate Equations and Units

Reaction:

\[ E + S \leftrightharpoons ES \rightarrow E + P\]

Reversible rates are k₊₁ and k_-1. The final irreversible step has rate k_cat.

Rate equation: \[ v = {V_{\max}[S] \over K_M + [S]} \]

Variable	Units	Meaning
v	min^-1	Rate of formation of product
V_max	min^-1	Maximum possible rate of product formation
[S]	M	Concentration of substrate
K_M	M	Concentration of substrate at 1/2 V_max

Plot v vs. [S]

Hover over the traces to see the exact coordinates!

Code

substrate_concentrations = collect(range(start=0, stop=1e-2, length=100))

function mm_curve(vmax, km)
    function mm(substrate_concentration)
        vmax * substrate_concentration / (km + substrate_concentration)
    end

    dP_dts = mm.(substrate_concentrations)
    ytick_vals = range(start=0, stop=maximum(dP_dts), length=5)
    ytick_labels = [@sprintf("%.3e", val) for val in ytick_vals]

    Dict(:dP_dts => dP_dts, :ytick_vals => ytick_vals, :ytick_labels => ytick_labels)
end

kms = [1e-3, 2e-3, 4e-3]
labels = ["Km=1e-3" "Km=2e-3" "Km=4e-3"]
curves = [mm_curve(1.0e-4, km) for km in kms]
ys = [curve[:dP_dts] for curve in curves]

xtick_vals = range(start=0, stop=maximum(substrate_concentrations), length=5)
xtick_labels = [@sprintf("%.3e", val) for val in xtick_vals]

ytick_vals = curves[1][:ytick_vals]
ytick_labels = curves[1][:ytick_labels]

plot(substrate_concentrations, ys, label=labels, xlims=(0, 1.1e-2), xlabel="[S] (M)", ylabel="v (M/min)", xticks=(xtick_vals, xtick_labels), yticks=(ytick_vals, ytick_labels), linewidth=3)