PK44 (Part 2) - Estimation of inhibitory constant Ki

1 Background

• Structural model - Estimation of inhibitory rate constant in a non-competitive enzyme inhibition model
• Number of subjects - 1
• Number of compounds - 1

2 Learning Outcome

This exercise demonstrates how to build non-competitive inhibitory models and understand the relationship between rate of metabolite formation and concentration

3 Objectives

To build a non-competitive inhibitory model and simulate the model for a single subject.

4 Libraries

Load the necessary libraries.

using PumasUtilities
using Random
using Pumas
using CairoMakie
using AlgebraOfGraphics
using DataFramesMeta
using CSV
using Dates

5 Model definition

Note the expression of the model parameters with helpful comments. The model is expressed with differential equations. Residual variability is an additive error model.

pk_44_nim = @model begin
    @metadata begin
        desc = "Non-Competitive Inhibitory Model"
        timeu = u"minute"
    end

    @param begin
        """
        Maximum metabolic rate (μM*gm/min)
        """
        tvvmax ∈ RealDomain(lower = 0)
        """
        Michaelis-Mentons constant (μmol/L)
        """
        tvkm ∈ RealDomain(lower = 0)
        """
        Inhibitory constant (μmol/L)
        """
        tvki ∈ RealDomain(lower = 0)
        Ω ∈ PDiagDomain(2)
        """
        Additive RUV
        """
        σ_add ∈ RealDomain(lower = 0)
    end

    @random begin
        η ~ MvNormal(Ω)
    end

    @covariates conc I

    @pre begin
        Vmax = tvvmax * exp(η[1])
        Km = tvkm * exp(η[2])
        Ki = tvki
        _conc = conc
        _I = I
    end

    @derived begin
        ## Noncompetitive Inhibition Model
        rate_nim = @. ((Vmax * _conc / (Km + _conc)) * (Ki / (Ki + _I)))
        """
        Metabolic rate (nmol/min/mg)
        """
        dv_rate_nim ~ @. Normal(rate_nim, σ_add)
    end
end

PumasModel
  Parameters: tvvmax, tvkm, tvki, Ω, σ_add
  Random effects: η
  Covariates: conc, I
  Dynamical system variables: 
  Dynamical system type: No dynamical model
  Derived: rate_nim, dv_rate_nim
  Observed: rate_nim, dv_rate_nim

6 Initial Estimates of Model Parameters

The model parameters for simulation are the following. Note that tv represents the typical value for parameters.

• Vmax - Maximum metabolic rate (μM*gm_protein/min)
• Km - Michaelis-Mentons constant (μmol/L)
• Ki - Inhibitory constant (μmol/L)
• I - Inhibitor concentration/Exposure

param = (
    tvvmax = 122.827,
    tvkm = 42.7053,
    tvki = 98.8058,
    Ω = Diagonal([0.01, 0.01]),
    σ_add = 2.256,
)

7 Creating a Dataset

The time, concentration data and exposure (I) will be used to estimate the rate of metabolite concentration

df_sub1 = map(
    i -> DataFrame(id = i, time = 1:1:1000, dv_rate_nim = missing, conc = 1:1:1000, I = 0),
    1:6,
)
df = vcat(DataFrame.(df_sub1)...)
df[!, :I] = ifelse.(df.id .== 2, 10, df.I)
df[!, :I] = ifelse.(df.id .== 3, 25, df.I)
df[!, :I] = ifelse.(df.id .== 4, 50, df.I)
df[!, :I] = ifelse.(df.id .== 5, 75, df.I)
df[!, :I] = ifelse.(df.id .== 6, 100, df.I)
df_sub1 = df
sub1 = read_pumas(
    df_sub1,
    observations = [:dv_rate_nim],
    covariates = [:conc, :I],
    event_data = false,
)

Population
  Subjects: 6
  Covariates: conc, I
  Observations: dv_rate_nim

8 Single-Subject Simulation

Simulate the rate of metabolite formation

Initialize the random number generator with a seed for reproducibility of the simulation.

Random.seed!(123)

sim_sub1 = simobs(pk_44_nim, sub1, param)

Simulated population (Vector{<:Subject})
  Simulated subjects: 6
  Simulated variables: rate_nim, dv_rate_nim

pkdata_44_2 = DataFrame(sim_sub1)
first(pkdata_44_2, 5)

5×14 DataFrame

Row	id	time	rate_nim	dv_rate_nim	evid	conc	I	Vmax	Km	Ki	_conc	_I	η_1	η_2
	String	Float64	Float64?	Float64?	Int64	Int64?	Int64?	Float64?	Float64?	Float64?	Int64?	Int64?	Float64	Float64
1	1	1.0	3.20261	3.70198	0	1	0	154.369	47.2009	98.8058	1	0	0.228566	0.100091
2	1	2.0	6.27503	7.51125	0	2	0	154.369	47.2009	98.8058	2	0	0.228566	0.100091
3	1	3.0	9.22504	11.6413	0	3	0	154.369	47.2009	98.8058	3	0	0.228566	0.100091
4	1	4.0	12.0598	12.2449	0	4	0	154.369	47.2009	98.8058	4	0	0.228566	0.100091
5	1	5.0	14.786	13.3071	0	5	0	154.369	47.2009	98.8058	5	0	0.228566	0.100091

9 Visualize Results

@chain pkdata_44_2 begin
    @rsubset :conc ∈ [1, 5, 10, 15, 26, 104, 251, 502, 1000]
    data(_) *
    mapping(
        :conc => "Concentration (μM)",
        :rate_nim => "Metabolic rate (nmol/min/mg protein)",
        color = :I => nonnumeric => "Exposure",
    ) *
    visual(ScatterLines, linewidth = 4, markersize = 12)
    draw(; figure = (; fontsize = 22), axis = (; xticks = 0:100:1000, yticks = 0:10:140))
end

@chain pkdata_44_2 begin
    @rsubset :conc ∈ [1, 5, 10, 15, 26, 104, 251, 502, 1000]
    data(_) *
    mapping(
        :conc => "log Concentration (μM)",
        :rate_nim => "Metabolic rate (nmol/min/mg protein)",
        color = :I => nonnumeric => "Exposure",
    ) *
    visual(ScatterLines, linewidth = 4, markersize = 12)
    draw(;
        figure = (; fontsize = 22),
        axis = (;
            xscale = log10,
            xtickformat = i -> (@. string(round(i; digits = 1))),
            yticks = 0:10:140,
        ),
    )
end

10 Conclusion

This tutorial showed how to build a non-competitive inhibitory model and perform a rate of metabolite formation versus concentration simulation.