PK44 (Part 1) - Estimation of inhibitory constant Ki

1 Background

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

2 Learning Outcome

This exercise demonstrates how to build a competitive inhibitory model and understand the relationship between rate of metabolite formation and concentration.

3 Objectives

To build a 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_cim = @model begin
    @metadata begin
        desc = "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
        ## Competitive Inhibiton Model
        rate_cim = @. Vmax * _conc / ((Km * (1 + (_I / Ki)) + _conc))
        """
        Metabolic rate (nmol/min/mg)
        """
        dv_rate_cim ~ @. Normal(rate_cim, 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_cim, dv_rate_cim
  Observed: rate_cim, dv_rate_cim

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 = 99.8039,
    tvkm = 11.3192,
    tvki = 5.05605,
    Ω = Diagonal([0.01, 0.01]),
    σ_add = 1.256,
)

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_cim = 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_cim],
    covariates = [:conc, :I],
    event_data = false,
)

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

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_cim, sub1, param)

Simulated population (Vector{<:Subject})
  Simulated subjects: 6
  Simulated variables: rate_cim, dv_rate_cim

pkdata_44_1 = DataFrame(sim_sub1)
first(pkdata_44_1, 5)

5×14 DataFrame

Row	id	time	rate_cim	dv_rate_cim	evid	conc	I	η_1	η_2	Vmax	Km	Ki	_conc	_I
	String?	Float64	Float64?	Float64?	Int64?	Int64?	Int64?	Float64?	Float64?	Float64?	Float64?	Float64?	Int64?	Int64?
1	1	1.0	8.26724	9.78751	0	1	0	-0.0261068	-0.0505607	97.2321	10.7611	5.05605	1	0
2	1	2.0	15.2388	16.1051	0	2	0	-0.0261068	-0.0505607	97.2321	10.7611	5.05605	2	0
3	1	3.0	21.1971	19.3164	0	3	0	-0.0261068	-0.0505607	97.2321	10.7611	5.05605	3	0
4	1	4.0	26.3481	25.2571	0	4	0	-0.0261068	-0.0505607	97.2321	10.7611	5.05605	4	0
5	1	5.0	30.8455	31.2996	0	5	0	-0.0261068	-0.0505607	97.2321	10.7611	5.05605	5	0

9 Visualize Results

@chain pkdata_44_1 begin
    @rsubset :conc ∈ [1, 5, 10, 15, 26, 104, 251, 502, 1000]
    data(_) *
    mapping(
        :conc => "Concentration (μM)",
        :rate_cim => "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 = 0:10:100))
end

@chain pkdata_44_1 begin
    @rsubset :conc ∈ [1, 5, 10, 15, 26, 104, 251, 502, 1000]
    data(_) *
    mapping(
        :conc => "log Concentration (μM)",
        :rate_cim => "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 = 0:10:100,
        ),
    )
end

10 Conclusion

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