Exercise PK44_2 - Estimation of inhibitory constant Ki

2021-09-06

Background

  • Structural model - Estimation of inhibitory rate constant in non-competitive enzyme inhibition model

  • Number of subjects - 1

  • Number of compounds - 1

Learning Outcome

  • To get an understanding about inhibitory models

  • To understand the relationship between rate of metabolite formation and concentration

Objectives

  • To analyze enzyme data by means of simultaneous nonlinear regression

Libraries

call the necessary 'libraries' to get started

using Random
using Pumas
using PumasUtilities
using CairoMakie
using AlgebraOfGraphics
using DataFramesMeta

Model - Non-Competitive Inhibitory Model

pk_44_nim  = @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
    ## 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 variables: 
  Derived: rate_nim, dv_rate_nim
  Observed: rate_nim, dv_rate_nim

Parameters

Parameters provided for simulation are as below.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([0.0,0.0]),
          σ_add  = 2.256)
(tvvmax = 122.827, tvkm = 42.7053, tvki = 98.8058, Ω = [0.0 0.0; 0.0 0.0], 
σ_add = 2.256)

Creating a Dataset

In this dataset we will be having time, concentration data and Exposure(I) which 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

Simulation

We will now simulate the rate of metabolite formation

Random.seed!(123)
sim_sub1 = simobs(pk_44_nim, sub1, param)
df44_2      = DataFrame(sim_sub1)

Visualization

Non-Competitive Inhibitory Model
@chain df44_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(axis = (;xticks = 0:100:1000, yticks = 0:10:100))        
end
@chain df44_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(axis = (;xscale = log10, yticks = 0:10:100))        
end