Generate Data from the Total QSSA (tQ) Model for Enzyme Kinetics

Simulate data from the total QSSA (tQ) model a refinement of the classical Michaelis-Menten enzyme kinetics ordinary differential equation described in (Choi, et al., 2017, DOI: 10.1038/s41598-017-17072-z). Consider the kinetic rate equation


                  kf
                 --->    kcat
          E + S  <---  C --->  E + P
                  kb

Usage

tQ_model_generate(time, kcat, kM, ET, ST, ...)

Arguments

time: numeric vector. Increasing time points (e.g. time[i] > time[i+1], for i in [1, ... n])
kcat: numeric value catalytic rate constant
kM: numeric value Michaelis rate constant
ET: numeric value total enzyme concentration
ST: numeric value total substrate concentration
...: additional arguments to deSolve::ode()

Value

run the tQ ordinary differential equation forwards starting with initial product concentration of 0 and specified kcat and kM parameters for the specified time steps.

Details

where the free enzyme (E) reversibly binds to the substrate (S) to form a complex (C) with forward and backward rate constants of kf and kb, which is irreversibly catalyzed into the product (P), with rate constant of kcat, releasing the enzyme to catalyze additional substrate. The total enzyme concentration is defined to be the ET := E + C. The total substrate and product concentration is defined to be ST := S + C + P. The Michaelis constant is the defined to be the kM := (kb + kcat) / kf. The kcat rate constant determines the maximum turn over at saturating substrate concentrations, Vmax := kcat * ET. The rate constants kcat and kM can be estimated by monitoring the product accumulation over time (enzyme progress curves), by varying the enzyme and substrate concentrations.

From (Choi, et al, 2017, equation 2, the total quasi-steady-state approximation (tQ) differential equation is defined by


  Observed data:
     M     = number of measurements        # number of measurements
     t[M]  = time                          # measured in seconds
     Pt[M] = product                       # product produced at time t
     ST    = substrate total concentration # specified for each experiment
     ET    = enzyme total concentration    # specified for each experiment

  Model parameters:
    kcat    # catalytic constant (min^-1)
    kM      # Michaelis constant ()

  ODE formulation:
    dPdt = kcat * (
             ET + kM + ST - Pt +
             -sqrt((ET + kM + ST - Pt)^2 - 2 * ET * (ST - Pt))) / 2

  initial condition:
     P := 0

In (Choi, et al. 2017) they prove, that the tQ model is valid when


    K/(2*ST) * (ET+kM+ST) / sqrt((ET+kM+ST+P)^2 - 4*ET(ST-P)) << 1,

where K = kb/kf is the dissociation constant.

References

Choi, B., Rempala, G.A. & Kim, J.K. Beyond the Michaelis-Menten equation: Accurate and efficient estimation of enzyme kinetic parameters. Sci Rep 7, 17018 (2017). https://doi.org/10.1038/s41598-017-17072-z

Examples

if (FALSE) { # \dontrun{
BayesPharma::tQ_model_generate(
  time = seq(0.1, 3, by = .5),
  kcat = 3,
  kM = 5,
  ET = 10,
  ST = 10) |>
  as.data.frame(col.names = c("time", "P_pred"))
} # }