Noise and Observable Parameters

It is common to measure an observable, such as a product P, with different experimental assays. This can lead to variation in the measurement noise parameter sigma between measurements for the same observable, and additionally, different assays might measure P on different relative scales, necessitating different scale and offset parameters. Measurement-specific parameters can be handled by defining noise and observable parameters. To demonstrate how, let us continue with the same enzyme kinetics model as in the Creating a PEtab Parameter Estimation Problem in Julia tutorial.

using Catalyst 
using Distributions
using PEtab

rn = @reaction_network begin
    @parameters se0
    @species SE(t) = se0  # se0 = initial value for S
    c1, S + E --> SE
    c2, SE --> S + E
    c3, SE --> P + E
end
state_map =  [:E => 1.0, :P => 0.0]
parameter_map = [:c1 => 1.0]

_c3 = PEtabParameter(:c3, scale=:log10)
_se0 = PEtabParameter(:c3, prior=LogNormal(1.0, 0.5), prior_on_linear_scale=true)
_c2 = PEtabParameter(:c2)
_sigma = PEtabParameter(:sigma)
_scale = PEtabParameter(:scale)
_offset = PEtabParameter(:offset)
parameters = [_c2, _c3, _se0, _sigma, _scale, _offset]

condition_c0 = Dict(:S => 5.0)
condition_c1 = Dict(:S => 2.0)
simulation_conditions = Dict("c0" => condition_c0, 
                             "c1" => condition_c1)

Now, to incorporate time-point-specific noise and measurement parameters into the observable formula, we encode them in the form observableParameter... and noiseParameters.

@unpack P, E, SE = rn
@parameters noiseParameter1_obs_P observableParameter1_obs_P observableParameter2_obs_P
obs_P = PEtabObservable(observableParameter1_obs_P * P + observableParameter2_obs_P, noiseParameter1_obs_P * P)
obs_Sum = PEtabObservable(E + SE, 3.0, transformation=:log)
observables = Dict("obs_P" => obs_P, 
                   "obs_Sum" => obs_Sum)

Note

noiseParameters and observableParameter must always be on the format observableParameter${n}_${observableId} and noiseParameter${n}_${observableId}with n starting from 1, in order to correctly map parameters when building the objective function.

Next, we provide values for these parameters in the measurement data. These parameters can either be specified as values or parameters that have been defined as a PEtabParameter. In the case of multiple noise or observable parameters for a measurement, they are delimited by a semicolon:

simulation_id (str)	obs_id (str)	time (float)	measurement (float)	observable_parameters (str\|float)	noise_parameters (str\|float)
c0	obs_P	1.0	0.7
c0	obs_Sum	10.0	0.1	scale;offset	sigma
c1	obs_P	1.0	1.0
c1	obs_Sum	20.0	1.5	1.0;1.0	4.0

Note, in case an observable (like obs_P) does not have noise-or observable-parameters we do not have to provide any values for it. For the first observation for observable obs_Sum the parameter scale maps to observableParameter1_obs_Sum while offset maps to observableParameter2_obs_Sum. In Julia, the measurement data would look like:

measurements = DataFrame(
    simulation_id=["c0", "c0", "c0", "c0"],
    obs_id=["obs_P", "obs_Sum", "obs_P", "obs_Sum"],
    time=[1.0, 10.0, 1.0, 20.0],
    measurement=[0.7, 0.1, 1.0, 1.5], 
    observable_parameters=[missing, "scale;offset", missing, "1.0;1.0"],
    noise_parameters=[missing, "sigma", missing, 4.0]
)

With this setup, you can create a PEtabODEProblem for model calibration:

petab_model = PEtabModel(
    rn, simulation_conditions, observables, measurements,
    parameters, state_map=state_map, parameter_map=parameter_map, verbose=true
)
petab_problem = PEtabODEProblem(petab_model)