Example: metaparameters

Metaparameters are a PEtab Select feature that enables model selection with model hypotheses that involve multiple parameters.

An example use case would be a model selection problem where one hypothesis is that a process with a Michaelis-Menten kinetic exists. Since the Michaelis-Menten kinetic \(\frac{V_{\mathrm{max}}\, s}{K_m + s}\) (parameters \(V_{\mathrm{max}}\) and \(K_m\), and state variable \(s\)) involves two parameters, both need to be set to be estimated simultaneously to include the process in the model, or set to 0 simultaneously to turn the process off and ensure the correct number of parameters are counted when computing model selection criteria such as the AIC.

We show an example of this below. As metaparameters only affect the PEtab Select files, we only show those.

Without metaparameters

As Vmax and Km need to be toggled on or off simultaneously, without metaparameters, we require two model space rows.

model_subspace_id	model_subspace_petab_yaml	k1	Vmax	Km	k3
M_mm_off	petab_problem.yaml	0;estimate	0	0	0;estimate
M_mm_on	petab_problem.yaml	0;estimate	estimate	estimate	0;estimate

With metaparameters

Instead, we can define a metaparameter mm in the problem YAML, which represents both Vmax and Km.

format_version: 1.0.0
criterion: AIC
method: forward
model_space_files:
- model_space.tsv
metaparameters:
  mm:
  - Vmax
  - Km

This enables concise specification in the model space table. Vmax and Km are assigned the same value as mm.

model_subspace_id	model_subspace_petab_yaml	k1	mm	k3
M	petab_problem.yaml	0;estimate	0;estimate	0;estimate

Notes

During model selection steps (e.g. with the forward and backward methods), a metaparameter is treated as a single parameter. This enables model selection in terms of selection of model hypotheses, rather than the individual parameters associated with hypotheses.

However, during computation of criteria such as the AIC, then the correct number of estimated parameters is used, i.e., with Vmax and Km in the example above if mm_R2 is set to be estimated.