PEtab data format specification

Format version: 2.0.0

This document explains the PEtab data format.

Purpose

Providing a standardized way for specifying parameter estimation problems in systems biology, especially for the case of Ordinary Differential Equation (ODE) models.

Scope

The scope of PEtab is the full specification of parameter estimation problems in typical systems biology applications. In our experience, a typical setup of data-based modeling starts either with (i) the model of a biological system that is to be calibrated, or with (ii) experimental data that are to be integrated and analyzed using a computational model. Measurements are linked to the biological model by an observation and noise model. Often, measurements are taken after some perturbations have been applied, which are modeled as derivations from a generic model (Figure 1A). Therefore, one goal was to specify such a setup in the least redundant way. Furthermore, we wanted to establish an intuitive, modular, machine- and human-readable and -writable format that makes use of existing standards.

A common setup for data-based modeling studies and its representation in PEtab.

Figure 1: A common setup for data-based modeling studies and its representation in PEtab.

Overview

The PEtab data format specifies a parameter estimation problem using a number of text-based files ( Tab-Separated Values (TSV)) (Figure 2), i.e.

  • A model

  • A measurement file to fit the model to [TSV]

  • A condition file specifying model inputs and condition-specific parameters [TSV]

  • An observable file specifying the observation model [TSV]

  • A parameter file specifying estimateable parameters and related information [TSV]

  • A grouping file that lists all of the files and provides additional information including employed extensions [YAML]

  • (optional) A simulation file, which has the same format as the measurement file, but contains model simulations [TSV]

  • (optional) A visualization file, which contains specifications how the data and/or simulations should be plotted by the visualization routines [TSV]

  • (optional) A mapping file, which allows mapping PEtab entity IDs to entity IDs in the model, which might not have valid PEtab IDs themselves [TSV]

Files constituting a PEtab problem

Figure 2: Files constituting a PEtab problem.

Figure 1B shows how those files relate to a common setup for data-based modeling studies.

The following sections will describe the minimum requirements of those components in the core standard, which should provide all information for defining the parameter estimation problem.

Extensions of this format (e.g. additional columns in the measurement table) are possible and intended. However, while those columns may provide extra information for example for plotting, downstream analysis, or for more efficient parameter estimation, they should not affect the estimation problem as such.

General remarks

  • All model entities, column names and row names are case-sensitive

  • Fields in “[]” are optional and may be left empty.

Changes from PEtab 1.0.0

PEtab 2.0.0 is a major update of the PEtab format. The main changes are:

  • Support for non-SBML models

  • Clarification and specification of various previously underspecified aspects (math expressions, overriding values in the condition table, etc.)

  • Support for extensions

  • Observable IDs are now allowed to be used in observable/noise formulas

Model definition

PEtab 2.0.0 is agnostic of specific model formats. A model file is referenced in the PEtab problem description (YAML) via its file name or a URL.

Condition table

The condition table specifies parameters, or initial values of species and compartments for specific simulation conditions (generally corresponding to different experimental conditions).

This is specified as a tab-separated value file in the following way:

conditionId

[conditionName]

modelEntityId1

modelEntityId${n}

STRING

[STRING]

NUMERIC|STRING

NUMERIC|STRING

e.g.

conditionId1

[conditionName1]

0.42

parameterId

conditionId2

Row- and column-ordering are arbitrary, although specifying conditionId first may improve human readability.

Additional columns are not allowed.

Detailed field description

  • conditionId [STRING, NOT NULL]

    Unique identifier for the simulation/experimental condition, to be referenced by the measurement table described below. Must consist only of upper and lower case letters, digits and underscores, and must not start with a digit.

  • conditionName [STRING, OPTIONAL]

    Condition names are arbitrary strings to describe the given condition. They may be used for reporting or visualization.

  • ${modelEntityId}

    Further columns may be the IDs of model entities that have globally unique IDs, such as parameters, species or compartments defined in the model to set condition-specific values. Only one column is allowed per ID. Values for these entities may be provided either as numeric values, or as IDs of globally unique entity IDs as defined in the model, the mapping table or the parameter table.

    Any non-NaN value will override the original values of the model, or if preequilibration was used, they will override the value obtained from preequilibration. A NaN value indicates that the original value of the model is to be used (when used in the preequilibration condition, or in the simulation condition if no preequilibration is used) or that the result of preequilibration is to be used (when used in the simulation condition after preequilibration).

    The value in the condition table either replaces the initial value or the value at all timepoints based on whether the model entity has a rate law assigned or not:

    • For model entities that have constant algebraic assignments (but not necessarily constant values), i.e, that do not have a rate of change with respect to time assigned and that are not subject to event assignments, the algebraic assignment is replaced statically at all timepoints. Examples for such model entities are the targets of SBML AssignmentRules.

    • For all other entities, e.g., those that are assigned by SBML RateRules, only the initial value can be assigned in the condition table. If an assignment of the rate of change with respect to time or event assignment is desired, the values of model entities that are used to define rate of change or event assignments must be assigned in the condition table. If no such model entities exist, assignment is not possible.

    If the model has a concept of species and a species ID is provided, its value is interpreted as amount or concentration in the same way as anywhere else in the model.

Measurement table

A tab-separated values files containing all measurements to be used for model training or validation.

Expected to have the following named columns in any (but preferably this) order:

observableId

[preequilibrationConditionId]

simulationConditionId

measurement

time

observableId

[conditionId]

conditionId

NUMERIC

NUMERIC|inf

(wrapped for readability)

[observableParameters]

[noiseParameters]

[parameterId|NUMERIC[;parameterId|NUMERIC][…]]

[parameterId|NUMERIC[;parameterId|NUMERIC][…]]

Additional (non-standard) columns may be added. If the additional plotting functionality of PEtab should be used, such columns could be

[datasetId]

[replicateId]

[datasetId]

[replicateId]

where datasetId is a necessary column to use particular plotting functionality, and replicateId is optional, which can be used to group replicates and plot error bars.

Detailed field description

  • observableId [STRING, NOT NULL, REFERENCES(observables.observableID)]

    Observable ID as defined in the observable table described below.

  • preequilibrationConditionId [STRING OR NULL, REFERENCES(conditionsTable.conditionID), OPTIONAL]

    The conditionId to be used for preequilibration. E.g. for drug treatments, the model would be preequilibrated with the no-drug condition. Empty for no preequilibration.

  • simulationConditionId [STRING, NOT NULL, REFERENCES(conditionsTable.conditionID)]

    conditionId as provided in the condition table, specifying the condition-specific parameters used for simulation.

  • measurement [NUMERIC, NOT NULL]

    The measured value in the same units/scale as the model output.

  • time [NUMERIC OR STRING, NOT NULL]

    Time point of the measurement in the time unit specified in the SBML model, numeric value or inf (lower-case) for steady-state measurements.

  • observableParameters [NUMERIC, STRING OR NULL, OPTIONAL]

    This field allows overriding or introducing condition-specific versions of output parameters defined in the observation model. The model can define observables (see below) containing place-holder parameters which can be replaced by condition-specific dynamic or constant parameters. Placeholder parameters must be named observableParameter${n}_${observableId} with n ranging from 1 (not 0) to the number of placeholders for the given observable, without gaps. If the observable specified under observableId contains no placeholders, this field must be empty. If it contains n > 0 placeholders, this field must hold n semicolon-separated numeric values or parameter names. No trailing semicolon must be added.

    Different lines for the same observableId may specify different parameters. This may be used to account for condition-specific or batch-specific parameters. This will translate into an extended estimation parameter vector.

    All placeholders defined in the observation model must be overwritten here. If there are no placeholders used, this column may be omitted.

  • noiseParameters [NUMERIC, STRING OR NULL, OPTIONAL]

    The measurement standard deviation or empty if the corresponding sigma is a model parameter.

    Numeric values or parameter names are allowed. Same rules apply as for observableParameters in the previous point.

  • datasetId [STRING, OPTIONAL]

    The datasetId is used to group certain measurements to datasets. This is typically the case for data points which belong to the same observable, the same simulation and preequilibration condition, the same noise model, the same observable transformation and the same observable parameters. This grouping makes it possible to use the plotting routines which are provided in the PEtab repository.

  • replicateId [STRING, OPTIONAL]

    The replicateId can be used to discern replicates with the same datasetId, which is helpful for plotting e.g. error bars.

Observable table

Parameter estimation requires linking experimental observations to the model of interest. Therefore, one needs to define observables (model outputs) and respective noise models, which represent the measurement process. Since parameter estimation is beyond the scope of SBML, there exists no standard way to specify observables (model outputs) and respective noise models. Therefore, in PEtab observables are specified in a separate table as described in the following. This allows for a clear separation of the observation model and the underlying dynamic model, which allows, in most cases, to reuse any existing SBML model without modifications.

The observable table has the following columns:

observableId

[observableName]

observableFormula

STRING

[STRING]

STRING

e.g.

relativeTotalProtein1

Relative abundance of Protein1

observableParameter1_relativeTotalProtein1 * (protein1 + phospho_protein1 )

(wrapped for readability)

[observableTransformation]

noiseFormula

[noiseDistribution]

[lin(default)|log|log10]

STRING|NUMBER

[laplace|normal]

e.g.

lin

noiseParameter1_relativeTotalProtein1

normal

Detailed field description

  • observableId [STRING]

    Unique identifier for the given observable. Must consist only of upper and lower case letters, digits and underscores, and must not start with a digit. This is referenced by the observableId column in the measurement table.

  • [observableName] [STRING, OPTIONAL]

    Name of the observable. Only used for output, not for identification.

  • observableFormula [STRING]

    Observation function as plain text formula expression. May contain any symbol defined in the SBML model (including model time time) or parameter table. In the simplest case just an SBML species ID or an AssignmentRule target. Additionally, any observable ID introduced in the observable table may be referenced, but circular definitions must be avoided.

    May introduce new parameters of the form observableParameter${n}_${observableId}, which are overridden by observableParameters in the measurement table (see description there).

  • observableTransformation [STRING, OPTIONAL]

    Transformation of the observable and measurement for computing the objective function. Must be one of lin, log or log10. Defaults to lin. The measurements and model outputs are both assumed to be provided in linear space.

  • noiseFormula [NUMERIC|STRING]

    Measurement noise can be specified as a numerical value which will default to a Gaussian noise model if not specified differently in noiseDistribution with standard deviation as provided here. In this case, the same standard deviation is assumed for all measurements for the given observable.

    Alternatively, some formula expression can be provided to specify more complex noise models. The formula may reference any uniquely identifiable model entity with PEtab-compatible identifier or any observable ID specified in the observable table.

    A noise model which accounts for relative and absolute contributions could, e.g., be defined as:

    noiseParameter1_observable_pErk + noiseParameter2_observable_pErk * observable_pErk
    

    with noiseParameter1_observable_pErk denoting the absolute and noiseParameter2_observable_pErk the relative contribution for the observable observable_pErk corresponding to species pErk. IDs of noise parameters that need to have different values for different measurements have the structure: noiseParameter${indexOfNoiseParameter}_${observableId} to facilitate automatic recognition. The specific values or parameters are assigned in the noiseParameters field of the measurement table (see above). Any parameters named noiseParameter${1..n}_${observableId} must be overwritten in the measurement table.

    Noise formulae can also contain observable parameter overrides, which are described under observableFormula in this table. An example is when an observable formula contains an override, and a proportional noise model is used, which means the observable formula also appears in the noise formula.

  • noiseDistribution [STRING: ‘normal’ or ‘laplace’, OPTIONAL]

    Assumed noise distribution for the given measurement. Only normally or Laplace distributed noise is currently allowed (log-normal and log-Laplace are obtained by setting observableTransformation to log, similarly for log10). Defaults to normal. If normal, the specified noiseParameters will be interpreted as standard deviation (not variance). If Laplace ist specified, the specified noiseParameter will be interpreted as the scale, or diversity, parameter.

Noise distributions

For noiseDistribution, normal and laplace are supported. For observableTransformation, lin, log and log10 are supported. Denote by \(y\) the simulation, \(m\) the measurement, and \(\sigma\) the standard deviation of a normal, or the scale parameter of a laplace model, as given via the noiseFormula field. Then we have the following effective noise distributions.

  • Normal distribution:

    \[\pi(m|y,\sigma) = \frac{1}{\sqrt{2\pi}\sigma}\exp\left(-\frac{(m-y)^2}{2\sigma^2}\right)\]
  • Log-normal distribution (i.e. log(m) is normally distributed):

    \[\pi(m|y,\sigma) = \frac{1}{\sqrt{2\pi}\sigma m}\exp\left(-\frac{(\log m - \log y)^2}{2\sigma^2}\right)\]
  • Log10-normal distribution (i.e. log10(m) is normally distributed):

    \[\pi(m|y,\sigma) = \frac{1}{\sqrt{2\pi}\sigma m \log(10)}\exp\left(-\frac{(\log_{10} m - \log_{10} y)^2}{2\sigma^2}\right)\]
  • Laplace distribution:

    \[\pi(m|y,\sigma) = \frac{1}{2\sigma}\exp\left(-\frac{|m-y|}{\sigma}\right)\]
  • Log-Laplace distribution (i.e. log(m) is Laplace distributed):

    \[\pi(m|y,\sigma) = \frac{1}{2\sigma m}\exp\left(-\frac{|\log m - \log y|}{\sigma}\right)\]
  • Log10-Laplace distribution (i.e. log10(m) is Laplace distributed):

    \[\pi(m|y,\sigma) = \frac{1}{2\sigma m \log(10)}\exp\left(-\frac{|\log_{10} m - \log_{10} y|}{\sigma}\right)\]

The distributions above are for a single data point. For a collection \(D=\{m_i\}_i\) of data points and corresponding simulations \(Y=\{y_i\}_i\) and noise parameters \(\Sigma=\{\sigma_i\}_i\), the current specification assumes independence, i.e. the full distributions is

\[\pi(D|Y,\Sigma) = \prod_i\pi(m_i|y_i,\sigma_i)\]

Parameter table

A tab-separated value text file containing information on model parameters.

This table must include the following parameters:

  • Named parameter overrides introduced in the conditions table, unless defined in the SBML model

  • Named parameter overrides introduced in the measurement table

and must not include:

  • Placeholder parameters (see observableParameters and noiseParameters above)

  • Parameters included as column names in the condition table

  • Parameters that are AssignmentRule targets in the SBML model

  • SBML local parameters

it may include:

  • Any SBML model parameter that was not excluded above

  • Named parameter overrides introduced in the conditions table

One row per parameter with arbitrary order of rows and columns:

parameterId

[parameterName]

parameterScale

lowerBound

upperBound

nominalValue

estimate

STRING

[STRING]

log10|lin|log

NUMERIC

NUMERIC

NUMERIC

0|1

(wrapped for readability)

[initializationPriorType]

[initializationPriorParameters]

[objectivePriorType]

[objectivePriorParameters]

see below

see below

see below

see below

Additional columns may be added.

Detailed field description

  • parameterId [STRING, NOT NULL]

    The parameterId of the parameter described in this row. This has to match the ID of a parameter specified in the SBML model, a parameter introduced as override in the condition table, or a parameter occurring in the observableParameters or noiseParameters column of the measurement table (see above).

  • parameterName [STRING, OPTIONAL]

    Parameter name to be used e.g. for plotting etc. Can be chosen freely. May or may not coincide with the SBML parameter name.

  • parameterScale [lin|log|log10]

    Scale of the parameter to be used during parameter estimation.

    lin

    Use the parameter value, lowerBound, upperBound, and nominalValue without transformation.

    log

    Take the natural logarithm of the parameter value, lowerBound, upperBound, and nominalValue during parameter estimation.

    log10

    Take the logarithm to base 10 of the parameter value, lowerBound, upperBound, and nominalValue during parameter estimation.

  • lowerBound [NUMERIC]

    Lower bound of the parameter used for estimation. Optional, if estimate==0. The provided value should be untransformed, as it will be transformed according to parameterScale during parameter estimation.

  • upperBound [NUMERIC]

    Upper bound of the parameter used for estimation. Optional, if estimate==0. The provided value should be untransformed, as it will be transformed according to parameterScale during parameter estimation.

  • nominalValue [NUMERIC]

    Some parameter value to be used if the parameter is not subject to estimation (see estimate below). The provided value should be untransformed, as it will be transformed according to parameterScale during parameter estimation. Optional, unless estimate==0.

  • estimate [BOOL 0|1]

    1 or 0, depending on, if the parameter is estimated (1) or set to a fixed value(0) (see nominalValue).

  • initializationPriorType [STRING, OPTIONAL]

    Prior types used for sampling of initial points for estimation. Sampled points are clipped to lie inside the parameter boundaries specified by lowerBound and upperBound. Defaults to parameterScaleUniform.

    Possible prior types are:

    • uniform: flat prior on linear parameters

    • normal: Gaussian prior on linear parameters

    • laplace: Laplace prior on linear parameters

    • logNormal: exponentiated Gaussian prior on linear parameters

    • logLaplace: exponentiated Laplace prior on linear parameters

    • parameterScaleUniform (default): Flat prior on original parameter scale (equivalent to “no prior”)

    • parameterScaleNormal: Gaussian prior on original parameter scale

    • parameterScaleLaplace: Laplace prior on original parameter scale

  • initializationPriorParameters [STRING, OPTIONAL]

    Prior parameters used for sampling of initial points for estimation, separated by a semicolon. Defaults to lowerBound;upperBound. The parameters are expected to be in linear scale except for the parameterScale priors, where the prior parameters are expected to be in parameter scale.

    So far, only numeric values will be supported, no parameter names. Parameters for the different prior types are:

    • uniform: lower bound; upper bound

    • normal: mean; standard deviation (not variance)

    • laplace: location; scale

    • logNormal: parameters of corresp. normal distribution (see: normal)

    • logLaplace: parameters of corresp. Laplace distribution (see: laplace)

    • parameterScaleUniform: lower bound; upper bound

    • parameterScaleNormal: mean; standard deviation (not variance)

    • parameterScaleLaplace: location; scale

  • objectivePriorType [STRING, OPTIONAL]

    Prior types used for the objective function during estimation. For possible values, see initializationPriorType.

  • objectivePriorParameters [STRING, OPTIONAL]

    Prior parameters used for the objective function during estimation. For more detailed documentation, see initializationPriorParameters.

Visualization table

A tab-separated value file containing the specification of the visualization routines which come with the PEtab repository. Plots are in general collections of different datasets as specified using their datasetId (if provided) inside the measurement table.

Expected to have the following columns in any (but preferably this) order:

plotId

[plotName]

[plotTypeSimulation]

[plotTypeData]

STRING

[STRING]

[LinePlot(default)|BarPlot|ScatterPlot]

[MeanAndSD(default)|MeanAndSEM|replicate;provided]

(wrapped for readability)

[datasetId]

[xValues]

[xOffset]

[xLabel]

[xScale]

[datasetId]

[time(default)|parameterOrStateId]

[NUMERIC]

[STRING]

[lin|log|log10|order]

(wrapped for readability)

[yValues]

[yOffset]

[yLabel]

[yScale]

[legendEntry]

[observableId]

[NUMERIC]

[STRING]

[lin|log|log10]

[STRING]

Detailed field description

  • plotId [STRING, NOT NULL]

    An ID which corresponds to a specific plot. All datasets with the same plotId will be plotted into the same axes object.

  • plotName [STRING, OPTIONAL]

    A name for the specific plot.

  • plotTypeSimulation [STRING, OPTIONAL]

    The type of the corresponding plot, can be LinePlot, BarPlot and ScatterPlot. Default is LinePlot.

  • plotTypeData [STRING, OPTIONAL]

    The type how replicates should be handled, can be MeanAndSD, MeanAndSEM, replicate (for plotting all replicates separately), or provided (if numeric values for the noise level are provided in the measurement table). Default is MeanAndSD.

  • datasetId [STRING, NOT NULL, REFERENCES(measurementTable.datasetId), OPTIONAL]

    The datasets which should be grouped into one plot.

  • xValues [STRING, OPTIONAL]

    The independent variable, which will be plotted on the x-axis. Can be time (default, for time resolved data), or it can be parameterOrStateId for dose-response plots. The corresponding numeric values will be shown on the x-axis.

  • xOffset [NUMERIC, OPTIONAL]

    Possible data-offsets for the independent variable (default is 0).

  • xLabel [STRING, OPTIONAL]

    Label for the x-axis. Defaults to the entry in xValues.

  • xScale [STRING, OPTIONAL]

    Scale of the independent variable, can be lin, log, log10 or order. The order value should be used if values of the independent variable are ordinal. This value can only be used in combination with LinePlot value for the plotTypeSimulation column. In this case, points on x axis will be placed equidistantly from each other. Default is lin.

  • yValues [observableId, REFERENCES(measurementTable.observableId), OPTIONAL]

    The observable which should be plotted on the y-axis.

  • yOffset [NUMERIC, OPTIONAL]

    Possible data-offsets for the observable (default is 0).

  • yLabel [STRING, OPTIONAL]

    Label for the y-axis. Defaults to the entry in yValues.

  • yScale [STRING, OPTIONAL]

    Scale of the observable, can be lin, log, or log10. Default is lin.

  • legendEntry [STRING, OPTIONAL]

    The name that should be displayed for the corresponding dataset in the legend and which defaults to the value in datasetId.

Mapping table

Mapping PEtab entity IDs to entity IDs in the model. This optional file may be used to reference model entities in PEtab files where the ID in the model would not be a valid identifier in PEtab (e.g., due to inclusion of blanks, dots, or other special characters).

The TSV file has two mandatory columns, petabEntityId and modelEntityId. Additional columns are allowed.

petabEntityId

modelEntityId

STRING

STRING

reaction1_k1

reaction1.k1

Detailed field description

  • petabEntityId [STRING, NOT NULL]

    A valid PEtab identifier that is not defined in any other part of the PEtab problem. This identifier may be referenced in condition, measurement, parameter and observable tables, but cannot be referenced in the model itself.

  • modelEntityId [STRING, NOT NULL]

    A globally unique identifier defined in the model, that is not a valid PEtab ID (see Identifiers).

    For example, in SBML, local parameters may be referenced as $reactionId.$localParameterId, which are not valid PEtab IDs as they contain a . character. Similarly, this table may be used to reference specific species in a BNGL model that may contain many unsupported characters such as ,, ( or .. However, please note that IDs must exactly match the species names in the BNGL-generated network file, and no pattern matching will be performed.

Extensions

Additional columns, such as Color, etc. may be specified. Extensions that define operations on multiple PEtab problems need to employ a single PEtab YAML file as entrypoint to the analysis. This PEtab file may leave all fields specifying files empty and reference the other PEtab problems in the extension specific fields.

Examples

Examples of the visualization table can be found in the Benchmark model collection, for example in the Chen_MSB2009 model.

YAML file for grouping files

To link the model, measurement table, condition table, etc. in an unambiguous way, we use a YAML file.

This file also allows specifying a PEtab version and employed PEtab extensions.

Furthermore, this can be used to describe parameter estimation problems comprising multiple models (more details below).

The format is described in the jsonschema, which allows for easy validation:

# For syntax see: https://json-schema.org/understanding-json-schema/index.html
#$schema: "https://json-schema.org/draft/2019-09/meta/core"
$schema: "http://json-schema.org/draft-06/schema"
description: PEtab parameter estimation problem config file schema

properties:

  format_version:
    anyof:
      - type: string
        #  (corresponding to PEP 440).
        pattern: ^([1-9][0-9]*!)?(0|[1-9][0-9]*)(\.(0|[1-9][0-9]*))*((a|b|rc)(0|[1-9][0-9]*))?(\.post(0|[1-9][0-9]*))?(\.dev(0|[1-9][0-9]*))?$
      - type: integer

    description: Version of the PEtab format

  parameter_file:
    oneOf:
    - type: string
    - type: array
    description: |
      File name (absolute or relative) or URL to PEtab parameter table
      containing parameters of all models listed in `problems`. A single
      table may be split into multiple files and described as an array here.
  problems:
    type: array
    description: |
      One or multiple PEtab problems (sets of model, condition, observable
      and measurement files). If different model and data files are
      independent, they can be specified as separate PEtab problems, which
      may allow more efficient handling. Files in one problem cannot refer
      to models entities or data specified inside another problem.
    items:

      type: object
      description: |
        A set of PEtab model, condition, observable and measurement
        files and optional visualization files.
      properties:

        model_files:
          type: object
          description: One or multiple models

          # the model ID
          patternProperties:
            "^[a-zA-Z_]\\w*$":
              type: object
              properties:
                location:
                  type: string
                  description: Model file name or URL
                language:
                  type: string
                  description: |
                    Model language, e.g., 'sbml', 'cellml', 'bngl', 'pysb'
              required:
                - location
                - language
          additionalProperties: false

        measurement_files:
          type: array
          description: List of PEtab measurement files.

          items:
            type: string
            description: PEtab measurement file name or URL.

        condition_files:
          type: array
          description: List of PEtab condition files.

          items:
            type: string
            description: PEtab condition file name or URL.

        observable_files:
          type: array
          description: List of PEtab observable files.

          items:
            type: string
            description: PEtab observable file name or URL.

        visualization_files:
          type: array
          description: List of PEtab visualization files.

          items:
            type: string
            description: PEtab visualization file name or URL.

        mapping_files:
          type: array
          description: List of PEtab mapping files.

          items:
            type: string
            description: PEtab mapping file name or URL.

      required:
        - model_files
        - observable_files
        - measurement_files
        - condition_files

  extensions:
    type: object
    description: |
      PEtab extensions being used.
    patternProperties:
      "^[a-zA-Z][\\-\\w]*$":

        type: object
        description: |
          Information on a specific extension
        properties:
          version:
            type: string
            pattern: ^([1-9][0-9]*!)?(0|[1-9][0-9]*)(\.(0|[1-9][0-9]*))*((a|b|rc)(0|[1-9][0-9]*))?(\.post(0|[1-9][0-9]*))?(\.dev(0|[1-9][0-9]*))?$
          required:
            type: bool
            description: |
              Indicates whether the extension is required for the
              mathematical interpretation of the problem.
        required:
          - version
          - required
      additionalProperties: true

    additionalProperties: false

required:
  - format_version
  - parameter_file
  - problems

Parameter estimation problems combining multiple models

Parameter estimation problems can comprise multiple models. For now, PEtab allows one to specify multiple models with corresponding condition and measurement tables, and one joint parameter table. This means that the parameter namespace is global. Therefore, parameters with the same ID in different models will be considered identical.

Math expressions syntax

This section describes the syntax of math expressions used in PEtab files, such as the observable formulas.

Supported symbols, literals, and operations are described in the following. Whitespace is ignored in math expressions.

Symbols

  • The supported identifiers are:

    • parameter IDs from the parameter table

    • model entity IDs that are globally unique and have a clear interpretation in the math expression context

    • observable IDs from the observable table

    • PEtab placeholder IDs in the observable and noise formulas

    • PEtab entity IDs in the mapping table

    • time for the model time

    • PEtab function names listed below

Identifiers are not supported if they do not match the PEtab identifier format. PEtab expressions may have further context-specific restrictions on supported identifiers.

  • The functions defined in PEtab are tabulated below. Other functions, including those defined in the model, remain undefined in PEtab expressions.

  • Special symbols (such as \(e\) and \(\pi\)) are not supported, and neither is NaN (not-a-number).

Model time

The model time is represented by the symbol time, which is the current simulated time, not the current duration of simulated time; if the simulation starts at \(t_0 \neq 0\), then time is not the time since \(t_0\).

Literals

Numbers

All numbers, including integers, are treated as floating point numbers of undefined precision (although no less than double precision should be used. Only decimal notation is supported. Scientific notation is supported, with the exponent indicated by e or E. The decimal separator is indicated by .. Examples of valid numbers are: 1, 1.0, -1.0, 1.0e-3, 1.0e3, 1e+3. The general syntax in PCRE2 regex is \d*(\.\d+)?([eE][-+]?\d+)?. inf and -inf are supported as positive and negative infinity.

Booleans

Boolean literals are true and false.

Operations

Operators

The supported operators are:

Supported operators in PEtab math expressions.

Operator

Precedence

Interpretation

Associativity

Arguments

Evaluates to

f(arg1[, arg2, ...])

1

call to function f with arguments arg1, arg2, …

left-to-right

any

input-dependent

()

1

parentheses for grouping
acts like identity


any single expression

argument

^

2

exponentiation
(shorthand for pow)
right-to-left

float, float

float

+
-
3
unary plus
unary minus
right-to-left
float
float

!

3

not

bool

bool

*
/
4
multiplication
division
left-to-right
float, float
float
+
-
5
binary plus, addition
binary minus, subtraction
left-to-right
float, float
float
<
<=
>
>=
6
less than
less than or equal to
greater than
greater than or equal to
left-to-right
float, float
bool
==
!=
6
is equal to
is not equal to
left-to-right
(float, float) or (bool, bool)
bool
&&
||
7
logical and
logical or
left-to-right
bool, bool
bool

,

8

function argument separator

left-to-right

any

Note that operator precedence might be unexpected, compared to other programming languages. Use parentheses to enforce the desired order of operations.

Operators must be specified; there are no implicit operators. For example, a b is invalid, unlike a * b.

Functions

The following functions are supported:

Supported functions
Function
Comment
Argument types
Evaluates to

pow(a, b)

power function b-th power of a

float, float

float

exp(x)

exponential function pow(e, x)
(e itself not a supported symbol,
but exp(1) can be used instead)

float

float

sqrt(x)

square root of x
pow(x, 0.5)

float

float

log(a, b)
log(x)
ln(x)
log2(x)
log10(x)
logarithm of a with base b
log(x, e)
log(x, e)
log(x, 2)
log(x, 10)
(log(0) is defined as -inf)
(NOTE: log without explicit
base is ln, not log10)

float[, float]

float

sin
cos
tan
cot
sec
csc

trigonometric functions

float

float

arcsin
arccos
arctan
arccot
arcsec
arccsc

inverse trigonometric functions

float

float

sinh
cosh
tanh
coth
sech
csch

hyperbolic functions

float

float

arcsinh
arccosh
arctanh
arccoth
arcsech
arccsch

inverse hyperbolic functions

float

float

piecewise(
true_value_1,
condition_1,
[true_value_2,
condition_2,]
[...]
[true_value_n,
condition_n,]
otherwise
)
The function value is
the true_value* for the
first true condition*
or otherwise if all
conditions are false.
*value*: all float or all bool
condition*: all bool

float

abs(x)

absolute value
piecewise(x, x>=0, -x)

float

float

sign(x)

sign of x
piecewise(1, x > 0, -1, x < 0, 0)

float

float

min(a, b)
max(a, b)
minimum / maximum of {a, b}
piecewise(a, a<=b, b)
piecewise(a, a>=b, b)

float, float

float

Boolean <-> float conversion

Boolean and float values are implicitly convertible. The following rules apply:

bool -> float: true is converted to 1.0, false is converted to 0.0.

float -> bool: 0.0 is converted to false, all other values are converted to true.

Operands and function arguments are implicitly converted as needed. If there is no signature compatible with the given types, Boolean values are promoted to float. If there is still no compatible signature, float values are demoted to boolean values. For example, in 1 + true, true is promoted to 1.0 and the expression is interpreted as 1.0 + 1.0 = 2.0, whereas in 1 && true, 1 is demoted to true and the expression is interpreted as true && true = true.

Identifiers

  • All identifiers in PEtab may only contain upper and lower case letters, digits and underscores, and must not start with a digit. In PCRE2 regex, they must match [a-zA-Z_][a-zA-Z_\d]*.

  • Identifiers are case-sensitive.

  • Identifiers must not be a reserved keyword (see below).

  • Identifiers must be globally unique within the PEtab problem. PEtab does not put any further restrictions on the use of identifiers within the model, which means modelers could potentially use model-format–specific (e.g. SBML) function names as identifiers. However, this is strongly discouraged.

Reserved keywords

The following keywords, case-insensitive, are reserved and must not be used as identifiers:

  • true, false: Boolean literals, used in PEtab expressions.

  • inf: Infinity, used in PEtab expressions and post-equilibration measurements

  • time: Model time, used in PEtab expressions.

  • nan: Undefined in PEtab, but reserved to avoid implementation issues.

  • PEtab math function names (Functions)