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AssignmentRule Class Reference

Detailed Description

Implementation of SBML's AssignmentRule construct.

The rule type AssignmentRule is derived from the parent class Rule. It is used to express equations that set the values of variables. The left-hand side (the attribute named "variable") of an assignment rule can refer to the identifier of a Species, SpeciesReference (in SBML Level 3), Compartment, or Parameter object in the model (but not a Reaction). The entity identified must have its "constant" attribute set to false. The effects of an AssignmentRule are in general terms the same, but differ in the precise details depending on the type of variable being set:

  • In the case of a species, an AssignmentRule sets the referenced species' quantity (whether a "concentration" or "amount") to the value determined by the formula in the MathML subelement "math". The unit associated with the value produced by the "math" formula should (in SBML Level 2 Version 4 and in SBML Level 3) or must (in SBML releases prior to Level 2 version 4) be equal to the unit associated with the species' quantity. Restrictions: There must not be both an AssignmentRule "variable" attribute and a SpeciesReference "species" attribute having the same value, unless the referenced Species object has its "boundaryCondition" attribute set to true. In other words, an assignment rule cannot be defined for a species that is created or destroyed in a reaction unless that species is defined as a boundary condition in the model.

  • (For SBML Level 3 only) In the case of a species reference, an AssignmentRule sets the stoichiometry of the referenced reactant or product to the value determined by the formula in "math". The unit associated with the value produced by the "math" formula should be consistent with the unit "dimensionless", because reactant and product stoichiometries in reactions are dimensionless quantities.

  • In the case of a compartment, an AssignmentRule sets the referenced compartment's size to the value determined by the formula in the "math" subelement of the AssignmentRule object. The overall units of the formula in "math" should (in SBML Level 2 Version 4 and in SBML Level 3) or must (in SBML releases prior to Level 2 version 4) be the same as the units of the size of the compartment.

  • In the case of a parameter, an AssignmentRule sets the referenced parameter's value to that determined by the formula in the "math" subelement of the AssignmentRule object. The overall units of the formula in the "math" subelement should (in SBML Level 2 Version 4 and in SBML Level 3) or must (in SBML releases prior to Level 2 version 4) be the same as the units defined for the parameter.

In the context of a simulation, assignment rules are in effect at all times, t $\geq$ 0. For purposes of evaluating expressions that involve the delay "csymbol" (see the SBML Level 2 specification), assignment rules are considered to apply also at t $\leq$ 0. Please consult the relevant SBML specification for additional information about the semantics of assignments, rules, and entity values for simulation time t $\leq$ 0.

A model must not contain more than one AssignmentRule or RateRule object having the same value of "variable"; in other words, in the set of all assignment rules and rate rules in an SBML model, each variable appearing in the left-hand sides can only appear once. This simply follows from the fact that an indeterminate system would result if a model contained more than one assignment rule for the same variable or both an assignment rule and a rate rule for the same variable.

Similarly, a model must also not contain both an AssignmentRule and an InitialAssignment for the same variable, because both kinds of constructs apply prior to and at the start of simulation time, i.e., t $\leq$ 0. If a model contained both an initial assignment and an assignment rule for the same variable, an indeterminate system would result.

The value calculated by an AssignmentRule object overrides the value assigned to the given symbol by the object defining that symbol. For example, if a Compartment object's "size" attribute value is set in its definition, and the model also contains an AssignmentRule object having that compartment's "id" as its "variable" value, then the "size" assigned in the Compartment object definition is ignored and the value assigned based on the computation defined in the AssignmentRule. This does not mean that a definition for a given symbol can be omitted if there is an AssignmentRule object for it. For example, there must be a Parameter definition for a given parameter if there is an AssignmentRule for that parameter. It is only a question of which value definition takes precedence.

General summary of SBML rules

In SBML Level 3 as well as Level 2, rules are separated into three subclasses for the benefit of model analysis software. The three subclasses are based on the following three different possible functional forms (where x is a variable, f is some arbitrary function returning a numerical result, V is a vector of variables that does not include x, and W is a vector of variables that may include x):
Algebraic:left-hand side is zero0 = f(W)
Assignment:left-hand side is a scalar:x = f(V)
Rate:left-hand side is a rate-of-change:dx/dt = f(W)
In their general form given above, there is little to distinguish between assignment and algebraic rules. They are treated as separate cases for the following reasons:
  • Assignment rules can simply be evaluated to calculate intermediate values for use in numerical methods. They are statements of equality that hold at all times. (For assignments that are only performed once, see InitialAssignment.)

  • SBML needs to place restrictions on assignment rules, for example the restriction that assignment rules cannot contain algebraic loops.

  • Some simulators do not contain numerical solvers capable of solving unconstrained algebraic equations, and providing more direct forms such as assignment rules may enable those simulators to process models they could not process if the same assignments were put in the form of general algebraic equations;

  • Those simulators that can solve these algebraic equations make a distinction between the different categories listed above; and

  • Some specialized numerical analyses of models may only be applicable to models that do not contain algebraic rules.
The approach taken to covering these cases in SBML is to define an abstract Rule structure containing a subelement, "math", to hold the right-hand side expression, then to derive subtypes of Rule that add attributes to distinguish the cases of algebraic, assignment and rate rules. The "math" subelement must contain a MathML expression defining the mathematical formula of the rule. This MathML formula must return a numerical value. The formula can be an arbitrary expression referencing the variables and other entities in an SBML model. Each of the three subclasses of Rule (AssignmentRule, AlgebraicRule, RateRule) inherit the the "math" subelement and other fields from SBase. The AssignmentRule and RateRule classes add an additional attribute, "variable". See the definitions of AssignmentRule, AlgebraicRule and RateRule for details about the structure and interpretation of each one.

Additional restrictions on SBML rules

An important design goal of SBML rule semantics is to ensure that a model's simulation and analysis results will not be dependent on when or how often rules are evaluated. To achieve this, SBML needs to place two restrictions on rule use. The first concerns algebraic loops in the system of assignments in a model, and the second concerns overdetermined systems.

A model must not contain algebraic loops

The combined set of InitialAssignment, AssignmentRule and KineticLaw objects in a model constitute a set of assignment statements that should be considered as a whole. (A KineticLaw object is counted as an assignment because it assigns a value to the symbol contained in the "id" attribute of the Reaction object in which it is defined.) This combined set of assignment statements must not contain algebraic loops—dependency chains between these statements must terminate. To put this more formally, consider a directed graph in which nodes are assignment statements and directed arcs exist for each occurrence of an SBML species, compartment or parameter symbol in an assignment statement's "math" subelement. Let the directed arcs point from the statement assigning the symbol to the statements that contain the symbol in their "math" subelement expressions. This graph must be acyclic. SBML does not specify when or how often rules should be evaluated. Eliminating algebraic loops ensures that assignment statements can be evaluated any number of times without the result of those evaluations changing. As an example, consider the set of equations x = x + 1, y = z + 200 and z = y + 100. If this set of equations were interpreted as a set of assignment statements, it would be invalid because the rule for x refers to x (exhibiting one type of loop), and the rule for y refers to z while the rule for z refers back to y (exhibiting another type of loop). Conversely, the following set of equations would constitute a valid set of assignment statements: x = 10, y = z + 200, and z = x + 100.

A model must not be overdetermined

An SBML model must not be overdetermined; that is, a model must not define more equations than there are unknowns in a model. An SBML model that does not contain AlgebraicRule structures cannot be overdetermined. LibSBML implements the static analysis procedure described in Appendix B of the SBML Level 3 Version 1 Core specification for assessing whether a model is overdetermined. (In summary, assessing whether a given continuous, deterministic, mathematical model is overdetermined does not require dynamic analysis; it can be done by analyzing the system of equations created from the model. One approach is to construct a bipartite graph in which one set of vertices represents the variables and the other the set of vertices represents the equations. Place edges between vertices such that variables in the system are linked to the equations that determine them. For algebraic equations, there will be edges between the equation and each variable occurring in the equation. For ordinary differential equations (such as those defined by rate rules or implied by the reaction rate definitions), there will be a single edge between the equation and the variable determined by that differential equation. A mathematical model is overdetermined if the maximal matchings of the bipartite graph contain disconnected vertexes representing equations. If one maximal matching has this property, then all the maximal matchings will have this property; i.e., it is only necessary to find one maximal matching.)

Rule types for SBML Level 1

SBML Level 1 uses a different scheme than SBML Level 2 and Level 3 for distinguishing rules; specifically, it uses an attribute whose value is drawn from an enumeration of 3 values. LibSBML supports this using methods that work with the enumeration values listed below.