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Improvements to the interpretation of reactions

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Image:Updated.gif on 2009-7-23.

Several of the changes in SBML Level 3 are the result of an improved and generalized understanding of reactions as processes. This understanding grew out of experience with Level 2. The (slight) generalization of the notion of reactions in Level 3 allows the reaction concept to be more correctly applicable to a broader range of processes beyond only biochemical reactions.

Classical interpretation of reactions and extent of reaction

Reactions are processes in which particular entities interact in some way. At some fine level of detail, the processes involve individual, discrete events. For example, when we say a biochemical reaction A + B → C takes place, what we mean is that entities we call "A" interact with entities we call "B" to bind together somehow and produce a new kind of entity we call "C".

The approach for describing and communicating how many of these events takes place per unit of time is up to humans to decide. Since it is generally not possible to measure directly the individual molecular events, the speed with which biochemical reactions take place is instead given in terms of what can be measured: the quantities of the substances affected per unit time. A common way of describing classical biochemical reactions is in terms of moles of substance: X moles of one species of substance interacts with Y moles of another, to yield Z moles of a third. Another common approach is to talk in terms of masses: P grams of one species interacts with Q grams of another to yield R grams of another.

Consider now the following single reaction taking place in a closed system:

νA A + νB B → νC C

The numbers represented by the Greek letter nu, νA, νB and νC, are the stoichiometric coefficients of substances A, B, and C, respectively. It follows from the law of definite proportions that the mass mi of a component i which is formed in a reaction is proportional to its molecular weight Mi and to its stoichiometric coefficient νi. Suppose we start the closed system above with a mass of A equal to mA0. What the law of definite proportion means is that if we measure the mass of A at some later time, it will be given by the following relationship:

mA − mA0 = νA MA ξ

where ξ is the extent of reaction. In fact, it turns out that for a closed system such as this,

mA − mA0 = νA MA ξ
mB − mB0 = νB MB ξ
mC − mC0 = νC MC ξ

where mB0 and mC0 represent the initial masses of mB and mC, respectively. In other words, ξ relates all of the quantities together. Rearranging the above gives

(mA − mA0)/MA = νA ξ
(mB − mB0)/MB = νB ξ
(mC − mC0)/MC = νC ξ

This is what leads to the classical interpretation of the extent as representing the degree to which a reaction has progressed. Since mi/Mi gives the number of moles of substance i, it follows that the unit of extent in this particular case is the mole.

Let ni = mi/Mi. Now if we differentiate the expressions above with respect to time, remembering that the initial masses are constants and thus drop out when differentiated, and rearrange the results slightly, we get

1   dnA           1   dnB           1   dnC          

 
    =    
 
    =    
 
    =    
νA     dt           νB     dt           νC     dt           dt


The rate of change of the extent is thus the rate of change of the quantities of A, B and C, measured in moles of substance per unit time.

This description applies to a single reaction in a closed system. If there are multiple simultaneous independent reactions taking place, each reaction will each have an extent variable. The same concepts can be extended to include this case, although we do not provide the equations here.

Classical relationship between reaction rate and extent

The rate of change of extent, dξ / dt, is equal to the classical rate law for a reaction.

A crucial detail that must be kept in mind, however, is that reaction rates are given in terms of concentrations (or more properly speaking, activities, which are usually, but not always, approximately equal to substance concentrations in dilute solutions). This necessarily¬†entails the inclusion of volume as a factor when dealing with rate laws and reaction rates; in turn, this entails the discussion of compartments in SBML. For the purposes of the current topic, we do not need to discuss these topics, but they are the subject of another page explaining other parts of Level 3 Core.

Generalizing the concept of reactions

In SBML, we need to view the reaction in a more general light, as a process in which events of a given kind take place over time, with biochemical reactions being just one kind of possible process. SBML Level 2 already supports this view, but it is not supported as well or as clearly as it could be. Importantly, Level 2 does not have a separate unit for extent; instead, the units of reaction rate are intertwined with Level 2's predefined unit substance, making a clear treatment of these two characteristics difficult when reactions do not involve substances. In Level 3, several changes described elsewhere will improve on this situation:

Conceptually, the notion of reaction and reaction extent in Level 3 are the same as the classical notion; they are simply widened to support more easily¬†the handling of entities other than substances and units other than mole. (This is actually possible to do in Level 2 too, although it entails implicit unit conversions and unobvious, awkward model construction. Eliminating these is the reason for the changes in this area in Level 3.) The key enablers are (1) the introduction of an explicit representation for the units of reaction extent, and (2) the introduction of an explicit conversion factor relating units of species in a reaction to the units of reaction extent. This avoids the problem in Level 2 that there is a single construct (the predefined unit substance) used as part of the definition of both reaction rate and global units of substance quantity. As described elsewhere, the units of reaction extent in Level 3 are defined by a global attribute on the <model> component. In addition:

  • In Level 3, the units of extent may be defined to be moles, but they may also be defined to be other measures, as appropriate for the problem at hand. The rate of change of extent of a reaction simply describes how many "things" are being acted upon per unit time by the reaction process, and "things" in Level 3 can be whatever makes sense in the context of the model.
  • To express how species quantities in a reaction process relate to the rate of the process (which is to say, the rate of change of extent per time), species will have an explicit conversion factor attribute. This allows a model to explicitly convert units of a reaction to the units of species quantity. Importantly, this allows treating certain situations, such as those involving mass, that in Level 2 must be handled by conflating unit conversions with the stoichiometry of reactants and products. In the Level 3, the stoichiometry of reactants and products are separated from the conversion factors.

References

  • The textbook by Prigione and Defay provides a clear treatment of reactions and reaction extent. In Chapter 1, section 8, they note that extent of reaction can be generalized, and they call the generalized version extent of change.
  • The article by Cvitas in J. Chem. Educ., 1999, summarizes succinctly the concepts of reaction extent and reaction rate. (However, the author states the unit of extent is mole, whereas here for SBML, we follow the Prigione and Defay approach of generalizing extent to apply to any kind of reactions, not just those involving substances.)

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