SBMLsqueezer: A CellDesigner plug-in to generate kinetic rate equations for biochemical networks

Background The development of complex biochemical models has been facilitated through the standardization of machine-readable representations like SBML (Systems Biology Markup Language). This effort is accompanied by the ongoing development of the human-readable diagrammatic representation SBGN (Systems Biology Graphical Notation). The graphical SBML editor CellDesigner allows direct translation of SBGN into SBML, and vice versa. For the assignment of kinetic rate laws, however, this process is not straightforward, as it often requires manual assembly and specific knowledge of kinetic equations. Results SBMLsqueezer facilitates exactly this modeling step via automated equation generation, overcoming the highly error-prone and cumbersome process of manually assigning kinetic equations. For each reaction the kinetic equation is derived from the stoichiometry, the participating species (e.g., proteins, mRNA or simple molecules) as well as the regulatory relations (activation, inhibition or other modulations) of the SBGN diagram. Such information allows distinctions between, for example, translation, phosphorylation or state transitions. The types of kinetics considered are numerous, for instance generalized mass-action, Hill, convenience and several Michaelis-Menten-based kinetics, each including activation and inhibition. These kinetics allow SBMLsqueezer to cover metabolic, gene regulatory, signal transduction and mixed networks. Whenever multiple kinetics are applicable to one reaction, parameter settings allow for user-defined specifications. After invoking SBMLsqueezer, the kinetic formulas are generated and assigned to the model, which can then be simulated in CellDesigner or with external ODE solvers. Furthermore, the equations can be exported to SBML, LaTeX or plain text format. Conclusion SBMLsqueezer considers the annotation of all participating reactants, products and regulators when generating rate laws for reactions. Thus, for each reaction, only applicable kinetic formulas are considered. This modeling scheme creates kinetics in accordance with the diagrammatic representation. In contrast most previously published tools have relied on the stoichiometry and generic modulators of a reaction, thus ignoring and potentially conflicting with the information expressed through the process diagram. Additional material and the source code can be found at the project homepage (URL found in the Availability and requirements section).


Extracting Specific Rate Laws from SBML Process Diagrams
SBMLsqueezer was designed to interpret process diagrams created by CellDesigner and to apply rate laws for each reaction depending on the context. This document describes how SBMLsqueezer extracts the desired information from process diagrams and gives an overview of all currently supported rate laws. Each rate law is explained with an example process diagram, the general formula and the result yielded by SBMLsqueezer. At the end of each section we discuss the Systems Biology Ontology (SBO) terms which correspond to the introduced rate law. The appendix of this document contains a table of all currently available SBO terms for mathematical expressions in cellular systems (Section A, page 25).
Models of biological systems consist of reacting species. The rate of change of each species' concentration is determined by the reactions it is involved in. This rate depends on the velocities of the reactions. Each reaction velocity can basically be influenced by two types of modifiers: inhibitors and activators. Inhibitors lower the reaction velocity whereas activators speed up or enable the reaction. Any interaction of modulators with the reaction must be reflected in the rate law. An activated molecule can be an activator but this is not necessarily true. Here activation means activation of a reaction, in contrast to the activation of a molecule, which is a reaction in which the non-activated form of the molecule turns into the active form.

Extracting Specific Rate Laws from SBML Process Diagrams
The graphical notation of process diagrams used by CellDesigner extends standard SBML [1] with additional information that can be interpreted to automatically assign appropriate rate laws to each reaction. CellDesigner allows including reaction-specific information to only a certain level of detail, thus several reaction mechanisms cannot be distinguished. For instance, it is not possible to include exact formulas for inhibition and activation as can be done for selected mechanisms in relevant text books like those of Cornish-Bowden or Segel [2,3]. Often the process diagrams do not show at which state the modifier affects the reaction, i. e., whether the inhibitor reacts with the first substrate or the second one, with the ES 1 , the ES 2 or with the EP 1 P 2 complex, and so on.
To overcome this difficulty, Liebermeister and Klipp defined a generic inhibition and activation term [4]. This function is a prefactor, that can be multiplied with a kinetic equation to introduce modification. SBMLsqueezer applies this prefactor also to rather more detailed rate laws like the random order, ping-pong or the ordered ternary-complex mechanism.
CellDesigner 4.0α supports specialized arrows for two types of activating modification: one for transcriptional and another one for translational activation. A specific arrow for activation of enzyme reactions has been available since version 4.0β: the trigger symbol and-depending on the contextthe symbol for physical stimulation. We reached the following accommodation: Besides the trigger and physical stimulation symbol, SBMLsqueezer also interprets an unknown catalysis arrow as an activation of the respective reaction (Table 1). This also allows modeling an activation with the α-version of CellDesigner 4.0. For a complete list of all symbols used in CellDesigner reference can be made to Kitano et al. [5] and the CellDesigner homepage [6]. The term "Modulation" was also introduced

Unknown Catalysis Activation
Trigger Activation

Physical Stimulation Activation
Modulation Activation and Inhibition by CellDesigner 4.0β. It specifies one of the two kinds of interplay between reaction and modulator. Hence its meaning is decidedly not clear. By considering such a modulator as both inhibitor and activator SBMLsqueezer assumes that in a later parameter optimization process an appropriate optimizer determines which role prevails. In some cases, certain rate laws are special cases of other ones that only differ in their parameter settings. In these cases SBMLsqueezer always assigns the most general equation to the reaction, driven by the assumption that in a later parameter optimization process an optimizer will find the correct solution. Alternatively, all rate laws created by SBMLsqueezer may also be modified manually using the designated CellDesigner dialog boxes.

Supported Kinetic Equations
This section gives a complete list of all currently supported kinetic formulas and shows examples in the graphical notation of CellDesigner [7,5,6]. For an up-to-date list, refer to the project web page http://www.ra.cs.uni-tuebingen.de/software/SBMLsqueezer [8].
According to the process diagram, it often remains unclear at which state of the reaction the inhibitor or activator binds to enzyme, substrate or some intermediate complex. As stated in Section 1, SBMLsqueezer applies a generic formula for inhibition and activation for many rate laws such as the generalized mass-action kinetics or the rather detailed ternary-complex mechanisms. Equation (1), which was defined by Liebermeister et al. [4] in the context of convenience kinetics (Section 2.6, page 15), gives the general formula for this prefactor of the desired rate equation: where S and p are vectors of the concentrations of all reacting species in the system or parameter values, respectively. The matrices N ± contain the absolute values of all positive or negative elements of the stoichiometric matrix N or zero otherwise [4]. The matrices W ± are derived from the modulation matrix W in a similar way [4]. The modulation functions read: ( As an alternative to this simplified approach, one has to include all possible parameters assuming for a single inhibitor that it potentially acts at each state during the reaction. If more detailed knowledge about the mechanism is known, the rate law generated by SBMLsqueezer may serve as an initial equation that can be modified manually. For reactions with two or more catalysts, one rate law will be generated for each catalyst. The rate law for this particular reaction is given as the sum of the rates of all participating catalysts. If the reaction is one of those, whose modification is modeled according to Equation (1), the whole rate law will be multiplied by the modification term f .
We also grant that the enzyme may be omitted from the process diagram for the sake of simplicity and clear arrangement of the reacting species. SBMLsqueezer therefore offers a checkbox asking whether all reactions should be considered as being enzyme-catalyzed. In this case, all factors [E] 0 · k cat ± are replaced by the parameters V m ± that hide the enzyme concentration and allow estimation of the whole factor by appropriate optimizers.
The context menu of SBMLsqueezer for single reactions considers RNA and asRNA, simple and unknown molecules, complexes, truncated as well as generic proteins, and receptors all as enzymes to allow the user to apply any possible kinetic formula to a certain reaction whereas the plug-in window provides user settings to restrict this list to a more detailed selection of possible enzymes.
In SBML every species has an identifier (ID) and may also have a name. The ID is an obligatory tag whereas the name may be empty [1,9]. The name is intended to be a biologically meaningful identifier, which can in some cases be very long. Since the ID is supposed to be a short systematic identifier, SBMLsqueezer uses the ID for its L A T E X export.

Generalized Mass-Action Kinetics
In SBMLsqueezer generalized mass-action kinetics utilizes Equation (1) to include modification effects. This approach has already been successfully applied [10]. Figure 1 depicts an example of a S1 P1 I1 Figure 1: Example of a reaction to be modeled using the generalized mass-action rate law reaction which is catalyzed by ion I 1 and thus cannot be modeled using enzyme kinetic approaches. This reaction may have an arbitrary mechanism in which the product P 1 acts as an inhibitor. Equation (5) shows the general formula for the reversible, and Equation (6) for the irreversible, case. The reaction velocity v j of reaction j depends on a vector of all reacting species S and a parameter vector p. Equation (7) gives the kinetic equation generated for the process in Figure 1: F j (S, p) is allowed to be any positive function [11]. Thus all kinetic equations presented in the remainder of this section are special forms of this general formula as they can be derived from Equation (5) by setting F j (S, p) appropriately. Because of the availability of more specific equations, SBMLsqueezer restricts F j in the following way: Generalized mass-action kinetics allows modeling of reactions with any number of reactant and product molecules. However, since reactions with more than two reactants are unlikely to take place [2], warnings will be displayed. SBMLsqueezer applies Equations (5) and (6) to all reactions which are not catalyzed by an enzyme or catalyzed by non-enzymes. Gene regulation (transcription) processes that are neither activated nor inhibited by other factors proceed at a constant rate (basal gene expression) and hence follow a zeroth order mass-action rate law. Another example of where SBMLsqueezer applies mass-action kinetics is in degradation processes.
The Systems Biology Ontology (SBO) defines several special cases of generalized mass-action kinetics ( Table 2). None of the formulas defined therein includes the presence of any modulators. Monoexpotential decay (SBO:0000333) as a special case of first order irreversible mass-action kinetics (SBO:0000049) is indirectly supported by SBMLsqueezer because the rate constant k +j can

Uni-Uni Michaelis-Menten Kinetics
be set to a value less than one. SBMLsqueezer was designed to create rate laws for continuous simulators and does not support any derivatives of irreversible mass-action kinetics, discrete scheme (SBO:0000166). Note that the discrete formulas SBO:0000140 1 , SBO:0000141 2 , SBO:0000143 3 and SBO:0000146 4 are formally identical to their corresponding continuous forms. All other special cases of the mass-action kinetics in SBO can be created by SBMLsqueezer. Whenever a mass-action rate law is applicable, SBMLsqueezer also offers selection of a zeroth order rate law (either for the forward or the reverse reaction).  The latter includes both constants k Ia and K Ib . This allows for optimization of the model to fit the parameters and to decide which kind of inhibition is the most appropriate one if it is not known, including the following three special cases, that are often of particular interest:

Uni-Uni Michaelis-Menten Kinetics
1. Competitive inhibition for 0 < k Ia < ∞ and K Ib → ∞ 2. Noncompetitive inhibition for 0 < k Ia = K Ib < ∞ 3. Uncompetitive inhibition for k Ia → ∞ and 0 < K Ib < ∞ A detailed explanation of the different kinds of modification can be found in "The Regulation of Cellular Systems" [11]. In addition to the well-described inhibition, we employ the activation prefactor of Equation (1). The general Michaelis-Menten equation is given in Equation (9) with its corresponding irreversible form in Equation (10) and the example generated for the process diagram in Figure 2(a) is written in
The SBO defines several special cases of Equation (10) and provides one special case of Equation (9) without inhibition (SBO:0000326). Activation is currently not considered in SBO. In the case of no modulation, Equation (10) This equation includes SBO:0000276 and SBO:0000277 if exactly two inhibitors lower the reaction velocity v j . The latter one applies when ∀i : K Iai = K Ibi , which depends on the parameter settings. Another special case of Equation (12) emerges for ∀i : K Iai j → ∞: this rate law then includes SBO:0000270 (competitive inhibition of irreversible unireactant enzymes by exclusive inhibition) and its derivatives SBO:0000271 and SBO:0000274 ( Table 2).
The competitive inhibition of irreversible unireactant enzymes by non-exclusive non-cooperative inhibitors (SBO:0000273) and its derivative SBO:0000267 constitute a special case of Equation (10) only if exactly one inhibitor interferes with the reaction, K Ib j → ∞ and the exponent m i = 1: Therefore, SBMLsqueezer offers this equation as an alternative for each irreversible reaction with one substrate molecule and more than one inhibitor. Activation is included using the prefactor from convenience kinetics. Reversible uni-uni reactions with more than one inhibitor are modeled using the following equation which makes use of Equation (1) and is not included in the SBO: In some cases a single enzyme reacts with two reactants. Depending on the sequence in which the reactants bind to the enzyme, we can distinguish two different reaction mechanisms. Additionally, convenience kinetics constitutes a third alternative when no information about the mechanism is available. For irreversible bi-uni enzyme reactions without modulation, Equation (32) gives an additional modeling alternative. A special case of this bi-uni reaction emerges if there is one reactant species that has a stoichiometric coefficient of two. Figure 3(a) shows a possible graphical representation of this type of reaction. Neither the random order mechanism nor the ordered mechanism for bi-uni reactions is currently defined in SBO. For both mechanisms we also apply the prefactor defined by convenience kinetics in Equation (1).

Random Order Mechanism
The reaction scheme of this mechanism is presented in Figure 3(b). For the sake of simplicity the inhibition mechanism is omitted from this scheme. Both substrates bind in arbitrary sequence to the enzyme. The general formula for this mechanism is given in Equation (15), and its irreversible form Figure 3: Bi-uni random order enzyme reaction mechanism is shown in Equation (16). The automatically generated equation to Figure 3(a) with respect to this mechanism can be found in Equation (17). For a derivation of this formula see Section 3. Figure 4(b) depicts the reaction scheme of the ordered bi-uni mechanism. Note that in this reaction mechanism the sequence in which the species react is fixed. Equation (18) gives the general formula S1 P1 E1 S2 (a) Bi-uni enzyme-catalyzed reaction The general reaction scheme of the random order mechanism for bi-bi reactions is given in Figure 6. The sequence in which the reactants bind to the enzyme and the products leave the enzyme complex is arbitrary. Equation (21) models the reversible reaction with this rapid-equilibrium random order Figure 6: Reaction scheme of the random order bi-bi mechanism ternary-complex mechanism [2, p. 169] whereas the irreversible alternative is given by Equation (22). The automatically derived equation for the example in Figure 5 is shown in Equation (23).

Ordered Mechanism
Due to the following relations among the Michaelis and inhibition constants, the constants k M j,S 1 and k M j,P 2 do not appear explicitly in Equation (21): Figure 7(b) presents the reaction scheme for the ordered bi-bi mechanism, which is also called the compulsory-order ternary-complex mechanism [2, pp. 166-168]. As in the bi-uni case (Section 2.3.2), the sequence, in which all reactants bind to the enzyme, is fixed. Furthermore, the products also leave the enzyme complex in a defined sequence. A special case of this reaction is given when there is just one reactant or just one product with the stoichiometry of two.

Ordered Mechanism
(a) Example of a bi-bi enzyme reaction The formula for a reversible reaction is given by Equation (26) whereas the corresponding irreversible form can be found in Equation (27). An example for a generated equation with respect to Figure 5 can be found in Equation (28).

Ping-Pong Mechanism
A special case of the ordered mechanism is the ping-pong reaction, whose scheme is presented in Figure 8. The reactants bind in a fixed sequence, and the products leave the enzyme complex in a specific succession. However, during the reaction, the enzyme passes through different states so that it can only react with the next reactant or set the next product free. This is why the mechanism is also called a substituted-enzyme mechanism. No corresponding bi-uni reaction exists because it would formally be equal to the ordered bi-uni mechanism. Equation (29) gives the general formula

Irreversible Non-Modulated Non-Interacting Reactant Enzymes
Irreversible enzyme-catalyzed reactions with more than one substrate can alternatively be modeled using the following equation if there is no modulator: In contrast to the formulas for reversible reactions, the number of products does not matter for rate laws of irreversible reactions. Figure 9 depicts an example of a compatible process diagram. Two

Convenience Kinetics and Thermodynamics
substrate molecules react to one product. Equation (33) gives the generated rate law for this example.
SBMLsqueezer offers the user the choice of selecting this equation (SBO:0000150) whenever the aforementioned conditions are fulfilled. Equation (32) also covers the special cases SBO:0000151 and SBO:0000152 for two or three substrate molecules, respectively. This rate law is a special case of convenience kinetics with distinct reactants, each with stoichiometry one and no modulation at all.

Convenience Kinetics and Thermodynamics
In their original work Liebermeister et al. published the convenience kinetics in two forms [4]: The prefactor f introduces the modifiers for activation or inhibition, to the kinetic equation and was defined in Equation (1). Equation (34) can be applied to any enzyme-catalyzed reaction. However, if the stoichiometric matrix N of the reaction system contains linearly dependent columns, i. e., N does not have full column rank, then at least one reaction is thermodynamically dependent on another. In this case, choosing the parameters of the equation while ignoring this dependency may fit given measurement data well but will violate the thermodynamic constraints of the system. Hence, Liebermeister et al.
This ensures that all newly introduced parameters are thermodynamically independent. Note that every k G i stands for molecule i regardless of the respective reaction, whereas every k V j is a parameter for reaction j and does not depend on any molecule. The Michaelis analog parameter k M ji depends on both reaction j and molecule i and thus links both parameters together. For a complete derivation see the original paper of Liebermeister et al. [4].
Because Equation (35) is more complicated and contains additional parameters, SBMLsqueezer uses the simpler formula whenever applicable. To ensure the thermodynamic correctness of the system, an implementation of the Gaussian algorithm, which computes the rank of a matrix, is invoked. If the stoichiometric matrix of the system has full column rank, there is no need to apply Equation (35). Otherwise SBMLsqueezer will assign every reaction to be modeled using convenience kinetics with Equation (35).
For all enzyme-catalyzed reactions independent of the mechanism, convenience kinetics may be an appropriate choice if the user lacks detailed biochemical knowledge. As stated before, reactions with more than two substrate molecules are unlikely to take place. SBMLsqueezer will show a warning message for such reactions. This number does not only stand for the number of different reactant species, but rather for the stoichiometry on the left hand side. For instance, the reaction will also be considered unrealistic. This warning is, however, user-defined and the equations can still be generated properly pursuant to the particular formula. In the case of the context menu, warnings are shown whenever the number of reacting species exceeds two. In an application of convenience kinetics to a mixed network together with uni-uni Michaelis-Menten equations, it was shown that convenience kinetics leads to reasonable results when fitted to in vivo data [12]. At the time of writing no form of convenience kinetics is included in the systems biology ontology (SBO, Table 2).

Thermodynamically Dependent Form
Equation (34) shows the thermodynamically dependent formula of convenience kinetics for reversible reactions. Equation (37) gives its corresponding irreversible form [4]. An example of a generated rate law is shown in Equation (38) for the bi-bi reaction presented in Figure 5

Thermodynamically Independent Form
As stated at the beginning of this Section (page 15), if there are linear dependencies within the stoichiometric matrix, SBMLsqueezer applies the thermodynamically independent form of convenience kinetics, which is shown in Equation (35) and for its corresponding irreversible form in Equation (39) 2.7. Hill Equation [4]. Equation (40) shows the generated independent form of the reaction example depicted in Figure 5,

Hill Equation
Gene regulation can also be modeled using CellDesigner. A common rate law to model those reactions is the Hill equation [11,13]. Figure 10 depicts one example of a process considered gene regulation. Gene s 1 is expressed and the RNA molecule s 2 assembles. This process is (transcriptionally) inhibited by the translation product, protein s 3 . The translation process is (translationally) activated by protein s 4 . Note that the concentration of gene s 1 remains unchanged during this process as the transcription does not change the state of the gene. SBMLsqueezer recognizes mistakes within the SBML file and sets the boundary conditions of genes to "true". Furthermore, SBMLsqueezer will show warnings if a transcription is, for instance, "translationally" activated. Since the release of CellDesigner 4.0β there have been two special arrows for the transitions described here: transcription and translation. To ensure backwards compability, SBMLsqueezer supports simple state transitions, even between genes and RNA as well as between RNA and proteins. However, if transcription and translation arrows are used, SBMLsqueezer will show a warning message if they are mixed up.
The general Hill equation is given in Equation (41). The formula for the translation example in Figure 10 can be found in Equation (42) and the rate of transcription generated according to Figure 10 is given by Equation (43). Note that the exponents w ± jm are defined according to the modulation matrices W ± in Section 2.6, page 15.
The constant k i distinguishes genes from other species: The SBO defines three forms of the Hill equation (SBO:0000192, SBO:0000195 and SBO:0000198). The form described here is the general microscopic form (SBO:0000195) where no inhibition is involved; see Table 2 for details. Both other types are special cases of this formula for appropriate parameter settings. If a gene regulation or translation reaction without an assigned activator or inhibitor occurs, Equation (42) formally becomes a zeroth order mass-action equation (Section 2.1, page 6).

Derivation of Predefined Kinetic Equations for Bi-Uni Reactions
This section shows the derivation of rate laws for the random order and the ordered bi-uni mechanisms using the King-Altman method [2]. This derivation is necessary, since for this special case the common literature does not provide appropriate equations to be used as a pre-computed formula in SBMLsqueezer [2,3,14].
The King-Altman method provides an algorithm to create rate laws even for complex enzymecatalyzed reaction mechanisms according to the quasi-steady-state approximation [2,14]. The algorithm roughly consists of five steps that will be explained in detail in the remainder of this section.

Ordered Bi-Uni Mechanism
Firstly, we derive the rate law for the ordered bi-uni mechanism. Note that the sequence, in which the reactants bind to the enzyme molecule, is fixed (Figure 4(b), page 10).

First Step
A polygon, whose arcs and vertices reflect the reaction mechanism, is charted ( Figure 11). Every vertex symbolizes one of the forms of the enzyme during the reaction. The edges mirror the transition between these forms. All charted transitions have to be first-order reactions. Second-order reactions must be given in pseudo-first-order form. The arrows, which constitute the edges of the diagram, Figure 11: Underlying reaction scheme for the ordered bi-uni mechanism are labeled with the rate constants, which are multiplied with entering ligands if necessary, for the corresponding transition.

S S S S S S S S S S S S S S
The master pattern drafts the reaction scheme as a rough structure. In this case we obtain a triangle ( Figure 12).  Each of these patterns contains exactly one edge fewer than the master pattern. We obtain three structures, each with two edges (Figure 13).

Third Step
In this step every single enzyme state is marked one time within each pattern and directed arcs replace the edges, each pointing towards the highlighted enzyme state. According to the resulting pattern, we determine an equation for the relative amount of each highlighted enzyme state ( Figure 14).
The denominator D equals the sum of all numerator terms of the equations in Figure 14 and reads Figure 14: Sub-patterns with their respective equations

Fourth
Step We now write the denominator as the product of coefficients in a way that all constants are ordered with respect to their concentration terms: where The rate law is then given as the sum of the rates to form a particular product decremented by the rates that reduce this product. In this ordered bi-uni mechanism there is only one step, in which P is produced. Hence, there is only one way to consume P again and the formula reads: D .

Fifth
Step The kinetic parameters are defined based on the coefficients determined in the fourth step. The Michaelis constants k M i are defined as the ratio of all constants of the substrate or product formation rate, minus the constants of the product or substrate formation rate, and the coefficient of the rates of all substrates or products.
The maximal activities for the forward and reverse reaction, V m + and V m − , are the quotient of the respective numerator coefficient and the coefficient of all substrates or products, respectively.
After some conversions these kinetic equations can be combined as follows: and Applying some more conversions and aggregation, the rate law then reads where we define the following constants: To accommodate this equation with the uni-uni Michaelis-Menten equation, we set V m This leads us to the following equation for a reversible ordered bi-uni mechanism: For the irreversible ordered bi-uni mechanism the formula for the rate law reads

Random Order Bi-Uni Mechanism
The random order mechanism is characterized by its arbitrary sequence, in which the reactants bind to the enzyme. The binding of every substrate is carried out independently of the others (Figure 3(a) on page 9). This must also be reflected in the corresponding rate law.
To derive an appropriate pre-computed formula for this mechanism, we again apply the five steps of the King-Altman method [14, pp. 126-131]. Here, we only briefly summarize the steps and omit a detailed explanation.

First Step
We chart the master pattern of the mechanism from the polygon that belongs to this particular reaction scheme ( Figure 15).

Second Step
Next we construct all possible substructures, each with exactly one edge fewer than the master pattern ( Figure 16).

Random Order Bi-Uni Mechanism
Step The denominator and the preliminary kinetic equation can now be written as: (77)

Fifth
Step To complete the derivation, we have to define the Michaelis constant. According to Segel, the random order bi-uni mechanism does not provide a hyperbole function when no substrate saturation is present [3]. Thus, a kinetic formula based on this mechanism cannot be linearized. Hence, we may combine several rate constants to Michaelis-like constants similar to what we applied in the derivation for the ordered bi-uni mechanism. However, these constants would not be in accordance with the definition of the Michaelis constant of the uni-uni formula [3]. Assuming fast steady-states between the ternary and the binary complexes EAB and EP, and also assuming all conversions to be fast, this equation can be simplified as follows: According to Cornish-Bowden, the squared terms in the numerator and denominator as well as the terms  [2]. This leads to the definition of the dissociation constants k I A , k I B and k I P for k M A , k M B and k M P . Since the sequence, in which the reactants bind to the enzyme, is arbitrary, we have k M A k I B = k I A k M B . Hence, the derived equation is valid for this mechanism and assumes an underlying rapid-equilibrium-random-mechanism. The constants of this equation are defined as follows: And the final equation then reads: For the irreversible random order bi-uni mechanism the above formula can be simplified to . (84)

A. Systems Biology Ontology of Mathematical Expressions
The following Table 2 provides an overview of all mathematical expressions (SBO:0000064), defined mainly by Nicolas Le Novère, Michael Hucka and Andrew Finney in the Systems Biology Ontology [15]. The SBO term "mathematical expression" contains the term "conservation law" (SBO:0000355) and has two sub-categories: "rate law" (SBO:0000001) and "obsolete mathematical expression" (SBO:0000005).
Here we only consider the category "rate law", whose terms are listed in the following table. The identifiers of internal nodes are written in bold face. The column "variables" contains both parameters and reacting species. The table lists the tree of SBO terms for rate laws, which are defined at the time of writing, serially. The entries in this table are not sorted with respect to the SBO identification number but to the order of their occurrence within the tree from top to bottom. The discrete scheme mass-action kinetics, which are currently not supported by SBMLsqueezer, are printed gray. Note that four of these are formally equivalent to their corresponding continuous rate laws (see Section 2.1 for details).
Also  0000086 First order forward, third order reverse with two products, reversible mass action kinetics, continuous scheme k f , k r , R, P 1 , P 2 k f · R − k r · P 2 1 · P 2 0000087 First order forward, third order reverse with three products, reversible mass action kinetics, continuous scheme k f , k r , R, P 1 , P 2 , P 3 k f · R − k r · P 1 · P 2 · P 3 0000088 Second order forward reversible mass action kinetics general category 0000089 Second order forward with one reactant reversible mass action kinetics general category 0000090 Second order forward with one reactant, zeroth order reverse, reversible mass action kinetics, continuous scheme k f , k r , R k f · R 2 − k r 0000091 Second order forward with one reactant, first order reverse, reversible mass action kinetics, continuous scheme k f , k r , R, P k f · R 2 − k r · P 0000092 Second order forward with one reactant, second order reverse, reversible mass action kinetics general category 0000093 Second order forward with one reactant, second order reverse with one product, reversible mass action kinetics, continuous scheme k f , k r , R, P k f · R 2 − k r · P 2 0000094 Second order forward with one reactant, second order reverse with two products, reversible mass action kinetics, continuous scheme k f , k r , R, P 1 , P 2 k f · R 2 − k r · P 1 · P 2 0000095 Second order forward with one reactant, third order reverse, reversible mass action kinetics general category SBO Term Variables Formula 0000096 Second order forward with one reactant, third order reverse with one product, reversible mass action kinetics, continuous scheme k f , k r , R, P k f · R 2 − k r · P 3 0000097 Second order forward with one reactant, third order reverse with two products, reversible mass action kinetics, continuous scheme k f , k r , R, P 1 , P 2 k f · R 2 − k r · P 2 1 · P 2 0000098 Second order forward with one reactant, third order reverse with three products, reversible mass action kinetics, continuous scheme k f , k r , R, P 1 , P 2 , P 3 k f · R 2 − k r · P 1 · P 2 · P 3 0000099 Second order forward with two reactants reversible mass action kinetics general category 0000100 Second order forward with two reactants, zeroth order reverse, reversible mass action kinetics, continuous scheme k f , k r , R 1 , R 2 k f · R 1 · R 2 − k r 0000101 Second order forward with two reactants, first order reverse, reversible mass action kinetics, continuous scheme k f , k r , R 1 , R 2 , P k f · R 1 · R 2 − k r · P 0000102 Second order forward with two reactants, second order reverse, reversible mass action kinetics general category 0000103 Second order forward with two reactants, second order reverse with one product, reversible mass action kinetics, continuous scheme k f , k r , R 1 , R 2 , P k f · R 1 · R 2 − k r · P 2 0000104 Second order forward with two reactants, second order reverse with two products, reversible mass action kinetics, continuous scheme k f , k r , R 1 , R 2 , P 1 , P 2 k f · R 1 · R 2 − k r · P 1 · P 2 0000105 Second order forward with two reactants, third order reverse, reversible mass action kinetics general category