Steady-state kinetic product inhibition experiments and complete steady-state kinetic rate equations were used to measure dissociation constants of NAD+, NMN, and AMP and eliminate the possibility that AMP is the second product released in an ordered mechanism. Determining by steady-state kinetics whether the release of sealed DNA and AMP products goes by an ordered (AMP last off) or RER mechanism was shown to require a product inhibition study using sealed DNA.
Although the 3-step kinetic mechanism of DNA ligase has long been known, the order of release of the last 2 products, sealed DNA and AMP, has rarely been investigated. Steady-state kinetic studies using product inhibition of bacteriophage T4 DNA ligase by AMP and PPi contributed to the conclusion that the enzyme has a Ping Pong mechanism and is non-processive (i.e. it dissociates from DNA with each catalytic cycle rather than moving along it). Teraoka et al.  concluded from AMP and PPi product inhibition studies of calf thymus DNA ligase that sealed DNA is released before AMP (Figure 1, top). It should be noted, however, that the authors did not address the possibility that the release of sealed DNA and AMP might be random (Figure 1, bottom) instead of ordered. Product inhibition by sealed DNA, which could have addressed this question, could not be investigated because no inhibition by sealed DNA was observed. Cooper and Rudolph , discussing the results of Teraoka et al. , pointed out that ordered release of sealed DNA then AMP would prevent product inhibition by sealed DNA within cells as long as the AMP concentration is low relative to its K I . Other investigators have measured the potency of inhibition of DNA ligases by AMP [11, 12] or its binding affinity . In no case, however, has the steady-state rate equation including product concentrations been published for the Bi Ter Ping Pong kinetic mechanism of DNA ligase.
In this paper, we derive the complete steady-state kinetic rate equations, including product terms, for the Bi Ter Ping Pong Uni-Uni Uni-Bi mechanisms in which the last 2 products are released in either an ordered or RER fashion (Additional file 1). We apply these equations to separate product inhibition studies with NMN and AMP to measure dissociation constants for NAD+, NMN, and AMP in the reaction of the NAD+-dependent DNA ligase from Haemophilus influenza. These studies allowed us to eliminate one of the 3 possible kinetic mechanisms for release of the sealed DNA and AMP products by this enzyme, namely release of AMP before sealed DNA. The complete rate equations demonstrate that a product inhibition study with sealed DNA, which was not feasible with our experimental system, would be required to determine by steady-state kinetics whether AMP is released after sealed DNA in an ordered mechanism or the two products are released in random order.
The availability of the complete steady-state kinetic rate equations for the Bi Ter Ping Pong mechanism of DNA ligase and a quantitative high-throughput FRET assay made it possible to use product inhibition studies with NMN and AMP to obtain 3 dissociation constants; Kia, Kip, and Kir for NAD+, NMN, and AMP, respectively; that could not be obtained by steady-state kinetics with substrates only. In addition, these equations and the product inhibition studies made it possible to eliminate from consideration one of the 3 possible kinetic mechanisms for release of the last 2 products, sealed DNA and AMP. The same approach could be used for ATP-dependent DNA ligases, substituting ATP for NAD+ and PPi for NMN.
To distinguish between the remaining 2 possible kinetic mechanisms for release of sealed DNA and AMP (and to obtain a value for Kiq if the mechanism were RER) by steady-state kinetics would require a product inhibition experiment with sealed DNA. This experiment was not possible with the DNA ligase FRET assay, which detects sealed DNA. A different assay format would be required to perform that experiment. Even with a suitable assay, however, the experiment may be impractical if there is insufficient inhibition by sealed DNA. Teraoka et al.  observed no inhibition of calf thymus ATP-dependent DNA ligase by sealed DNA. It is reasonable to expect that the affinity of DNA ligase for sealed DNA at any step in its catalytic mechanism would be negligible considering the enormous amount of nick-free DNA present in the bacterial cytoplasm or eukaryotic nucleus. Otherwise, DNA ligase could experience severe product inhibition by the sealed DNA.
The suggestion by Cooper and Rudolph  that ordered release of sealed DNA then AMP would prevent product inhibition by sealed DNA within cells as long as the AMP concentration is low relative to its K I does not apply in the case of H. influenzae NAD+-dependent DNA ligase because the K I for AMP is likely to be similar to the cytoplasmic AMP concentration. Therefore, it is more likely that product inhibition by sealed DNA is prevented by a lack of affinity of the enzyme for sealed DNA. This means of preventing inhibition by sealed DNA would also be effective if sealed DNA and AMP were released by an RER mechanism.
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.
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).
To simulate the dynamic behavior of these biochemical networks, kinetic equations have to be associated with each reaction. If the reaction mechanism is known, the kinetic formula can be derived from generalized mass-action kinetics . Otherwise, a generic kinetic equation can be utilized, such as the recently published convenience rate law . The derived formulas can be assigned to each reaction in several ways, for instance by manual input of C-like strings or through the selection of kinetic equations from predefined lists. In any case, the inserted formulas should be in agreement with the SBML and SBGN representation of the model. Hence, either the user is required to assure this consistency or an automated procedure is needed which assigns kinetic equations consistently with the SBGN representation.
To bridge the gap between the SBGN and systems of kinetic equations, SBMLsqueezer was developed. This CellDesigner plug-in allows one to specify the quantitative dynamics of biological networks, i.e., metabolic, signal transduction and gene regulatory networks based on the SBGN. Thereby, it distinguishes between different reaction types such as transcription, translation or state transition, between different species, e.g., simple molecules, proteins, genes or mRNA as well as between different regulatory modes like activation or inhibition. SBMLsqueezer takes all this information into account and provides a contextual selection of possible formulas for each particular reaction within the model. For each reaction in these networks a specific or a generic kinetic equation can be applied. The resulting equations are written directly into the SBML file as Math ML  strings such that the user can simulate the model directly within CellDesigner. Earlier approaches towards automatic equation generation like Cellerator [16, 17] require the user to generate a new model while choosing rate equations for each reaction step-by-step from a predefined list of kinetics. In other approaches the same type of generic equation is assigned to each reaction . The framework COPASI  supports the import of SBML files, and allows the user to select a kinetic formula for each reaction from a drop-down list. This drop-down list is generated in accordance with the stoichiometry and the number of modulators. JDesigner [19, 20] provides a graphical representation and also allows the selection of rate laws for each reaction from a limited list.
Non-enzyme state transition reactions are modeled through generalized mass-action kinetics. Whenever this equation can be applied, SBMLsqueezer also offers the zeroth order forward or reverse mass-action kinetics, depending on the reversibility property of the reaction. SBMLsqueezer covers all special cases of this type of equation defined in the SBO besides a few irreversible rates for discrete simulation. It also allows for non-integer stoichiometries. 153554b96e