C A mathematical technique for maintaining unit consistency in a kinetic law with variable stoichiometry

(Appendix contributed by Chris Myers, University of Utah.)

Section 4.11.3 on p. 71 describes how the stoichiometry of a **SpeciesReference** can be changed as part of a simulation. This can be useful in some cases, such as reactions where the stoichiometry depends upon pH.

However, it can be difficult to get the **KineticLaw** of such reactions to maintain the correct units as the stoichiometry of the reaction changes. For example, let us assume that we are modeling the following set of chemical reactions:

$$A + A \underset{k_r}{\overset{k_f}{\rightleftharpoons}} A_2$$

$$A_2 + A \underset{k_r}{\overset{k_f}{\rightleftharpoons}} A_3$$

...

$$A_{n-1} + A \underset{k_r}{\overset{k_f}{\rightleftharpoons}} A_n$$

where we would like to allow $n$ to be a variable. In order to allow for this, we can approximate the above set of equations with a single equation as shown below using a *quasi-steady state approximation* (namely that the species $A_2, \dots, A_{n-1}$ have a short lifetime).

$$nA \underset{k_r}{\overset{k_r \cdot K_{eq}^{n-1}}{\rightleftharpoons}} A_n$$

where $K_{eq}$ is equal to $k_f/k_r$. The rate law for the above reaction is:

$$k_r \cdot K_{eq}^{n-1} \cdot A^n - k_r \cdot A_n$$

Let us assume that all species $A$, $A_2$, ..., $A_n$ have units of mole, $k_f$ is in units of $(\text{mole} \cdot \text{second})^{-1}$, and $k_r$ is in units of second$^{-1}$. Therefore, $K_{eq}$ is in units of mole$^{-1}$ which makes the rate law above have units of mole/second as desired, regardless of the value of $n$, the stoichiometry of $A$.
Acknowledgments

The development of SBML was originally funded by the Japan Science and Technology Agency (JST) under
the ERATO Kitano Symbiotic Systems Project during the years 2000–2003. From 2003 to the present, funding
for development of SBML and associated software such as libSBML and the SBML Test Suite has been
provided chiefly by the National Institute of General Medical Sciences (USA) via grant numbers GM070923
and GM077671. Additional grant funding has in the past been provided by National Human Genome Research
Institute (USA); the International Joint Research Program of NEDO (Japan); the JST ERATO-SORST
Program (Japan); the Japanese Ministry of Agriculture; the Japanese Ministry of Education, Culture, Sports,
Science and Technology; the BBSRC e-Science Initiative (UK); the DARPA IPTO Bio-Computation Program
(USA); the Army Research Office’s Institute for Collaborative Biotechnologies (USA); and the Air Force
Office of Scientific Research (USA).

Additional support has been or continues to be provided by the following institutions, either directly for activities related to SBML or indirectly by supporting the work of present and past SBML Editors: the Beckman Institute at the California Institute of Technology (USA), EML Research gGmbH (Germany), the University of Heidelberg (Germany), the European Molecular Biology Laboratory’s European Bioinformatics Institute (UK), the Molecular Sciences Institute (USA), the University of Hertfordshire (UK), the University of Newcastle (UK), the Systems Biology Institute (Japan), and the Virginia Bioinformatics Institute (USA). Fengkai Zhang was supported by the intramural program of NIAID, NIH.

The following individuals served as past SBML Editors and authors of SBML specifications. Their efforts
helped shape what SBML is today:

• Hamid Bolouri

• Andrew M. Finney

• Herbert M. Sauro

SBML was first conceived at the JST/ERATO-sponsored *First Workshop on Software Platforms for Systems Biology*, held in April, 2000, at the California Institute of Technology in Pasadena, California, USA. The participants collectively decided to begin developing a common XML-based declarative language for representing models. The development and evolution of the Systems Biology Markup Language has continued ever since. Many discussions are archived online in the mailing list/forums area of http://sbml.org; many more discussions took place during meetings and workshops (a list of which is also available at http://sbml.org).

SBML Level 3 has benefitted from so many contributions, large and small, by so many people who constitute the international SBML Forum, that we regret it has become infeasible to list individuals by name. We thank everyone who has participated in SBML's development throughout the years, and we hope that this latest specification before you is a good step forward in SBML's continued evolution.
References

Abramowitz, M. and Stegun, I. A., editors (1977). *Mathematical Functions: With Formulas, Graphs, and Mathematical Tables*. Dover Publications Inc.

Ausbrooks, R., Buswell, S., Carlisle, D., Dalmas, S., Devitt, S., Diaz, A., Froumentin, M., Hunter, R., Ion, P., Kohlhase, M., Miner, R., Poppelier, N., Smith, B., Soiffer, N., Sutor, R., and Watt, S. (2003). Mathematical Markup Language (MathML) Version 2.0 (second edition): W3C Recommendation 21 October 2003. Available via the World Wide Web at http://www.w3.org/TR/2003/REC-MathML2-20031021/.

Ball, K., Kurtz, T. G., Popovic, L., and Rempala, G. (2006). Asymptotic analysis of multiscale approximations to reaction networks. *Annals of Applied Probability*, 16(4):1925–1961.

Biron, P. V. and Malhotra, A. (2000). XML Schema part 2: Datatypes (W3C candidate recommendation 24 October 2000). Available via the World Wide Web at http://www.w3.org/TR/xmlschema-2/.

Bray, T., D. Hollander, D., and Layman, A. (1999). Namespaces in XML. W3C 14-January-1999. Available via the World Wide Web at http://www.w3.org/TR/1999/REC-xml-names-19990114/.

Bray, T., Paoli, J., Sperberg-McQueen, C. M., Maler, E., and Yergeau, F. (2004). Extensible markup language (XML) 1.0 (third edition), W3C recommendation 4-February-2004. Available via the World Wide Web at http://www.w3.org/TR/2004/REC-xml-20040204.

Bureau International des Poids et Mesures (2006). The International System of Units (SI) 8th edition (2006). Available via the World Wide Web at http://www.bipm.org

Chartrand, G. (1977). *Introductory Graph Theory*. Dover Publishing, Inc., New York.

Courtot, M., Juty, N., Knpfer, C., Waltemath, D., Zhukova, A., Drger, A., Dumontier, M., Finney, A., Golebiewski, M., Hastings, J., Hoops, S., Keating, S., Kell, D. B., Kerrien, S., Lawson, J., Lister, A., Lu, J., Machne, R., Mendes, P., Pocock, M., Rodriguez, N., Villeger, A., Wilkinson, D. J., Wimalaratne, S., Laibe, C., Hucka, M., and Le Novre, N. (2011). Controlled vocabularies and semantics in systems biology. *Mol Syst Biol*, 7:543.

DCMI Usage Board (2005). DCMI Metadata Terms. Available online via the World Wide Web at the address http://www.dublincore.org/documents/dcmi-terms/.

Dublin Core Metadata Initiative (2005). Dublin Core metadata initiative. Available via the World Wide Web at http://dublincore.org/.

Eriksson, H.-E. and Penker, M. (1998). *UML Toolkit*. John Wiley & Sons, New York.

Evans, T. W., Gillespie, C. S., and Wilkinson, D. J. (2008). The SBML discrete stochastic models test suite. *Bioinformatics*, 24:285–286.

Fallside, D. C. (2000). XML Schema part 0: Primer (W3C candidate recommendation 24 October 2000). Available via the World Wide Web at http://www.w3.org/TR/xmlschema-0/.

Gillespie, D. (1977). Exact stochastic simulation of coupled chemical reactions. *J. Phys. Chem.*, 81:2340–2361.

Gillespie, D. (1992). A rigorous derivation of the chemical master equation. *Physica A*, 188:404–425.

Harold, E. R. and Means, E. S. (2001). *XML in a Nutshell*. O'Reilly & Associates.

Hedley, W. J., Nelson, M. R., Bullivant, D., Cuellar, A., Ge, Y., Grehlinger, M., Jim, K., Lett, S., Nickerson, D., Nielsen, P., and Yu, H. (2001). CellML specification. Available online via the World Wide Web at https://www.cellml.org/specification.

Hopcroft, J. E. and Karp, R. M. (1973). An $n^{5/2}$ algorithm for maximum matchings in bipartite graphs. *SIAM Journal on Computing*, 2(4):225–231.
Hucka, M., Finney, A., Sauro, H. M., and Bolouri, H. (2001). Systems Biology Markup Language (SBML) Level 1: Structures and facilities for basic model definitions. Available via the World Wide Web at http://www.sbml.org/Documents/Specifications.

Hucka, M., Finney, A., Sauro, H. M., Bolouri, H., Doyle, J. C., Kitano, H., Arkin, A. P., Bornstein, B. J., Bray, D., Cornish-Bowden, A., Cuellar, A. A., Dronov, S., Gilles, E. D., Ginkel, M., Gor, V., Goryanin, I. I., Hedley, W. J., Hodgman, T. C., Hofmeyr, J.-H., Hunter, P. J., Juty, N. S., Kasberger, J. L., Kremling, A., Kummer, U., Le Novère, N., Loew, L. M., Lucio, D., Mendes, P., Minch, E., Mjolsness, E. D., Nakayama, Y., Nelson, M. R., Nielsen, P. F., Sakurada, T., Schaff, J. C., Shapiro, B. E., Shimizu, T. S., Spence, H. D., Stelling, J., Takahashi, K., Tomita, M., Wagner, J., and Wang, J. (2003). The Systems Biology Markup Language (SBML): A medium for representation and exchange of biochemical network models. *Bioinformatics*, 19(4):524–531.

Jacobs, I. (2004). World Wide Web Consortium process document. Available via the World Wide Web at http://www.w3.org/2004/02/Process-20040205/.

Juty, N., Le Novère, N., and Laibe, C. (2012). Identifiers.org and MIRIAM Registry: Community resources to provide persistent identification. *Nucleic Acids Research*, 40(December 2011):580–586.

Kokkelink, S. and Schwänzl, R. (2002). Expressing qualified Dublin Core in RDF/XML. Available via the World Wide Web at http://dublincore.org/documents/dcq-rdf-xml/index.shtml.

Lassila, O. and Swick, R. (1999). Resource description framework (RDF) model and syntax specification. Available via the World Wide Web at http://www.w3.org/TR/REC-rdf-syntax/.

Le Novère, N., Finney, A., Hucka, M., Bhalla, U., Campagne, F., Collado-Vides, J., Crampin, E. J., Halstead, M., Klipp, E., Mendes, P., Nielsen, P., Sauro, H., Shapiro, B., Snoep, J. L., Spence, H. D., and Wanner, B. L. (2005). Minimum information requested in the annotation of biochemical models (MIRIAM). *Nature Biotechnology*, 23:1509–1515.

Mohr, P. J., Taylor, B. N., and Newell, D. B. (2008). CODATA Recommended Values of the Fundamental Physical Constants: 2006. *Reviews of Modern Physics*, 80:633–731.

O'Brien, E. J., Lerman, J. A., Chang, R. L., Hyduke, D. R., and Palsson, B. Ø. (2013). Genome-scale models of metabolism and gene expression extend and refine growth phenotype prediction. *Molecular systems biology*, 9(1):693.

Oestereich, B. (1999). *Developing Software with UML: Object-Oriented Analysis and Design in Practice*. Addison-Wesley Publishing Company.

Pemberton, S., Austin, D., Axelsson, J., Celik, T., Dominiak, D., Elenbaas, H., Epperson, B., Ishikawa, M., Matsui, S., McCarron, S., Navarro, Peruvemba, S., Relyea, R., Schnitzenbaumer, S., and Stark, P. (2002). XHTML™ 1.0 the Extensible HyperText Markup Language (second edition): W3C Recommendation 26 January 2000, revised 1 August 2002. Available via the World Wide Web at http://www.w3.org/TR/xhtml1/.

Perreault, S. (2011). vcard format specification. Available via the World Wide Web at https://tools.ietf.org/html/rfc6350.

Thompson, H. S., Beech, D., Maloney, M., and Mendelsohn, N. (2000). XML Schema part 1: Structures (W3C candidate recommendation 24 October 2000). Available online via the World Wide Web at the address http://www.w3.org/TR/xmlschema-1/.

Unicode Consortium (1996). *The Unicode Standard, Version 2.0*. Addison-Wesley Developers Press, Reading, Massachusetts.

W3C (2000a). Naming and addressing: URIs, URLs, .... Available online via the World Wide Web at http://www.w3.org/Addressing/.
1

W3C (2000b). W3C's math home page. Available via the World Wide Web at http://www.w3.org/Math/.

2

W3C (2004a). RDF/XML syntax specification (revised). Available online via the World Wide Web at http://www.w3.org/TR/rdf-syntax-grammar/.

3

4

5

W3C (2004b). Resource description framework (RDF). Available online via the World Wide Web at the address http://www.w3.org/RDF/.

6

Wilkinson, D. J. (2006). *Stochastic Modelling for Systems Biology*. Chapman & Hall/CRC.

7

Wolf, M. and Wicksteed, C. (1998). Date and time formats. Available online via the World Wide Web at http://www.w3.org/TR/NOTE-datetime.

8

9

Zwillinger, D., editor (1996). *Standard Mathematical Tables and Formulae*. CRC Press LLC, 30th edition.