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xCellerator: A Mathematica package for biological modeling

xCellerator is a Mathematica package designed to aide biological modeling via the automated conversion of chemical reactions into ODEs and their subsequent solution via numerical integration.

installation instructions for the base package are given here.

The software can be downloaded from here

xCellerator is really a family of computer programs to learn les tables de multiplication. Components include:

xlr8r - Reaction translation to ODE and numerical solution. The reaction syntax is backwards compatiable with Cellerator but is more general.
Cellzilla - Simulation of a Cellerator reaction network on a two-dimensional template.
SSA - Stochastic simulation usings Gillespie's stochastic simulation algorithm.
xSSA - extended SSA algorithm for stochastic simulation of most Cellerator reactions (not just mass action).
CelleratorML/CellzillaML - an XML-based text file format for saving Cellerator and Cellzilla models.
SBML support through xlr8r2SBML and SBML2xlr8r plugins and MathSBML.
py[xlr8r] - A Mathematica-free implementation of Cellerator in Python. (under development! Not all features have been implmented yet.)
In addition there are a number of other programs that are compatible with xCellerator that can be used to extend its functionality. These include:

Sigmoid - an online database of Cellerator models and a downloadable database manager, Sigmoid Model Explorer that can be used to query the database and perform simulations using xCellerator.
kmech - a Cellerator based enzyme mechanism language. Since xCellerator release 0.88 kMech has been fully integrated into the xCellerator distribution.
RedoxMech - a Cellerator based oxidoreductase modeling framework.
mPower - Voronoi, Delaunay, and power diagram computation in n-dimensions via QHULL and regtet.
Cambium - a mathematical process language for describing developmental models; the plugin provides support for model conversion. (in development)
Cellerator is a similar program, that also converts reactions to odes, implemented by the same team of developers. It is not as general as xlr8r, and less efficient. Cellerator has a more restrictive use and distribution license.
For all inquiries please contact Bruce (-dot-) E (-dot-) Shapiro (-at-) csun (-dot-) edu.

xCellerator User's Guide

xCellerator Quick Start

For a quick start,

  1. Download and install xlr8r. Instructions are given here.
  2. Download the Quick Start Notebook from the link at the end of this list of instructions. In order to manipulate it in Mathematica you should right-click and select "save file" or "save link as .." option. Make sure to save the file with file type ".nb".
  3. Double click on your downloaded copy of QuickStart.nb. If Mathematica is properly installed correctly on your system it should open in Mathematica. If it opens in another application (such as Word or a text editor) then quit and open Mathematica directly, then find your copy of the file QuickStart.nb using "File" -> "Open .." to import it into Mathematica.
  4. Execute the cells one line at a time by selecting each one and pressing Shift-Enter to see what happens, or to evaluate them all at once select "Evaluation"->"Evaluate Notebook"
  5. Change some of the parameters and initial conditions to see what happens.
  6. For more information on any function look at the online documentation at
  7. Try building your own models.

Example (Download QuickStart.nb as Mathematica Notebook)

define a model

It might help to open the xCellerator palette (from the Palettes menu, above)

This is a model of a simple enzymatic transition that is initiated by an input signal “input” and the product is degraded at a constant rate d.

Test the model by running a simulation.

All parameters that are not set will default to a value of 1. In the following simulation, the parameter d=0 but all other parameters such as k1,k2, k3,k4 are automatically set to 1. This is useful for testing a model, but eventually you will have to tune your parameters to the correct values.

All initial conditions (values of variables such as S, P, X, etc at t=0) will default to zero. In the following simulation, the initical value of “input” and “X” are 1, and the initial values of S and P are both zero, since they are not specified.

When running your own simulations you should take special care here to make sure that the units of all of your parameters and initial conditions are consistent, e.g., if you mix nanoMoles with microMoles, make sure to apply to appropriate conversion factors to your values.

We use the frozen option to keep the enzyme X fixed. This can also be done by adding a reaction {Ø⇔X,kf,kr} with kf/kr set to the desired steady state value. However, if we use the reaction, we will have to wait for the reaction to reach steady state; the frozen option puts it there immediately.

Instead of specifying an input for the reactang S, we want to keep it zero until such time that an input signal turns it on. We define a stimulus function:

run a simulation using the stimulus function for the variable “input”

use runPlot instead of automatic plotting to give more control over plots.

Use manipulate to play with some of the parameters and initial conditions.


Arabidopsis thaliana Shoot Apical Meristem
Wuschel Expression (published model)
Jönsson H, Heisler M, Reddy GV, Agrawal V, Gor V, Shapiro BE, Mjolsness E, Meyerowitz EM (2005) "Modeling the organization of the Wuschel domain in the shoot apical meristem," Bioinformatics 21(S1): i232-i240 View Reference

Belousov-Zhabotinski Reaction (published models)
Prigogene L Lefever R (1968) Symmetry breaking instabilities in dissipative systems. II. J. Chem. Phys 48, 1695-1700. View Reference
Oregonator (FIeld-Noyes Model):
Field RJ, Körös E, Noyes RM (1972) Oscillations in Chemical Systems. II. Thorough Analysis of Temporal Oscillations in the Bromate-Cerium-Malonic Acid System. JACS, 94(25):8649-8664. View Reference

ClO2-I-MA Oscillators
Lengyel, Rabai, Epstein. "Experimental and Modeling Study of Oscillations in the Chlorine Dioxide-Iodine-Malonic Acid Reaction." Journal of the Am. Chem. Soc. (1990) 112:9104-9110
XML for Modification of Lengyel reduced model to avoid 5th order reaction (example included in nb file above)
Circadian Rhythms (Published Models)
(1) Vilar JMG, Kueh HY, Barkai N, Leibler S, (2002) "Mechanisms of noise resistance in genetic oscillators," PNAS, 99(9):5988-5992. View Reference
(2) Barkai N, Leibler S (2000) "Circadian clocks limited by noise," Nature, 403:267-268. View Reference

Drosophila Model
Leloup JC, Goldbeter A (1999) Chaos and birhythmicity in a mdoel for circadian oscillations of the PER and TIM proteins in Drosophila. J Theor Biol 198:445-459. View Reference

Dominoes & Clocks (Published Models)
Gonze D & Goldbeter A (2000) A model for a network of phosphorylation-dephosphorylation cycles displaying the dynamics of dominoes and clocks, J. Theor. Biol; 210:167-186.
Murray AW & Kirschner MW (1989) Dominoes and clocks: the union of two views of the cell cycle, Science: 246:614-621.

G-Protein Activation (Textbook Example)
Based on figure 11.7 ,page 403 of G.Fain (1999) Molecular and Cellular Physiology of Neurons, Harvard University Press

Lineage Determination (Published Model)
Chickarmane V, Peterson C (2008) " A Computational Model for Understanding Stem Cell, Trophectoderm and Endoderm Lineage Determination." PLoS ONE 3(10): e3478. Journal web site. [doi/10.1371/journal.pone.0003478]
MAP-Kinase Cascades (Published Models)
Kholodenko, Boris N. (2000) "Negative feedback and ultrasensitivity can bring about oscillations in the mitogen-activated protein kinase cascades." Eur J Biochem 267:1583-8. View Reference

Re-interpretation of Kholodenko model using Mass-action rather than Michaelis-Menten Kinetics

MAPK Cascade in Solution Models
(1) Levchenko, A., Bruck, J., Sternberg, P.W. (2000). Scaffold proteins may biphasically affect the levels of mitogen-activated protein kinase signaling and reduce its threshold properties. PNAS, 97(11):5818–5823. View Reference.
(2) Shapiro, B, Leevchenko A., Mjolsness E. (2001) Automatic Equation Generation for Signal Transduction with Applications to MAP-Kinase, in Foundations of Systems Biology, Edited by H. Kitano, MIT Press.

Mass Action Reactions (Demonstration Moddel)
Illustration of simple mass action arrows in xCellerator.

Mass Action Oscillations (Published Moddel)
Alfred Lotka "Undamped Oscillations from the Law of Mass Action" Journal of the American Chemical Society 49:1595-1599 (1920).
Mitotic Oscillators (Published Models)
Goldbeter, A. A minimal cascade model for the mitotic oscillator involving cyclin and cdc2 kinase. (1991) PNAS, 88:9107-1101 (1991).View Reference

Gardner TS, Dolnik M, Colllins JJ (1998) "A theory for controlling cell cycle dynamics using a reversibly binding inhibitor." PNAS 95:14190-14195. View Reference

Tyson JJ (1991) "Modeling the cell division cycle: cdc2 and cyclin interactions." View Reference (6 variable model)

Novak B, Tyson JJ (1997) "Modeling the control of DNA replication in fission yeast," PNAS, 94, pp. 9147–9152. View Reference

NF-κβ (Nuclear Factor Kappa Beta)
Hoffmann A, Levchnko A, Scott M, Baltimore D (2002) Science, 298(5596):1241-1245. View Reference

Elowitz MB, Leibler S (2000) A synthetic oscillatory network of transcriptional regulators. Nature. 403:335-338. View Reference

Ring Oscillators (classroom demonstration models)
GRN Model

Hill Function Model

Mass-Action Model

Michaelis-Menten Model (2-Stage)

Michaelis-Menten Model (1-Stage)

Michaelis-Menten Model (damped)

Citing Cellerator and Related Packages

Citation for Cellzilla:
Shapiro BE, Meyerowitz E and Mjolsness E. Using Cellzilla for Plant Growth Simulations at the Cellular Level. (2013) Front. Plant Sci. 4:408 [doi: 10.3389/fpls.2013.00408]

Citation for kMech:
Yang CR, Shapiro BE, Mjolsness ED, Hatfield GW. An enzyme mechanism language for the mathematical modeling of metabolic pathways. Bioinformatics 21(6): 774-780 (2005). [doi: 10.1093/bioinformatics/bti068 ]

Citation for MathSBML:
Shapiro BE, Hucka M, Finney A, Doyle J. MathSBML: a package for manipulating SBML-based biological models. Bioinformatics 20(16):2829-2831 (2004) [doi: 10.1093/bioinformatics/bth271]

Citation for Cellerator:
Shapiro BE, Levchenko A, Meyerowitz EM, Wold BJ, Mjolsness ED. Cellerator: extending a computer algebra systems to include biochemical arrows for signal transduction simulations. Bioinformatics 19(5):677-678 (2003) [doi: 10.1093/bioinformatics/btg042]

Selected Publications that Cite Cellerator

This list includes a sampling of papers from the literature that that cite Cellerator and Cellzilla as being used to complete their research. For further information please contact the authors directly.
Chang I, Baldi P. A Unifying Kinetic Framework for Modeling Oxidoreductase Catalyzed Bioenergetic Reactions (2013) Bioinformatics, 29(10):1299-1307 [DOI 10.1093/bioinformatics/btt140]

Costa, P.R., Acencio, M.L., and Lemke, N. Cooperative RNA Polymerase Molecules Behavior on a Stochastic Sequence-Dependent Model for Transcription Elongation PLoS One. 2013; 8(2): e57328 [doi: 10.1371/journal.pone.0057328]

Ang Li, Meng Chen, Ting-Xin Jiang, Ping Wu, Qing Nie, Randall Widelitz, and Cheng-Ming Chuong, Shaping organs by a wingless-int/Notch/nonmuscle myosin module which orients feather bud elongation PNAS. 2013; 110:E1452-E1461. [doi:10.1073/pnas.1219813110]

Compani B, Su T, Chang I, Cheng I, Shah KH, Whisenant T, Dou Y, Bergmann A, Cheong R, Wold B, Bardwell L, Levchenko A, Paldi P, Mjolsness E. A Scalable and Integrative System for Pathway Bioinformatics and Systems Biology. Adv. Exp. Medicine and Biology, 680(6):523-534 (2010) [DOI: 10.1007/978-1-4419-5913-3_58]

Pepke S, Kinzer-Ursem T, Mihalas M, Kennedy MB. A Dynamic Model of Interactions of Ca2+, Calmodulin, and Catalytic Subunits of Ca2+/Calmodulin-Dependent Protein Kinase II. PLOS Computational Biology 6(2):e1000675 (2010) [doi:10.1371/journal.pcbi.1000675]

Cui J, Kaandoop JA, Lloyd CA. Simulating in vitro transcriptional response of zinc homeostasis system in Escherichia coli. BMC Systems Biology 2:89 (2008) [doi:10.1186/1752-0509-2-89]

Yang CR. An enzyme-centric approach for modelling non-linear biological complexity. BMC Systems Biology, 2:70 (2008) [doi: 10.1186/1752-0509-2-70]

Nadji TS, Yang CR, Shapiro BE, Hatfield GW, Mjolsness ED. Application of a generalized MWC model for the mathematical simulation of metabolic pathways regulated by allosteric enzymes. J. Bioinformatcs and Computational Biology 4(2):35-355 (2006).[DOI: 10.1142/S0219720006001862]

Gor V, Shapiro BE, Jonsson H, Heisler M, Reddy GV, Meyerowitz EM, Mjolsness E. A Software Architecture for Developmental Modeling in Plants: The Computable Plant Project In Bioinformatics of Genome Regulation and Structure (ed: N, Kolchanov, R. Hofestaedt, L. Milanesi), Springer (2005) [ISBN 8181283775]

Fernando CT, Rowe J. Hebbian learning in a simple gene circuit in. Proc. 9th Annual Conference on Genetic and Evolutionary Computing (GECCO), pp. 426. Full Paper at CiteSeer []

Hicks MJ, Yang CR, Kotlajich MV, Hertel KJ. Linking Splicing to Pol II Transcription Stabilizes Pre-mRNAs and Influences Splicing Patterns. PLOS Biology 4(6):e147 (2006)[doi: 10.1371/journal.pbio.0040147]

Podkolodny NL, Podkolodnaya NN, Miginsky DS, Poplavsky AS, Likhoshvai VA, Compani B, Mjolsness E. An Integration of the Descriptions of Gene Networks and Their Models Presented in SIGMOID (Cellerator) and GENENET . In: Bioinformatics of Genome Regulation and Function, BGRS 2006, Volume 3, pp. 86-90

Cheng J, Scharenbroich L, Baldi P, Mjolsness E. Sigmoid: a software infrastructure for pathway bioinformatics and systems biology. IEEE Intelligent Systems 20(3):68-75 [DOI 10.1109/MIS.2005.51]

Cheong R, Bergmann A, Werner SL, Regal J, Hoffmann A, Levchenko A. Transient I-kappa-B Kinase Activity Mediates Temporal NF-kappa-B Dynamics in Response to a Wide Range of Tumor Necrosis Factor-alpha Doses J. Biol. Chem. 281(5):2945-2950 (2006). [doi: 10.1074/jbc.M510085200]

Hilioti ZA, Gallagher DA, Low-Nam ST, Ramaswamy P, Gajer P, Kingsbury TJ, Birchwood CJ, Levchenko A, Cunningham KW. GSK-3 kinases enhance calcineurin signaling by phosphorylation of RCNs . Genes. Dev. 18(1):35-47 (2003) [doi: 10.1101/gad.1159204]

Shapiro BE, Levchenko A, Mjolsness E Automatic Model Generation for Signal Transduction with Applications to MAP-Kinase Pathways in Foundations of Systems Biology (ed: H. Kitano), MIT Press, pp. 145-162 (2002) [ISBN 0-262-11266-3]

Selected Posters

(ISMB-2007/Vienna - 2.2 MB pdf). Mathematica Platforms for Modeling in Systems Biology: Recent Developments in MathSBML and Cellerator, Shapiro BE, Lu J, Hucka M, Mjolsness E.

(ICSB-2006/Yokohama - 2.9 MB png). Systems Biollogy Software Support in Mathematica: Recent Developments in Cellerator, Shapiro BE, Vorobyov A, Murakami JG, Mjolsness ED.

(ICSB-2004/Heidelberg - 2.4 MB pdf). MathSBML: A Mathematica Package for Systems Biology, Shapiro BE, Hucka M, Finney A.

(PSB-2004/Hawaii - 1.9 MB pdf). MathSBML and Systems Biology Simulations, Shapiro BE, Hucka M, Finney A.

(ICSB-2001/Pasadena - 2.4 MB pdf). Developmental Simulations with Cellerator, Shapiro BE, Mjolsness ED.

(SIAM-2001/San Diego - 1.2 MB pdf). Automatic Equation Generation for Signal Transduction Modeling, Shapiro BE, Levchenko A, Mjolsness E.