8 July 2009 - James Hengenius

    Dynamic models describe the evolution of systems over time.  Models may be treated discretely or continuously in both time and state-space, and may be simulated deterministically (typically by numerical integration of differential equations) or stochastically by the Gillespie algorithm.  

    Stochastic methods are computationally expensive and infeasible when large numbers of molecules are modeled.  Recent hybrid approaches combine deterministic integration for high concentration species and stochastic algorithms for events tracked at the molecular level (e.g., protein-DNA binding or ion channel transitions).

     Mathematica Interactive Scripts

    Mathematica has a "cell-based" GUI.  Code is written in nested collapsible cells and executed individually by clicking on text anywhere a cell and typing [Shift]+[Enter].  I've collapsed certain cells so that only the interactive simulation interface is available.  To uncollapse everything and look under the hood, click on the borders along the right edge of the window.

    Mathematica can be accessed from Purdue's GoRemote software site.

    • Lotka-Volterra Model [NB file] 
    • Logistic Growth Model [NB file]
    • Goldbeter's Cyclin-CDK Model [NB file] 

    Papers for Review

    Only highlighted titles are required reading for class.  All others are supplemental.

    Modeling Background

    • Alon U.  An Introduction to Systems Biology.  Chapman & Hall/CRC, 2006.  (Chapter 1)
    • Nowak MA.  Evolutionary Dynamics: Exploring the Equations of Life.  Belknap Press, 2006.  (Chapter 1)
    • Peck SL.  Simulation as experiment: a philosophical reassessment for biological modeling.   Trends in Ecology and Evolution 2004, 19(10):530. Link
    • Szallasi Z, Stelling J, Periwal V.  System Modeling in Cellular Biology: From Concepts to Nuts and Bolts.  The MIT Press, 2006. (Chapters 1-3)

    Deterministic Models

    • Edelstein-Keshet L.  Mathematical Models in Biology.  SIAM, 2005.
    • Goldbeter A.  A minimal cascade model for the mitotic oscillator involving cyclin and cdc2 kinase. PNAS 1991, 88(20):9107.  Link
      • Contains a simple ODE model of CDK activity and common analyses of ODE models.
    • Gotelli NJ.  A Primer of Ecology.  Sinauer Associates, 2001.
    • Gupta S, Bisht SS, Kukreti R, Jain S, Brahmachari SK.  Boolean network analysis of a neurotransmitter signaling pathway.  Journal of Theoretical Biology 2007, 244(3):463.  Link
    • Gurney  WSC, Nisbet RM.  Ecological Dynamics.  Oxford University Press, 1998.
    • Hodgkin AL, Huxley AF.  A quantitative description of membrane current and its application to conduction and excitation in nerve.  Journal of Physiology 1952, 117:500.  Link
    • Hoppensteadt FC, Keller JB.  Synchronization of periodical cicada emergences.  Science 1976, 194(4262):335.  Link
    • May RM.  Simple mathematical models with very complicated dynamics.  Nature 1976, 261:459.  Link
    • Michaelis L, Menten ML.  The kinetics of the inversion effect.  Biochemische Zeitschrift 1913, 49:333.  Link
    • Strogatz S.  Nonlinear Dynamics and Chaos.  Perseus Books Publishing, 1994.
    • Turing AM.  The Chemical Basis of Morphogenesis.  Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences 1952, 237(641):37.  Link

    Stochastic Models

    • Delbruck M.  Statistical Fluctuations in Autocatalytic Reactions. Journal of Chemical Physics 1940, 8:120. Link
    • Gillespie DT. Exact Stochastic Simulation of Coupled Chemical Reactions. Journal of Physical Chemistry 1977, 81(25):2340. Link
    • Gillespie DT.  Stochastic Chemical Kinetics.  Mathematical Biosciences Institute at Ohio State University: Workshop 2: Regulatory Networks,2004.  Slides and Video.
    • Gillespie DT.  Stochastic Simulation of Chemical Kinetics.  Annual Review of Physical Chemistry 2007, 58:35.  Link
    • Hänggi P.  Stochastic resonance in biology. How noise can enhance detection of weak signals and help improve biological information processing.  Chemphyschem 2002, 3(3):285.  Link
    • Higham DJ.  Modeling and Simulating Chemical Reactions.  SIAM Review 2008, 50(2):347.  Link
    • Karmakar R, Bose I.  Stochastic model of transcription factor-regulated gene expression.  Physical Biology 2006, 3:200.  Link
    • Lok L.  The need for speed in stochastic simulation.  Nature Biotechnology 2004, 22(8):1017.  Link
    • Rao CV, Wolf DM, Arkin AP.  Control, exploitation and tolerance of intracellular noise.  Nature 2002, 420:231.  Link
    • Rathinam M, Petzold L, Cao Y, Gillespie DT. Stiffness in stochastic chemically reacting systems: the implicit tau-leaping method.  Journal of Chemical Physics 2003, 119:12784.  Link
    • Salwinski L, Eisenberg D.  In silico simulation of biological network dynamics.  Nature Biotechnology 2004, 22:1017.  Link
    • Samoilov M, Plyasunov S, Arkin AP.  Stochastic amplification and signaling in enzymatic futile cycles through noise-induced bistability with oscillations.  PNAS 2005, 102(7):2310.  Link
    •  Taylor HM, Karlin S.  An Introduction to Stochastic Modeling.  Academic Press, 1998.
    • Wilkinson DJ.  Stochastic Modelling for Systems Biology.  Chapman & Hall/CRC, 2006. (Chapter 6: Chemical and Biochemical Kinetics)
      • Describes conversion of ODEs to stochastic models, the Gillespie algorithm (aka exact stochastic simulation algorithm/SSA), and the chemical master equation.  The last element of thr matrix on page 140 should be -1.  Pages 160-161 are optional.
      • I didn't realize how much Petri nets were glossed over.  
        • P is a vector of n species.
        • T is a vector of m reactions.
        • Pre is an m by n integer matrix of the input stoichiometry of each reaction (ie, the i,j-th element of Pre denotes the number of molecules of species pj of P consumed by the ti reaction of T).
        • Post is an m by n integer matrix of the output stoichiometry of each reaction (ie, the i,j-th element of Post denotes the number of molecules of species pj of P produced by the ti reaction of T).
        • M, marking, is a length n vector of initial conditions of the species. 
        • The stoichiometry matrix S = (Post - Pre)'.
        • S(i) is the ith column of S, or the balanced stoichiometry of the ith reaction

    Stochastic-Deterministic Equivalence

    • Kurtz TG.  The Relationship between Stochastic and Deterministic Models for Chemical Reactions.  Journal of Chemical Physics 1972, 57:2976.  Link
    • Stamatakis M, Mantzaris NV.  Comparison of Deterministic and Stochastic Models of the lac Operon Genetic Network.  Biophysical Journal 2009, 96(3):887.  Link
    • Vellela M, Qian H.  A Quasistationary Analysis of a Stochastic Chemical Reaction: Keizer’s Paradox.  Bulletin of Mathematical Biology 2007, 69:1727.  Link

    Hybrid Models 

    • Alfonsi A, Cancès E, Turinici G, Di Ventura B, Huisinga W.  Adaptive simulation of hybrid stochastic and deterministic models for biochemical systems.  ESAIM: Proc. 2005, 14:1.  Link
    • Rüdiger S, Shuai JW, Huisinga W, Nagaiah C, Warnecke G, Parker I, Falcke M.  Hybrid Stochastic and Deterministic Simulations of Calcium Blips.  Biophysical Journal 2007, 93(6):1847.  Link
    • Salis H, Kaznessis Y.  Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions.  Journal of Chemical Physics 2005, 122(5):54103.  Link
    • Zak DE, Gonye GE, Schwaber JS, Doyle FJ III.  Importance of input perturbations and stochastic gene expression in the reverse engineering of genetic regulatory networks: insights from an identifiability analysis of an in silico network.  Genome Research 2003, 13(11):2396.  Link
      • Couples intracellular signaling (deterministic) with gene expression (stochastic) and analyzes effects on system sensitivities.

     

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    Files 3

    FileSizeDateAttached by 
     goldbeter 1991.nb
    Goldbeter Cyclin-CDK model
    64.57 kB15:39, 10 Jul 2009jhengeniActions
     logisticgrowth.nb
    Logistic Growth model
    21.13 kB14:51, 10 Jul 2009jhengeniActions
     lv pred prey.nb
    Lotka-Volterra model
    13.25 kB16:45, 9 Jul 2009jhengeniActions
    Viewing 1 of 1 comments: view all
    Hmm, the book chapter refreshed my memory after 5-6 years. And it is a really good basic intro! I really like it! +1!

    Other two paper are small, +1 again! LOL
    Posted 17:34, 5 Jul 2009
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