Xs systems extended s systems algebraic differential automata for modeling cellular behavior
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HiPC 2002 12 19 2002. xS-systems: eXtended S-systems & Algebraic Differential Automata for Modeling Cellular Behavior. ¦ Bud Mishra Professor (Cold Spring Harbor Laboratory) Professor of CS & Mathematics (Courant, NYU) With M. Antoniotti, A. Policriti and N. Ugel. Why Systems Biology?.

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xS-systems: eXtended S-systems & Algebraic Differential Automata for Modeling Cellular Behavior

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HiPC 2002

12 19 2002

xS-systems:eXtended S-systems & Algebraic Differential Automata for Modeling Cellular Behavior


Bud Mishra

Professor (Cold Spring Harbor Laboratory)

Professor of CS & Mathematics (Courant, NYU)

With M. Antoniotti, A. Policriti and N. Ugel

Why Systems Biology?

  • It is not uncommon to assume certain biological problems to have achieved a cognitive finality without rigorous justification.

  • Rigorous mathematical models with automated tools for reasoning, simulation, and computation can be of enormous help to uncover

    • cognitive flaws,

    • qualitative simplification or

    • overly generalized assumptions.

  • Some ideal candidates for such study would include:

    • prion hypothesis

    • cell cycle machinery

    • muscle contractility

    • processes involved in cancer (cell cycle regulation, angiogenesis, DNA repair, apoptosis, cellular senescence, tissue space modeling enzymes, etc.)

    • signal transduction pathways, and many others.


Systems Biology

Combining the mathematical rigor of numerology with the predictive power of astrology.








Interpretive Biology

Computational Biology

Integrative Biology



Computational/Systems Biology

How much of reasoning about biology can be automated?


Reversible Reaction



Single splitting reaction

generating two products X2 and X3, in stoichiometric proportion.




Divergence Branch Point: Degradation processes of X1 into X2 and X3 are independent




Single synthetic reaction

involving two source components X1 and X2, in stoichiometric proportion.



Convergence Branch Point: Degradation processes of X1 and X2 into X3 are independent




Graphical Representation






The conversion of X1 into X2 is modulated by X3




The conversion of X1 into X2 is modulated by an inhibitor X3




Graphical Representation

The reaction between X1 and X2 requires coenzyme X3 which is converted to X4






Phosphorylase a




Phosphoglucose isomerase



An Artificial Clock

  • Three proteins:

    • LacI, tetR & l cI

    • Arranged in a cyclic manner (logically, not necessarily physically) so that the protein product of one gene is rpressor for the next gene.

      LacI!:tetR; tetR! TetR

      TetR!:l cI; l cI !l cI

      l cI!:lacI; lacI! LacI

Cycles of Repression

  • The first repressor protein, LacI from E. coli inhibits the transcription of the second repressor gene, tetR from the tetracycline-resistance transposon Tn10.

  • Protein product in turn TetR from tetR inhibits the expression of a third gene, cI from l phage.

  • Finally, CI inhibits lacI expression,

    completing the cycle.

Biological Model

  • Standard molecular biology: Construct

    • A low-copy plasmid encoding the repressilator and

    • A compatible higher-copy reporter plasmid containing the tet-repressible promoter PLtet01 fused to an intermediate stability variant of gfp.










Cascade Model: Repressilator?

dx2/dt = a2 X6g26X1g21 - b2 X2h22

dx4/dt = a4 X2g42X3g43 - b4 X4h44

dx6/dt = a6 X4g64X5g65 - b6 X6h66

X1, X3, X5 = const

SimPathica System

Modal Logic Queries

SimPathica:Trace Analysis System


Canonical Forms

Systems of Differential Equations

  • dXi/dt

    = (instantaneous) rate of change in Xi at time t

    = Function of substrate concentrations, enzymes, factors and products:

    dXi/dt = f(S1, S2, …, E1, E2, …, F1, F2,…, P1, P2,…)

  • S-systems result in Non-linear Time-Invariant DAE System.

General Form

  • dXi/dt = Vi+(X1, X2, …, Xn) – Vi-(X1, X2, …, Xn):

    • Where Vi+(¢) term represents production (or accumulation) rate of a particular metabolite and Vi-(¢) represent s depletion rate of the same metabolite.

  • Generalizing to n dependent variables and m independent variables, we have:

    dXi/dt =

    Vi+(X1, X2, …, Xn, U1, U2, …, Um)

    – Vi-(X1, X2, …, Xn, U1, U2, …, Um):

Canonical Forms

S-System Automaton AS

  • S-System Automata Definition:

    • Combine snapshots of the IDs (“instantaneous descriptions”) of the system to create a possible world model

    • Transitions are inferred from “traces” of the system variables:

  • Definition:Given an S-systems S, the S-system automaton AS associated to S is 4-tuple AS = (S, D, S0, F), where S µD1£L£DW is a set of states, Dµ S £ S is the binary transition relation, and S0, F ½ S are initial and final states respectively. ð

  • Definition: A trace of an S-system automaton AS is a sequence s0, s1, …, sn,…, such that s02 S0, D(si, si+1), 8 i = 0.ð

Trace Automaton

Simple one-to-one construction of the

“trace” automata AS for an S-system S

Collapsing Algorithm

Collapsed Automata

The effects of the collapsing construction of the

“trace” automata AS for an S-system S

Purine Metabolism

Purine Metabolism

  • Purine Metabolism

    • Provides the organism with building blocks for the synthesis of DNA and RNA.

    • The consequences of a malfunctioning purine metabolism pathway are severe and can lead to death.

  • The entire pathway is almost closed but also quite complex. It contains

    • several feedback loops,

    • cross-activations and

    • reversible reactions

  • Thus is an ideal candidate for reasoning with computational tools.

Simple Model

Biochemistry of Purine Metabolism

  • The main metabolite in purine biosynthesis is 5-phosphoribosyl-a-1-pyrophosphate (PRPP).

    • A linear cascade of reactions converts PRPP into inosine monophosphate (IMP). IMP is the central branch point of the purine metabolism pathway.

    • IMP is transformed into AMP and GMP.

    • Guanosine, adenosine and their derivatives are recycled (unless used elsewhere) into hypoxanthine (HX) and xanthine (XA).

    • XA is finally oxidized into uric acid (UA).

Biochemistry of Purine Metabolism

  • In addition to these processes, there appear to be two “salvage” pathways that serve to maintain IMP level and thus of adenosine and guanosine levels as well.

  • In these pathways, adenine phosphoribosyltransferase (APRT) and hypoxanthine-guaninephosphoribosyltransferase (HGPRT) combine with PRPP to form ribonucleotides.

Purine Metabolism

<?xml version="1.0" ?>-<map xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="map.xsd">-<substrate><id>1</id><concentration>5</concentration><name>PRPP</name></substrate>-<substrate><id>2</id><concentration>100</concentration><name>IMP</name></substrate>-<substrate><id>3</id><concentration>2500</concentration><name>Ado</name></substrate>-<substrate><id>4</id><concentration>425</concentration><name>GMP</name></substrate>-


XML Description




Variation of the initial concentration of PRPP does not change the steady state.(PRPP = 10 * PRPP1) implies steady_state()

This query will be true when evaluated against the modified simulation run (i.e. the one where the initial concentration of PRPP is 10 times the initial concentration in the first run – PRPP1).

Persistent increase in the initial concentration of PRPP does cause unwanted changes in the steady state values of some metabolites.

If the increase in the level of PRPP is in the order of 70% then the system does reach a steady state, and we expect to see increases in the levels of IMP and of the hypoxanthine pool in a “comparable” order of magnitude.Always (PRPP = 1.7*PRPP1) implies steady_state()




Consider the following statement:


(Always (PRPP = 1.7 * PRPP1) implies steady_state() and Eventually

(Always(IMP < 2 * IMP1))and Eventually (Always

(hx_pool < 10*hx_pool1)))

where IMP1 and hx_pool1 are the values observed in the unmodified trace. The above statement turns out to be false over the modified experiment trace..

In fact, the increase in IMP is about 6.5 fold while the hypoxanthine pool increase is about 60 fold.

Since the above queries turn out to be false over the modified trace, we conclude that the model “over-predicts” the increases in some of its products and that it should therefore be amended



Final Model

Purine Metabolism


  • This change to the model allows us to reformulate our query as shown below:

    Always(PRPP > 50 * PRPP1implies(steady_state() and Eventually(IMP > IMP1) and Eventually(HX < HX1) and Eventually(Always(IMP = IMP1)) and Eventually(Always(HX = HX1))

  • An (instantaneous) increase in the level of PRPP will not make the system stray from the predicted steady state, even if temporary variations of IMP and HX are allowed.


Related Publications

  • “Simulating Large Biochemical and Biological Processes and Reasoning about their Behavior." (with M. Antoniotti, F. Park, A. Policriti and N. Ugel), ICSB, Sweden, 2003.

  • "Foundations of a Query and Simulation System for the Modeling of Biochemical and Biological Processes." (with M. Antoniotti, F. Park, A. Policriti and N. Ugel), The Pacific Symposium on Biocomputing (PSB 2003), Hawaii, January 3-7, 2003.

  • "Model Building and Model Checking for Biochemical Processes," (with M. Antoniotti, A. Policriti and N. Ugel), Cell Biochemistry and Biophysics (CBB), Humana Press, 2003, (In press)

  • "A Symbolic Approach to Modelling Cellular Behaviour," International Conference On High Performance Computing, HiPC 2002, December 18-21, Bangalore, India, 2002. (In Press)

  • "Wild by Nature," (with M. Wigler), Science, 296: 1407-1408, 24 May 2002.

The End


http://cs.nyu.edu/faculty/mishra/ http://bioinformatics.cat.nyu.edu/

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