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Title: Principia


1
Principia Biologica
Design Principles in Biological Systems
2
Bud Mishra
  • Professor of Computer Science, Mathematics and
    Cell Biology
  • Courant Institute, NYU School of Medicine, Tata
    Institute of Fundamental Research, and Mt. Sinai
    School of Medicine

3
(No Transcript)
4
Robert Hooke
  • Robert Hooke (1635-1703) was an experimental
    scientist, mathematician, architect, and
    astronomer. Secretary of the Royal Society from
    1677 to 1682,
  • Hooke was considered the Englands Da Vinci
    because of his wide range of interests.
  • His work Micrographia of 1665 contained his
    microscopical investigations, which included the
    first identification of biological cells.
  • In his drafts of Book II, Newton had referred to
    him as the most illustrious HookeClarissimus
    Hookius.
  • Hooke became involved in a dispute with Isaac
    Newton over the priority of the discovery of the
    inverse square law of gravitation.

5
Hooke to Halley
  • Huygens Preface is concerning those
    properties of gravity which I myself first
    discovered and showed to this Society and years
    since, which of late Mr. Newton has done me the
    favour to print and publish as his own
    inventions.

6
Newton to Halley
  • Now is this not very fine? Mathematicians that
    find out, settle do all the business must
    content themselves with being nothing but dry
    calculators drudges another that does nothing
    but pretend grasp at all things must carry away
    all the inventions
  • I beleive you would think him a man of a strange
    unsociable temper.

7
Newton to Hooke
  • If I have seen further than other men, it is
    because I have stood on the shoulders of giants
    and you my dear Hooke, have not."
  • Newton to Hooke

8
Image Logic
  • The great distance between
  • a glimpsed truth and
  • a demonstrated truth
  • Christopher Wren/Alexis Claude Clairaut

9
MicrographiaPrincipia
10
Micrographia
11
The Brain the Fancy
  • The truth is, the science of Nature has already
    been too long made only a work of the brain and
    the fancy. It is now high time that it should
    return to the plainness and soundness of
    observations on material and obvious things.
  • Robert Hooke. (1635 - 1703), Micrographia 1665

12
Principia
13
Induction Hypothesis
  • Truth being uniform and always the same, it is
    admirable to observe how easily we are enabled to
    make out very abstruse and difficult matters,
    when once true and genuine Principles are
    obtained.
  • Halley, The true Theory of the Tides, extracted
    from that admired Treatise of Mr. Issac Newton,
    Intituled, Philosophiae Naturalis Principia
    Mathematica, Phil. Trans. 226445,447.
  • This rule we must follow, that the argument of
    induction may not be evaded by hypotheses.

Hypotheses non fingo.I feign no
hypotheses.Principia Mathematica.
14
Morphogenesis
15
Alan Turing 1952
  • The Chemical Basis of Morphogenesis, 1952,
    Phil. Trans. Roy. Soc. of London, Series B
    Biological Sciences, 2373772.
  • A reaction-diffusion model for development.

16
A mathematical model for the growing embryo.
  • A very general program for modeling
    embryogenesis The model is a simplification
    and an idealization and consequently a
    falsification.
  • Morphogen is simply the kind of substance
    concerned in this theory in fact, anything that
    diffuses into the tissue and somehow persuades
    it to develop along different lines from those
    which would have been followed in its absence
    qualifies.

17
Diffusion equation
first temporal derivative rate
second spatial derivative flux
a/ t Da r2 a
a concentration Da diffusion constant
18
Reaction-Diffusion
  • a/ t f(a,b) Da r2 a f(a,b) a(b-1) k1
  • b/ t g(a,b) Db r2 b g(a,b) -ab k2

Turing, A.M. (1952).The chemical basis of
morphogenesis. Phil. Trans. Roy. Soc. London B
237 37
19
Reaction-diffusion an example
A2B ! 3B B ! P
B extracted at rate F, decay at rate k
A fed at rate F
Pearson, J. E. Complex patterns in simple
systems. Science 261, 189-192 (1993).
20
Reaction-diffusion an example
21
Genes 1952
  • Since the role of genes is presumably catalytic,
    influencing only the rate of reactions, unless
    one is interested in comparison of organisms,
    they may be eliminated from the discussion

22
Crick Watson 1953
23
Genome
  • Genome
  • Hereditary information of an organism is encoded
    in its DNA and enclosed in a cell (unless it is a
    virus). All the information contained in the DNA
    of a single organism is its genome.
  • DNA molecule can be thought of as a very long
    sequence of nucleotides or bases
  • S A, T, C, G

24
The Central Dogma
  • The central dogma(due to Francis Crick in 1958)
    states that these information flows are all
    unidirectional
  • The central dogma states that once information'
    has passed into protein it cannot get out again.
    The transfer of information from nucleic acid to
    nucleic acid, or from nucleic acid to protein,
    may be possible, but transfer from protein to
    protein, or from protein to nucleic acid is
    impossible. Information means here the precise
    determination of sequence, either of bases in the
    nucleic acid or of amino acid residues in the
    protein.

Transcription
Translation
DNA
RNA
Protein
25
RNA, Genes and Promoters
  • A specific region of DNA that determines the
    synthesis of proteins (through the transcription
    and translation) is called a gene
  • Originally, a gene meant something more
    abstract---a unit of hereditary inheritance.
  • Now a gene has been given a physical molecular
    existence.
  • Transcription of a gene to a messenger RNA, mRNA,
    is keyed by a transcriptional activator/factor,
    which attaches to a promoter (a specific sequence
    adjacent to the gene).
  • Regulatory sequences such as silencers and
    enhancers control the rate of transcription

26
The Brain the Fancy
  • Work on the mathematics of growth as opposed to
    the statistical description and comparison of
    growth, seems to me to have developed along two
    equally unprofitable lines It is futile to
    conjure up in the imagination a system of
    differential equations for the purpose of
    accounting for facts which are not only very
    complex, but largely unknown,What we require at
    the present time is more measurement and less
    theory.
  • Eric Ponder, Director, CSHL (LIBA), 1936-1941.

27
Axioms of Platitudes -E.B. Wilson
  1. Science need not be mathematical.
  2. Simply because a subject is mathematical it need
    not therefore be scientific.
  3. Empirical curve fitting may be without other than
    classificatory significance.
  4. Growth of an individual should not be confused
    with the growth of an aggregate (or average) of
    individuals.
  5. Different aspects of the individual, or of the
    average, may have different types of growth
    curves.

28
Genes for Segmentation
  • Fertilization followed by cell division
  • Pattern formation instructions for
  • Body plan (Axes A-P, D-V)
  • Germ layers (ecto-, meso-, endoderm)
  • Cell movement - form gastrulation
  • Cell differentiation

29
PI Positional Information
  • Positional value
  • Morphogen a substance
  • Threshold concentration
  • Program for development
  • Generative rather than descriptive
  • French-Flag Model

30
bicoid
  • The bicoid gene provides an A-P morphogen
    gradient

31
gap genes
  • The A-P axis is divided into broad regions by gap
    gene expression
  • The first zygotic genes
  • Respond to maternally-derived instructions
  • Short-lived proteins, gives bell-shaped
    distribution from source

32
Transcription Factors in Cascade
  • Hunchback (hb) , a gap gene, responds to the
    dose of bicoid protein
  • A concentration above threshold of bicoid
    activates the expression of hb
  • The more bicoid transcripts, the further back hb
    expression goes

33
Transcription Factors in Cascade
  • Krüppel (Kr), a gap gene, responds to the dose
    of hb protein
  • A concentration above minimum threshold of hb
    activates the expression of Kr
  • A concentration above maximum threshold of hb
    inactivates the expression of Kr

34
Segmentation
  • Parasegments are delimited by expression of
    pair-rule genes in a periodic pattern
  • Each is expressed in a series of 7 transverse
    stripes

35
Pattern Formation
  • Edward Lewis, of the California Institute of
    Technology
  • Christiane Nuesslein-Volhard, of Germany's
    Max-Planck Institute
  • Eric Wieschaus, at Princeton
  • Each of the three were involved in the early
    research to find the genes controlling
    development of the Drosophila fruit fly.

36
The Network of Interaction
  • Legend
  • WGwingless
  • HHhedgehog
  • CIDcubitus iterruptus
  • CNrepressor fragment of CID
  • PTCpatched
  • PHpatched-hedgehog complex

37
Completenessvon Dassow, Meir, Munro Odell,
2000
  • We used computer simulations to investigate
    whether the known interactions among segment
    polarity genes suffice to confer the properties
    expected of a developmental module.
  • Using only the solid lines in earlier figure
    we found no such parameter sets despite extensive
    efforts.. Thus the solid connections cannot
    suffice to explain even the most basic behavior
    of the segment polarity network
  • There must be active repression of en cells
    anterior to wg-expressing stripe and something
    that spatially biases the response of wg to Hh.
    There is a good evidence in Drosophila for wg
    autoactivation

38
Completeness
  • We incorporated these two remedies first (light
    gray lines). With these links installed there are
    many parameter sets that enable the model to
    reproduce the target behavior, so many that they
    can be found easily by random sampling.

39
Model Parameters
40
Complete Model
41
Complete Model
42
Is this your final answer?
  • 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.

43
Computational Systems Biology
44
Systems Biology
Combining the mathematical rigor of numerology
with the predictive power of astrology.
Cyberia
Numerlogy
Astrology
Numeristan
HOTzone
Astrostan
Infostan
Interpretive Biology
Computational Biology
Integrative Biology
Bioinformatics
BioSpice
45
Why do we need a tool?
We claim that, by drawing upon mathematical
approaches developed in the context of dynamical
systems, kinetic analysis, computational theory
and logic, it is possible to create powerful
simulation, analysis and reasoning tools for
working biologists to be used in deciphering
existing data, devising new experiments and
ultimately, understanding functional properties
of genomes, proteomes, cells, organs and
organisms.
Simulate Biologists! Not Biology!!
46
Reasoning and Experimentation
Comparison
Hypotheses
Revision
Symbolic Analysis Reachability Analysis Simulation
Temporal Logic Verification
47
Simpathica is a modular system
Canonical Form
  • Characteristics
  • Predefined Modular Structure
  • Automated Translation from Graphical to
    Mathematical Model
  • Scalability

48
Glycolysis
Glycogen
P_i
Glucose-1-P
Glucose
Phosphorylase a
Phosphoglucomutase
Glucokinase
Glucose-6-P
Phosphoglucose isomerase
Fructose-6-P
Phosphofructokinase
49
Formal Definition of S-system
50
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

Leibler et al., Guet et al., Antoniotti et al.,
Wigler Mishra
51
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.

52
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

53
SimPathica System
54
Simpathica System
Model Simulation
Model Building
Model Checking
55
Symbolic Analysis
Invariant F(s(t))
f(s(t), s(tD t), D t)
Invariant F(s(tD t))
F(s) m X. X(s(t)) Æ f(s(t), s(t D t), D
t) ) X(s(tD t))
56
Algebraic Approaches
57
Differential Algebra
58
Example System
59
Input-Output Relations
60
Obstacles
61
Simpler Computational Models
  • Kripke Models/Discrete Event Systems
  • Hybrid Automata
  • Their Connection to
  • Turing Machines
  • Real Turing Machines

62
Kripke Structure
  • Formal Encoding of a Dynamical System
  • Simple and intuitive pictorial representation of
    the behavior of a complex system
  • A Graph with nodes representing system states
    labeled with information true at that state
  • The edges represent system transitions as the
    result of some action

63
Computation Tree
  • Finite set of states Some are initial states
  • Total transition relation every state has at
    least one next state i.e. infinite paths
  • There is a set of basic environmental variables
    or features (atomic propositions)
  • In each state, some atomic propositions are true

64
Buchï Automata
65
Hybrid Automata
66
Thermostat
67
Intuition
68
Semantics
69
Engineered Systems
70
Chemotaxis
  • Escherichia coli has evolved a strategy for
    responding to a chemical gradient in its
    environment
  • It detects the concentration of ligands through a
    number of receptors
  • It reacts by driving its flagella motors to alter
    its path of motion.
  • Either it runs moves in a straight line by
    moving its flagella counterclockwise (CCW), or it
    tumbles randomly change its heading by moving
    its flagella clockwise (CW).
  • The response is mediated through the molecular
    concentration of CheY in a phosphorylated form,
    which in turn is determined by the bound ligands
    at the receptors that appear in several forms.
  • The more detailed pathway involves other
  • CheB (either with phosphorylation or without, B_p
    and B_0),
  • CheZ (Z),
  • bound receptors (LT) and
  • unbound receptors (T)
  • Their continuous evolution is determined by a set
    of differential algebraic equations derived
    through kinetic mass action formulation.

71
Non-Stochastic Chemotaxis
72
Back to Turing
  • Definition A Turing Machine is a 7-tuple
  • T (Q, S, G, ?, q0, qaccept, qreject), where
  • Q is a finite set of states
  • S is the input alphabet, where ? ? S
  • G is the tape alphabet, where ? ? G and S ? G
  • ? Q ? G ? Q ? G ? L,R
  • q0 ? Q is the start state
  • qaccept ? Q is the accept state
  • qreject ? Q is the reject state, and qreject ?
    qaccept

73
Decidability
  • A TM decides a language if it accepts all strings
    in the language and rejects all strings not in
    the language
  • A language is called decidable or recursive if
    some TM decides it
  • There are languages over 0,1 that are not
    decidable
  • By the Church-Turing Thesis, it also implies that
    there are things that computers inherently cannot
    do
  • Emperor has NO MIND!!

74
Proof by Diagonalization
  • Theorem There is no onto function from the
    positive integers to the real numbers between 0
    and 1 (exclusive)
  • Proof

Valid Programs
String of Characters, N
1 if n-th digit of f(n) ? 1
n-th digit of r
0 otherwise
f(n) ? r for all n
Injective functions N a R
75
Semi-decidable
  • ATM (M,w) M is a TM that accepts string w
  • Theorem ATM is semi-decidable (r.e.) but NOT
    decidable
  • Define TM U as follows
  • On input (M,w), U runs M on w. If M ever accepts,
    accept. If M ever rejects, reject.
  • ATM is undecidable
  • Assume machine H decides ATM

76
But Not decidable
  • Construct a new TM D as follows on input M, run
    H on (M,M) and output the opposite of H

D
Reject if M accepts M Accept if M does not
accept M
D
D( M )
D
D
D
77
Questions of Interest
  • Controllability
  • Assume that the system is at the origin
    initially. Can we find a control signal so that
    the state reaches a given position at a fixed
    time?
  • Observability
  • Can the state x be determined from observations
    of the output y over some time interval.
  • Reachability A computationally simpler problem
  • Can we determine what states are reached as the
    system evolves autonomously or under a class of
    control signal.
  • HALTING Problem
  • Can the system reach a designated state at some
    time and then stay there?

78
Decision problems
79
Dynamics
  • Replacing differential equations by equivalent
    dynamics

80
Michaels Form
  • Let FxV(T) X Dyn(v)X, X, T Æ Inv(v)X
  • A Hybrid automaton is in Michael's form if
  • FxV is lower semi-continuous
  • For each t 2 IXV the set FxV(t) is closed and
    convex
  • where IXV is the largest 0, t) such that FxV(t)
    ¹ , 8 t 2 0, t).

81
Reachability
The path ph must not be infinite!!
82
Two New Models
83
First Order Theory of Reals
  • Tarski's theorem says that the first-order theory
    of reals with , , , and gt allows quantifier
    elimination. Algorithmic quantifier elimination
    implies decidability.
  • Every quantifier-free formula composed of
    polynomial equations and inequalities, and
    Boolean connectives defines a semialgebraic set.
    Thus a set S is semi-algebraic if

84
SaCoRe
  • Hybrid Automatas inclusion dynamics,
    approximated by semi-algebraic formula. DynX,X,
    T Semialgebraic Set
  • A more realistic approximation, for time
    invariant systems
  • DynX, X, h
  • ¼ X X X F(X,0) h d, d lt e,
  • for a suitably chosen
  • e F(X,0) h2/2! F(X,0) h3/3! L

85
Another Example Biological Pattern Formation
  • Embryonic Skin Of The South African Claw-Toed
    Frog
  • Salt-and-Pepper pattern formed due to lateral
    inhibition in the Xenopus epidermal layer

86
Delta-Notch Signalling
Physically adjacent cells laterally inhibit each
others ciliation (Delta production)
87
Delta-Notch Pathway
  • Delta binds and activates its receptor Notch in
    neighboring cells (proteolytic release and
    nuclear translocation of the intracellular domain
    of Notch)
  • Activated Notch suppresses ligand (Delta)
    production in the cell
  • A cell producing more ligands forces its
    neighboring cells to produce less

88
Pattern formation by lateral inhibition with
feedback a mathematical model of Delta-Notch
intercellular signallingCollier et al.(1996)
Rewriting
Where
Collier et al.
89
One-Cell Delta-Notch Hybrid Automaton
Ghosh et al.
90
Two-Cell Delta-Notch System
Cell 1
Cell 2
16 Discrete States
91
System PropertiesTrue Approximate
92
State Reachability
93
State Reachability
94
Impossibility Of Reaching Wrong Equilibrium
95
Hybrid Hierarchy
96
Decidability
  • Classical theory of computation and complexity
    analysis centered around the binary Turing
    machine is not sufficient to fully characterize
    problems involving real-valued mathematics
  • BSS Machine the more general real Turing
    machine that has exact rational operations and
    comparison of real numbers built-in as atomic
    operations represented as maps

97
Real Turing Machine
  • Can we imagine a more Powerful Computing Device
    that captures biology better?
  • BSS (Blum-Shub-Smale) Machine.

98
The instructions of a BSS Machine
99
Undecidability Of The Mandelbrot Set
  • The Mandelbrot set is not decidable over R. This
    follows from the fact that the Mandelbrot set
    cannot be the countable union of semi-algebraic
    sets over R as its boundary has complex
    mathematical properties
  • Complement of

100
Logic Model-Checking
101
Deciphering Design Principles in a Biological
Systems
  • Step 1. Formally encode the behavior of the
    system as a hybrid automaton
  • Step 2. Formally encode the properties of
    interest in a powerful logic
  • Step 3. Automate the process of checking if the
    formal model of the system satisfies the formally
    encoded properties using Model Checking

102
Temporal Logic
  • First Order Logic Time is an explicitly
    quantified variable
  • Propositional Modal logic was invented to
    formalize modal notions and suppress the
    quantified variables with operators possibly
    P and necessarily P (similar to eventually
    and henceforth)

103
First Order Logic
  • A first order logic is given by a set of function
    symbols and a set of predicate symbols.
  • Each function or predicate symbol comes with an
    arity, which is natural number.
  • Function symbols of arity 0 are known as constant
    symbols.
  • Terms are recursively defined by
  • variables are terms, and
  • if ti are terms for i1,...,n and f is an n-ary
    function symbol, then f(t1,...,tn) is a term.
  • Formulas are recursively defined by
  • if ti are terms for i1,...,n and p is an n-ary
    predicate symbol, then p(t1,...,tn) is a formula.
  • if P and Q are formulas, then P is a formula,
    P Æ Q is a formula, P Ç Q is a formula, P ! Q is
    a formula, and P Q is a formula.
  • if P is a formula and x a variable, then 8 x P
    and 9 x P are formulas.

104
First Order Logic
  • Theorem It is undecidable whether a first order
    logic formula is provable (or true under all
    possible interpretations).
  • Encode a Turing Machine and ask for a solution to
    the halting problem Whenever M has a transition
    from state q to state r, reading a, writing b,
    and moving right, the formula
  • 8 x 8 y fq(x,ay) ) fr(bx,y)
  • holds. Here x and y are variables. Likewise, if M
    has a transition from state q to state r, reading
    a, writing b, and moving left, the formulas
  • 8 x 8 y fq(cx,ay) ) fr(x,cby)
  • hold for every choice of a letter c.
  • Now consider the formula
  • fq0( l,w) T ) 8 x 8 y fq-acc(x,y).

105
Branching versus Linear Time
  • Temporal Logic
  • Short hand for describing the way properties of
    the system change with time
  • Time is implicit
  • Linear-time Only one possible future in a moment
  • Look at individual computations
  • Branching-time It may be possible to split to
    different courses depending on possible futures
  • Look at the tree of computations

Time is Linear
Time is Branching
106
Computation Tree Logic (CTL)
  • Branching Time temporal logic interpreted over
    an execution tree where branching denotes
    non-deterministic actions
  • Explicitly quantify over two modes the path and
    the time
  • Each time we talk about a temporal property, we
    also specify whether it is true on all possible
    paths or whether it is true on at least one path
    - Path quantifiers
  • A for all future paths
  • E for some future path

107
Semantics for CTL
  • For p?AP
  • s ² p ? p ? L(s) s ² ?p ? p ? L(s)
  • s ² f Æ g ? s ² f and s ² g
  • s ² f Ç g ? s ² f or s ² g
  • s ² EX f ? ? ?hs0s1... i from s s1 ² f
  • s ² E(f U g) ? ? ? hs0s1... i from s
  • ?j?0 sj ² g and ?i 0? i ?j
    si ² f
  • s ² EG f ? ? ? hs0s1... i from s ?i ? 0 si ² f

108
Some CTL Operators
AF g
EG g
EF g
AG g
109
CTL
  • A path quantifier can be followed by an arbitrary
    number of temporal operators
  • There are properties expressible in CTL but not
    in LTL and vice-versa
  • LTL, CTL are contained in CTL

110
CTL Model-Checking
  • Straight-forward approach Recursive descent on
    the structure of the query formula
  • Label the states with the terms in the formula
  • Proceed by marking each point with the set of
    valid sub-formulas
  • Global algorithm
  • Iterate on the structure of the property,
    traversing the whole of the model in each step
  • Use fixed point unfolding to interpret Until

111
Naïve CTL Model-Checker
112
Other Model Checking Algorithms
  • LTL Model Checking Tableu-based
  • CTL Model Checking Combine CTL and LTL Model
    Checkers
  • Symbolic Model Checking
  • Binary Decision Diagram
  • OBDD-based model-checking for CTL
  • Fixed-point Representation
  • Automata-based LTL Model-Checking
  • SAT-based Model Checking
  • Algorithmic Algebraic Model Checking
  • Hierarchical Model Checking

113
Complexity Comparison
  • Size of transition system n
  • Size of temporal logic formula m
  • Worst Case Comparison
  • CTL linear - O(nm)
  • LTL exponential n 2O(m)
  • For an LTL formula that can also be expressed in
    ?CTL, LTL model-checking can be done in a time
    linear in the size of the formula
  • LTL is PSPACE complete Hamiltonian Path problem
    can be reduced to an LTL Model Checking problem
  • Fp1 Æ Fp2 Æ Fp3 Æ..
  • G (p1! XG p1) Æ G(p2! XG p2) Æ.

114
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.

115
Simple Model
116
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).

117
Purine Metabolism
118
Queries
  • 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
    1.7PRPP1) implies steady_state()

TRUE
TRUE
119
Queries
  • Consider the following statement
  • Eventually
  • (Always (PRPP 1.7 PRPP1)impliessteady_state(
    )and Eventually
  • Always(IMP lt 2 IMP1))and Eventually
    (Always (hx_pool lt 10hx_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

False
120
Final Model
121
Purine Metabolism
122
Continuous-Time Logics
  • Linear Time
  • Metric Temporal Logic (MTL)
  • Timed Propositional Temporal Logic (TPTL)
  • Real-Time Temporal Logic (RTTL)
  • Explicit-Clock Temporal Logic (ECTL)
  • Metric Interval Temporal Logic (MITL)
  • Branching time
  • Real-Time Computation Tree Logic (RTCTL)
  • Timed Computation Tree Logic (TCTL)

Alur et al,
123
TCTL Syntax And Semantics
124
TCTL
125
TCTL One-Step Until
  • q can be reached within one step of the hybrid
    system and p holds until that point in the
    transition
  • p continuously holds until some intermediate
    point immediately followed by q being true
  • p or q holding all along that one step of the
    hybrid system and q being true at the end of the
    one-step evolution
  • Discrete time model-checking next state
    operator X
  • Continuous-mode single-step until operator

126
T-µ CALCULUS Syntax
127
Until T- µ Fixpoint
  • s2 is true now or
  • s1 holds for one-step on some path after which
    s2 holds or
  • s1 holds for one-step on some path after which
    s1 holds for one more step on some path after
    which s2 holds or
  • and so on..

128
TCTL Model Checking
  • Only Until requires computation
  • Until Iterative computation of one-step Until
  • Least fixpoint computation

129
Semi-Decidability Of TCTL
  • Global time variable
  • Allows interpretation of the TCTL operators
    freeze (z.X) and subscripted until (Ua)
  • While one-step until is decidable, the fixpoint
    is not guaranteed to converge
  • So TCTL is semi-decidable

130
Mandelbrot Hybrid Automaton
Let
Then
Reachability Query
131
Solution
  • Bounded Model Checking
  • Fully O-minimal Systems for Dense CTL
  • Constrained Systems
  • Linear Systems for Dense CTL
  • O-minimal for Dense CTL
  • SACoRe (Semi algebraic Constrained Reset) for
    TCTL
  • IDA (Independent Dynamics Automata) for TCTL

132
GamesParikh, Pauly, van Benthem
  • Game frames are based on the notion of a
    strategic game form a tuple h N, Si I 2 N o,
    Si consisting of
  • a (non-empty) set of players N
  • a family of sets of actions (or strategies) Si
    for each player i 2 N
  • a set of states S
  • an outcome function o Õ Si ! S which associates
    with every strategy profile (tuple of strategies,
    one for each of the players) an outcome state in
    S.

133
MGF MGM
  • A multi-player game frame (MGF)
  • for a set of players N is a pair (S, g) where S
    is a non-empty set of states and g S ! GSN is a
    mapping associating a strategic game form with
    each state in S.
  • A multi-player game model (MGM)
  • for a set of players N over a set of
    propositions P is a triple M (S, g, v) where
    (S, g) is a multi-player game frame and v S !
    2P is a valuation labeling each state from S with
    the set of propositions that are true at that
    state.

134
Coalition Game Logic
  • Coalition logic (CL), is a propositional modal
    logic with a family of (non-normal) modalities
    C C µ N where N is a fixed set of players
  • Intuitively, C f means that the coalition C can
    enforce an outcome state satisfying f.
  • The formulae of CL are defined recursively
  • f p f f1 Ç f2 C f

135
Semantics
  • The semantics of CL can be given in terms of
    truth at a state of an MGM M (S, g, v) via the
    clauses
  • M,q p iff p 2 v (q) for atomic propositions p
  • M,q Cf iff there is a strategy profile sC
    such that for every strategy profile sN-C, M,
    oq(sC, sN-c) f.

136
HookeThursday 25 May 1676
  • Damned Doggs.
  • Vindica me deus.
  • Commenting on
  • Sir Nicholas Gimcrack character in
  • The Virtuoso, a play by Thomas Shadwell.

137
Hooke
  • So many are the links, upon which the true
    Philosophy depends, of which, if any can be
    loose, or weak, the whole chain is in danger of
    being dissolved
  • it is to begin with the Hands and Eyes, and to
    proceed on through the Memory, to be continued by
    the Reason
  • nor is it to stop there, but to come about to
    the Hands and Eyes again, and so, by a continuall
    passage round from one Faculty to another, it is
    to be maintained in life and strength.

138
Hookein the Royal Society, 26 June 1689
  • I have had the misfortune either not to be
    understood by some who have asserted I have done
    nothing
  • Or to be misunderstood and misconstrued (for
    what ends I now enquire not) by others
  • And though many things I have first Discovered
    could not find acceptance yet I finde there are
    not wanting some who pride themselves on
    arrogating of them for their own
  • But I let that passe for the present.

139
The end
140
The end
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