Principia

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Principia Biologica
Design Principles in Biological Systems
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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

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(No Transcript)
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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.

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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.

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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.

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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

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Image Logic

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

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MicrographiaPrincipia
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Micrographia
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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

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Principia
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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.
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Morphogenesis
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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.

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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.

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Diffusion equation
first temporal derivative rate
second spatial derivative flux
a/ t Da r2 a
a concentration Da diffusion constant
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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
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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).
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Reaction-diffusion an example
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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

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Crick Watson 1953
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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

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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
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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

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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.

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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.

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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

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PI Positional Information

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

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bicoid

• The bicoid gene provides an A-P morphogen
  gradient

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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

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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

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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

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Segmentation

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

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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.

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The Network of Interaction

• Legend
• WGwingless
• HHhedgehog
• CIDcubitus iterruptus
• CNrepressor fragment of CID
• PTCpatched
• PHpatched-hedgehog complex

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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

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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.

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Model Parameters
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Complete Model
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Complete Model
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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.

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Computational Systems Biology
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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
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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!!
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Reasoning and Experimentation
Comparison
Hypotheses
Revision
Symbolic Analysis Reachability Analysis Simulation
Temporal Logic Verification
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Simpathica is a modular system
Canonical Form

• Characteristics
• Predefined Modular Structure
• Automated Translation from Graphical to
  Mathematical Model
• Scalability

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

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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

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SimPathica System
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Simpathica System
Model Simulation
Model Building
Model Checking
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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))
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Algebraic Approaches
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Differential Algebra
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Example System
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Input-Output Relations
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Obstacles
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Simpler Computational Models

• Kripke Models/Discrete Event Systems
• Hybrid Automata
• Their Connection to
• Turing Machines
• Real Turing Machines

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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

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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

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Buchï Automata
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Hybrid Automata
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Thermostat
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Intuition
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Semantics
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Engineered Systems
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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.

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Non-Stochastic Chemotaxis
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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

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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!!

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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
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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

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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
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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?

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Decision problems
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Dynamics

• Replacing differential equations by equivalent
  dynamics

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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).

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Reachability
The path ph must not be infinite!!
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Two New Models
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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

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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

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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

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Delta-Notch Signalling
Physically adjacent cells laterally inhibit each
others ciliation (Delta production)
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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

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Pattern formation by lateral inhibition with
feedback a mathematical model of Delta-Notch
intercellular signallingCollier et al.(1996)
Rewriting
Where
Collier et al.
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One-Cell Delta-Notch Hybrid Automaton
Ghosh et al.
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Two-Cell Delta-Notch System
Cell 1
Cell 2
16 Discrete States
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System PropertiesTrue Approximate
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State Reachability
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State Reachability
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Impossibility Of Reaching Wrong Equilibrium
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Hybrid Hierarchy
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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

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Real Turing Machine

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

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The instructions of a BSS Machine
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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

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Logic Model-Checking
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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

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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)

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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.

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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).

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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
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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

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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

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Some CTL Operators
AF g
EG g
EF g
AG g
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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

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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

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Naïve CTL Model-Checker
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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

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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.

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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.

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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).

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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
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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