Cell Talk

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Cell Talk
SIAM Meeting 10 04 2002

• Bud Mishra
• Professor of CS Mathematics (Courant, NYU)
• Professor (Cold Spring Harbor Laboratory)

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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, he is remembered for the discovery
  of the proportional relationship of the extension
  of a spring and the force applied to produce that
  extension.
• His work Micrographia of 1665 contained his
  microscopical investigations, which included the
  first identification of biological cells.
• Hooke became involved in a dispute with Isaac
  Newton over the priority of the discovery of the
  inverse square law of gravitation. Although he
  communicated some form of inverse square law to
  Newton, modern opinion is that credit for the law
  of universal gravitation must go to Newton.
• Aubrey held his ability in high regard "He is
  certainly the greatest Mechanick this day in the
  World."

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Newton Hooke

• 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.
• And particularly that of the oval figure of the
  Earth which was read by me to this Society about
  27 years since upon the occasion of the carrying
  the pendulum clocks to sea and at two other times
  since, though I have had the ill fortune not to
  be heard, and I conceive there are some present
  that may very well remember and do know that Mr.
  Newton did not send up that addition to his book
  till some weeks after I had read and showed the
  experiments and demonstration thereof in this
  place and had answered the reproachful letter of
  Dr. Wallis from Oxford.

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Newton 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

• Rule IV. In experimental philosophy we are to
  look upon propositions collected by general
  induction from phenomena as accurately or very
  nearly true, notwithstanding any contrary
  hypotheses that may be imagined, 'till such time
  as other phenomena occur, by which they may
  either be made more accurate, or liable to
  exceptions
• 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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Genome in Detail
The Human Genome at Four Levels of Detail. Apart
from reproductive cells (gametes) and mature red
blood cells, every cell in the human body
contains 23 pairs of chromosomes, each a packet
of compressed and entwined DNA (1, 2).
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DNA Structure.
The four nitrogenous bases of DNA are arranged
along the sugar- phosphate backbone in a
particular order (the DNA sequence), encoding all
genetic instructions for an organism. Adenine (A)
pairs with thymine (T), while cytosine (C) pairs
with guanine (G). The two DNA strands are held
together by weak bonds between the bases.
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The Central Dogma

• The intermediate molecule carrying the
  information out of the nucleus of an eukaryotic
  cell is RNA, a single stranded polymer.
• RNA also controls the translation process in
  which amino acids are created making up the
  proteins.
• 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.

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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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Gene Expression

• When genes are expressed, the genetic information
  (base sequence) on DNA is first transcribed
  (copied) to a molecule of messenger RNA, mRNA.
• The mRNAs leave the cell nucleus and enter the
  cytoplasm, where triplets of bases (codons)
  forming the genetic code specify the particular
  amino acids that make up an individual protein.
• This process, called translation, is accomplished
  by ribosomes (cellular components composed of
  proteins and another class of RNA) that read the
  genetic code from the mRNA, and transfer RNAs
  (tRNAs) that transport amino acids to the
  ribosomes for attachment to the growing protein.

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Regulation of Gene Expns

• Motifs (short DNA sequences) that regulate
  transcription
• Promoter
• Terminator
• Motifs that modulate transcription
• Repressor
• Activator
• Antiterminator

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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

• Fertilisation 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

positive interacions
negative interacions
mRNA
proteins
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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
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Model Parameters
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Model
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Complete Model
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Complete Model
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S-system
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Graphical Representation
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Graphical Representation
The reaction between X1 and X2 requires coenzyme
X3 which is converted to X4
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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,)
• E.g. Michaelis-Menten for substrate S product
  P
• dS/dt - Vmax S/(KM S)
• dP/dt Vmax S/(KM S)

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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)

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Canonical Forms

• S-systems result in Non-linear Time-Invariant DAE
  System.
• Note that Given a system of equations with f and
  g being arbitrary rational functions, we can
  transform the system into a set of Differential
  Binomial Equation System with Linear Constraints
• dxi/dt a x1a1L xnan - b x1b1L xnbn
• g1 x1 L gn xn 0

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Transformation I

• Assume that an equation is given as
• dx/dt p(x(t), u(t))/q(x(t), u(t))
• A rational function. p q are polynomials
• p(x(t), u(t)) a1 m1 L ak mk - b1 p1 - L -
  bl pl
• where ms and ps are power-products with
  arbitrary power. as and bs are positive-valued.
• dx/dt p(x(t), u(t)) y(t)-1,
• dc/dt q(x(t), u(t)) y(t),
• c 0.

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Transformation II

• dx/dt a1 m1 L ak mk - b1 p1 - L - bl pl
• (a1 m1 w(t)/k) L (ak mk w(t)/k)
• (b1 p1 - w(t)/l) - L - (bl pl - w(t)/l)
• Equivalent System
• x(t) - g1(t) - L - gk(t) gk1(t) L gkl(t)
  0
• dgi/dt ai mi w(t)/k , 1 i k
• dgj/dt b1 p1 - w(t)/l , k1 j kl

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Canonical Forms
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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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How Stable is This???
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Synergy of Tools

• Exploit the special structure of Pathways Models
  and of traces' to create a synergy among
  different conceptual components
• ODEs (XS-systems canonical form)
• Temporal Logic
• Time Series Analysis
• Symbolic Mathematics

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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
• DefinitionGiven an S-systems S, the S-system
  automaton AS associated to S is 4-tuple AS (S,
  D, S0, F), where S µ D1 L DW is a set of
  states, D µ 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 s0 2 S0,
  D(si, si1), 8 i 0.ð

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Trace Automaton
Simple one-to-one construction of the trace
automata AS for an S-system S
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State Collapse

• Definition The relation Rd holds between two
  states sk X(t k q) and skj X(t (kj) q),
• iff 8 i 2 1, , nm,
• dX/dt(t k q) - dX/dt(t (kj) q) d. ð

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Collapsing Algorithm
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Collapsed Automata
The effects of the collapsing construction of
the trace automata AS for an S-system S
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SimPathica System
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Modal Logic Queries
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SimPathicaTrace Analysis System
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SimPathica
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Computational Differential Algebra
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Algebraic Approaches
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State Space Description
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Input-Output Relation
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Differential Algebra
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Related Problems
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Example System
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Input-Output Relations
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Membership Problem
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Obstacles
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Some Remarks

• Many problems of Kinetic modeling lead naturally
  to formulation in Differential Algebra!
• Yet, most problems in Differential Algebra remain
  to be solved satisfactorily!!
• Many of the tools developed in the algebraic
  setting (e.g., Gröbner bases, elimination theory,
  etc.) do not generalize.
• Complexity and solvability questions pose
  intriguing and challenging problems for applied
  mathematicians and computer scientists!!

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Isaac Newton
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The End

• http//www.cs.nyu.edu/mishra
• http//bioinformatics.cat.nyu.edu
• Valis, Gene Grammar, NYU MAD, Cell Simulation,

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Other Ongoing Projects

• OPTICAL MAPPING
• Single Molecule Genomics Optical Mapping,
  Optical Sequencing RFLP Haplotyping
• (In collaboration with Univ. Wisc. funded by
  NCI)
• Valis Bioinformatic
• Environment Language
• (Funded by DOE NYSTAR)
• ROMA (Representational Oligonucleotide Microarray
  Analysis)
• Microarray-based Genome Mapping--
• (In collaboration with CSHL funded by NCI/NIH)
• Expression Data Analysis
• (In collaboration with NYU Biology funded by
  NSF MHHI)
• Cell Informatics
• (Funded by DARPA Airforce)

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Optical Mapping

• Sizing Error
• (Bernoulli labeling, absorption cross-section,
  PSF)
• Partial Digestion
• False Optical Sites
• Orientation
• Spurious molecules, Optical chimerism, Calibration

Image of restriction enzyme digestedYAC clone
YAC clone 6H3, derived from human chromosome 11,
digested with the restriction endonuclease Eag I
and Mlu I, stained with a fluorochrome and imaged
by fluorescence microscopy.
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Optical MappingInterplay between Biology and
Computation
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Mapping the DAZ locus on Y Chromosome
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Gentig MapDeinococcus radiodurans
Nhe I map of D.radiodurans generated by Gentig
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E. coli Shotgun Map
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Gentig MapsPlasmodium falciparum

• A. Gap-free consensus BamHI NheI maps for all
  14 chromosomes.
• B.BamHI map
• C. NheI map
• D.NheI map of Chromosome 3 displayed by ConVEx

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P. Falciparum c14 Alignment
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HaplotypingOutput of the RFLP Phasing Algorithm
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Valis
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Valis Architecture
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Valis Screenshot
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Wild by NatureA simple metastable biological
circuit.

• The combinatorial genetic network depicted
  consists of
• Two mutually inhibiting repressor genes, A and B,
  which are modulated by two small molecule
  inducers, X1 and X2.
• The gene products encode the state of the system,
  and the inducers act as inputs to the network.
• The state of B encodes the output (Y).
• This network has two stable states (output is
  "high" or "low"), but also a metastable state
  (output assumes an intermediate state between
  "high" and "low") that is achieved by withdrawing
  both inputs simultaneously.
• For these reasons, the network is also extremely
  sensitive to the relative order in which the
  inputs arrive and, thus, is unpredictable.

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SimPathica System
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Modal Logic Queries
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SimPathicaTrace Analysis System
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Nitrogen Pathway
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NYU MAD
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Data Analysis in NYU MAD
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Cocultivation Experiments

• The pairing of factor and receptors are largely
  unknown.
• The consequences of most factor-receptor
  interactions are unknown.
• These pairings and their consequences are
  explored by cell cocultivation experiments.
• We examine cell type A and B alone, and when
  cocultured (A c B)We examine the genes
  expressed by cells using DNA microarrays, that
  quantitate tens of thousands of genes
  simultaneously.

• Cells signal through communication proteins
• Many communication proteins fall into two
  classes
• Extracellular factors and
• External receptors.
• Factor-receptor interactions occur in pairs and
  influence the genes and proteins that cells
  express.
• Factors and receptors are encoded by genes, about
  a thousand of each class.
• Only a few of each class are expressed in cells
  of a particular type.

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Gene Induction upon Co-culture
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New transcriptional states upon co-cultivation of
RD-2 (sarcoma) and DLD-1(carcinoma) cell lines

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People

• Marco Antoniotti
• Sr. Res. Scientist (CS, Courant)
• Simulation System//Shift
• Archisman Rudra
• Sr. Res. Scientist (CS, Courant)
• Genome Grammar//CAML
• Raoul Daruwalla
• Jr. Res. Scientist (CS, Courant)
• Urn Models for Genome Evolution
• Will Casey
• Jr. Res. Scientist (CS, Courant)
• Haplotyping
• Salvatore Paxia
• Jr. Res. Scientist (CS, Courant)
• Software Environment//Valis
• Saurabh Sinha
• Postdoc (Courant Rockefeller)
• Detecting CIS elements

• Marc Rejali
• Jr. Res. Scientist (CS, Courant)
• Microarray Data Analysis//MAD
• Jiawu Feng
• Jr. Res. Scientist (CS, Courant)
• Microarray Data Analysis//MAD
• Nadia Ugel
• Jr. Res. Scientist (CS, Courant)
• XS system
• Maoyen Chi
• Jr. Res. Scientist (CSHL)
• Cocultivation
• Joe McQuown
• Jr. Res. Scientist (Stat, NYU)
• Statistical Analysis
• Graduate Students
• Ken Chang (Biology, CSHL)
• Joe West (Biology, CSHL)
• Joey Zhou (Biology, NYU)

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Visitors Collaborators

• VISITORS
• Alberto Policriti
• Computer Science,
• University of Udine, Italy
• Haim Wolfson
• Computer Science,
• Tel Aviv University, Israel
• Chris Wiggins
• Physics Applied Mathematics
• Columbia University, USA
• Frank Park
• Control Theory
• University of Seoul, South Korea
• Naomi Silver
• Computer Science
• Marco Isopi
• Applied Mathematics
• Italy

• COLLABORATORS
• Mike Wigler
• Cold Spring Harbor Lab
• Larry Norton
• Memorial Sloan-Kettering
• Misha Gromov Ale Carbone
• IHES Courant
• Jack Schwartz
• Courant Institute
• Tom Anantharaman Dave Schwartz
• University of Wisconsin
• Steve Burakoff
• Skirball Institute
• Harel Weinstein, Ravi Iyengar Bob Desnick
• Mt Sinai School of Medicine
• Mike Seoul
• Bioarrays