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Title: Cell Talk


1
Cell Talk
NYU, Department of Computer Science 09 19 2003
  • Bud Mishra
  • Professor of CS Mathematics (Courant, NYU)
  • Professor (Cold Spring Harbor Laboratory)
  • Professor (Tata Institute of Fundamental
    Research, Adjunct)
  • Professor of Human Genetics (Mt Sinai School of
    medicine)

2
(No Transcript)
3
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.
  • Aubrey held his ability in high regard "He is
    certainly the greatest Mechanick this day in the
    World. In his drafts of Book II, Newton had
    referred to him as the most illustrious
    HookeClarissimus Hookius. Hooke was
    considered the Englands Da Vinci because of
    his wide range of interests.

4
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.

5
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
  • Should a man who thinks himself knowing loves
    to know it in correction instructing others,
    come to you when you are busy, notwithstanding
    your excuse, press discourse upon you through
    his own mistakes correct you multiply
    discourses then make this use of it, to boast
    that he taught you all he spake oblige you to
    acknowledge it cry out injury injustice if
    you do not
  • I beleive you would think him a man of a strange
    unsociable temper.

6
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

7
Image Logic
  • The great distance between
  • a glimpsed truth and
  • a demonstrated truth
  • Christopher Wren/Alexis Claude Clairaut

8
MicrographiaPrincipia
9
Micrographia
10
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

11
Principia
12
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.
13
Morphogenesis
14
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.

15
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.

16
Diffusion equation
first temporal derivative rate
second spatial derivative flux
a/ t Da r2 a
a concentration Da diffusion constant
17
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
18
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).
19
Reaction-diffusion an example
20
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

21
Crick Watson 1953
22
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

23
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
24
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

25
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.

26
Axioms of Platitudes -E.B. Wilson
  • Science need not be mathematical.
  • Simply because a subject is mathematical it need
    not therefore be scientific.
  • Empirical curve fitting may be without other than
    classificatory significance.
  • Growth of an individual should not be confused
    with the growth of an aggregate (or average) of
    individuals.
  • Different aspects of the individual, or of the
    average, may have different types of growth
    curves.

27
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

28
PI Positional Information
  • Positional value
  • Morphogen a substance
  • Threshold concentration
  • Program for development
  • Generative rather than descriptive
  • French-Flag Model

29
bicoid
  • The bicoid gene provides an A-P morphogen
    gradient

30
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

31
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

32
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

33
Segmentation
  • Parasegments are delimited by expression of
    pair-rule genes in a periodic pattern
  • Each is expressed in a series of 7 transverse
    stripes

34
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.

35
The Network of Interaction
  • Legend
  • WGwingless
  • HHhedgehog
  • CIDcubitus iterruptus
  • CNrepressor fragment
  • of CID
  • PTCpatched
  • PHpatched-hedgehog
  • complex

positive interacions
negative interacions
mRNA
proteins
36
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

37
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.

38
Model Parameters
39
Complete Model
40
Complete Model
41
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.

42
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
43
ComputationalSystems Biology
How much of reasoning about biology can be
automated?
44
Graphical Representation
45
Graphical Representation
The reaction between X1 and X2 requires coenzyme
X3 which is converted to X4
46
Glycolysis
Glycogen
P_i
Glucose-1-P
Glucose
Phosphorylase a
Phosphoglucomutase
Glucokinase
Glucose-6-P
Phosphoglucose isomerase
Fructose-6-P
Phosphofructokinase
47
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

48
Cycles of Repression
  • The first repressor protein, LacI from E. coli
    inhibits the transcription of the second
    repressor gene, tetR from the tetracycline-resista
    nce transposon Tn10, whose protein product in
    turn inhibits the expression of a third gene, cI
    from l phage.
  • Finally, CI inhibits lacI expression,
  • completing the cycle.

49
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.

50
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

51
SimPathica System
52
Simpathica Movies
..\..\Simpathica Movies
53
Canonical Forms Model Building
54
Systems of Differential Equations
  • dXi/dt
  • (instantaneous) rate of change in Xi at time t
  • Function of substrate concentrations, enzymes,
    factors and products
  • dXi/dt f(S1, S2, , E1, E2, , F1, F2,, P1,
    P2,)
  • S-systems result in Non-linear Time-Invariant DAE
    System.

55
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)

56
Canonical Forms
57
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

58
Trace Automaton
Simple one-to-one construction of the trace
automata AS for an S-system S
59
Collapsing Algorithm
60
Collapsed Automata
The effects of the collapsing construction of
the trace automata AS for an S-system S
61
Temporal Logic Model Checking
62
Models of Modal LogicKripke Structure
63
Kripke Structure
64
CTL
65
Temporal Modes
66
Syntax
67
Semantics
68
Least Fixed Point Characterization
69
Model Checking
70
Bisimulation
71
Bisimulation
Theoretical Computer Science, 2003
72
Bisimulation Lemma
73
Example
74
Equations in Canonical Form
75
Structure of the Collapsed System
76
Computational Differential Algebra
77
Algebraic Approaches
78
State Space Description
79
Input-Output Relation
80
Differential Algebra
81
Example System
82
Input-Output Relations
83
Membership Problem
84
Obstacles
85
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!!

86
Purine Metabolism
87
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.

88
Simple Model
89
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).

90
Biochemistry of Purine Metabolism
  • In addition to these processes, there appear to
    be two salvage pathways that serve to maintain
    IMP level and thus of adenosine and guanosine
    levels as well.
  • In these pathways, adenine phosphoribosyltransfera
    se (APRT) and hypoxanthine-guanine
    phosphoribosyltransferase (HGPRT) combine with
    PRPP to form ribonucleotides.

91
Purine Metabolism
92
XML Description
. . .
  •   lt?xml version"1.0" ?gt - ltmap
    xmlnsxsi"http//www.w3.org/2001/XMLSchema-instan
    ce" xsinoNamespaceSchemaLocation"map.xsd"gt-
    ltsubstrategt  ltidgt1lt/idgt   ltconcentrationgt5lt/concen
    trationgt   ltnamegtPRPPlt/namegt   lt/substrategt-
    ltsubstrategt  ltidgt2lt/idgt   ltconcentrationgt100lt/conc
    entrationgt   ltnamegtIMPlt/namegt   lt/substrategt-
    ltsubstrategt  ltidgt3lt/idgt   ltconcentrationgt2500lt/con
    centrationgt   ltnamegtAdolt/namegt   lt/substrategt-
    ltsubstrategt  ltidgt4lt/idgt   ltconcentrationgt425lt/conc
    entrationgt   ltnamegtGMPlt/namegt   lt/substrategt-
  • ltsynthesisgt  ltreactant1gt1lt/reactant1gt  
    ltreactant2gt8lt/reactant2gt   ltproductgt2lt/productgt  
    ltpower_function1gt1.1lt/power_function1gt  
    ltrate1gt12.570lt/rate1gt   ltpower_function2gt0.48lt/pow
    er_function2gt   ltrate2gt12.570lt/rate2gt -
    ltmodulationgt  ltenzymegt2lt/enzymegt  
    ltpower_function_enzymegt-0.89lt/power_function_enzym
    egt   lt/modulationgt  lt/synthesisgt- ltoutputgt 
    ltreactantgt11lt/reactantgt   ltpower_functiongt2.21lt/po
    wer_functiongt   ltrategt0.00008744lt/rategt  
    lt/outputgt  lt/mapgt

93
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
94
Queries
  • Consider the following statement
  • Eventually
  • (Always (PRPP 1.7 PRPP1) implies
    steady_state() and Eventually
  • (Always(IMP lt 2 IMP1)) and Eventual
    ly (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
95
Final Model
96
Purine Metabolism
97
Query
  • This change to the model allows us to reformulate
    our query as shown below
  • Always(PRPP gt 50 PRPP1 implies (steady_stat
    e() and Eventually(IMP gt IMP1) and
    Eventually(HX lt HX1) and Eventually(Always(IMP
    IMP1)) and Eventually(Always(HX HX1))
  • An (instantaneous) increase in the level of PRPP
    will not make the system stray from the predicted
    steady state, even if temporary variations of IMP
    and HX are allowed.

TRUE
98
Time FrequencyRAS Pathways
99
Feedback in Biochemical Pathways
  • Iyengar and Bhalla analyze a complex pathway
    Science vol. 283, 1999
  • The pathway presents a feedback loop involving
    PKC, MAPK, and Ras
  • Bistability plot of PKC vs MAPK concentrations
  • A is the active point, B is the basal point, and
    T is the threshold point
  • A and B are the stable states

100
PLC?-PKC and Ras-Raf-MAPK Pathways
101
PLC?-PKC and Ras-Raf-MAPK Pathways
The trajectories of MAPK and PKC change if EGF
stimulus is provided. A 6000sec EGF stimulus is
provided at 5 different levels (1,2,3,5 and
7nM). The two different modes for MAPK and PKC
are observed.
102
Orthonormal Bases and Projection
  • Behavior of a biological process can be described
    by the trajectory of abundance of a particular
    molecule or reactant
  • Time series functions their approximate
    representation in terms of an M-dimensional
    vector in a Euclidean space.
  • Projection of the time series.
  • The most typical as well as robust behaviors of
    the system are determined by the sets of time
    series functions giving rise to unique clusters.

103
Orthonormal Bases and Projection
  • With a suitably chosen orthonormal bases
  • g1, g2,,gM, ,
  • the function can be expressed as a linear
    combination
  • f(t)åi11 h f, gi i gi
  • and yields a sufficiently good approximation
  • fM(t)åi1M h f, gi i gi ¼ f(t), k f(t)
    fM(t)k C M-a.
  • Note that for a well chosen orthonormal bases and
    e gt 0,
  • 9M, d k f1,M(t) f2,M(t)k lt d
  • ) k f1(t) f2(t)k lt d 2 C M-a lt e.
  • Projection P f a h f, gi i i1M

104
Multi-resolutionTime-Frequency Analysis
Time-frequency activity induced by EGF. Two
clusters are formed, corresponding to two levels
of EGF stimulus low (2nm, red o symbols) and
high (5nm, blue x symbols) levels of
MAPK/PKC. As the stimulus is applied, the system
shows higher activity along the chosen
discriminating vectors at high EGF stimulus. In
the relaxation phase, the system shows higher
time-frequency activity after the withdrawal of
the lower stimulus, as it is relaxing to the
stability points that existed before stimulation,
thereby reversing the effect of EGF stimulus.
After the 5nm stimulus, on the other hand, the
systems active components are relaxing to higher
concentration levels (memory effect), thereby
displaying lower activity along the chosen
time-frequency vectors.
105
Multi-resolutionTime-Frequency Analysis
Loss of memory caused by breaking the feedback
loop involving RAS, MAPK, etc. in the RAS
pathway.
106
Multi-resolutionTime-Frequency Analysis
107
Multi-resolutionTime-Frequency Analysis
108
Multi-resolutionTime-Frequency Analysis
109
Time FrequencyCell Cycle
110
The Cell Cycle
G1
start
cell division
Cdk
Cdk
Cdk
Cyclin
S
M (anaphase)
APC
APC
finish
G2
M (metaphase)
111
The Cell Cycle
  • The chromosome cycle is divided into four
    classes
  • G1, S, G2, M
  • During S phase, a new copy of each chromosome is
    synthesized
  • During M phase (mitosis) the sister chromatids
    are separated so that each daughter cell receives
    a copy of each chromosome.
  • G1 and S-G2-M are separated by two transitions
  • start finish
  • Cell cycle events are controlled by a network of
    molecular signals
  • The central components are Cdks (cyclin-dependent
    protein kinases), cyclin molecules and APC
    (anaphase-promoting complex)..

112
The Cell Cycle
  • In the G1 phase Cdk activity is low as cyclin
    mRNA synthesis is inhibited and cyclin protein is
    degraded rapidly
  • At start, cyclin synthesis is induced and cyclin
    degradation is inhibited, causing a rise in Cdk
    activity that persists throughout S-G2-M phase
  • High Cdk activity is needed for DNA replication,
    chromosome condensation and spindle assembly.
  • At finish, the proteins needed for APC complex is
    activated.
  • APC consists of core complex of a dozen
    polypeptides plus two auxiliary proteins Cdc20
    and Cdh1.
  • Together, Cdc20 and Cdh1 label cyclins for
    degradation at telophase, thus returning the cell
    to G1.

113
The interaction ofCyclin B/Cdk and Cdh1/APC
  • dCycB/dt
  • k1 (k2 k2Cdh1)CycB
  • dCdh1/dt
  • (k3 k3 A) (1-Cdh1)/ (J31 Cdh1)
    k4 m CycBCdh1/ (J4 Cdh1)
  • A pair of nonlinear ODE (ordinary differential
    equations) describing the biochemical reactions
    at the center.

114
Yeast Cell Cycle Regulations
115
Simulation of Yeast Cell Cycle
116
Simulation of Yeast Cell Cycle
117
Simulation of Yeast Cell Cycle
118
Analysis of Experimental Traces(1)
The trace is loaded
119
Analysis of Experimental Traces (2)
Queries can be asked to the system
120
Simulated Yeast Cell Cycle.
  • Plots of Cdh1 and CycB with respect to two
    dominant time-frequency modes reflect the
    distance of the initial conditions from the two
    stable states

121
Simulated Yeast Cell Cycle of wild type vs.
CKI/SK double mutant
  • The points corresponding to the trajectories of
    the double mutant (o symbols) are much more
    scattered than those corresponding to the wild
    type (x symbols), indicating that, although in
    double mutant the oscillations of the cell cycle
    is restored, the system is less stable than the
    wild type.

122
NYU SIM
123
NYUSIM Trace Database
  • Time course data need to be classified according
    to various criteria
  • Parameters
  • Algorithmic descriptions
  • References to model used
  • The objective is to avoid the directory dump
    effect and to provide a standardized way to
    access time-series biological data

124
NYUSIM/JDesigner
Data produced with JDesigner/SBW is readily
inserted in NYUSIM
125
NYUSIM/Simpathica
Simpathica/XSSYS can access the NYUSIM DB and
analyze data produced by several sources.
126
NYU BioWAVE
  • A tool for classification of time course data

127
NYUBIOWAVE Example
  • The set of functions used to test the system
  • 30 Beta functions with different parameters
  • 10 Step functions with different amplitude,
    sharpness and shift

128
NYUBIOWAVE Example
  • The same set of functions from another viewpoint

129
NYUBIOWAVE Classification
  • The Matlab NYUBIOWAVE interface showing the
    classification of the step functions in a single
    (amplitude normalized) group

130
NYUBIOWAVE Classification
  • Classification of bell-shaped Beta functions by
    NYUBIOWAVE

131
C elegans
132
Caenorhabditis elegans(C. elegans)
  •                                                 
                                                      
                                                      
           
  • An organism of exactly 959 cells.
  • Two things are known about C. elegans
  • the complete sequence of its DNA, and
  • what every one of its 959 cells does.
  • There are many things we don't know about C.
    elegans. One is the answer to the question since
    all 959 cells come from one original cell, how
    does each of the 959 cells decide what sort of
    cell to become

133
Worm Guys
seminal discoveries concerning the genetic
regulation of organ development and programmed
cell death. By establishing and using the
nematode Caenorhabditis elegans as an
experimental model system, possibilities were
opened to follow cell division and
differentiation from the fertilized egg to the
adult. The discoveries are important for
medical research and have shed new light on the
pathogenesis of many diseases.
                    
                    
                       
134
Germ Line Cells in C. elegans
135
More C. elegans
136
Modeling Stem Cells Processes
  • Mathematical Models
  • Population Models for differentiation - i.e.
    'state transitions'
  • Diffusion Models and preliminary regulatory
    models for proliferation, differentiation and
    self-renewal
  • Model Types
  • Differential Equations (usually differential
    algebraic equations - DAE) Models

137
Stem Cells Proliferation and Differentiation
  • Stem Cells Division/Proliferation (Morrison et.
    al. Cell 88, 287-298, Feb 1997)

138
Queue Model
a
b2
N1
g
b1
N2
N3
139
Markov Model
h N1, N2-1, N32i
b2
g
a
h N1, N2, N3 i
h N1, N2, N3 -1i
h N11, N2, N3 i
b1
h N1-1, N22, N3i
140
Simulation (without aging)
N2
N3
N1
N1N2N3
141
Simulation (with aging)
N1
N2
N3
N1N2N3
142
ODE Model
  • Model with three generations
  • dN1/dt a(t) - b1 IN1 1
  • dN2/dt 2 b1 IN1 1 - b2 IN2 1
  • dN3/dt 2 b2 IN2 1 - g IN3 1

143
Solution to the ODEs
without aging
with aging
144
4 Generations..
N4
N1 N2 N3 N4
N1
N3
N2
145
More to Come
  • Image Processing
  • Experiments
  • Hypothesis Testing
  • Statistical Algorithms
  • Narrowed down two possible hypotheses
  • Most likely, the combinatorial model is incorrect
  • Physical model (Protein gradient vs. Prostheses)

146
Stochastic Aggregate Discrete Model
  • To introduce stochastic effects in our
    simulations, we developed a stochastic aggregate
    model based on a Finite/Hybrid Automata System
  • The simulation proceeds through a sequence of
    transitions, which increment or decrement the
    number of Stem Cells (Ns) or Committed
    Progenitors (Np)
  • The simulation was carried out with any Hybrid
    System Simulation tool, e.g.
  • LambdaSHIFT (Simsek, UC Berkeley, 2000)
  • Charon, (Alur et al, Upenn, 2000)

147
Stem Cell Finite/HybridState Automata
Asymmetric Division
Ns no. of Stem Cells Np no. of Progenitor
Cells
Np Np 1
Ns Ns -1
Symmetric Division 2S
die
quiescent
Ns Ns 1
Np Np 2 Ns Ns -1
SymmetricDivision 2P
148
SpatialSim Tool
  • The SpatialSim interface allows you to create
    specialized Stem Cell population simulations (2D)

149
Spatial Simulation Description (contd...)
  • 2D/3D Spatial Grid (3D grid visualization in
    development)
  • Local Rules (e.g)
  • apoptosis for Stem Cells

as / (1 Stem Cell Neighbors)
150
Spatial Simulation Description (contd...)
  • Other local rules
  • Symmetric subdivision
  • Asymmetric subdivision
  • Migration
  • Differentiation

151
Spatial SimulationGillespie-like Engine
  • The simulation engine achieves its efficiency by
    adopting a standard Poisson process assumption
    regarding the probability of concurrent events.
  • At each simulation step a single cell is randomly
    chosen and made evolve
  • This is similar to the Gillespie simulations of
    Chemical reactions

152
People
  • Marco Antoniotti
  • Sr. Res. Scientist (CS, Courant)
  • Simulation System//Simpathica
  • Archisman Rudra
  • Sr. Res. Scientist (CS, Courant)
  • Genome Grammar//Copy Number Analysis
  • Raoul Daruwala
  • Sr. Res. Scientist (CS, Courant)
  • Copy Number Analysis// Learning
  • Salvatore Paxia
  • Sr. Res. Scientist (CS, Courant)
  • Software Environment//Valis
  • Vera Cherepinsky
  • Sr. Res. Scientist (CS, Courant)
  • Software Environment//Valis
  • Gilad Lerman
  • Sr. Res. Scientist (Mathematics, Courant)
  • Multi-strip Algorithms// Normalization
  • Paolo Barbano
  • Saurabh Sinha
  • Postdoc (Courant Rockefeller)
  • Detecting CIS elements
  • Marc Rejali
  • Sr. Res. Scientist (CS, Courant)
  • Microarray Data Analysis//MAD
  • Nadia Ugel
  • Jr. Res. Scientist (CS, Courant)
  • Simpathica//Stem Cell Models
  • Marina Spivak
  • Jr. Res. Scientist (Biology CS, Courant)
  • Simpathica//Stem Cell Models
  • Joe McQuown
  • Jr. Res. Scientist (Stat, NYU)
  • Statistical Analysis
  • Graduate Students
  • Joey Zhou (Biology, NYU)
  • Bing Sun (Computer Science, NYU)
  • Jerry Huang (Biology, NYU)

153
Visitors Collaborators
  • VISITORS
  • Alberto Policriti
  • Computer Science,
  • University of Udine, Italy
  • Pasquale Cainiello
  • Computer Science,
  • University of LAquilla, Italy
  • Haim Wolfson
  • Computer Science,
  • Tel Aviv University, Israel
  • Chris Wiggins
  • Physics Applied Mathematics
  • Columbia University, USA
  • Franz Winkler
  • Mathematics
  • Johann Kepler University, Austria
  • COLLABORATORS
  • Mike Wigler, Rob Lucito Yuri Lazebnik
  • Cold Spring Harbor Lab
  • Misha Gromov Ale Carbone
  • IHES Courant
  • Amir Pnueli
  • Minerva Center Courant
  • Steve Burakoff
  • Skirball Institute
  • Harel Weinstein, Ravi Iyengar Bob Desnick
  • Mt Sinai School of Medicine
  • Sanjoy Mitter Dimitri Beretskas
  • MIT
  • Charles Cantor Jim Collins
  • Boston Univ
  • Mike Seoul
  • Bioarrays
  • VISITORS
  • Frank Park
  • Control Theory
  • University of Seoul, S. Korea
  • Naomi Silver
  • Computer Science
  • Marco Isopi
  • Applied Mathematics
  • Italy
  • Ilya Nemenman
  • Physics Neurosicience
  • ITP, California
  • David Harel
  • Computer Science
  • Weizmann Institute, Israel
  • Carla Piazza
  • Computer Science Mathematics
  • Universita Ca Foscari di Venezia,

154
Blakes Newton
  • Newton says Doubt
  • Aye thats the way to make all Nature out,
  • Doubt Doubt dont believe without experiment
  • --William Blake,
  • On the Virginity of the Virgin Mary Johanna
    Southcott,
  • (1757-1827)

155
The End
  • http//www.cs.nyu.edu/mishra
  • http//bioinformatics.cat.nyu.edu
  • Valis, Gene Grammar, NYU MAD, Cell Simulation,

156
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)

157
Optical Mapping
158
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.
159
Optical MappingInterplay between Biology and
Computation
160
Y
  • From a genes point of view, reshuffling is a
    great restorative
  • The Y, in its solitary state disapproves of such
    laxity. Apart from small parts near each tip
    which line up with a shared section of the X, it
    stands aloof from the great DNA swap. Its genes,
    such as they are, remain in purdah as the
    generations succeed. As a result, each Y is a
    genetic republic, insulated from the outside
    world. Like most closed societies it becomes both
    selfish and wasteful. Every lineage evolves an
    identity of its own which, quite often, collapses
    under the weight of its own inborn weaknesses.
  • Celibacy has ruined mans chromosome.
  • Steve Jones, Y The descent of Men, 2002.

161
Mapping the DAZ locus on Y Chromosome
162
Gentig MapDeinococcus radiodurans
Nhe I map of D.radiodurans generated by Gentig
163
E. coli Shotgun Map
164
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

165
P. Falciparum c14 Alignment
166
HaplotypingOutput of the RFLP Phasing Algorithm
167
Array Mapping
168
Measuring distances
  • A one dimensional Buffons needle problem.
  • Take two points on a line, and drop unit-length
    needles of some color.
  • The probability that the two points will have
    different colors monotonically increases with the
    distance between these two points
  • as distance increases from 0 to 1
  • attains a fixed value for all distances konger
    than 1.
  • One can generalize by considering
  • More than two pointsP points.
  • Dropping a small set of bichromatic needles

p
p
p
Distance ¼ 3/6 0.5
169
The Experiments
cX coverage subsample
cX coverage subsample
  • Probes are points
  • BACs are needles
  • Hybridization on an array simulates dropping the
    bichromatic needles

M
High Coverage BAC Library
cX coverage subsample
cX coverage subsample
170
Final Estimator
171
Given Inferred Probe Positions
172
Copy Number
173
Amplifications Deletions
174
ROMA.Tumor Vs. Normal
  • Copy number can be measured by computing the fold
    changes
  • Yellow Copy number unchanged
  • Red Amplification (More tumor material than
    normal)
  • Green Deletion (Less tumor material than normal)

175
BglII Representation (3)
176
Copy Number Fluctuation
177
Detecting Amplifications Deletions
178
VALIS
179
Valis
180
Valis Architecture
181
Valis Screenshot
182
NYU MAD
183
Nitrogen Pathway
184
NYU MAD
185
Data Analysis in NYU MAD
186
Shrinkage Estimators
187
JSEJames Stein Estimator
188
Simulation
189
ROC Curve
190
False Positives and Negatives
191
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