Remembrance of

1
Remembrance of Experiments Past
2
Bud Mishra

• Professor of Computer Science, Mathematics and
  Cell Biology
• Courant Institute, NYU School of Medicine, Tata
  Institute of Fundamental Research, and Mt. Sinai
  School of Medicine

3
Ludwig Joseph Johann Wittgenstein
4
When Wittgenstein told Russell that he wanted to
leave Cambridge and go to Norway, Russell tried
to dissuade him I said it would be dark, and he
said he hated daylight. I said it would be
lonely, and he said he prostituted his mind
talking to intelligent people. I said he was mad,
and he said God preserve him from sanity. (God
certainly will.)
5
Wittgenstein Brown Book

• Augustine, in describing his learning of
  language, says that he was taught to speak by
  learning the names of things
• Suppose a man describes a game of chess, without
  mentioning the existence and operations of the
  pawns. His description of the game as a natural
  phenomenon will be incomplete. On the other hand,
  we may say that he has completely described a
  simpler game.

6
An Augustinian Game

• Four Characters Analyzed using Mathematical Logic
• Can this be applied to Time-Series
  Gene-Expression Data?

7
Female, Sings, Overweight
X1 X2 X3 X4
Male, Talks, Thin
Male, Sings, Overweight
Female, Sings, Underweight
8
X4
X3
X2
X1
X3
X1
X4
X2
9
X4
X3
X2
X1
X3
X1
X4
X2
10
X4
X3
X2
X1
X3
X1
X4
X2
11
X2 X4
X3 X4
X1 X2
Finale
X3
X1
12
X2 X4
O, FLS
X3 X4
X1 X2
Finale
X3
O, FLS
O, FLS
O
O, FLS
X1
O, FLS
13
X2 X4
O, FLS O UFLS
X3 X4
X1 X2
Finale
X3
O, FLS O UFLS
O, FLS
O
O, FLS
X1
O, FLS O UFLS
14
X2 X4
O, FLS O UFLS
X3 X4
X1 X2
Finale
X3
O, FLS O UFLS
O, FLS O UFLS
O
O, FLS
X1
O, FLS O UFLS
15
It aint over til the fat lady sings
X2 X4
O, FLS O UFLS
X3 X4
X1 X2
Finale
X3
O, FLS O UFLS
O, FLS O UFLS A O UFLS
O
O, FLS
X1
O, FLS O UFLS
16
Augustine

• The good Christian should beware of
  mathematicians and all those who make empty
  prophecies. The danger already exists that
  mathematicians have made a covenant with the
  devil to darken the spirit and confine man in the
  bonds of Hell.

17
A Picture
Biological System Part-List Functional
Relations
Measurements
Regulatory, Metabolic Signaling Relations
Recomputation
Redescription
Descriptive Relation
Numerical Traces
KRIPKE MODELS
Invariants
18
Typical Analysis

• Time-Course data set
• Example
• Fibroblast response to Serum
• Iyer et al, Science (1999)
• Cluster by global expression pattern
• Manually look for genes of interest

19
GOALIEGene Ontology Algorithmic Logic for
Information Extraction

• Assign biological vocabulary to expression data
• Piece together into Kripke-structure model
• Support query, inference, and comparative
  assessment tasks
• Present descriptive summaries

20
Examples
21
Reconstructing Temporal Invariants

• GOALIE aims to reconstruct temporal invariants
  from time course experiments
• Several illustrations follow

22
Example Datasets

• Spellman cell cycle dataset (4 cycles alpha
  factor, Cdc15, elutration, Cdc28)
• Tu et al. dynamic organization of cell cycle (3
  cycles of 12 time points each)
• Results show significant potential to recover
  common trends in cell cycle control
• 18 functions common to all Spellman cycles
• 67 functions common to all Tu cycles
• 16 functions common to all

23
Invariants
G0
G1(I)
M
G1(II)
S
G2
24
Example Gantt Chart
25
Yeast-Cell Cycle DataSpellman et al.
26
Chasing Time
27
Reconstructing Temporal Models

• Cluster time windows to compress gene expression
  values but preserve information about category
  membership
• Clusters are modeled using Gaussian/vMF
  distributions category memberships are modeled
  multinomial
• Stochastic approximation algorithm for
  iteratively re-assigning genes to clusters

28
Host-Pathogen Interaction

• Pre-Apoptosis 6 time points data analysis
• Six time-point data at 2h, 4h, 6h, 8h, 12h, 24h
• Kidney cells treated with SEB (anthrax)
• Control untreated cells
• Data from Jett-Lab (Walter-Reed)

29
Hypothesized pathway
30
Clusters by P.C., TJU
31
GOALIE GO Algorithmic Logic for Invariant
Extraction
Clusters connection treeEach level a window
Micro-array accessions
GO categories
Cluster Information
Connection information
Clusters information
32
GOALIE GO Algorithmic Logic for Invariant
Extraction
GO categories describing source cluster but not
destination
GO categoriesdescribing destination cluster
but not source
GO categories shared with destination cluster
GO categories describing genes in source cluster
33
GOALIE SEB Analysis Preliminary Results

• Time Course Window 1 to Time Course Window 2
  Connection 19 to 218.By inspecting the first
  cluster in the first window (Cluster19), we
  note that one of the connection to the cluster2
  in the second window (Cluster218) is labeled
  (among many others) by the GO categories
  circulation (GO0008015), and by the category
  negative regulation of heart rate (GO0045822).
  This represents a constant set of biological
  processes shared by this cluster chain,
  traversing Cluster 317, to Cluster 413.
• Time Course Window 1 to Time Course Window 2
  Connection 19 to 26.The connection between
  Cluster 19 and Cluster 26 is interesting
  because it shows how the category regulation of
  lymphocyte proliferation (GO0050670) becomes
  activated in the next time-window (Cluster 26),
  while the categories antigen presentation and
  antigen processing became inactive. This should
  indicate that some of the genes in the clusters
  start a response to the pathogen in the second
  time point.
• Goalie movies\GOALIEshort.avi

34
Framework Outline

• Language
• Ontology
• Controlled Vocabulary
• Logic Models
• Kripke Structure
• Temporal Logic
• Model Checking
• Model Building
• Model Building Hidden Kripke Models (HKM)
• Information Bottleneck
• Invariants Redescriptions
• Labeling with Propositions
• Statistical Significance

• Examples
• Yeast Cell Cycle
• Host-Pathogen Interaction
• Life Cycle of a Parasite
• Cancer Initiation and Progression
• Implementation

35
Language Ontology
36
The Gene Ontology (GO) Consortium

• Ashburner et al. Nature Genetics 25 25-29.
  http//www.geneontology.org
• GO provides three structured networks of defined
  terms to describe gene product attributes
• Molecular Function Ontology (7304 terms as of
  April 5, 2004) the tasks performed by
  individual gene products examples are
  carbohydrate binding and ATPase activity
• Biological Process Ontology (8517 terms) broad
  biological goals, such as mitosis or purine
  metabolism, that are accomplished by ordered
  assemblies of molecular functions
• Cellular Component Ontology (1394 terms)
  subcellular structures, locations, and
  macromolecular complexes examples include
  nucleus, telomere, and origin recognition complex

37
Gene Ontology

• Systematic Hierarchical

38

• Note that GOALIE is not intimately tied to any
  particular ontology. It can be customized to work
  with other ontologies e.g., MeSH, UMLS, MetaCYC,
  Reactome
• Thus GOALIE also provides a way to compare
  different ontologies

From the GO web site. The path back to each
ontology from a gene. We will call each term in
a path a split.
39
Redescription

• A redescription mining algorithm was used to
  relate concerted clusters of gene expression to
  membership in GO taxonomical categories. Prior
  work (e.g., host-pathogen interactions in plant)
  has shown how this algorithm identifies
  functionally coherent subsets of genes, yielding
  testable biological hypotheses.

40
Logic Model Checking
41
Task Verify Temporal Properties of a Reactive
System

• Formally encode the behavior of the system

42
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

43
Kripke Structure Example

• There is a 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

44
Task Verify Temporal Properties of a Reactive
System

• Formally encode the behavior of the system
• Formally encode the properties of interest

45
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)
• Temporal Logic
• Short hand for describing the way properties of
  the system change with time
• Time is implicit

46
Branching versus Linear Time

• 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
47
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

48
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

49
Some CTL Operators
AF g
EG g
EF g
AG g
50
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

51
Task Verify Temporal Properties of a Reactive
System

• Formally encode the behavior of the system
• Formally encode the properties of interest
• Automate the process of checking if the formal
  model of the system satisfies the formally
  encoded properties

52
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

53
Naïve CTL Model-Checker
54
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

55
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.

56
Story generation

• Temporal Logic formulae can be rendered in
  English.
• Temporal Logic formulae can be generated
  automatically (with care).
• Each formula can be tested against a set of
  datasets differences can then be noted.

57
Cell Cycle Story Generation Results (HTML
rendering)

• Report on "Test Experiment WT, 1 Mutant, 2
  Mutants.
• The results refer to the following datasets
• The first dataset is named "Experiment/Yeast
  Dataset WT".
• The second dataset is named "Experiment/Yeast
  Dataset Mut1".
• The third dataset is named "Experiment/Yeast
  Dataset mut2".

• CDH1 less than or equal to 1.0071783 will
  always hold until CDH1 activates CYCB, is true
  in the first dataset, is true in the second
  dataset, and is false in the third dataset.
• CDH1 represses CYCB implies CYCB is greater than
  or equal to 0.65, is false in the first
  dataset, is true in the second dataset, and is
  true in the third dataset.
• eventually, CDH1 is less than or equal to CYCB,
  is false in the first dataset, is true in the
  second dataset, and is true in the third
  dataset.

58
Model Building
59
Hidden Kripke Model

• Hidden Kripke Model
• Reconstruction via ontology based redescription
  of time-sliced clusters of time-course
  measurements (arrays)
• Information Bottle Neck Parsimony
• Example Kripke Models
• Spellmans Yeast Cell Cycle
• SEB host-pathogen data from WRAIR
• P. falciparum dataset Bozdech et al, 1(1)085
• Subset of Genome Module Map dataset Segal et al

60
Information Bottleneck

• The computational principle uses Information
  Bottleneck Theory
• Di the random variable samples are the rows in
  submatrix Di
• A sample corresponds to a gene expression vector
  of size (n - k).
• Using the GO biological process assignments for
  each gene, we posit a joint distribution P(DiO)
  where O is a second random variable whose sample
  space is the process ontology.
• Cluster Di into disjoint clusters (without using
  O)Effectively, identifying a third random
  variable Xi such that the mutual information
  between Di and Xi I(DiXi) is minimized.
• Further, relate the conditional distribution P(O
  Xi) with
• P(O Xi1), 1 i lt k, and with P(O
  Xi-1), 1 lt i k. In the information bottleneck
  framework, the mutual information terms I(O XiO
  Xi1) and I(O XiO Xi-1) must be
  simultaneously maximized.

61
Information Bottleneck

• To construct the Hidden Kripke Model, the
  clusters and cluster-edges must optimize the
  mutual information terms
• A variational approach with a Lagrange multiplier
  that captures the tradeoff between these
  objectives
• minimize
• I(DiXi) - b1 I(O Xi O Xi1) - b2 I(O XiO
  Xi-1)
• Notice that, conditional on Di, O is independent
  of Xi. This suggests an EM-style alternating
  algorithm
• First cluster each Di, identify connections
  across clusters in neighboring time points
• Use these connections to derive new constraints
  on clustering, and re-cluster.

62
GOALIE Data Flow
63
Invariants Redescriptions
64
Statistical Tests

• How do we associate a term (a GO category) to a
  cluster?
• Fisher Exact Test
• Used to determine whether or not nonrandom
  associations between two categorical variables
  exist
• Null hypothesis generally assumes that there is
  no association
• Actually a multivariate generalization of the
  hypergeometric test
• Results of the test are p-values
• Binomial Test
• Used for large data sets
• Results are approximated p-values
• Previous works
• GOminer, GOStats, CLASSIFI, GeneMAPP, FATIGO

65
Implementation of Fishers Exact Test

• Let there exist an m n observation matrix with
  entries aij
• sum both the columns and rows and calculate
• Add these sums to attain the total sum (first
  figure)
• Calculate the conditional probability of getting
  the actual matrix given the particular row and
  column sums (second figure)

66
Implementation Continued

• Once the conditional probability is found, all
  possible matrices of non-negative integers
  consistent with the row and column sums must be
  found
• For each new matrix, the associated conditional
  probability must be calculated using the previous
  formula, where the sum of these probabilities
  must be 1
• To compute the p-value of the test, the tables
  must then be ordered by some criterion that
  measures dependence
• Those tables that represent equal or greater
  deviation from independence than the observed
  table are the ones whose probabilities are added
  together to determine the p-value.
• This p-value is then compared to the original
  alpha-level to determine statistical significance

67
The Binomial Test and Drawbacks of the Fisher
Exact Test

• For large mn matrices, the Fisher Exact test
  becomes unwieldy and incredibly difficult to
  compute
• The binomial test provides a substitute for this
  test in the presence of large amounts of data
• The test measures whether the proportion of two
  categorical dependent variables significantly
  differs from a hypothesized proportion
• The result is only an approximate p-value, but
  requires less computational time

68
Controls over the False Discovery Rate

• In order to avoid a lot of spurious positives,
  the a-level needs to be lowered to account for
  the number of comparisons being performed.
• Corrections to avoid a large amount of type II
  errors
• The Bonferroni correction, a simple yet highly
  conservative approach
• The Benjamini-Hochberg Procedure

69
FDR Controls Continued

• Benjamini and Hochberg Correction
• This correction is becoming increasingly popular
  and provides just one alternative to the
  Bonferroni Correction
• It provides strong control over the rate of false
  discovery under positive regression dependency of
  the null hypothesis

70
The Benjamini-Hochberg Procedure

• Consider testing a set of hypotheses H1,.Hm
  based on corresponding p-values P1,,Pm
• Also, impose an ordering on the p-values such
  that P1 P2 . Pm and denote by H(i) the null
  hypothesis corresponding to P(i)
• Define the following multiple testing procedure
• Let k be the largest i for which P(i) lt (i/m)q
  where q is some predefined limit
• Then reject all H(i) i 1, 2, , k
• For independent test statistics and for any
  configuration of false null hypotheses, the above
  procedure controls the false discovery rate at q

71
GOALIE Architecture
The overall GOALIE architecture. Several Common
Lisp modules have been developed to take care of
bits and pieces of the system. Several libraries
were also used in the process. Core is ANSI,
User Interface is CAPI/Lispworks.
72
Other Genomic Data Analysis Tools
73
BiNGO

• Java based tool that works in conjunction with
  Cytoscape, a software platform for visualizing
  molecular interaction networks
• determines statistically overrepresented
  categories in a set of genes
• Allows for two statistical tests
• Hypergeometric test (nearly equivalent to the
  Fisher Exact Test)
• Binomial test
• Currently allows for only the most widely
  used/basic test corrections
• Bonferroni correction
• Benjamini Hochberg correction

74
BiNGO

• BiNGO produces a color-coded graph which
  visualizes the GO categories that were found
  significantly over-represented
• Size of the nodes is proportional to the number
  of genes in the test set which are annotated to
  that node
• The color of the node represents the strength of
  the p-value
• White not significantly over-represented
• All others are colored on a scale ranging from
  yellow to dark orange
• Dark orange represents p-levels that are 5 orders
  of magnitude smaller than the chosen significance
  level

75
Other Aspects of BiNGO

• Allows for both Standard and Custom Annotations
  and Ontologies
• Is able to provide both GO and GOSlim ontologies
• Graphs can be saved and manipulated

76
GoMiner

• A program package that calculates the enrichment
  or depletion of categories with genes that have
  changed expression
• Takes as imput two lists
• The total set on the array
• The subset that the user flags as interesting
• Displays the genes within the framework of the
  Gene Ontology hierarchy as a directed aclyclic
  graph and a tree structure

77
GoMiner

• Incorporates several external data resources
  including LocusLink, PubMed, MedMiner, GeneCards,
  the NCBIs Structure Database, and BioCarta
• Utilizes a two-sided Fisher Exact Test to
  determine the association between categories
• Null Hypothesis for each category, there is no
  difference between the proportion of flagged
  genes that fall into the category and the
  proportion of flagged genes that do not fall into
  the category
• Currently uses the Bonferroni Correction but are
  working on other approaches

78
CLASSIFI (Cluster Assignment for Biological
Inference)

• Data-mining tool that identifies significant
  co-clustering of genes with similar functional
  properties
• Uses a Hypergeometric Distribution Test to
  determine statistical significance
• Orders ontologies based on the p-values
  determined from the test
• Raw data must already be filtered, normalized,
  and clustered

79
CLASSIFI

• Does not work well on small data sets
• User must perform p-value corrections
• Authors suggest the Bonferroni Correction
• Generates 3 files
• A GO file that contains the results from the
  automated annotation
• An output file that contains all of the
  enumerated variables that were used in the
  hypergeometric test and the probability results
  from the calculation
• A Top file that lists the GO IDs that give the
  lowest p-value in each of the gene clusters

80
dChip

• Supports Go Ontology
• Allows time course data
• Clusters not connected through time
• Can filter data by gene annotations

81
STEMShort Time-series Expression Miner

• Limited to 8 time points (algorithm tailored
  specifically to small datasets)
• Clusters not connected through time
• Can compare experiments
• Uses Gene Ontology database

82
Bozdech et al. P. falciparum analysis with GOALIE

• A new, very preliminary, analysis with GOALIE
• CAVEAT the analysis was done in few hours
  without inputs from a biologist
• Datasets from
• "The Transcriptome of the Intraeythrocytic
  Developmental Cycle of Plasmodium Falciparum", Z.
  Bozdech, M. Llinas, B. L. Pulliam, E. D. Wong, J.
  Zhu, and J. L. DeRisi, PLOS Biology,
  1(1)085100.
• Dataset chosen for analysis was s003, which is
  described as the overview dataset.
• GO associations were downloaded from the GO site
• (http//www.geneontology.org/GO.current.annotation
  s.shtml)
• Annotations with GeneDB from Sanger Institute

83
Data preparation

• The P. falciparum dataset contains 48 time-points
• The analysis presented, arbitrarily chose to
  group the first 34 into windows of size 8 with
  two time-points overlapping
• Each time window was clustered separately using a
  k-means algorithm from the Genesis tool (Sturn et
  al., 2003-2005)

84
Examples Follow

• The examples show chains where many DNA and RNA
  manipulations processes appear
• The examples also show processes more
  mechanical in nature, like cell-cell adhesion

85
Cell-cell adhesion process

• The cell-cell adhesion process becomes active
  between the second and third time window and
  remains active until the last one
• The genes involved then start acting together
  with other involved in tRNA modification and drug
  susceptibility/resistance

86
tRNA modification

• tRNA modification processes become active
  between the second and the third time window

87
DNA processes

• In this last example we note how DNA
  (methylation and topological change) processes
  become active while drug susceptibility/resistance
  and cell-cycle related activities stop

88
Dana Scott Advice on Modal Logic

• We have to consider not only of the flow of time
  but also of alternative courses of events. No,
  come to think of it, that is not the answer
  either, for that only makes the individual
  concept t fatter but not more amusing. Or
  maybe it does.
• (Oh my, I see that much more thought and
  experimentation are needed to make the ideas into
  something useful. In any case I feel that a
  precise and general semantical framework is
  essential, and that is, as I have been trying to
  indicate, now available.)

89
Dana Scott Advice on Modal Logic

• Postscript (December, 1969)
• This paper was written very hastily in the latter
  part of May, 1968. The haste is apparent and the
  style intolerable I find it now very painful
  reading.
• Dana Scott, Advice on Modal Logic. In
  Philosophical Problems in Logic Some Recent
  Developments, K. Lambert (ed.), pp. 143173,
  Dordrecht Reidel, 1970.

90
Processes involved in Lumen Formation Breast
Cancer
Architecture of Mammary Tissue Brugge-Lab
(Harvard)
91
Epithelial Cell Morphogenesis
92
Genes under transcriptional regulation during
lumen morphogenesis
93
Hypothesis Generation withGOALIE Analysis

• What triggers the selective apoptosis of inner
  cells?
• One hypothesis is that loss of adhesion can
  induce epithelial cell apoptosis. If true, what
  is the key player linking the two processes?
• How were the different morphogenic changes
  coordinated?
• What are the metabolic processes being regulated
  during the Morphagenesis? How are they
  coordinated with the major events (polarity,
  growth arrest, selective cell death)?
• What is the difference between the cell adhesion
  program in 3D growth and 2D culture?

94
DEMO
95
(No Transcript)
96
Quine Epistemology Naturalized

• Philosophers have rightly despaired of
  translating everything into observational and
  logico-mathematical terms. Carnap and the other
  logical positivists of the Vienna Circle had
  already pressed the term metaphysics into
  pejorative use, as connoting meaninglessness and
  the term epistemology was next. Wittgenstein
  and his followers, mainly at Oxford, found a
  residual philosophical vocation in therapy in
  curing philosophers of the delusion that there
  were epistemological problems.

97
How to convert static network maps into dynamic
mathematical models How to predict protein
function ab initio How to build hierarchical
models across multiple scales of time space
How to reduce complex multi- dimensional models
to underlying principles
Glycolysis
SIMPATHICA
98
SimPathica System
99
Simpathica is a multi-functional system
100
Simpathica is a modular system
Canonical Form

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

101
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.

102
Simple Model
103
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).

104
Purine Metabolism
105
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
106
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
107
Final Model
108
Purine Metabolism
109
A Tragedy, a Scandal or a Challenge

• The lack of real contact between mathematics and
  biology is either a tragedy, a scandal or a
  challenge, it is hard to decide which.
• Gian-Carlo Rota, (1986, in Discrete thoughts)

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111
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Thanks to

• NYU Bioinformatics
• Group
• Anantharaman
• Antoniotti
• Daruwala
• Mishra
• Paxia

• Collaborators
• Bhardwaj
• Brugge
• Cordoza
• Cantor
• Demidov
• Dynlacht
• Fitch
• Gimzewski
• Gromov
• Hubbard

• Cheng
• Gill
• Heymann
• Ionita
• Kleinberg
• Lord
• Mysore
• Rudkevich
• Sun
• Waltman

• Jett
• Lazebnik
• Ostrer
• Pagano
• Parida
• Ramakrishnan
• Reed
• Sobel
• States
• Teitell
• Zhou

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The end