Boolean and Probabilistic Boolean Networks as Models of Genomic Regulation

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Boolean and Probabilistic Boolean Networks as
Models of Genomic Regulation

• Ivan Ivanov
• Department of Veterinary Physiology and
  Pharmacology,
• Genomic Signal Processing Lab
• Texas AM University
• gsp.tamu.edu/people/ivan.html

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The Central dogma in cell biology
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Model based scientific approach

• Mathematical models allow for a formal and
  unified description of physical phenomena

Experiment design
Mathematical Model
Data
Biology
Experiment
Inference
Any model that allows prediction could be
considered as a mathematical model
Prediction
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Goals

• Must incorporate rule-based dependencies between
  genes
• Rule-based dependencies may constitute important
  biological information
• Must allow to systematically study global network
  dynamics
• In particular, individual gene effects on
  long-run network behavior
• Must be able to cope with uncertainty
• Small sample size, noisy measurements, robustness
• Must permit quantification of the relative
  influence and sensitivity of genes in their
  interactions with other genes
• This allows us to focus on individual (groups of)
  genes

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Regulatory diagram for the activation of the
tumor-suppressor protein p53
Vogelstein, B., Lane, D. Levine, A. Surfing the
p53 network. Nature 408, 307-310 (2000)
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Challenges

• Biological systems function in exceedingly
  parallel, nonlinear and extraordinarily
  integrated fashion
• Presence of protein-DNA feedback loops (negative
  or positive)
• Availability and quality of data
• Model selection
• Fine/Continuous or Coarse/Discrete

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cDNA microarray
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• Given
• Genes communicate/interact via the proteins they
  encode
• Protein production (transcription and
  translation) is controlled by a multitude of
  biochemical reactions which are in turn
  influenced by many internal or external to the
  cell factors.
• Assumption
• Gene expression Xj of a particular gene i is a
  random function xj(t, w) of the cell internal and
  external environment.
• Goal
• A good mathematical model for the dynamical
  behavior of the genes

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Biochemical interactions network
Metabolic space
Metabolite 1
Metabolite 2
Microarrays
Protein space
Biochemical model
Protein 2
Protein 4
Complex 3-4
Protein 1
Protein 3
Gene 4
Gene 2
Biological phenomena
Relationship
Gene 3
Gene 1
Gene space
Variable
From Brazhnik et. Al. Gene networks how to put
the function in genomics, TRENDS in
Biotechnology, 20 (11), 2002
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Discrete Models

• Faithful representation of upregulated/expressed
  and downregulated/repressed gene activity
• Filter the noise in data
• Dynamical behavior can be clearly related to some
  underlying biological phenomena
• Fine details like protein concentrations or
  kinetics of reactions cannot be captured

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Gene Networks Inference
Biochemical interaction network
Projection to the gene space
Gene 4
Gene 2
Gene 3
Gene Regulatory Network Model
Gene 1
Gene space
From Brazhnik et. Al. Gene networks how to put
the function in genomics, TRENDS in
Biotechnology, 20 (11), 2002
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Considerations

• Can it explain all the biological process?
• NO
• (Definition of context adding back the other
  layers)
• Can we understand better the physical phenomena?
• YES
• (Kauffman, attractorsphenotype, logical rules,
  etc)
• It is an useful model?
• YES
• (we can make some good predictions)

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Example of Cell Cycle Regulation
Logic diagram AND gate outputs
cdk2 p21/WAF1 is the input for a NOT gate NAND
gate outputs Rb
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Discussion

• How to derive a discrete representation of the
  biological process in a consistent way ??
• How to define the quality of a mathematical model
  to describe the biological model ??
• Obs here I dont use the word data. What is
  important is the model, and the data is a way to
  estimate its parameters !

Data
Biology
Model
Experiments
Parameters estimation
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Boolean Network (BN)
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Model Boolean functions

• Activity of gene 1 (promoter) promotes the
  activation of gene 3, unless gene 2 is active
  (repressor).

Gene 1
?
Gene 3
Gene 2
G1 G2 Y(G1,G2)
0 0 0
0 1 0
1 0 1
1 1 0
A possible Boolean function to represent this
biological relationship
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Note

• The Boolean function model is for the biological
  model, NOT for the observed data !!!
• Each binary function mimics the biological
  behavior with some degree of fitness.
• The quality of this fitness can be measured via
  an error measure
• There is always an optimal binary function, that
  best fits the biological model.

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Inference of Boolean Functions

• Boolean relationship between genes can be
  estimated from microarray data.

Experiment 2
Experiment 3
Experiment 1
Experiment 4
Experiment 2
Experiment 3
Experiment 1
Experiment 4
Experiment 5
Experiment 5
Experiment 6
Experiment 6
Examples A B
C Experiment 1 0 0 1 Experiment 2
0 1 0 Experiment 3 1 1
0 Experiment 4 1 1 1 Experiment 5
1 1 1 Experiment 6 0 0 1
Gene A
Gene A
0
0
1
1
1
0
Gene B
Gene B
0
1
1
1
1
0
Gene C
Gene C
1
0
0
1
1
1
Gene D
Gene D
1
0
0
0
1
1
A
B
Boolean function fc for C A B C 0 0
1 0 1 0 1 0 X 1 1 1
fC
C
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Error measure for binary functions

• How good is this function ? to model the
  relationship between G1,G2 and G3 ?
• The quality of the function ? depends on the
  joint distribution of G1,G2 and G3
• In the same way, if the constant function is
  defined by ?0c

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Optimal Function

• Between all possible Boolean functions ?, one of
  them has the minimal error, as predictor of G3
  from G1 and G2. This function is called ?opt.
• ? ?opt ? ? ? for any other Boolean function ?
• If G1 and G2 are good predictors of G3, then the
  relationship between them will be captured by
  ?opt and ? ?opt will be small.
• The optimal constant predictor is called ?0-opt.
  (there are only 2 possible constant predictors 0
  and 1).
• If G3 is almost constant, then ??0-opt will be
  small.

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Coefficient of determination

• The Coefficient of Determination (CoD) of the
  pair of genes G1 and G2 as predictors of the gene
  G3 is given by the relative improvement in the
  prediction when using the optimal predictor ?opt
  over the optimal constant predictor ?0-opt.
• The CoD depends ONLY on the joint distribution of
  G1,G2 and G3.

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Probabilistic Boolean Network (PBN)
PBN (BN1, , BNk, p1, , pk, p, q) 0 lt p
lt 1 - probability of switching context 0 lt
pi lt 1 probability for BNi being used 0 lt q
lt 1 probability of gene flipping Context
Which BN is used for the next
transition the regime in which
the cell operates/functions Gene
flipping mutation rate
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Basic Building Block of a PBN
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p1
q p2
x1 x2 x3 f1 f2 f3
0 0 0 1 0 0
0 0 1 0 0 0
0 1 0 1 1 1
0 1 1 0 1 1
1 0 0 1 0 0
1 0 1 0 0 0
1 1 0 1 0 0
1 1 1 0 0 0
x1 x2 x3 f1 f2 f3
0 0 0 0 0 1
0 0 1 0 0 1
0 1 0 1 0 0
0 1 1 0 1 0
1 0 0 1 0 0
1 0 1 0 0 0
1 1 0 0 1 0
1 1 1 0 0 1
p
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Context Switching
X2
X2
X3
X3
p
X1
X1
p1 q
p2
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Attractors in PBNs

• Attractors in the Boolean Networks should
  correspond to cellular types (Kauffman)

• PBNs are formed by a family of Boolean Networks
• Steady-state analysis of the PBN may be
  meaningful for classification based on
  gene-expression data
• Relationships between steady-state distribution
  and the attractors of the Boolean Networks allow
  structural analysis of the Network

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Dynamics of PBNs with perturbations

• Perturbations are added to the model to assure
  the existence of a steady-state distribution
• Perturbations move the system from the actual
  state to a close state
• The system behaves like a deterministic Boolean
  Network until a perturbation or change of
  function occurs

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Dynamics of PBNs with perturbations
The same Boolean Network being used
Time
In a basin
In the Attractor
Change of function or perturbation
Next change of function or perturbation
The system reaches the Attractor
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Steady-state analysis

• In the long run, the system is expected to stay
  in the attractors of the Boolean Networks

From the same initial point the system can
transition to two different regions (attractors)
depending on the Boolean Function being used
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Modeling of real genetic regulatory systems using
PBNs

• PBNs with p 1 and q 0 are equivalent to
  Dynamic Bayesian Networks (DBNs) Lähdesmäki, H.,
  Relationships between probabilistic Boolean
  networks and dynamic Bayesian networks as models
  of gene regulatory networks, In Workshop on
  Discrete Models for Genetic Regulatory Networks,
  Texas AM University, College Station, TX
  November 5-6, 2003.
• Bayesian optimization of connectivity X. Zhou,
  X. Wang, R. Pal, I. Ivanov, Michael Bittner and
  E. Dougherty, A Bayesian Connectivity-based
  Approach to Constructing PGRs, Bioinformatics
  V.20 no 17, pp 2918-2927, 2004.

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Applications of PBNs

• Problem (A) Study a long-run characteristics
  of a given dynamical system
• Inverse Problem (B) Generate a BN/PBN with a
  prescribed dynamical behavior
• Control policies for reaching a desirable steady
  state distribution

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Melanoma Application

• Microarray data
• 31 malignant melanoma samples
• 6971 unique genes on the array
• 7 genes of interest WNT5A, pirin, S100P,
  RET1, MART1, HADHB, STC2
• Binarization of the gene expression profiles
• 18 distinct data states
• Suboptimal PGRN generation (using MSE distance)
• 10 attractor sets selected according to the
  data frequency
• 2 lt size of each attractor set lt 5
• 100 BNs generated for each attractor set
• 10 BNs selected to form PGRN, p q .001

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Mathematics is biologys next microscope, only
better biology is mathematics next physics,
onlybetter. - J.E. Cohen Mathematics is
biologys next microscope, only better biology
is mathematics next physics, only better, PLOS
Biology 2 (2004) No.12.
CAN BIOLOGY LEAD TO NEW THEOREMS? B. Sturmfels,
Department of Mathematics, Univ. of California,
Berkeley, CA 94720, USA
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Estimation of the CoD for G1,G2 and G3.
Microarray
Example of Ternary Expression Matrix
Exp1 Exp2 Exp3 Exp4 Exp5 Exp6 Exp7 Exp8
G1 1 1 0 1 1 1 1 1
G2 0 1 1 1 0 1 0 1
G3 0 1 0 1 0 1 1 0

Estimation of the optimal functions ?opt and
?0-opt for G1,G2 as predictors of G3
Estimated CoD for G1,G2 as predictors of G3
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Estimation of ??opt for G1,G2 and G3 from the
data
Ternary Expression Matrix for G1,G2 and G3
Exp1 Exp2 Exp3 Exp4 Exp5 Exp6 Exp7 Exp8
G1 1 1 0 1 1 1 1 1
G2 0 1 1 1 0 1 0 1
G3 0 1 0 1 0 1 0 0
Splitting of the matrix in Training and Test sets
TRAIN Exp1 Exp2 Exp3 Exp4
G1 1 1 0 1
G2 0 1 1 1
G3 0 1 0 1
TEST Exp5 Exp6 Exp7 Exp8
G1 1 1 1 1
G2 0 1 0 1
G3 0 1 0 0
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Estimation of ??opt for G1,G2 and G3 from the
data
More frequent value computed from data (X
denotes a non- observed configuration)
Generalization to fill non-observed configurations
TRAIN Exp1 Exp2 Exp3 Exp4
G1 1 1 0 1
G2 0 1 1 1
G3 0 1 0 1
Statistical Inference of the optimal function
?opt.
G1 G2 ?opt(G1,G2) ?opt(G1,G2)
0 0 X 0
0 1 0 0
1 0 0 0
1 1 1 1
TEST Exp5 Exp6 Exp7 Exp8
G1 1 1 1 1
G2 0 1 0 1
G3 0 1 0 0
Estimation of the error of ??opt from test set
1 mistake on 4 ? ??opt 0.25
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Estimation of ??0-opt for G1,G2 and G3 from the
data
Frequencies of possible values of G3 on train data
TEST Exp1 Exp2 Exp3 Exp4
G1 1 1 0 1
G2 0 1 1 1
G3 0 1 0 1
Statistical Inference of the optimal function
?0-opt.
G3 Frequency
0 2
1 2
?0-opt. 1 (use heuristic) (most frequent
observed value for G3)
TEST Exp5 Exp6 Exp7 Exp8
G1 1 1 1 1
G2 0 1 0 1
G3 0 1 0 0
Estimation of the error of ??opt from test set
3 mistakes on 4 ? ??0-opt 0.75
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Estimation of the CoD for G1,G2 and G3 from the
data
??0-opt 0.75
??opt 0.25
The error is reduced in a 66
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Estimation of the CoD for G1,G2 and G3.

• The previous process is repeated 1000 times, with
  different random splitting of the set in training
  and test sets.
• The estimated value for the CoD is the average of
  the 1000 values of ?.
• If we want to know the predictive power of other
  pair of genes, say G4,G5, over G3, we must repeat
  the whole process
• G1,G2 ? G3 ? ?312
• G4,G5 ? G3 ? ?345

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Result

• Determination of the predictive genetic network