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Probabilistic BooleanNetworks as Models of
Gene RegulatoryNetworks:
Inference, Simulation, Intervention
Ilya ShmulevichUniversity of Texas
M. D. Anderson Cancer Center
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Outline of the PresentationBiological motivation
l Why study genetic networks?
Requirements of our modelsl What questions do we want to answer?
Boolean Formalism
Biological Implications
Probabilistic Boolean Networks
l Dynamics, Uncertainty, Graphical Models,Influence and Sensitivity of Genes, Perturbation,
Intervention, Sensitivity Analysis
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Biological Motivation
Were in the era of holistic biology.Massive amounts of biological data
await interpretation:
l this calls for formal modeling andcomputational methods;
l it opens up a window on dynamical andfunctional characteristics (physiology) of anorganism and disease progression.
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Biological Motivation
Genes are not independent.
They regulate each other and act collectively.This collective behavior can be observedusing microarrays.
Interest is shifting to temporal, genome-wideexpression profiling.l Such studies benefit from microarray technology.
l e.g. clustering, PCA, multidimensional scaling,network inference.
The interrelationships among genes
constitute gene regulatory networks.
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Genetic Network Models: Goals
Must incorporate rule-based dependencies betweengenesl Rule-based dependencies may constitute important
biological information.Must allow to systematically study global networkdynamicsl In particular, individual gene effects on long-run network
behavior.
Must be able to cope with uncertaintyl Small sample size, noisy measurements, robustness
Must permit quantification of the relative influenceand sensitivity of genes in their interactions with othergenesl This allows us to focus on individual (groups of) genes.
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Boolean FormalismStudies give rise to qualitative phenomena,as observed by experimentalists.
Studied systems exhibit multiple steadystates and switchlike transitions betweenthem.
It is experimentally shown that such systemsare robust to exact values of kineticparameters of individual reactions.
For practical approximation, gene regulatorynetworks have been treated with a Booleanformalism (i.e. ON/OFF).
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Boolean Formalism
Boolean idealization enormously
simplifies the modeling task.We want to study the collectiveregulatory behavior without specificquantitative details.
Boolean networks qualitatively capture
typical genetic behavior.
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Example
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Basic Structure of
Boolean Networks
1 means active/expressed0 means inactive/unexpressedA B
X
Boolean function
A B X
0 0 1
0 1 1
1 0 0
1 1 1
In this example, two genes (A and B) regulate gene X. In
principle, any number of input genes are possible.
Positive/negative feedback is also common (and necessaryfor homeostasis).
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Dynamics of Boolean
NetworksA B C D E F Time
0 1 1 0 1 0
1
A
1
B
0
C
1
D
1
E
0
F
At a given time point, all the genes form a genome-wide
gene activity pattern (GAP) (binary string of length n ).
Consider the state space formed by all possible GAPs.
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State Space of Boolean Networks
Similar GAPs lie closetogether.
There is an inherentdirectionality in the statespace.
Some states are
attractors (orlimit-cycleattractors). The systemmay alternate betweenseveral attractors.
Other states aretransient. Picture generated using the program DDLab.
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Implications for biology[see Huang, J. Mol. Med., 77, 469-480, 1999 formore details on the next 5 slides]
Equate cellular states with attractors.
Many different stimuli can lead to the same cellularstate (differentiation, growth, apoptosis). Thus, in realcells, these states correspond to attractors.
l e.g. radiation, chemotherapy.These attractor states are stable under minimalperturbations (this corresponds to flipping some bitsin the GAP).l most perturbations cause the network to flow back to the
attractor.
l some genes are more important (master genes) andchanging their activation can cause the system to transitionto a different attractor.
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Implications for biology
This stability is physiologically important itallows the cell to maintain its functional statewithin the tissue even under perturbations.
Nevertheless, cells do switch states, e.g. fromquiescence to growth, usually when certaingenes are affected by extracellular signals.
The cell translates such signals into specificalterations of genes/proteins.l cell surface receptors are wired to master
switches and are good targets for manipulation.
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Implications for biology
Hysteresis: a change in the systems statecaused by a stimulus is not changed backafter the stimulus is withdrawn.l Network simulations support this kind of
memory.
l It may also account for the fact that adaptivechanges are often preserved through many cell
division generations.l Stability and hysteresis could explain inheritance
of gene expressions (without physical fixation ofinformation in DNA).
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Tumorigenesis
Disturbance of the balance betweenattractors could be caused by mutationsaffecting the wiring or activation of importantgenes.l for example, stabilizing the growth state could lead
to tumorigenesis.
l such mutations change the size of the basins of
attraction.l since the state space is finite, an increase of one
basin of attraction leads to a decrease of another,
say, differentiation.
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Drug discoveryMost research has focused on the linear
paradigm.l manipulation of individual molecular targets
Robustness of attractor states explains why
single-gene perturbations have had littlesuccess on the macroscopic level.
Because of hysteresis, the off genes might
not be good targets for reversing pathologicaleffects.
We must rethink the functions of genes: to
regulate the dynamics of attractors.
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Drug discovery
The goal should be to push a tumor cellout of the growth attractor and into
apoptosis or differentiation attractor.
l to accomplish this, we have to intervene
with specific lever points. How to identify
them?
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Probabilistic Boolean Networks
(PBN)
Share the appealing rule-based properties ofBoolean networks.
Robust in the face of uncertainty.Dynamic behavior can be studied in thecontext of Markov Chains.
l Boolean networks are just special cases.Close relationship to Bayesian networksl Explicitly represent probabilistic relationships
between genes.Allows quantification of influence of genes onother genes.
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Boolean networks are
inherently deterministic
Conceptually, the regularity ofgenetic function and
interaction is not due to hard-
wired logical rules, but rather
to the intrinsic self-organizingstability of the dynamical
system.
Additionally, we may want to
model an open system with
inputs (stimuli) that affect the
dynamics of the network.
From an empirical viewpoint,the assumption of only one
logical rule per gene may
lead to incorrect conclusions
when inferring these rulesfrom gene expression
measurements, as the latter
are typically noisy and the
number of samples is small
relative to the number of
parameters to be inferred.
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Basic structure of PBNs
x'i
f1(i)
f2(i)
fl(i)(i)
x1
x2
x3
xn
c1(i )
c2(i )
cl(i)(i)
If we have several good
competing predictors(functions) for a given gene
and each one has
determinative power,
dont put all our faith in oneof them!
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Dynamics
Dynamics of PBNs can bestudied using Markov Chaintheory. From the Boolean
functions, we can computel transition probabilities
l stationary distribution
l steady-state distribution (if itexists)
We can ask the question:
In the long run, what is theprobability that some givengene(s) will be ON/OFF?
000111
110
101
100
011
010
001
1
1
1
1
P4
P3
P2
P1
P2+P4
P1+P
3
P2+P4
P1+P31
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Random Gene Perturbations
Genes can sometimes change valuewith a small probabilityp.
l The genome is not a closed system genes can be activated/inhibited due to
mutagens, heat stress, etc.
{ }
{ } [ ] pEii
n
===
gg
g
1Pr
1,0on vectorPerturbati
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Random Gene Perturbations
If no genes are perturbed, the standardnetwork transition function will be used.
Observation:
l Forp > 0, the Markov chain corresponding
to the PBN is ergodic.l Thus, the steady-state distribution exists.
l Convergence partially depends onp.
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Transition Probabilities
Uncertainty: Relationship to
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Uncertainty: Relationship to
Bayesian networks
Bayesian networks are graphical models thatrepresent probabilistic relationships between
variables.l They explicitly represent dependencies and
independencies between variables.
l They specify a probability distribution.
l Marriage between machine learning (rule-basedsystems) and uncertainty in AI.
l Naturally allow to select a model, from a set of
competing models, that best explains theexpression data.
PBN d B i N t k
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PBNs and Bayesian Networks
Bayesian networks are inherently static,although dynamic generalizations have beenproposed. However, the process of learning
the model structure & parameters isintractable (NP-hard).
Bayesian networks are not rule-based.
PBNs retain the attractive properties ofBayesian networks (e.g. probabilisticdependencies, model selection), but are rule-
based and inherently dynamic.The basic building blocks of BayesianNetworks (conditional probabilities) can be
obtained from PBNs.
I fl f
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Influence of genes
Some genes are more equal than others (indetermining the value of a target gene)
In a Boolean function, some variables have
greater determinative power on the output.Influence is defined in terms of the partialderivative of the Boolean function and the
underlying joint probability distribution of theinputs (efficient spectral methods exist)
PBNs naturally allow us to computeinfluences between (sets of) genesl genes with a high influence would make
potentially good targets for intervention.
Example
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Example
( ) 321321 xxxxxxf +=
Influence
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Influence
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Influence and Sensitivity
We can easily define the influence of agene on another (set of) gene(s), in the
PBN framework.
We can also define the sensitivityof agene (definition omitted here).
l Biologically, this represents the stability, orin some sense, the autonomy of a gene.
L t I fl
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Long-term Influence
0 10 20 30 40 50 60 70 80 90 1000.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
time-step
I2(x
1)
I2(x
2)
I2(x
3)
Intervention
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Intervention
One of the key goals of PBN modeling is thedetermination of possible intervention targets(genes) such that the network can be
persuaded to transition into a desired stateor set of states.
Clearly, perturbation of certain genes is more
likely to achieve the desired result than that ofsome other genes.
Our goal, then, is to discover which genes are
the best potential lever points in the senseof having the greatest possible impact ondesired network behavior.
Example
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Example
000111
110
101
100
011
010
001
1
1
1
1
P4
P3
P2
P1
P2+P4
P1+P
3
P2+P4
P1+P31
Clearly, the choice
in this simple
example should be
genex1.
Intervention
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Intervention
The problem of intervention is posed as:reaching a desired state as early as
possible.
We use first passage times
Same Example as Before
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Same Example as Before
0 2 4 6 8 10 12 14 16 18 200
0.05
0.1
0.15
0.2
0.25
K0
(011)
(101)
(110)
There are several
possibilities: find the gene
that
minimizes the mean first
passage time
maximizes the
probability of reaching a
particular state before a
certain fixed time
minimizes the timeneeded to reach a certain
state with a given fixed
probability.
Sensitivity of Stationary
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Distributions to Gene
Perturbations
What is the effect of perturbations on
long-term network behavior?
Similar problems have been addressed
in perturbation theory of stochasticmatrices.
Using recent results by Cho & Meyer(2000), we can show
Sensitivity Result
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Sensitivity Result
One important implication is that if a particular state of a PBN can be
easily reached from other states, meaning that the mean first passage
times are small, then its steady-state probability will be relatively
unaffected by perturbations. Such sets of states, if we hypothesize themto correspond to some functional cellular states, are thus relatively
insensitive to random gene perturbations.
C t k i
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Current work in
progressRobust inference of PBNs from data.
Designing small sub-networks from data.
Steady-state analysis using MCMC-type
methods.l Diagnosing convergence, establishing a priori
bounds on convergence using the structure of the
network.
Manipulating network structure to alter long-
term behavior in a desired way.
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39Subnetwork generated by Dr. Ronaldo Hashimoto
C t k i
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Current work in
progressRobust inference of PBNs from data.
Designing small sub-networks from data.
Steady-state analysis using MCMC-type
methods.l Diagnosing convergence, establishing a priori
bounds on convergence using the structure of the
network.Manipulating network structure to alter long-
term behavior in a desired way.
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