probabilistic boolean networks

8/4/2019 Probabilistic Boolean Networks

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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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probabilistic boolean networks

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