U-curve Search for Biological States Characterization and Genetic - - PowerPoint PPT Presentation

U-curve Search for Biological States Characterization and Genetic Network Design

Marcelo Ris – Universidade de São Paulo – Instituto de Matemática e Estatística Junior Barrera – Universidade de São Paulo – Instituto de Matemática e Estatística Helena Brentani - Hospital do Câncer, Fundação Antônio Prudente

Outline

• Introduction
• Feature selection problem
• U-curve search algorithm
• Characterization of biological

states

• Genetic network design
• Application

• Introduction
• Feature selection problem
• U-curve search algorithm
• Characterization of biological

states

• Genetic network design
• Application

• Biological Problems
• P1. Biological states characterization
• P2. Genetic Network Design
• Gene expression data
• P1. States samples
• P2. Time-course samples
• Mathematical approach

– Feature Selection Problem – State of the art: heuristic optimizations – U-curve algorithm

• Introduction
• Feature selection problem
• U-curve search algorithm
• Characterization of biological

states

• Genetic network design
• Application

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• Problem

– find the subset A that optimizes the cost function – Ex: mean conditional entropy minimization (cost function) – Exponential

• Search Space

– Complete boolean lattice of order n – Each node represents a possible candidate A – Cost function: estimated for each node – Find the node with the minimum cost

Boolean Lattice of order 4 4-element chain is emphasized

• Heuristics: SFS, SFFS

– Incremental – Does not search all the candidates space – Could not obtain the “best” result

• Ex: 2 elements alone turns the result worse, but

together improves it a lot

• Introduction
• Feature selection problem
• U-curve search algorithm
• Characterization of biological

states

• Genetic network design
• Application

• U-curve property of Ê[H(Y|X)]

Ë[H(Y|X)] |A|

– Why ? – Estimation composed by:

• Real measure – decreases from H(Y) to the real value

E[H(Y|X)]

• Estimation error – increases as more attributes are

added to X

– For a fixed number of samples – For any chain of the search space – Ê[H(Y|X)] forms an U-curve

• Features of the algorithm

– Branch-and-Bound: go through the whole space without having to visit all the candidates – Stochastic

– Some definitions:

• U-cost Boolean Lattice
• Local minimum
• Exhausted minimum
• Global minimum

• Search space

characterized by:

– Upper Bound List – Lower Bound List

• An element is reachable if

there is a chain from an upper or lower list element

• At each step:

– Select with some probability a beginning list – Select an aleatory element from this list – Build a chain iteratively:

• Inserts to the chain an

aleatory reachable adjacent to the last

• ne
• Stop, when the cost
• f the last element is

greater than the last

• ne

0000 0100 0110 1110

10 7 9 6

Prune Procedure

• Additional Procedures

– Minimum exhausting

• Avoid more than one visit to the same candidate
• Using a stack

– Pruning elements from an element E

• Upper bound list – remove elements U’s that contain E, and

inserts elemets reachable from U that not contain E

• Lower bound list – remove elements L’s that are contained

in E, and inserts elemets reachable from L that is not contained in E

• Introduction
• Feature selection problem
• U-curve search algorithm
• Characterization of biological

states

• Genetic network design
• Application

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• Introduction
• Feature selection problem
• U-curve search algorithm
• Characterization of biological

states

• Genetic network design
• Application

• Dynamical Systems

– State: vector x – Transition function – x[t+1] = (x[t])

• Stochastic Process

– Stochastic transition function

• Next State – aleatory vector realization

– Ex: Markov Chain (X|Y , 0 )

• Time-discrete, finite-size vector, finite domain
• Aleatory state sequence

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Almost Deterministic

Expression (Gene 1) time Expression (Gene 2) time Expression (Gene 3) time

. . . . . . . . . . . . . . . . . .

Expression (Gene n) time

] 1 [ x ] 2 [ x ] 3 [ x ] 4 [ x ] 5 [ x ] 6 [ x ] 7 [ x

] 9 [ x

] 10 [ x ] 11 [ x ] 12 [ x ] 13 [ x ] 1 [ − m x ] [m x

.... Expression Measurement Techniques

Time-Course Gene Expression Data

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• Introduction
• Feature selection problem
• U-curve search algorithm
• Characterization of biological

states

• Genetic network design
• Application

• Application to design a estrogen

regulated network

– 21 target genes estrogen regulated – Time-course gene expression (Ed Liu et al.)

• MCF-7 cells treated with estrogen
• 16 microarray experiments in 24 hours:

– each hour: 8 first hours – each 2 hours: 16 last hours

– Quantization (Barrera et al.)

• 3 levels {-1, 0, 1}

– For each target gene

• 15x21 matrix

– 20 first columns – gene expressions on the first 15 experiments – Last column – gene target expression on the last 15 experiments

– The algorithm returns between the 20 genes the subset that best predict the target

THBS1 PGR CXCL12 STC2 SERP INE1 NRIP1 FZD8 CRABP2 BMP7 IL6ST REA EPHA4 CD7 JAK1 CTSD JDP1 CCNG2 ABCA3 GATA3 NMA IGFBP4

Thanks!!!