1 applications of symbolic logic to gene regulation systems department of computer science and...

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Applications of Symbolic Logic to Applications of Symbolic Logic to Gene Regulation SystemsGene Regulation Systems

Department of Computer Science and Information Department of Computer Science and Information Engineering of National Chung-Cheng UniversityEngineering of National Chung-Cheng University

Speaker : Chuang-Chieh Lin

Computation Theory Laboratory in National Chung-Cheng UniversityComputation Theory Laboratory in National Chung-Cheng University 22

Introduction to MyselfIntroduction to Myself Chuang-Chieh LinChuang-Chieh Lin 林莊傑林莊傑 Education BackgroundEducation Background

B.S. Department of Mathematics, National Cheng-Kung University, B.S. Department of Mathematics, National Cheng-Kung University, September 1998 – June 2002.September 1998 – June 2002.

M.S. Department of Computer Science and Information Engineering, M.S. Department of Computer Science and Information Engineering, National Chi-Nan University, September 2002 – June 2004. National Chi-Nan University, September 2002 – June 2004.

Advisor Advisor (2002 – 2004)(2002 – 2004) Professor R. C. T. LeeProfessor R. C. T. Lee

ResearchResearch BiocomputingBiocomputing

• Sequence AssemblySequence Assembly• Evolutionary TreesEvolutionary Trees• Gene Networks <recently>Gene Networks <recently>

Computational GeometryComputational Geometry Other topics in the field of Computer AlgorithmsOther topics in the field of Computer Algorithms


OutlineOutline

Introduction and MotivationsIntroduction and Motivations Symbolic Logic and the Resolution-Principle Symbolic Logic and the Resolution-Principle

MethodMethod Boolean Gene Regulatory NetworkBoolean Gene Regulatory Network The State Determination ProblemThe State Determination Problem The Implicit Interaction Finding Problem The Implicit Interaction Finding Problem Previous WorkPrevious Work Future WorkFuture Work


Genes are known as specific regions on a DNA Genes are known as specific regions on a DNA sequence, and they carry information for sequence, and they carry information for manufacturing proteins. manufacturing proteins.

A A genomegenome is all the DNA in an organism, including is all the DNA in an organism, including its genes.its genes.

DNA is made up of four similar chemicals (called DNA is made up of four similar chemicals (called bases and abbreviated A, T, C, and G) that are bases and abbreviated A, T, C, and G) that are repeated millions or billions of times throughout a repeated millions or billions of times throughout a genome. The human genome has 3 billion pairs of genome. The human genome has 3 billion pairs of bases.bases.

Introduction and MotivationsIntroduction and Motivations


Human genome sequencing was the most important Human genome sequencing was the most important target of target of Human Genome ProjectHuman Genome Project (HGP) which begun (HGP) which begun formally in 1990. formally in 1990.

However, after the human genome sequencing was However, after the human genome sequencing was completed, the postgenomic era and the age of completed, the postgenomic era and the age of functional genomics have arrived. functional genomics have arrived.

One aspect of functional genomics is One aspect of functional genomics is the understanding the understanding of how genes are expressed or regulatedof how genes are expressed or regulated which is which is critically important to finding ways to fight diseases.critically important to finding ways to fight diseases.

It has been found by scientists that diseases are often It has been found by scientists that diseases are often related to how genes are expressed and regulated. related to how genes are expressed and regulated.


To study genes, we have to understand To study genes, we have to understand gene gene expressionsexpressions, which are the processes that hereditary , which are the processes that hereditary information of genes transforms into mRNA or information of genes transforms into mRNA or proteins. We also can call the gene expression of a proteins. We also can call the gene expression of a gene “gene “statestate”.”.

We say that a gene is We say that a gene is activatedactivated if its process of if its process of making mRNA or a protein is executedmaking mRNA or a protein is executed ; otherwise, we ; otherwise, we say that a gene is say that a gene is inhibitedinhibited. Hereafter, we say that the . Hereafter, we say that the gene expression or the state of a gene gene expression or the state of a gene AA denotes denotes whether whether AA is activated or inhibited. is activated or inhibited.


Gene A Gene B Gene C Gene D Gene EDNA

Protein

P Ptranscription

factorprotein kinase

catalyze

catalyze

protein phosphatase

phosphorylated protein

transcription factor

Through the graph above, we know that each gene’s expression may affect other genes’ expressions. Actually, such affections include activations, inhibitions, etc.


Suppose we have “gene Suppose we have “gene AA activates gene activates gene BB”, we obtain ”, we obtain if gene if gene AA is activated, gene is activated, gene BB will be activated and if will be activated and if gene gene AA is not activated, gene is not activated, gene BB won’t be activated. won’t be activated.

Similarly, we can obtain that if gene Similarly, we can obtain that if gene AA is activated, gene is activated, gene BB will be inhibited and if gene will be inhibited and if gene AA is not activated, gene is not activated, gene BB will be activated from “gene will be activated from “gene AA inhibits gene inhibits gene BB”.”.

A Bactivate

A Binhibit


We say that “We say that “AA is inhibited” is the same as “ is inhibited” is the same as “AA is not is not activated”, and “activated”, and “AA is activated” is the same as “ is activated” is the same as “AA is is not inhibited”.not inhibited”.

Hence, we may consider the interactions and gene Hence, we may consider the interactions and gene expressions as formulas in symbolic logic.expressions as formulas in symbolic logic.

Now, let us go to get familiar with symbolic logic Now, let us go to get familiar with symbolic logic first.first.


Symbolic LogicSymbolic Logic

For symbolic logic, the symbols, such as For symbolic logic, the symbols, such as AA, , BB and and CC, , are called are called atomsatoms..

FormulasFormulas are defined recursively as follows: are defined recursively as follows: An atom is a formula.An atom is a formula. If If GG is a formula, then is a formula, then GG is also a formula. is also a formula. If If GG and and HH are formulas, then are formulas, then GG HH, , GG HH, , GG HH and and GG

HH are formulas, where are formulas, where , , , , and and dente “or”, “and”, dente “or”, “and”, “imply” and “if and only if ” respectively.“imply” and “if and only if ” respectively.

All formulas are generated by applying the above three All formulas are generated by applying the above three rules. rules.


For example, For example,

““AA”, “”, “BB”, “”, “CC” are all formulas.” are all formulas. ““AA BB” and “” and “BB CC” are both formulas.” are both formulas. ““((AA BB)” and)” and ““((AA BB)) BB CC” are both formulas.” are both formulas.


We define that an atom or the negation of an atom is We define that an atom or the negation of an atom is a a literalliteral. .

For example, For example, AA, , BB, , CC are all literals. are all literals.

Suppose we have formulas Suppose we have formulas FF11, , FF22, …, , …, FFnn, then , then FF11 FF22 … … FFnn is called the is called the disjunctiondisjunction of of FF11, , FF22, …, , …, FFnn while while FF11 FF22 … … FFnn is called the is called the conjunctionconjunction of of FF11, , FF22, …, , …, FFnn. .


A disjunction of literals is called a A disjunction of literals is called a clauseclause. For . For example, example, AA BB, , XX YY ZZ are both clauses. are both clauses.

A formula A formula FF is said to be in a is said to be in a conjunctive normal conjunctive normal formform if and only if if and only if FF has the form has the form FF11 FF22 … … FFnn , , nn 1, where each 1, where each FFii is a clause, is a clause, i i = 1, 2, …, = 1, 2, …, nn. .

For example, (For example, (AA BB CC) ) ( (PP QQ RR)) is a formula in a is a formula in a conjunctive normal form. conjunctive normal form. AA ( (QQ RR) is also a ) is also a formula in a conjunctive normal formformula in a conjunctive normal form..


An An interpretationinterpretation of of GG is an assignment of truth is an assignment of truth values to values to AA11, , AA22, …, , …, AAnn in which every in which every AAii, 1, 1 ii nn, is , is assigned either assigned either TT or or FF, but not both. A formula is , but not both. A formula is said to be said to be validvalid if and only if it is true under all its if and only if it is true under all its interpretations, while a formula is said to be interpretations, while a formula is said to be inconsistentinconsistent if and only if it is false under all its if and only if it is false under all its interpretations.interpretations.

For example,For example,

““XX YY XX”” is valid.is valid. ““XX XX”” is inconsistent.is inconsistent.


Given formulas Given formulas FF11, , FF22, …, , …, FFnn and a formula and a formula GG, , GG is is said to be a said to be a logical consequencelogical consequence of of FF11, , FF22, …, , …, FFnn if and if and only if whenever only if whenever FF11 FF22 … … FFn n is true then is true then GG is also is also true. That is, true. That is, GG is a logical consequence of is a logical consequence of FF11, , FF22, …, , …, FFnn if and only if the formula ( if and only if the formula (FF11 FF22 … … FFnn) ) GG is is valid. valid.

The resolution-principle methodThe resolution-principle method is a method for is a method for deducing logical consequences from a given set of deducing logical consequences from a given set of clauses. We define the resolution principle method as clauses. We define the resolution principle method as follows. follows.


The Resolution-Principle MethodThe Resolution-Principle Method

For any two clauses For any two clauses CC11 and and CC22, if there is a literal , if there is a literal LL11 in in CC11 that that is complementary to a literal is complementary to a literal LL22 in in CC22, then delete , then delete LL11 and and LL22 from from CC11 and and CC22 respectively, and construct the disjunction of respectively, and construct the disjunction of the remaining clauses. The constructed clause is a logical the remaining clauses. The constructed clause is a logical consequence of consequence of CC11 and and CC22. .

For example, For example,


Through what we have discussed previously, how a Through what we have discussed previously, how a gene regulates the other genes may be simply gene regulates the other genes may be simply represented in symbolic logic. represented in symbolic logic.

A Bactivate

A Binhibit

For example,


Note that we can also transfer the following case into Note that we can also transfer the following case into formulas in symbolic logic.formulas in symbolic logic.

A

D

activate

B

C

E

Finhibit

activa

te

inhibit

inhibit


In this thesis, “In this thesis, “AA” stands for “gene ” stands for “gene AA is activated” while is activated” while ““AA” stands for “gene ” stands for “gene AA is not activated”, that is, “gene is not activated”, that is, “gene AA is inhibited”. is inhibited”.

For “For “AA BB”, “”, “AA BB”, “”, “AA BB” and “” and “AA BB”, we have ”, we have the following explanations. the following explanations.

““AA BB” means “If ” means “If AA is activated, is activated, BB will be activated.” will be activated.” ““AA BB” means “If ” means “If AA is inhibited, is inhibited, BB will be activated.” will be activated.” ““AA BB” means “If ” means “If AA is activated, is activated, BB will be inhibited.” will be inhibited.” ““AA BB” means “If ” means “If AA is inhibited, is inhibited, BB will be inhibited.” will be inhibited.”


Note that Note that AA BB is equivalent to is equivalent to AA B.B.

Similarly, Similarly, AA BB is equivalent to is equivalent to AA BB,, AA BB is equivalent to is equivalent to AA BB AA BB is equivalent to is equivalent to AA BB..

Next, we are going to introduce a graphic model Next, we are going to introduce a graphic model representing a system of given genes and the representing a system of given genes and the regulations between them.regulations between them.


Boolean Gene Regulatory Boolean Gene Regulatory NetworkNetwork

A Boolean gene regulatory network is shown as follows.A Boolean gene regulatory network is shown as follows. Genes Genes AA, , BB and and CC are called are called key regulatorskey regulators because no genes because no genes

can affect each of them.can affect each of them.

A

D

B F

E

C

G

–

+AND

–+

––

+

AND


After the Boolean gene regulatory network is given, After the Boolean gene regulatory network is given, we can consider two problems related to this graph we can consider two problems related to this graph model.model.

The State Determination ProblemThe State Determination Problem

The Implicit Interaction Finding ProblemThe Implicit Interaction Finding Problem

To simplify our discussion, we abbreviate “the To simplify our discussion, we abbreviate “the Boolean gene regulatory network” to “Boolean gene regulatory network” to “the Boolean the Boolean networknetwork”. ”.


The State Determination ProblemThe State Determination Problem Assume that we are given the states of key regulators, determine Assume that we are given the states of key regulators, determine

other genes’ states. other genes’ states.

Given:Given: A Boolean network and the states of key regulators A Boolean network and the states of key regulators Output:Output: All genes’ states All genes’ states

A

D

B F

E

C

G

–

+AND

–+

– –+

AND

0

1 1

0: inhibited

1: activated


We can determine all genes’ states, that is, activated We can determine all genes’ states, that is, activated or inhibited, by or inhibited, by the depth-first-search methodthe depth-first-search method or or the the resolution-principle methodresolution-principle method..

Note that we don’t consider any Boolean network Note that we don’t consider any Boolean network with cycles or self-loops. In addition, the Boolean with cycles or self-loops. In addition, the Boolean gates here we use are only AND gates.gates here we use are only AND gates.


By the depth-first-search method:By the depth-first-search method:

A

D

B F

E

C

G

–

+AND

–+

– –+

AND

0

1 1

Stage 0:

Key regulators: A, B, C


A

D

B F

E

C

G

–

+AND

–+

– –+

AND

0

1 10

Stage 1:

1

0

1


A

D

B F

E

C

G

–

+AND

–+

– –+

AND

0

1 10

Stage 2:

1

0

1


A

D

B F

E

C

G

–

+AND

–+

– –+

AND

0

1 10

Stage 3:

1

0

1


By the resolution-principle method:By the resolution-principle method:

A

D

B F

E

C

G

–

+AND

–+

– –+

AND

0

1 10

1

0

1

A

B C

and


…(1)…(2)…(3)…(4)…(5)…(6)…(7)…(8)…(9)…(10)…(11)

A …(12) B …(13) C …(14)

Key regulators

Original Boolean network


(7)&(14) (7)&(14) GG ………………… ………………… (15)(15)(1)&(12) (1)&(12) BB FF DD ………... ………... (16)(16)(13)&(16) (13)&(16) FF DD ……………... ……………... (17)(17)(5)&(13) (5)&(13) FF ………….......... ………….......... (18)(18)(17)&(18) (17)&(18) DD …………………. …………………. (19)(19)(9)&(17) (9)&(17) CC EE FF ……….. ……….. (20)(20)(14)&(20) (14)&(20) EE FF ……………… ……………… (21)(21)(18)&(21) (18)&(21) EE …………….…… …………….…… (22)(22)


AA InhibitedInhibited

BB ActivatedActivated

CC ActivatedActivated

DD ActivatedActivated

EE ActivatedActivated

FF InhibitedInhibited

GG InhibitedInhibited

The result can be summarized as follows.


This problem must be able to be solved based upon This problem must be able to be solved based upon Lemma 1 and Theorem 1 as follows.Lemma 1 and Theorem 1 as follows.

Lemma 1Lemma 1 A Boolean gene regulatory network which is free of cycles and A Boolean gene regulatory network which is free of cycles and free of self loops has at lease one node whose indegree, that is, free of self loops has at lease one node whose indegree, that is, the number of other genes that inhibits or activates it directly, the number of other genes that inhibits or activates it directly, is equal to 0.is equal to 0.

Theorem 1Theorem 1Assume that a Boolean gene regulatory network G and the Assume that a Boolean gene regulatory network G and the states of all key regulators in G are given, then the states of all states of all key regulators in G are given, then the states of all the nodes G can be all determined.the nodes G can be all determined.


Lemma 1 and Theorem 1 are easy to be proved. Here Lemma 1 and Theorem 1 are easy to be proved. Here we omit the detail of the proofs.we omit the detail of the proofs.

Now, let us go to discuss the other problem: the Now, let us go to discuss the other problem: the implicit interaction finding problem.implicit interaction finding problem.


The Implicit Interaction Finding ProblemThe Implicit Interaction Finding Problem

The implicit interaction finding problem is to derive The implicit interaction finding problem is to derive more interactions which are previously unknown from more interactions which are previously unknown from a given Boolean gene regulatory network.a given Boolean gene regulatory network.

Given:Given: A Boolean network A Boolean network Output:Output: Implicit interactions in the Boolean network Implicit interactions in the Boolean network

A

B

–+

D –

C

+

AND


(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

A

B

–+

D –

C

+

AND


(2)&(4)

(1)&(3)

(3)&(7)

(13)

By applying the resolution principle method, we have

(11)(12)(13)(14)(15)

A

B

–+

D –

C

+–

AND

A

B

–+

D –

C

+–

AND


Previous WorkPrevious Work In the analysis of gene regulation systems, a lot of results In the analysis of gene regulation systems, a lot of results

are related to constructing graphic gene regulatory are related to constructing graphic gene regulatory networks.networks.

For instance, For instance, Andreas WagnerAndreas Wagner proposed a method to proposed a method to reconstruct a gene regulatory network with core structure reconstruct a gene regulatory network with core structure from given perturbation data.from given perturbation data.

[W2001][W2001] How to Reconstruct a Large Genetic Network from n Gene Perturbations in How to Reconstruct a Large Genetic Network from n Gene Perturbations in fewer than nfewer than n22 Easy Steps, Wagner, A., Bioinformatics, Vol. 17, No. 12, 2001, pp. 1183- Easy Steps, Wagner, A., Bioinformatics, Vol. 17, No. 12, 2001, pp. 1183-1197.1197.

Note that a Note that a perturbationperturbation is an experimental manipulation is an experimental manipulation performed on a gene.performed on a gene.


0: 2 161:2:3: 0 2 5 8 12 14 164:5: 0 2 12 14 166: 0 2 5 12 14 167: 2 8 178:9: 0 1 2 5 6 10 12 14 15 16 18 2010: 0 1 2 5 6 12 14 16 18 2011: 0 2 5 6 12 14 16 18 2012: 0 2 14 1613: 8 1714: 0 2 1615: 0 2 1616: 217: 818:19: 820: 0 2 5 6 12 14 16 18

perturbation-list:

Corresponding graph G will be very complicated, so we omit it here.


1

2 3

4

5

6

7

8 109

11

12

13

14

15

16

18

19

20

17

0

0: 161:2:3: 2 5 84:5: 126: 5 127: 2 178:9: 10 1510: 1 2011: 2012: 1413: 8 1714: 015: 016: 217: 818:19: 820: 6 18

The modified perturbation-list

Corresponding graph G


Future WorkFuture Work

The identification problemThe identification problem

Other topics on biocomputing and computer Other topics on biocomputing and computer algorithmsalgorithms


Given a set of genes and a set of results of perturbations performed on the genes. The identification problem is to determine whether there exists only one Boolean network consistent with the given data.

Akutsu et al. have shown that exponential perturbations are needed to identify the unique Boolean network.

[AKMM98] Identification of Gene Regulatory Networks by Strategic Gene Disruptions and Gene Overexpressions, Akutsu, T., Kuhara, S., Maruyama, O. and Miyano, S., Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, 1998, pp. 695-702.

The Identification ProblemThe Identification Problem


Gene ExpressionGene Expression

AA BB CC DD EE FF GG HH II JJ KK LL MM NN XX11 XX22

Normal ConditionNormal Condition 11 00 11 11 11 11 00 00 11 11 11 00 00 11 11 11

Disruption of Disruption of AA 00 11 11 00 00 00 11 11 11 00 11 00 00 11 11 11

Overexpression of Overexpression of BB 11 11 11 11 00 11 11 00 11 11 11 00 00 11 11 11

This Boolean network is consistent with the given data. However, we still have to test if there exists another Boolean network consistent with the given data.

A B

G

E–+

–

F

++I–H

J

––

M

–KC

+

+DX2

X1

+

+

OR

AND

NI

+

Gene NameGene Name

perturbationsperturbations

Note that Boolean gates, including OR, AND, XOR, etc., are allowed in the solutions to this problem.

Thank you.

1 applications of symbolic logic to gene regulation systems department of computer science and...

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