1 Applications of Symbolic Logic to Gene Regulation Systems Department of Computer Science and Information Engineering of National Chung-Cheng University Speaker : Chuang-Chieh Lin

Computation Theory Laboratory in National Chung-Cheng University 2 Introduction to Myself  Chuang-Chieh Lin 林莊傑  Education Background B.S. Department of Mathematics, National Cheng-Kung University, September 1998 – June B.S. Department of Mathematics, National Cheng-Kung University, September 1998 – June M.S. Department of Computer Science and Information Engineering, National Chi-Nan University, September 2002 – June M.S. Department of Computer Science and Information Engineering, National Chi-Nan University, September 2002 – June  Advisor (2002 – 2004) Professor R. C. T. Lee Professor R. C. T. Lee  Research Biocomputing Biocomputing Sequence AssemblySequence Assembly Evolutionary TreesEvolutionary Trees Gene Networks Gene Networks Computational Geometry Computational Geometry Other topics in the field of Computer Algorithms Other topics in the field of Computer Algorithms

Computation Theory Laboratory in National Chung-Cheng University 3 Outline  Introduction and Motivations  Symbolic Logic and the Resolution-Principle Method  Boolean Gene Regulatory Network  The State Determination Problem  The Implicit Interaction Finding Problem  Previous Work  Future Work

Computation Theory Laboratory in National Chung-Cheng University 4  Genes are known as specific regions on a DNA sequence, and they carry information for manufacturing proteins.  A genome is all the DNA in an organism, including its genes.  DNA is made up of four similar chemicals (called bases and abbreviated A, T, C, and G) that are repeated millions or billions of times throughout a genome. The human genome has 3 billion pairs of bases. Introduction and Motivations

Computation Theory Laboratory in National Chung-Cheng University 5  Human genome sequencing was the most important target of Human Genome Project (HGP) which begun formally in  However, after the human genome sequencing was completed, the postgenomic era and the age of functional genomics have arrived.  One aspect of functional genomics is the understanding of how genes are expressed or regulated which is critically important to finding ways to fight diseases.  It has been found by scientists that diseases are often related to how genes are expressed and regulated.

Computation Theory Laboratory in National Chung-Cheng University 6  To study genes, we have to understand gene expressions, which are the processes that hereditary information of genes transforms into mRNA or proteins. We also can call the gene expression of a gene “state”.  We say that a gene is activated if its process of making mRNA or a protein is executed ; otherwise, we say that a gene is inhibited. Hereafter, we say that the gene expression or the state of a gene A denotes whether A is activated or inhibited.

Computation Theory Laboratory in National Chung-Cheng University 7 Gene AGene BGene CGene DGene E DNA Protein P P transcription factor protein kinase 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.

Computation Theory Laboratory in National Chung-Cheng University 8  Suppose we have “gene A activates gene B”, we obtain if gene A is activated, gene B will be activated and if gene A is not activated, gene B won’t be activated.  Similarly, we can obtain that if gene A is activated, gene B will be inhibited and if gene A is not activated, gene B will be activated from “gene A inhibits gene B”. AB activate AB inhibit

Computation Theory Laboratory in National Chung-Cheng University 9  We say that “A is inhibited” is the same as “A is not activated”, and “A is activated” is the same as “A is not inhibited”.  Hence, we may consider the interactions and gene expressions as formulas in symbolic logic.  Now, let us go to get familiar with symbolic logic first.

Computation Theory Laboratory in National Chung-Cheng University 10 Symbolic Logic  For symbolic logic, the symbols, such as A, B and C, are called atoms.  Formulas are defined recursively as follows: An atom is a formula. An atom is a formula. If G is a formula, then  G is also a formula. If G is a formula, then  G is also a formula. If G and H are formulas, then G  H, G  H, G  H and G  H are formulas, where , ,  and  dente “or”, “and”, “imply” and “if and only if ” respectively. If G and H are formulas, then G  H, G  H, G  H and G  H are formulas, where , ,  and  dente “or”, “and”, “imply” and “if and only if ” respectively. All formulas are generated by applying the above three rules. All formulas are generated by applying the above three rules.

Computation Theory Laboratory in National Chung-Cheng University 11  For example, “A”, “B”, “C” are all formulas. “A”, “B”, “C” are all formulas. “A  B” and “B  C” are both formulas. “A  B” and “B  C” are both formulas. “  (A  B)” and “  (A  B)  B  C” are both formulas. “  (A  B)” and “  (A  B)  B  C” are both formulas.

Computation Theory Laboratory in National Chung-Cheng University 12  We define that an atom or the negation of an atom is a literal. For example, A,  B, C are all literals.  Suppose we have formulas F 1, F 2, …, F n, then F 1  F 2  …  F n is called the disjunction of F 1, F 2, …, F n while F 1  F 2  …  F n is called the conjunction of F 1, F 2, …, F n.

Computation Theory Laboratory in National Chung-Cheng University 13  A disjunction of literals is called a clause. For example, A  B,  X  Y  Z are both clauses.  A formula F is said to be in a conjunctive normal form if and only if F has the form F 1  F 2  …  F n, n  1, where each F i is a clause, i = 1, 2, …, n. For example, (A  B   C)  (P   Q  R) is a formula in a conjunctive normal form. A  (  Q  R) is also a formula in a conjunctive normal form.

Computation Theory Laboratory in National Chung-Cheng University 14  An interpretation of G is an assignment of truth values to A 1, A 2, …, A n in which every A i, 1  i  n, is assigned either T or F, but not both. A formula is said to be valid if and only if it is true under all its interpretations, while a formula is said to be inconsistent if and only if it is false under all its interpretations.  For example, “  X  Y  X” is valid. “  X  X” is inconsistent. “  X  X” is inconsistent.

Computation Theory Laboratory in National Chung-Cheng University 15  Given formulas F 1, F 2, …, F n and a formula G, G is said to be a logical consequence of F 1, F 2, …, F n if and only if whenever F 1  F 2  …  F n is true then G is also true. That is, G is a logical consequence of F 1, F 2, …, F n if and only if the formula (F 1  F 2  …  F n )  G is valid.  The resolution-principle method is a method for deducing logical consequences from a given set of clauses. We define the resolution principle method as follows.

Computation Theory Laboratory in National Chung-Cheng University 16 The Resolution-Principle Method  For any two clauses C 1 and C 2, if there is a literal L 1 in C 1 that is complementary to a literal L 2 in C 2, then delete L 1 and L 2 from C 1 and C 2 respectively, and construct the disjunction of the remaining clauses. The constructed clause is a logical consequence of C 1 and C 2.  For example,

Computation Theory Laboratory in National Chung-Cheng University 17  Through what we have discussed previously, how a gene regulates the other genes may be simply represented in symbolic logic. AB activate AB inhibit For example,

Computation Theory Laboratory in National Chung-Cheng University 18  Note that we can also transfer the following case into formulas in symbolic logic. A D activate B C E F inhibit activate inhibit

Computation Theory Laboratory in National Chung-Cheng University 19  In this thesis, “A” stands for “gene A is activated” while “  A” stands for “gene A is not activated”, that is, “gene A is inhibited”.  For “A  B”, “  A  B”, “A   B” and “  A   B”, we have the following explanations. “A  B” means “If A is activated, B will be activated.” “A  B” means “If A is activated, B will be activated.” “  A  B” means “If A is inhibited, B will be activated.” “  A  B” means “If A is inhibited, B will be activated.” “A   B” means “If A is activated, B will be inhibited.” “A   B” means “If A is activated, B will be inhibited.” “  A   B” means “If A is inhibited, B will be inhibited.” “  A   B” means “If A is inhibited, B will be inhibited.”

Computation Theory Laboratory in National Chung-Cheng University 20  Note that A  B is equivalent to  A  B. Similarly,  A  B is equivalent to A  B, A   B is equivalent to  A   B A   B is equivalent to  A   B  A   B is equivalent to A   B.  A   B is equivalent to A   B.  Next, we are going to introduce a graphic model representing a system of given genes and the regulations between them.

Computation Theory Laboratory in National Chung-Cheng University 21 Boolean Gene Regulatory Network  A Boolean gene regulatory network is shown as follows.  Genes A, B and C are called key regulators because no genes can affect each of them. A D B F E C G – + AND – + – – +

Computation Theory Laboratory in National Chung-Cheng University 22  After the Boolean gene regulatory network is given, we can consider two problems related to this graph model. The State Determination Problem The State Determination Problem The Implicit Interaction Finding Problem The Implicit Interaction Finding Problem  To simplify our discussion, we abbreviate “the Boolean gene regulatory network” to “the Boolean network”.

Computation Theory Laboratory in National Chung-Cheng University 23 The State Determination Problem  Assume that we are given the states of key regulators, determine other genes’ states. Given: A Boolean network and the states of key regulators Given: A Boolean network and the states of key regulators Output: All genes’ states Output: All genes’ states A D B F E C G – + AND – + – – : inhibited 1: activated

Computation Theory Laboratory in National Chung-Cheng University 24  We can determine all genes’ states, that is, activated or inhibited, by the depth-first-search method or the resolution-principle method.  Note that we don’t consider any Boolean network with cycles or self-loops. In addition, the Boolean gates here we use are only AND gates.

Computation Theory Laboratory in National Chung-Cheng University 25  By the depth-first-search method: A D B F E C G – + AND – + – – Stage 0: Key regulators: A, B, C

Computation Theory Laboratory in National Chung-Cheng University 26 A D B F E C G – + AND – + – – Stage 1: 1 0 1

Computation Theory Laboratory in National Chung-Cheng University 27 A D B F E C G – + AND – + – – Stage 2: 1 0 1

Computation Theory Laboratory in National Chung-Cheng University 28 A D B F E C G – + AND – + – – Stage 3: 1 0 1

Computation Theory Laboratory in National Chung-Cheng University 29  By the resolution-principle method: A D B F E C G – + AND – + – – AA B C and

Computation Theory Laboratory in National Chung-Cheng University 30 …(1) …(2) …(3) …(4) …(5) …(6) …(7) …(8) …(9) …(10) …(11)  A … (12) B … (13) C … (14) Key regulators Original Boolean network

Computation Theory Laboratory in National Chung-Cheng University 31 (7)&(14)  G ………………… (15) (1)&(12)   B  F  D ………... (16) (13)&(16)  F  D ……………... (17) (5)&(13)   F ………… (18) (17)&(18)  D ………………….(19) (9)&(17)   C  E  F ……….. (20) (14)&(20)  E  F ………………(21) (18)&(21)  E …………….……(22)

Computation Theory Laboratory in National Chung-Cheng University 32 AInhibited BActivated CActivated DActivated EActivated FInhibited GInhibited  The result can be summarized as follows.

Computation Theory Laboratory in National Chung-Cheng University 33  This problem must be able to be solved based upon Lemma 1 and Theorem 1 as follows.  Lemma 1 A Boolean gene regulatory network which is free of cycles and free of self loops has at lease one node whose indegree, that is, the number of other genes that inhibits or activates it directly, is equal to 0.  Theorem 1 Assume that a Boolean gene regulatory network G and the states of all key regulators in G are given, then the states of all the nodes G can be all determined.

Computation Theory Laboratory in National Chung-Cheng University 34  Lemma 1 and Theorem 1 are easy to be proved. Here we omit the detail of the proofs.  Now, let us go to discuss the other problem: the implicit interaction finding problem.

Computation Theory Laboratory in National Chung-Cheng University 35 The Implicit Interaction Finding Problem  The implicit interaction finding problem is to derive more interactions which are previously unknown from a given Boolean gene regulatory network. Given: A Boolean network Given: A Boolean network Output: Implicit interactions in the Boolean network Output: Implicit interactions in the Boolean network A B – + D – C + AND

Computation Theory Laboratory in National Chung-Cheng University 36 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) A B – + D – C + AND

Computation Theory Laboratory in National Chung-Cheng University 37 (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 + –

Computation Theory Laboratory in National Chung-Cheng University 38 Previous Work  In the analysis of gene regulation systems, a lot of results are related to constructing graphic gene regulatory networks.  For instance, Andreas Wagner proposed a method to reconstruct a gene regulatory network with core structure from given perturbation data. [W2001] How to Reconstruct a Large Genetic Network from n Gene Perturbations in fewer than n 2 Easy Steps, Wagner, A., Bioinformatics, Vol. 17, No. 12, 2001, pp  Note that a perturbation is an experimental manipulation performed on a gene.

Computation Theory Laboratory in National Chung-Cheng University 39 0:2 16 1: 2: 3: : 5: : : : 9: : : : : : : :2 17:8 18: 19:8 20: perturbation-list: Corresponding graph G will be very complicated, so we omit it here.

Computation Theory Laboratory in National Chung-Cheng University :16 1: 2: 3: : 5:12 6:5 12 7:2 17 8: 9: : :20 12:14 13: :0 15:0 16:2 17:8 18: 19:8 20:6 18 The modified perturbation-list Corresponding graph G

Computation Theory Laboratory in National Chung-Cheng University 41 Future Work  The identification problem  Other topics on biocomputing and computer algorithms

Computation Theory Laboratory in National Chung-Cheng University 42   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 The Identification Problem

Computation Theory Laboratory in National Chung-Cheng University 43 Gene Expression ABCDEFGHIJKLMN X1X1X1X1 X2X2X2X2 Normal Condition Disruption of A Overexpression of B 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 – K C + + D X2X2 X1X1 + + OR AND NI + Gene Name perturbations Note that Boolean gates, including OR, AND, XOR, etc., are allowed in the solutions to this problem.

Thank you.