୯ ҥ Ҭ ೯ ε Ꮲ
ႝᐒᆶڋπำس
ᅺγፕЎ
ሇ҆ᆶПᛶឦߙຂೈϐ୷ӢፓᆛၡǺ
ׯؼࠠ୷Ӣᄽᆉݤ
Genetic Regulatory Network of Yeast / Xenopus Frog Egg
Ǻ
Improved Genetic Algorithm
ࣴ ز ғǺڬ ৎ ፣
ࡰᏤ௲Ǻ బ റγ
ֆ ۩ ࣔ റγ
ሇ҆ᆶПᛶឦߙຂೈϐ୷ӢፓᆛၡǺׯؼࠠ୷Ӣᄽᆉݤ
Genetic Regulatory Network of Yeast / Xenopus Frog EggǺ
Improved Genetic Algorithm
ࣴ ز ғǺڬৎ፣
ࡰᏤ௲Ǻబ റγ
ֆ۩ࣔ റγ
Student
ǺChia-Hsien Chou
AdvisorǺDr. Tsu-Tian Lee
Dr. Shinq-Jen Wu
୯ ҥ Ҭ ೯ ε Ꮲ ႝᐒᆶڋπำس
ᅺγፕЎ
A Thesis
Submitted to Department of Electrical and Control Engineering College of Electrical and Computer Engineering
National Chiao Tung University in partial Fulfillment of the Requirements
for the Degree of Master
in
Electrical and Control Engineering July 2006
Hsinchu, Taiwan, Republic of China
ሇ҆ᆶПᛶឦߙຂೈϐ୷ӢፓᆛၡǺׯؼࠠ୷Ӣᄽᆉݤ
ᏢғǺڬৎ፣
ࡰᏤ௲Ǻబ റγ
ֆ۩ࣔ റγ
୯ҥҬ೯εᏢႝᐒᆶڋπำسȐࣴز܌ȑᅺγ
ᄔ
ा
ճҔׯؼࠠ୷ӢᄽᆉݤǴଞჹሇ҆ᆶПᛶឦߙຂೈϩձ୷Ӣፓᆛ ၡ modified power-law model ᆶ S-system ࡌኳǶᙖҗሇ҆ჴᡍکПᛶឦMichaelis-Menten model܌ளډޑਔ໔ᗺၗ૽ግǴаᕇڗന٫ϯޑୖ
ኧǴӢࣁ modified power-law model ک S-system ёаమཱӦඔॊ୷Ӣӧғԋ ܈ϸᔈਔࢂϯբҔᗋࢂڋբҔǴ᠙ܭٿޣޑՅǴჹሇ҆ޑಒ झຼයᆶӧПᛶឦߙຂೈಒझຼයύޑԖํϩڋǴૈܴᕕځ୷Ӣޑϸ ᔈǶڀԖᎂ౽բҔޑׯؼࠠ୷Ӣᄽᆉݤόՠёаډӄୱཛྷ൨Ǵᗋё аᆒमЬကޑཷۺڗளന٫ޑঁᡏǶനಖளډޑ୷Ӣፓᆛၡёаගٮ๏ ғނᏢৎӧሇ҆کПᛶឦߙຂೈಒझຼයБय़׳ుΕޑࣴزǶ
Genetic Regulatory Network of Yeast / Xenopus Frog EggǺ
Improved Genetic Algorithm
Student
ǺChia-Hsien Chou
Advisor
ǺDr. Tsu-Tian Lee
Dr. Shinq-Jen Wu
Department (Institute) of Electrical and Control Engineering
National Chiao Tung University
Abstract
An improved genetic algorithm is proposed to achieve gene regulatory network modeling of Xenopus frog egg in S-system and yeast in modified power-law model respectively. Via the time-course datasets from experiment of yeast and Michaelis-Menten model of Xenopus, the optimal parameters are learned. The modified power-law model and S-system can clearly describe activative and inhibitory interaction between genes as generating and consuming process. We concern cell cycle of yeast and the mitotic control in cell cycle of Xenopus frog egg to realize gene reactions. The proposed improved genetic algorithm can achieve global search with migration and keep the best individual with elitism operation. The generated gene regulatory networks can provide biological researchers for further experiments in yeast and Xenopus frog egg cell cycle.
Acknowledgements
२Ӄךाགᖴబਠߏکֆ۩ࣔԴৣǴӧ೭ٿԃ๏ךޑፌፌ௲ᇧǴӧ ࣴزБय़όսெӦ௲ך٤ᢀۺၟБݤǴ٬ךᕇؼӭǴᏢғόയགᐟǴ Ψࡐགᖴύ҅εᏢϯπسޑЦ௲ᆶᇞᓪᏢߏ๏ϒךჹғނၗૻࣴ زޑ٤ࡰ௲ǴᗋԖֆࡹၰᏢߏၟךଆவ٣୯ࣽीฝࣴزǴΨ๏ϒךࡐ ӭޑႴᓰᆶ௲ᏤǴߚதགᖴдॺǴڐշךٿԃޑࣴزғఱǶ ӧፐǴჴᡍ࠻ޑᏢߏЦ╨ᗶǴǴ݅ਤӹǴ݅ࢩǴֆܲڻǴֆ ۘ᎔аϷ৪⊭ਤᔅշךှ،ӧঅፐਔ܌ၶډޑୢᚒǴᗋԖځдჴᡍ࠻ޑӕᏢ ݅ࡏֻǴࡿሎǴमቺӧঅፐਔεৎϕ࣬ፕᔅԆǴΜϩགᖴǶԜѦǴ׳ Ԗᒘӭܻ϶ޑᆒઓЍǴࣗ܈ჴ፦ڐշǴךωૈճֹԋᏢǴࡸᅅǴ ёૈۘԖ٤҂ගϷޑܻ϶ॺǴӧԜ֡ٳठᖴǶ നࡕाགᖴךޑР҆ڬᓪϡᆶमǴךޑۊۊڬ܃ۇǴךޑڬৎ ѶǴᗋԖךޑζ϶ጰۏ։ǴऩόࢂдॺߏΦаٰޑЍǴόёૈԖךϞϺޑ λλԋ݀ǶContents
ᄔा ...i
Abstract ...ii
Acknowledgments ...iii
Contents ...iv
List of tables ...vi
List of figures ...vii
1 Introduction ...1
1.1 Research Background ...1
1.2 Literature Discussion ...1
1.3 Content Organization ...3
2 Biological Systems ...4
2.1 Yeast cell cycle ...4
2.2 M phase control of Xenopus frog egg ...6
3 Improved Genetic Algorithm ...10
3.1 Introduction ...10
3.2 IEDO ...11
3.3 Reproduction ...12
3.4 Crossover ...13
3.5 Probability Mutation and Elitism ...14
3.6 Migration ...15
3.7 Fitness ...16
4 Modeling and Simulation Results ...18
4.2 M phase control of Xenopus frog egg ...21
5 Conclusion ...30
References ...31
List of tables
List of figures
2.1 The four phases of cell cycle ...4
2.2 The concentrations of cyclin, unphosphorylated cyclin-Cdc2 and Tyr-15 phosphorylated cyclin-Cdc2 ...8
2.3 The concentrations of doubly phosphorylated cyclin-Cdc2 and Thr-161 phosphorylated cyclin-Cdc2 ...8
2.4 The concentrations of Cdc25 and IE ...9
2.5 The concentrations of Wee1 and APC ...9
3.1 Flow chart for improved genetic algorithm ...10
3.2 Population definition ...11
3.3 Rule of reproduction ...13
3.4 Crossover operation ...14
3.5 Probability mutation operation ...15
4.1 The gene regulatory network of the generated modified power-low system ...20 4.2 Training dataset-1 ...21 4.3 Training dataset-2 ...22 4.4 Training dataset-3 ...22 4.5 Training dataset-4 ...23 4.6 Cyclin evolution ...25
4.7 Unphosphorylated cyclin-Cdc2 evolution ...25
4.9 Doubly phosphorylated cyclin-Cdc2 evolution ...26
4.10 Thr-161 phosphorylated cyclin-Cdc2 evolution ...27
4.11 Cdc25 evolution ...27
4.12 Wee1 evolution ...28
4.13 IE evolution ...28
Chapter 1
Introduction
1.1 Research Background
As the rapid development in cDNA microarray technologies, time-course gene dataset becomes available day by day. Hence, the construction of gene regulatory networks and signal transduction cascades for complicated biological systems has come of age. In order to approximate biological behavior for controlling metabolic/biological reaction, more and more experiments are set up for achievement of quantitative control. After post-genomic era, new scientific and technological methods on the biotechnology such as microarray technology are developed to bring massive biological knowledgeable dataset. Now system biologists are trying to describe biochemical phenomenon via mathematical model. With the mathematical model, we can realize the detailed genes-genes interaction, simulate the gene regulatory network and predict gene behavior.
1.2 Literature Discussion
Numerous models are proposed to describe the gene network such as Boolean network, Bayesian network, Michaelis-Menten model, and S-system. Boolean network is to reconstruct gene regulatory network via Boolean function and
express gene relationship in graphical way [1, 2], which distinguish gene states to be INPUT and OUTPUT. At any time points, the state values of chosen INPUT genes are set to be 1 and 0 for non-chosen genes; the states values are given in the similar way as OUTPUT. Further, Bayesian network can also the probabilistic relationships of genes [2, 3]; joint probability distributions among genes are calculated to construct the graphical model. Michaelis-Menten model is nonlinear differential equations to describe the metabolic concentration in the biological system [4]. S-system is another nonlinear differential type expressed in power-law formalism [5, 6]. S-system describes gene regulation not only in mathematical description but also can further express into graphical form to show the activatory and inhibitory operation directly. Each equation is composed by synthesis and degradation flux; and the activation and inhibition relationship are shown in positive and negative kinetic order, respectively. In this paper, we shall develop the general S-system and another reformed nonlinear differential system, which is a modified power-low model modified from equation in [7, 8], to find out the gene regulatory network of yeast cell cycle and Xenopus cell cycle M phase control from microarray dataset, respectively.
However, the construction of such a highly nonlinear equation is a tough work. Chen and the authors first reform the nonlinear differential equation into linear form and then resolve it via linear algebra [7]. In these years, some researchers are devoted to infer gene regulatory network with various intelligent computation technologies such as hybrid differential evolution, genetic algorithm, genetic programming, ..., etc. Wang use Hybrid differential evolution and genetic algorithm to obtain the global optimal solution for highly nonlinear system and
various biochemical system [9, 10]. Kikuchi and coauthors use a genetic algorithm to transform parameters into individuals first and solve optimal parameters via evolution procedure [11]. Sakamoto and coauthors use genetic programming to develop the gene regulatory network in a tree form [12]. In this work, we shall adopt improved genetic algorithm [13] to infer the gene regulatory networks of yeast cell cycle in modified power-law model and Xenopus frog egg cell cycle in S-system. Improved evolutionary direction operator (IEDO), migration operation and elitism are combined into genetic program for global optimal, fast and best-optional searching. The input/output datasets, yeast cell cycle dataset [14], generated from Michaelis-Menten model of mitotic control in Xenopus frog eggs [15, 16], are used to train the genetic networks for searching the optimal parameters of the corresponding modified power-law model and S-system, respectively.
1.3 Content Organization
This paper is organized as follows: the biological systems, yeast cell cycle and cell cycle M phase control model of the Xenopus frog egg, are described in Chapter 2. Improved genetic algorithm is shown in Chapter 3. Chapter 4 shows the modeling and simulation results. Chapter 5 is the conclusion.
Chapter 2
Biological System
2.1 Yeast cell cycle
Research in cell cycle is very important not only for realizing cell reproduction but also for realizing cancer development. Figure 2.1 is the cell cycle that includes four phases (G1ÆSÆG2ÆM). Cell grows up in G1 phase, produces RNA and synthesizes protein. During S phase, DNA is duplicated to produce two similar daughter cells. The cell continues to grow and produce protein and prepares to enter M phase during G2 phase. As DNA replication is completed, the cell enters M phase and divides.
Table 2.1 The genes of the Yeast cell cycle.
Gene Description
CDC28 Catalytic subunit of the main cell cycle cyclin-dependent kinase CLN3
role in cell cycle START; involved in G(sub)1 size control;
G1/S-specific cyclin, interacts with Cdc28p protein kinase to control events at START
SWI4
Involved in cell cycle dependent gene expression; both Swi4p and Swi6p are required for the in vivo protection of the SCB sequences at any cell cycle stage
SWI6 Involved in cell cycle dependent gene expression MBP1 transcription factor
FUS3 Required for the arrest of cells in G(sub)1 in response to pheromone and cell fusion during conjugation
FAR1 Inhibitor of Cdc28p/Cln1p and Cdc28p/Cln2p complexes involved in cell cycle arrest for mating; Factor arrest protein
CLN1 G1 cyclin; role in cell cycle START CLN2 G1 cyclin; role in cell cycle START
SIC1 P40 inhibitor of Cdc28p-Clb5 protein kinase complex CLB5
role in DNA replication during S phase; additional functional role in formation of mitotic spindles along with Clb3 and Clb4; B-type cyclin involved in S-phase initiation
CLB6 role in DNA replication during S phase; B-type cyclin involved in S-phase initiation
CDC6
Protein involved in initiation of DNA replication; Protein that regulates initiation of DNA replication through binding to origins of replication at the end of mitosis, directing the assembly of MCM proteins and the pre-replication complex
CDC20 Cell Division Cycle; Required for onset of anaphase; adaptor for APC GRR1 F box protein with several leucine rich repeats
CDC4
Init. of DNA synthesis & spindle pole body separation; dispensable for both mitotic and meiotic spindle pole body dupl.; essential for mitotic but not premeiotic DNA synth.; wt levels of synaptonemal complexes and intragenic recombination
The yeast cell cycle gene expression data is collected by Spellman [14]. The dataset were covered six experimental conditions (CLN2; CLN3; ALPH, CDC15, CDC28 and ELU). We use the gene time-course data from experimental condition CDC28, which contains 24 time points. We concentrate in G1 and S phase and set up our dataset from the sub network of yeast cell cycle pathway. The descriptions and datasets are available at http://cellcycle-www.stanford.edu. This dataset in Table 2.1 involves 16 genes.
2.2 M phase control of Xenopus frog egg
Michaelis-Menten model are concerned to describe the mitotic control in cell-cycle of Xenopus frog egg [15, 16],
1 1 2 1 3 1 2 5 2 2 25 3 3 1 3 2 25 2 3 4 , (1) , (2) , pp wee cak wee cak pp x k k x k x x k x k k k x k x k x x k x k k k x k x 4 5 25 2 4 3 5 2 2 5 25 4 5 6 6 6 6 6 (3) , (4) , (5) 1 , 1 wee pp cak cak pp wee a b a b x k x k k k x k x x k x k k k x k x k x x k x x K x K x 7 5 7 7 7 7 5 8 8 8 8 8 9 (6) 1 , (7) 1 1 , (8) 1 f e e f g h g h k x k x x x K x K x k x x k x x K x K x k x 8 9 9 9 9 1 , (9) 1 c d c d x x k x K x K xwith
2 2 9 2 2 7 25 25 6 25 25 , , ,wee wee wee wee
k V x V V k V x V V k V x V V c cc c cc c cc c cc c
where xi, i = 1, 2, ···, 9, are the concentrations or activities of cyclin,
unphosphorylated cyclin-Cdc2, Tyr-15 phosphorylated cyclin-Cdc2, doubly phosphorylated cyclin-Cdc2, Thr-161 phosphorylated cyclin-Cdc2 activated by four enzymes Cdc25, Wee1, IE and APC, respectively. Eqs. (1) ~ (4) describe four phosphorylation states of the cyclin-Cdc2 dimer. The Thr-161 phosphorylated cyclin-Cdc2 represents the M phase promoting factor (MPF). The concentration of MPF can control the phases during cell cycle. For instance, high concentration of MPF can make the cell to divide to two child cells.
The numerical solution of Michaelis-Menten model is shown as Figure 2.2 ~ 2.5. The parameter values Ka=0.1, Kb=1.0, Kc=0.01, Kd=1.0, Ke=0.1, Kf=1.0,
Kg=0.01, Kh=0.01, ka=2.0, kb=0.1, kc=0.13, kd=0.13, ke=2.0, kf=0.1, kg=2.0, kh=0.15, k1=0.01, k3=0.5, kcak=0.64, kpp=0.004, V25c 0.017 , , , , , 25 0.17 Vcc 2 0.005
0 20 40 60 80 100 120 140 160 180 200 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 time [min] c o nc en tr a ti o n Cyclin Unphosphorylated cyclin-Cdc2
Tyr-15 phosphorylated cyclin-Cdc2
Figure 2.2 The concentrations of cyclin, unphosphorylated cyclin-Cdc2 and
Tyr-15 phosphorylated cyclin-Cdc2.
0 20 40 60 80 100 120 140 160 180 200 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 time [min] c o nc en tr a ti o n Thr-161 phosphorylated cyclin-Cdc2
doubly phosphorylated cyclin-Cdc2
Figure 2.3 The concentrations of doubly phosphorylated cyclin-Cdc2 and
0 20 40 60 80 100 120 140 160 180 200 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time [min] c o nc en tr a ti o n Cdc25 IE
Figure 2.4 The concentrations of Cdc25 and IE.
0 20 40 60 80 100 120 140 160 180 200 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time [min] c o nc en tr a ti o n Wee1 APC
Chapter 3
Improved Genetic Algorithm
3.1 Introduction
Based on general genetic algorithm, improved genetic algorithm includes improved evolutionary direction operator to speed the searching; migration operator to escape from bogging down into local solution and elitism to keeping the best to be passed down always. The flow chart is shown in Figure 3.1.
The algorithm initializes a population first. The definition of population is shown in Figure 3.2. The chromosome in individual represents parameter. And then, we evaluate the fitness value of each individual in the population and keep the better individuals and eliminates the worse individuals though evolution procedures.
Figure 3.2 Population definition.
3.2 IEDO
Three preferred fitness value and its associated individuals are chosen to decide the evolutionary direction in the population. Improved evolutionary direction operator (IEDO) has the ability for both local and global search synchronously.
optimal solution. Besides, the IEDO operator can avoid bogging down in the local optimal solution, which conventional GA is usually stuck into.
After completing all fitness values of individuals, we choose preferred fitness values denote Fb, Fs, Ft and its associated individuals denote Ib, Is, It. And
then, the new individual denote Iideo is calculated as
(10) (11) s t 1 1 2 2 iedo b b b max min iedo iedo I = I + r * D * ( I - I ) + r * D * ( I - I ), I = max(min(I ,I ),I ),
where r1 and r2 are two random numbers, r1, r2[0, 1]; D1 and D2 are the
magnitude of two evolutionary directions to be 1; Imax and Imin are the upper and
lower bound, respectively. The new fitness value Fnew of the Iideo is calculated; if
the Fnew is better than one of the three preferred fitness values, it would be
replaced that.
3.3 Reproduction
The probability of reproduction directly depends on the fitness of the individuals. The individual with better fitness has high probability of reproduction. In contrast, the individual with worse fitness has low probability of reproduction. The rule of reproduction is shown in Figure 3.3.
Figure 3.3 Rule of reproduction.
3.4 Crossover
Two-point crossover operation shown in Figure 3.4 is adopted and operates according to crossover probability, which involves selection of two crossover cut-points randomly and then exchanges the chosen two cut-points genes of parent individuals to generate two child individuals. And further, randomly select one of the new individuals to replace father or mother individuals.
Figure 3.4 Crossover operation.
3.5 Probability Mutation and Elitism
Different from the conventional GA to choose only one gene in individual randomly for mutation operation, all genes in individuals are chosen and their mutation probability are assigned by the designer for exchanging their original values in Figure 3.5. This will bring excessive diversity in population and hence may fail to converge to temperately optimal solution. Therefore, we adopt elitism operator to decrease this effect. Elitism operator is to keep the best individual to survive for each generation and hence to ensure good characteristic to pass down
always.
Figure 3.5 Probability mutation operation.
3.5 Migration
To wider the search space, a migration operator is done to get a new and diverse population. The degree of population diversity Ș is to check if the migration should be performed. 2 _ 1 1 1 0, if , (12) 1, otherwise , (13) _ ( 1) ij bj ij bj Dim I NP ij i j x x temp x temp Dim I NP
H
K
°° ® ° °¯ u¦ ¦
respectively the j-th chromosome in the i-th individual and the best individual; NP is the number of individual; Dim I is the dimension of individual; Ș is in the range between 0 and 1. İ1[0,1] is a tolerance-threshold of population diversity
for migration; if Ș is small than İ1, migration operate to generate a new
chromosome as follows.
,min 2 ,m 1 ,m j j x in ,max ,min ax 2 , if , (14) , ortherwise bj j bj bj j j ij bj bj x x x r x x r x x x x r x ° ° ® ° °¯ u ! uwhere xj,max and xj,min are the upper and lower bound of the j-th chromosome,
respectively. The r1 and r2 are two random numbers, r1, r2[0, 1].
3.6 Fitness
Every individual is evaluated by its fitness value, which keeps the better individuals and eliminates the worse individuals. We adopt two different methods to evaluate fitness. The fitness of an individual in yeast cell cycle is defined as
0 0 2 1 1 , (15) ei i i j fitness X t j t X t j t kinetic order ' '¦¦
¦
-1 n Nwhere the Xi(t) is approximate value of the i-th variable at time t, Xei(t) is original
The fitness of an individual in Xenopus M phase control is defined as
_ -1 Dim I N 2 0 0 1 1 -, (16) _ ei i i j X t j t X t j t fitness Dim I N ' ' u¦ ¦
where N is number of time points; Xei(t) and Xi(t) are experiment value and
Chapter 4
Modeling and Simulation Results
4.1 Yeast cell cycle
The mathematical model in yeast cell cycle is approximated from the model adopted in [7, 8].
( ) ( ) ( ), =1,2, , , (17)
i i i i
X t G t O X t i n
where Gi(t) is the transcription rate, Ȝi is the self-degradation rate and n is the
number of the variable, Xi(t) is the concentration of the i-th gene at time t. Gi(t) is
a nonlinear function,
^
`
1 ( ) , (18) 1 exp , , i ij j j G t a u tD
ª«E G
J
º» ¬ ¼¦
m 1 0 ( , , ) . (19) ( ) 0 j j j j j j j j j t t t u t t t tE
G
E
G
E
G
E
G
E G
E
G
E
E
G
G
E
G
° ° ° ° ® ° ° ° °¯ d d d d d tWe use a power-law function Vi(t) to approximate the nonlinear function Gi(t) in
1 ( ) fij( ), (20) i i j j V t O
n X t kwhere Ȝi is the rate constant and fij is kinetic order. Further, the degradation term
in Eq. (17) is replaced by ȖiXii( )t to emphasize how a gene reacts itself. The
kinetic orders, fij and Ȗi, can be positive or negative; positive kinetic orders
indicate activating influences, but negative kinetic orders mean inhibition. In other words, the following modified power-law dynamic model is proposed.
1 ( ) ( , ) ( ) ( ) ( ) ( ), =1,2, , , (21) i ij i i i i i i n f k i j i i j X t f V t X t X t X t i n
J
O
J
X P kwhere n is the number of the variables; the vector X in Eq. (21) indicates all genes in the yeast cell cycle; the vector P in Eq. (21) consists the rate constants, Ȝi and Ȗi, and kinetic orders, fij and ki. According to 16 genes of yeast cell cycle,
there are 16 differential equations with 288 parameters.
Figure 4.1 is the pathway for the modified power-low model whose fitness is 1.1104688E-02. Black lines represent activation reaction and red lines represent inhibition reaction. The start point of the lines is the reactant and the end point is the product. For instance, the concentration of CDC28 increases rapidly as the concentrations of MBP1, CLN1, CDC6, CDC20 and GRR1 increase; however, the concentration of CDC28 decreases rapidly as the concentrations of CLN3, SWI4, FUS3, FAR1, CDC4, CLB6 andCLB6 increase.
Figure 4.1 The gene regulatory network of the generated modified power-low
4.2 M phase control of Xenopus frog egg
We shall generate the training dataset from the Eqs. (1) ~ (9) to generate the corresponding S-system model of the frog cell cycle to further realize the gene-gene inhibitory and activatory operation for gene and enzyme synthesis and decomposition. The training dataset is shown in Figure 4.2 ~ 4.5.
0 10 20 30 40 50 60 70 80 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 time [min] c o nc en tr at io n Cyclin Unphosphorylated cyclin-Cdc2
Tyr-15 phosphorylated cyclin-Cdc2
0 10 20 30 40 50 60 70 80 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 time [min] c o nc en tr at io n Thr-161 phosphorylated cyclin-Cdc2
doubly phosphorylated cyclin-Cdc2
Figure 4.3 Training dataset-2.
0 10 20 30 40 50 60 70 80 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time [min] c o nc en tr at io n Cdc25 IE
0 10 20 30 40 50 60 70 80 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time [min] c o nc en tr at io n Wee1 APC
Figure 4.5 Training dataset-4.
S-system use power-law flux to describe the synergism and saturation of the biological system, 1 1 , for 1,2,..., , (22) ij ij g h i i j i j j j x
D
n xE
n x i nwhere xi is the state variable or reactant; n is the number of xi.Įi is the production
rate-constant and ȕi is the degradation rate-constant; both can be positive or zero.
gij and hij, are kinetic orders; their values can be positive to indicate activating
influences or negative to denote inhibition. We now construct our S-system structure for Xenopus frog egg as
19 19 11 g 11 h (23) g h x
D
x xE
x x 23 25 26 27 29 23 26 27 29 21 22 22 32 33 34 36 37 1 1 1 9 1 1 9 5 7 7 2 2 1 2 3 6 9 2 2 3 6 9 7 3 3 2 3 4 6 9 (24) g g g g g h h h h g g h g g g g g x x x x x x x x x x x x x x x x x x x xD
E
D
39 32 33 36 37 39 43 44 45 46 47 49 44 45 46 47 49 52 54 55 56 57 59 54 55 56 57 59 7 3 2 3 6 9 5 7 5 7 4 4 3 4 6 9 4 4 6 9 5 5 2 4 5 6 7 9 5 4 5 6 7 9 (25) (26) g h h h h h g g g g g g h h h h h g g g g g g h h h h h x x x x x x x x x x x x x x x x x x x x x x x x x x x x xE
D
E
D
E
65 66 65 66 75 77 75 77 85 88 5 5 6 6 6 6 6 7 7 5 7 7 5 7 5 8 8 8 8 (27) (28) (29) g g h h g g h h g g x x x x x x x x x x x x xD
E
D
E
D
E
85 88 98 99 98 99 5 8 9 9 8 9 9 8 9 (30) (31) h h g g h h x x xD
x xE
x xNote that since concentrations of x1, x2, x3, x4and x5 are too small as compared to
other variables. According to Eqs. (23) ~ (31), there are 83 parameters to estimate. The scale-up operation is adopted to normalize all states variables to a computation reasonable range to improve the computation error. Another test data from Michaelis-Menten model is used to demonstrate the performance of the improved genetic algorithm program, Figure 4.6 ~ 4.14 is the simulation results with the estimated fitness 3.0766122E-08 for N=80,000, Dim_I=83. The low fitness value ensures the good fitting of the simulation results with the datasets and also guarantees the reliability of the generated S-system. From the constructed S model, we can realize the interaction between various genes in Xenopus frog egg. For instance, the concentration of x2 increases rapidly as
concentrations of x1, x3 and x5 increase; the concentration of x2 decreases rapidly
0 20 40 60 80 100 120 140 160 180 200 0.013 0.014 0.015 0.016 0.017 0.018 0.019 0.02 x1 (Cyclin) time [min] c o nc en tr at io n Test data S-system
Figure 4.6 Cyclin evolution.
0 20 40 60 80 100 120 140 160 180 200 0.006 0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.014 0.015 x2 (Unphosphorylated cyclin-Cdc2) time [min] c o nc en tr at io n Test data S-system
0 20 40 60 80 100 120 140 160 180 200 0 1 2 3 4 5 6 7x 10
-3 x3 (Tyr-15 phosphorylated cyclin-Cdc2)
time [min] c o nc en tr at io n Test data S-system
Figure 4.8 Tyr-15 phosphorylated cyclin-Cdc2 evolution.
0 20 40 60 80 100 120 140 160 180 200 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
x4 (doubly phosphorylated cyclin-Cdc2)
time [min] c o nc en tr at io n Test data S-system
0 20 40 60 80 100 120 140 160 180 200 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 x5 (Thr-161 phosphorylated cyclin-Cdc2) time [min] c o nc en tr at io n Test data S-system
Figure 4.10 Thr-161 phosphorylated cyclin-Cdc2 evolution.
0 20 40 60 80 100 120 140 160 180 200 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 x6 (Cdc25) time [min] c o nc en tr at io n Test data S-system Figure 4.11 Cdc25 evolution.
0 20 40 60 80 100 120 140 160 180 200 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 x7 (Wee1) time [min] c o nc en tr at io n Test data S-system
Figure 4.12 Wee1 evolution.
0 20 40 60 80 100 120 140 160 180 200 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x8 (IE) time [min] c o nc en tr at io n Test data S-system Figure 4.13 IE evolution.
0 20 40 60 80 100 120 140 160 180 200 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x9 (APC) time [min] c o nc en tr at io n Test data S-system
Chapter 5
Conclusion
Improved genetic algorithm technique, which involves IEDO, migration operation and Elitism, is adopted to construct mathematical models for yeast and Xenopus frog egg. The gene time-course data form experiment of yeast cell cycle is used to construct gene regulatory network in modified power-law model. The training and test dataset are generated from Michaelis-Menten metabolic model of mitotic cell-cycle control of Xenopus frog egg in S-system. The two proposed gene regulatory networks reveal activatory and inhibitory operations for gene/enzyme synthesis and decomposition. Hence, the networks can provide biological researchers for further experiments in yeast and Xenopus frog egg cell cycle.
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