Post on 22-Mar-2016
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Lecture 4: Metabolism
•Reaction system as ordinary differential equations•Reaction system as stochastic process
Introduction
Metabolism is the process through which living cells acquire energy and building material for cell components and replenishing enzymes.
Metabolism is the general term for two kinds of reactions: (1) catabolic reactions –break down of complex compounds to get energy and building blocks, (2) anabolic reactions—construction of complex compounds used in cellular functioning
How can we model metabolic reactions?
What is a Model?Formal representation of a system using--Mathematics--Computer program
Describes mechanisms underlying outputs
Dynamical models show rate of changes with time or other variable
Provides explanations and predictions
Typical network of metabolic pathways
Reactions are catalyzed by enzymes. One enzyme molecule usually catalyzes thousands reactions per second (~102-107)
The pathway map may be considered as a static model of metabolism
Dynamic modeling of metabolic reactions is the process of understanding the reaction rates i.e. how the concentrations of metabolites change with respect to time
An Anatomy of Dynamical Models
DiscreteTime
DiscreteVariables
ContinuousVariables
Deterministic--No Space -- -- Space --
Stochastic
--No Space -- -- Space --
Finite StateMachines
Boolean Networks;Cellular Automata
Discrete Time Markov Chains
Stochastic Boolean Networks;Stochastic Cellular Automata
Iterated Functions;Difference Equations
Iterated Functions;Difference Equations
Discrete Time Markov Chains
Coupled Discrete Time Markov Chains
Continuous Time
DiscreteVariables
ContinuousVariables
Boolean Differential Equations
Ordinary Differential Equations
Coupled Boolean Differential Equations
Partial Differential Equations
Continuous Time Markov Chain
Stochastic Ordinary Differential Equations
Coupled Continuous Time Markov Chains
Stochastic Partial Differential Equations
Differential equations
Differential equations are based on the rate of change of one or more variables with respect to one or more other variables
An example of a differential equation
Source: Systems biology in practice by E. klipp et al
An example of a differential equation
Source: Systems biology in practice by E. klipp et al
Source: Systems biology in practice by E. klipp et al
An example of a differential equation
Schematic representation of the upper part of the Glycolysis
Source: Systems biology in practice by E. klipp et al.
The ODEs representing this reaction system
Realize that the concentration of metabolites and reaction rates v1, v2, …… are functions of time
ODEs representing a reaction system
The rate equations can be solved as follows using a number of constant parameters
The temporal evaluation of the concentrations using the following parameter values and initial concentrations
Notice that because of bidirectional reactions Gluc-6-P and Fruc-6-P reaches peak earlier and then decrease slowly and because of unidirectional reaction Fruc1,6-P2 continues to grow for longer time.
The use of differential equations assumes that the concentration of metabolites can attain continuous value.But the underlying biological objects , the molecules are discrete in nature.When the number of molecules is too high the above assumption is valid.But if the number of molecules are of the order of a few dozens or hundreds then discreteness should be considered.Again random fluctuations are not part of differential equations but it may happen for a system of few molecules.The solution to both these limitations is to use a stochastic simulation approach.
Stochastic Simulation
Stochastic modeling for systems biologyDarren J. Wilkinson2006
Molecular systems in cell
Molecular systems in cell[ ]: concentration of ith object
[m1(in)] [m2] [m3]
[m4]
[m5]
[m1(out)]
[r1] [r2] [r3] [r ４ ]
[p1][p2]
[p3]
[p4]
Molecular systems in cellcj: cj’: efficiency of jth process
[m1(in)] [m2] [m3]
[m4]
[m5]
[m1(out)]
[r1] [r2] [r3] [r ４ ]
[p1][p2]
[p3]
[p4]
c1
c2
c3 c4
c5c6
c7
c8
c9
c10
c11
c12
c13
Molecular systems for small molecules in cell
[m1(in)] [m2] [m3]
[m4]
[m5]
[m1(out)]
c1
c2
c3 c4
c5h1=c1 [m1(out)] h2=c2 [m1(in)]
h4=c5 [m2]
h3=c3 [m2] h5=c4 [m3]
c2 p1 ,r1
c5 p3 ,r3
c3 p2 ,r2 c4 p4 ,r4
Stochastic selection of reaction based on(h1, h2, h3, h4, h5)
Molecular systems for small molecules in cell
[m1(in)] [m2] [m3]
[m4]
[m5]
[m1(out)]=100
c1
c2
c3 c4
c5h1=c1 [m1(out)] = 100 c1
h2=c2 [m1(in)]
h4=c5 [m2]
h3=c3 [m2] h5=c4 [m3]
c2 p1 ,r1
c5 p3 ,r3
c5 p2 ,r2 c4 p4 ,r4
Stochastic selection of reaction based on(100 c1, h2, h3, h4, h5)Reaction 1
Molecular systems for small molecules in cell
[m1(in)]=1[m2]=0
[m3]=0
[m4]=0
[m5]=0
[m1(out)]=99
c1
c2
c3 c4
c5h1=c1 [m1(out)]= 99 c1
h2=c2 [m1(in)]= 1 c2
h4=c5 [m2]=0
h3=c3 [m2]=0
h5=c4 [m3]=0
Stochastic selection of Reaction based on (99 c1, 1 c2, 0, 0, 0) Reaction 1
Molecular systems for small molecules in cell
[m1(in)]=2[m2]=0
[m3]=0
[m4]=0
[m5]=0
[m1(out)]=98
c1
c2
c3 c4
c5h1=c1 [m1(out)]= 98 c1
h2=c2 [m1(in)]= 2 c2
h4=c5 [m2]=0
h3=c3 [m2]=0
h5=c4 [m3]=0
Stochastic selection of Reaction based on (98 c1, 2 c2, 0, 0, 0) Reaction 1
Molecular systems for small molecules in cell
[m1(in)]=3[m2]=0
[m3]=0
[m4]=0
[m5]=0
[m1(out)]=97
c1
c2
c3 c4
c5h1=c1 [m1(out)]= 97 c1
h2=c2 [m1(in)]= 3 c2
h4=c5 [m2]=0
h3=c3 [m2]=0
h5=c4 [m3]=0
Stochastic selection of Reaction based on (97 c1, 3 c2, 0, 0, 0) Reaction 2
Molecular systems for small molecules in cell
[m1(in)]=2[m2]=1
[m3]=0
[m4]=0
[m5]=0
[m1(out)]=97
c1
c2
c3 c4
c5h2=c2 [m1(in)]= 2 c2
h4=c5 [m2]=1 c5
h3=c3 [m2]=1 c3
h5=c4 [m3]=0
h1=c1 [m1(out)]= 97 c1
Stochastic selection of Reaction based on (97 c1, c2, 1 c3, 0, 1 c5) Reaction 1
Molecular systems for small molecules in cell
[m1(in)]=3 [m2]=1[m3]=0
[m4]=0
[m5]=0
[m1(out)]=96
c1
c2
c3 c4
c5h1=c1 [m1(out)]= 97 c1
h2=c2 [m1(in)]= 3 c2
h4=c5 [m2]=1 c5
h3=c3 [m2]=1 c3
h5=c4 [m3]=0
Stochastic selection of Reaction(96 c1, 3 c2, 1 c3, 0, 1 c5)Reaction 3
Molecular systems for small molecules in cell
[m1(in)]=3 [m2]=0[m3]=1
[m4]=0
[m5]=0
[m1(out)]=96
c1
c2
c3 c4
c5h1=c1 [m1(out)]= 97 c1
h2=c2 [m1(in)]= 3 c2
h4=c5 [m2]=0
h3=c3 [m2]=0
h5=c4 [m3]=1 c4
Stochastic selection of Reaction based on (96 c1, 3 c2, 0, 1 c4 , 0)…
Input data
[m1(in)] [m2] [m3]
[m4]
[m5]
[m1(out)]
c1
c2
c3 c4
c5
c1m1(out) m1(in)
c2m1(in) m2
c3m2 m3 m3 m5
c4
m2 m5
c5
[m1(out)] [m1(in)] [m2] [m3] [m4] [m5]Initial concentrations
Reaction parameters and Reactions
Gillespie AlgorithmStep 0: System Definitionobjects (i = 1, 2,…, n) and their initial quantities: Xi(init) reaction equations (j=1,2,…,m)
Rj: m(Pre)j1 X1 + ...+ m(Pre)
jn Xn = m (Post) j1 X1 +...+ m (Post)
jnXn
reaction intensities: ci for Rj
Step 4: Quantities for individual objects are revised base on selected reaction equation[Xi] ← [Xi] – m (Pre)
s + m(Post)s
Step 1: [Xi]Xi(init)
Step 2: hj: :probability of occurrence of reactions based on cj (j=1,2,..,m) and [Xi] (i=1,2,..,n)
Step 3: Random selection of reaction Here a selected reaction is represented by index s.
Gillespie Algorithm (minor revision)
Step 0: System Definitionobjects (i = 1, 2,…, n) and their initial quantities Xi(init) reaction equations (j=1,2,…,m)Rj: m(Pre)
j1 X1 + ...+ m(Pre)jn Xn = m (Post)
j1 X1 +...+ m (Post) jnXn
reaction intensities: ci for Rj
Step 4: Quantities for individual objects are revised base on selected reaction equation X’i = [Xi] – m (Pre)
s + m(Post)s
Step 1: [Xi]Xi(init)
Step 2: hj: :probability of occurrence of reactions based on cj (j=1,2,..,m) and [Xi] (i=1,2,..,n)
Step 3: Random selection of reaction Here a selected reaction is represented by index s.
X’i 0 No
Step 5: [Xj] X’i
YesX’i Xi
max No
Yes
Software: Simple Stochastic Simulator1.Create stoichiometric data file and initial condition file
Reaction Definition: REQ**.txtR1 [X1] = [X2]R2 [X2] = [X1]
Reaction Parameter ci [X1] [X1] [X2] [X2]R1 1 1 0 0 1R2 1 0 1 1 0
Stoichiometetric data and ci: REACTION**.txt
ci is set by user
[X1] 100 0[X2] 100 0
Initial condition: INIT**.txt
max number (for ith object, max number is set by 0 for ith , [Xi]0 Initial quantitiy
Objects used are assigned by [ ] .
Software: Simple Stochastic Simulator2. Stochastic simulation
Stoichiometetric data and ci: REACTION**.txt
Initial condition: INIT**.txt
Reaction Parameterc: 1.0 1.0//time [X1] [X2]0.00 100.0 100.00.0015706073545097992 101.0 99.00.015704610011372147 100.0 100.00.01670413203960951 101.0 99.0….….
Simulation results: SIM**.txt
0
50
100
150
0 10 20 30 40 50
[X1][X2]
Example of simulation results# of type of chemicals =2
01002003004005006007008009001000
0 2 4 6 8
[X1][X2]
[X1][X2] c=1, [X1]=1000, [X2]=0
[X1][X2] [X2][X1]c1=c2=1[X1]=1000
01002003004005006007008009001000
0 1 2 3 4 5 6 7 8 9 10
[X1][X2]
# of type of chemicals =3
[X1][X2][X3], [X1]=1000, c=1
01002003004005006007008009001000
0 2 4 6 8 10
[X1][X2][X3]
[X1] [X2][X3], [X1]=1000, c=1
01002003004005006007008009001000
0 5 10 15 20
[X1][X2][X3]
[X1][X2][X3], [X1]=1000, c=1
01002003004005006007008009001000
0 2 4 6 8 10
[X1][X2][X3]
[X1][X2][X3],[X1]=1000, c=1
01002003004005006007008009001000
0 2 4 6 8
[X1][X2][X3]
loop reaction [X1][X2][X3][X1], [X1]=1000, c=1
01002003004005006007008009001000
0 2 4 6 8 10
[X1][X2][X3]
Representation of Reaction3. Gene Expression and RegulationTranscription (prokaryotes)
promoter gene
RNAP
mRNA
promoter + RNAP promoter ・ RNAP
promoter + RNAP + genepromoter ・ RNAP
# of free promoter is generally 0 (promoter ・ RNAP) or 1 !
Stochastic simulation
0
5
10
0 2 4 6 8 10[promoter][RNAP]
[promoter.RNAP]
[gene]
3. Gene Expression and RegulationTranscription (prokaryotes)
Representation of Reaction3. Gene Expression and RegulationTranscription (prokaryotes)
promoter1 gene
RNAP
mRNA1
promoter1 + RNAP promoter1 ・ RNAPpromoter1 + RNAP + mRNA1promoter1 ・ RNAP
# of free promoter is 0 (promoter ・ RNAP) or 1 !
promoter2 gene
RNAP
mRNA2
promoter2 + RNAP promoter2 ・ RNAPpromoter2 + RNAP + mRNA2promoter2 ・ RNAP
Stochastic Simulation1. Stoichiometric chemical reaction
Reaction Data
[X1] 2[X1]c1
[X1] + [X2] 2[X2]c2
[X2]c3
Stochastic modeling for systems biologyDarren J. Wilkinson2006
Representation of ReactionData Set
[X1] 2[X1]c1
[X1] + [X2] 2[X2]c2
[X2] Φc3
Reaction Data Initial Condition
[X1]= X1(init)
[X2]= X2(init)
Example 2 EMP
glcK ATP + [D-glucose] -> ADP + [D-glucose-6-phosphate]glcK ATP + [alpha-D-glucose] -> ADP + [D-glucose-6-phosphate]pgi [D-glucose-6-phosphate] <-> [D-fructose-6-phosphate]pgi [D-fructose-6-phosphate] <-> [D-glucose-6-phosphate]pgi [alpha-D-glucose-6-phosphate] <-> [D-fructose-6-phosphate]pgi [D-fructose-6-phosphate] <-> [alpha-D-glucose-6-phosphate] pfk ATP + [D-fructose-6-phosphate] -> ADP + [D-fructose-1,6-bisphosphate]fbp [D-fructose-1,6-bisphosphate] + H(2)O -> [D-fructose-6-phosphate] + phosphatefbaA [D-fructose-1,6-bisphosphate] <-> [glycerone-phosphate] + [D-glyceraldehyde-3-phosphate]fbaA [glycerone-phosphate] + [D-glyceraldehyde-3-phosphate] <-> [D-fructose-1,6-bisphosphate]tpiA [glycerone-phosphate] <-> [D-glyceraldehyde-3-phosphate]tpiA [D-glyceraldehyde-3-phosphate] <-> [glycerone-phosphate]gapA [D-glyceraldehyde-3-phosphate] + phosphate + NAD(+) -> [1,3-biphosphoglycerate] + NADH + H(+)gapB [1,3-biphosphoglycerate] + NADPH + H(+) -> [D-glyceraldehyde-3-phosphate] + NADP(+) + phosphatepgk ADP + [1,3-biphosphoglycerate] <-> ATP + [3-phospho-D-glycerate]pgk ATP + [3-phospho-D-glycerate] <-> ADP + [1,3-biphosphoglycerate]pgm [3-phospho-D-glycerate] <-> [2-phospho-D-glycerate]pgm [2-phospho-D-glycerate] <-> [3-phospho-D-glycerate]eno [2-phospho-D-glycerate] <-> [phosphoenolpyruvate] + H(2)Oeno [phosphoenolpyruvate] + H(2)O <-> [2-phospho-D-glycerate]
Example 2 EMP
D-glucose alpha-D-glucose
D-fructose-6-phosphatealpha-D-glucose-6-phosphate
[D-fructose-1,6-bisphosphate]
[D-glyceraldehyde-3-phosphate]
D-glucose-6-phosphate
[glycerone-phosphate]
[1,3-biphosphoglycerate]
[3-phospho-D-glycerate]
[2-phospho-D-glycerate]
[phosphoenolpyruvate]