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]