Lecture 4: Metabolism

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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 (~
10
2
-
10
7
)


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

Discrete

Time

Discrete

Variables

Continuous

Variables

Deterministic

--
No Space
--

--

Space
--

Stochastic

--
No Space
--

--

Space
--

Finite State

Machines

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

Discrete

Variables

Continuous

Variables

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 biology

Darren J. Wilkinson

2006

Molecular systems in cell

Molecular systems in cell

[ ]: concentration of
i
th object

[m
1(in)
]

[m
2
]

[m
3
]

[m
4
]

[m
5
]

[m
1(out)
]

[r
1
]

[r
2
]

[r
3
]

[r

]

[p
1
]

[p
2
]

[p
3
]

[p
4
]

Molecular systems in cell

c
j
:
c
j’
: efficiency of
j
th process

[m
1(in)
]

[m
2
]

[m
3
]

[m
4
]

[m
5
]

[m
1(out)
]

[r
1
]

[r
2
]

[r
3
]

[r

]

[p
1
]

[p
2
]

[p
3
]

[p
4
]

c
1

c
2

c
3

c
4

c
5

c
6

c
7

c
8

c
9

c
10

c
11

c
12

c
13

Molecular systems for small molecules in cell

[m
1(in)
]

[m
2
]

[m
3
]

[m
4
]

[m
5
]

[m
1(out)
]

c
1

c
2

c
3

c
4

c
5

h
1
=c
1
[m
1(out)
]

h
2
=c
2
[m
1(in)
]

h
4
=c
5
[m
2
]

h
3
=c
3
[m
2
]

h
5
=c
4
[m
3
]

c
2


p
1

,r
1


c
5


p
3

,r
3


c
3


p
2

,r
2


c
4


p
4

,r
4


Stochastic selection of reaction based on(
h
1
,
h
2
,
h
3
,
h
4
,
h
5
)

Molecular systems for small molecules in cell

[m
1(in)
]

[m
2
]

[m
3
]

[m
4
]

[m
5
]

[m
1(out)
]=100

c
1

c
2

c
3

c
4

c
5

h
1
=c
1
[m
1(out)
]

= 100 c
1

h
2
=c
2
[m
1(in)
]

h
4
=c
5
[m
2
]

h
3
=c
3
[m
2
]

h
5
=c
4
[m
3
]

c
2


p
1

,r
1


c
5


p
3

,r
3


c
5


p
2

,r
2


c
4


p
4

,r
4


Stochastic selection of reaction based on(
100

c
1
,
h
2
,
h
3
,
h
4
,
h
5
)


Re慣瑩潮‱

Molecular systems for small molecules in cell

[m
1(in)
]=1

[m
2
]=0

[m
3
]=0

[m
4
]=0

[m
5
]=0

[m
1(out)
]=99

c
1

c
2

c
3

c
4

c
5

h
1
=c
1
[m
1(out)
]

= 99 c
1

h
2
=c
2
[m
1(in)
]

= 1 c
2

h
4
=c
5
[m
2
]

=0

h
3
=c
3
[m
2
]

=0

h
5
=c
4
[m
3
]

=0

Stochastic selection of Reaction based on (
99 c
1
,
1 c
2
,
0
,
0
,
0
)



Re慣瑩an 1

Molecular systems for small molecules in cell

[m
1(in)
]=2

[m
2
]=0

[m
3
]=0

[m
4
]=0

[m
5
]=0

[m
1(out)
]=98

c
1

c
2

c
3

c
4

c
5

h
1
=c
1
[m
1(out)
]

= 98

c
1

h
2
=c
2
[m
1(in)
]

= 2 c
2

h
4
=c
5
[m
2
]

=0

h
3
=c
3
[m
2
]

=0

h
5
=c
4
[m
3
]

=0

Stochastic selection of Reaction based on (
98 c
1
,
2 c
2
,
0
,
0
,
0
)



Re慣瑩an 1

Molecular systems for small molecules in cell

[m
1(in)
]=3

[m
2
]=0

[m
3
]=0

[m
4
]=0

[m
5
]=0

[m
1(out)
]=97

c
1

c
2

c
3

c
4

c
5

h
1
=c
1
[m
1(out)
]

= 97

c
1

h
2
=c
2
[m
1(in)
]

= 3 c
2

h
4
=c
5
[m
2
]

=0

h
3
=c
3
[m
2
]

=0

h
5
=c
4
[m
3
]

=0

Stochastic selection of Reaction based on (
97 c
1
,
3 c
2
,
0
,

0
,
0
)



Re慣瑩an 2

Molecular systems for small molecules in cell

[m
1(in)
]=2

[m
2
]=1

[m
3
]=0

[m
4
]=0

[m
5
]=0

[m
1(out)
]=97

c
1

c
2

c
3

c
4

c
5

h
2
=c
2
[m
1(in)
]

= 2 c
2

h
4
=c
5
[m
2
]

=1 c
5

h
3
=c
3
[m
2
]

=1 c
3

h
5
=c
4
[m
3
]

=0

h
1
=c
1
[m
1(out)
]

= 97

c
1

Stochastic selection of Reaction based on (
97 c
1
,

c
2
,
1 c
3
,
0
,
1 c
5
)




Re慣瑩an 1

Molecular systems for small molecules in cell

[m
1(in)
]=3

[m
2
]=1

[m
3
]=0

[m
4
]=0

[m
5
]=0

[m
1(out)
]=96

c
1

c
2

c
3

c
4

c
5

h
1
=c
1
[m
1(out)
]

= 97

c
1

h
2
=c
2
[m
1(in)
]

= 3 c
2

h
4
=c
5
[m
2
]

=1 c
5

h
3
=c
3
[m
2
]

=1 c
3

h
5
=c
4
[m
3
]

=0

Stochastic selection of Reaction(
96 c
1
,
3 c
2
,
1 c
3
,

0
,
1 c
5
)


Re慣瑩an″

Molecular systems for small molecules in cell

[m
1(in)
]=3

[m
2
]=0

[m
3
]=1

[m
4
]=0

[m
5
]=0

[m
1(out)
]=96

c
1

c
2

c
3

c
4

c
5

h
1
=c
1
[m
1(out)
]

= 97

c
1

h
2
=c
2
[m
1(in)
]

= 3 c
2

h
4
=c
5
[m
2
]

=0

h
3
=c
3
[m
2
]

=0

h
5
=c
4
[m
3
]

=1 c
4

Stochastic selection of Reaction based on (
96 c
1
,
3 c
2
,
0
,
1
c
4

, 0
)




Input data

[m
1(in)
]

[m
2
]

[m
3
]

[m
4
]

[m
5
]

[m
1(out)
]

c
1

c
2

c
3

c
4

c
5

c
1

m
1(out)

m
1(in)

c
2

m
1(in)

m
2

c
3

m
2

m
3

m
3

m
5

c
4

m
2

m
5

c
5

[m
1(out)
]

[m
1(in)
]

[m
2
]

[m
3
]

[m
4
]

[m
5
]

Initial concentrations

Reaction parameters and Reactions

Gillespie Algorithm

Step 0
: System Definition

objects (
i

= 1, 2,…,
n
) and their initial quantities:
X
i
(init)

reaction equations (
j
=1,2,…,
m
)


R
j
:
m
(Pre)
j1
X
1

+ ...+ m
(Pre)
jn

X
n

=

m
(Post)

j1

X
1

+...+ m
(Post)

jn
X
n



reaction intensities:
c
i

for R
j

Step 4: Quantities for individual objects are revised base on selected reaction
equation


[
X
i
] ← [X
i
]


m

(Pre)
s

+
m
(Post)
s

Step 1:

[
X
i
]

X
i
(init)

Step 2:

h
j
: :
probability of occurrence of reactions

based on
c
j

(
j
=1,2,..,m) and [
X
i
] (
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 Definition

objects (
i

= 1, 2,…,
n
) and their initial quantities
X
i
(init)

reaction equations (
j
=1,2,…,
m
)

R
j
:
m
(Pre)
j1
X
1

+ ...+ m
(Pre)
jn

X
n

=

m
(Post)

j1

X1 +...+ m
(Post)

jn
X
n


reaction intensities:
c
i

for R
j

Step 4: Quantities for individual objects are revised base on selected reaction
equation

X’
i

= [X
i
]


m

(Pre)
s

+
m
(Post)
s

Step 1:

[
X
i
]

X
i
(init)

Step 2:

h
j
: :
probability of occurrence of reactions

based on
c
j

(
j
=1,2,..,m) and [
X
i
] (
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: [
X
j
]

X’
i

Yes

X’
i



X
i
max

No

Yes

Software: Simple Stochastic Simulator

1.Create stoichiometric data file and initial condition file

Reaction Definition: REQ**.txt

R1

[X1] = [X2]

R2

[X2] = [X1]

Reaction Parameter


ci

[X1]

[X1]

[X2]

[X2]

R1



1

1

0

0

1

R2



1

0

1

1

0

Stoichiometetric data and c
i
: REACTION**.txt

c
i

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 Simulator

2. Stochastic simulation

Stoichiometetric data and c
i
: REACTION**.txt

Initial condition: INIT**.txt

Reaction Parameter

c:

1.0

1.0

//

time



[X1]

[X2]

0.00



100.0

100.0

0.0015706073545097992

101.0

99.0

0.015704610011372147

100.0

100.0

0.01670413203960951

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

0
100
200
300
400
500
600
700
800
900
1000
0
2
4
6
8
[X1]
[X2]
[X1]

[X2]

c=1, [X1]=1000, [X2]=0

[X1]

[X2] [X2]

[X1]

c1=c2=1

[X1]=1000

0
100
200
300
400
500
600
700
800
900
1000
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

0
100
200
300
400
500
600
700
800
900
1000
0
2
4
6
8
10
[X1]
[X2]
[X3]
[X1]


[X2]

[X3], [X1]=1000, c=1

0
100
200
300
400
500
600
700
800
900
1000
0
5
10
15
20
[X1]
[X2]
[X3]
[X1]

[X2]

[X3], [X1]=1000, c=1

0
100
200
300
400
500
600
700
800
900
1000
0
2
4
6
8
10
[X1]
[X2]
[X3]
[X1]

[X2]

[X3],[X1]=1000, c=1

0
100
200
300
400
500
600
700
800
900
1000
0
2
4
6
8
[X1]
[X2]
[X3]
loop reaction [X1]

[X2]

[X3]

[X1], [X1]=1000,
c=1

0
100
200
300
400
500
600
700
800
900
1000
0
2
4
6
8
10
[X1]
[X2]
[X3]
Representation of Reaction

3. Gene Expression and Regulation

Transcription (prokaryotes)

promoter

gene

RNAP

mRNA

promoter + RNAP

promoter

RNAP

promoter + RNAP + gene

promoter

RNAP


# of free promoter is generally 0 (promoter

R乁N⤠潲‱‡o

Stochastic simulation

0

5

10

0

2

4

6

8

10

[promoter]

[RNAP]

[promoter.RNAP]

[gene]

3. Gene Expression and Regulation

Transcription (prokaryotes)

Representation of Reaction

3. Gene Expression and Regulation

Transcription (prokaryotes)

promoter1

gene

RNAP

mRNA1

promoter1 + RNAP

promoter1

RNAP

promoter1 + RNAP + mRNA1

promoter1

RNAP


# of free promoter is 0 (promoter

R乁N⤠潲oㄠ℠

promoter2

gene

RNAP

mRNA2

promoter2
+ RNAP

promoter2

RNAP

promoter2
+ RNAP + mRNA2

promoter2

RNAP

Stochastic Simulation

1. Stoichiometric chemical reaction

Reaction Data

[X
1
]

2[X
1
]

c
1

[X
1
] +
[X
2
]

2[X
2
]

c
2

[X
2
]

c
3

Stochastic modeling for systems biology

Darren J. Wilkinson

2006

Representation of Reaction

Data Set

[X
1
]

2[X
1
]

c
1

[X
1
] + [X
2
]

2[X
2
]

c
2

[X
2
]

Φ

c
3

Reaction Data

Initial Condition

[X
1
]= X
1
(init)

[X
2
]= X
2
(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] + phosphate

fbaA

[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(+) + phosphate

pgk

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)O

eno

[phosphoenolpyruvate] + H(2)O <
-
> [2
-
phospho
-
D
-
glycerate]

Example 2 EMP

D
-
glucose


alpha
-
D
-
glucose

D
-
fructose
-
6
-
phosphate

alpha
-
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]