Team:NTU-Singapore/Modelling/ODE

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<div id="maincontent" style="margin-top:200px;">
<div id="maincontent" style="margin-top:200px;">
-
=ODEs used in modeling=
+
='''ODEs used in modeling'''=
-
The following equations shows the break down of the different equations that will be used in this modeling exercise. By understanding this section, it would make the understanding of the system of ODEs used
+
In this section, we will have a short discussion on the ODEs used for Deterministic Modeling in our iGem project.
 +
For the modeling exercise, we use mainly simple linear ODEs to describe the system we have at hand. These ODEs are used to describe processes such as constitutive transcription/ translation etc. Nonlinear equations are used for more complex processes such as regulated transcription. The choice of these equations are in hope that they can describe the system to a relative degree of accuracy and yet be simple enough to be solved. This is very much an engineering compromise between a rigorous description and getting an answer that is good enough which is required.
 +
Our systems are based on the assumptions that Lumped parameter models are sufficient to describe them. <br>
 +
Lumped parameter models are those in which spatial variations are ignored and the dependent variables are uniform throughout the whole system. The only variations are those with respect to time leading to mathematical description in terms of the ODEs used.
 +
 +
 +
The following equations show the break down of the different equations that will be used in this modeling exercise. By understanding this section, it would make the understanding of the system of ODEs used in our iGEm modeling exercise.
 +
[[Image:Modeling cartoon.gif|center]]
 +
 +
---------------------------------------------------------------------------------------------------------------
==Constant synthesis & Linear Synthesis==
==Constant synthesis & Linear Synthesis==
-
[[Image:Linear_synthesis.JPG‎ |100px|Linear Synthesis]]
+
[[Image:Linear_synthesis.JPG‎|100px|Linear Synthesis]]<br>
*Simple ode to describe constant synthesis
*Simple ode to describe constant synthesis
Line 38: Line 47:
==Phosphorylation and Dephosphorylation==
==Phosphorylation and Dephosphorylation==
Assumptions:
Assumptions:
-
*Linear kintic rate laws apply only if XT is much less than the Michaelis constants of both kinase and phosphotase
+
*Linear kinetic rate laws apply only if XT is much less than the Michaelis constants of both kinase and phosphotase
[[Image:Phosphorylation.JPG|200px|Phosphorylation and Dephosphorylation]]<br>
[[Image:Phosphorylation.JPG|200px|Phosphorylation and Dephosphorylation]]<br>
-
XT : total cost of X protein in phosphorylated and unphosphorylated form<br>
+
{|border="0"|
-
S : protein kinase concentration<br>
+
|XT :
-
k2 : accounts for protein phosphotase<br>
+
|total cost of X protein in phosphorylated and unphosphorylated form
 +
|-
 +
|S :
 +
|protein kinase concentration
 +
|-
 +
|k2 :
 +
|accounts for protein phosphotase
 +
|}
*Modeled after simple linear kinetics
*Modeled after simple linear kinetics
Line 51: Line 67:
[[Image:Regulated1.JPG‎|260px|Regulated1]]<br>
[[Image:Regulated1.JPG‎|260px|Regulated1]]<br>
-
 
+
{|border="0"|
-
[P]: Protein Formed<br>
+
|[P]:
-
µ: Repression, µ=0;<br> Activation, µ=1<br>
+
|Protein Formed
-
K: Hill Constant Value of input that gives 50% response<br>
+
|-
-
n: Hill coefficient Slope of signal-response curve at this input signal<br>
+
|µ:
-
d: degradation of protein<br>
+
|Repression, µ=0;<br> Activation, µ=1
-
k1: basal gene expression<br>
+
|-
-
k: signal-dependent gene expression<br>
+
|K:
-
a: correlation between k1 and k, 0<a<1<br>
+
|Hill Constant Value of input that gives 50% response
 +
|-
 +
|n:
 +
|Hill coefficient Slope of signal-response curve at this input signal
 +
|-
 +
|d:
 +
|degradation of protein
 +
|-
 +
|k1:
 +
|basal gene expression
 +
|-
 +
|k:
 +
|signal-dependent gene expression
 +
|-
 +
|a:
 +
|correlation between k1 and k, 0<a<1
 +
|-
 +
|}
[[Image:Regulated2.JPG‎|280px|Regulated2]]<br>
[[Image:Regulated2.JPG‎|280px|Regulated2]]<br>
This ODE attempts to capture characteristics of the mRNA dynamics<br>
This ODE attempts to capture characteristics of the mRNA dynamics<br>
For our modeling, all our detection systems activates some form of transcription. Therefore µ=1 in all cases for our modeling exercise.
For our modeling, all our detection systems activates some form of transcription. Therefore µ=1 in all cases for our modeling exercise.
-
[[Image:Regulated3.JPG‎|220px|Regulated3]]<br>
+
[[Image:Regulated3.JPG‎|240px|Regulated3]]<br>
-
 
+
=ODE system used in model=
=ODE system used in model=
 +
The system can be viewed as two parts. The first part comprises of lactose induced production of colicin E7 and the immunity protein. The second part comprises of a detection mechanism that produces the lysis protein upon the detection of both Iron ions and Ai-2 ( Autoinducer 2).
==Lactose controlled production of E7 + Imm==
==Lactose controlled production of E7 + Imm==
-
===Variables===
+
'''Variables'''
-
*LacI = A
+
{|border="0"|
-
*Lactose =B
+
|LacI =
-
*E7 = C
+
|A
 +
|-
 +
|Lactose =
 +
|B
 +
|-
 +
|E7 =
 +
|C
 +
|}
-
# LacI production
+
===LacI production===
-
#* Transcription of LacI gene mRNA
+
* '''Transcription of LacI gene mRNA'''
[[Image:Lac1.JPG‎ |200px|Lac1]]<br>
[[Image:Lac1.JPG‎ |200px|Lac1]]<br>
-
[MA] : LacI mRNA concentration<br>
+
{|border="0"|
-
k1A : kinetic constant of transcription<br>
+
|[MA]:
-
U : system input<br>
+
|LacI mRNA concentration
-
d1A : degradation constant for mRNA<br>
+
|-
 +
|k1A:
 +
|kinetic constant of transcription
 +
|-
 +
|U:
 +
|system input
 +
|-
 +
|d1A:
 +
|degradation constant for mRNA
 +
|}
-
#* Translation of LacI Protein
+
* '''Translation of LacI Protein'''
[[Image:Lac2.JPG‎ |200px|Lac2]]<br>
[[Image:Lac2.JPG‎ |200px|Lac2]]<br>
-
[PA] : Protein concentration<br>
+
{|border="0"|
-
k2A : kinetic constant of translation<br>
+
|[PA]:
-
d2A : degradation constant for Protein<br>
+
|Protein concentration
-
 
+
|-
-
#Complex formation between LacI and Lactose
+
|k2A:
 +
|kinetic constant of translation
 +
|-
 +
|d2A:
 +
|degradation constant for Protein
 +
|}
 +
===Complex formation between LacI and Lactose===
[[Image:Complex1.JPG‎ |300px|Complex]]<br>
[[Image:Complex1.JPG‎ |300px|Complex]]<br>
-
[ComAB] : Complex of LacI and Lactose<br>
+
{|border="0"|
-
K3AB : Rate constant of complex formation<br>
+
|[ComAB] :
-
[PA]o : Protein concentration of LacI at the time of Lactose addition<br>
+
|Complex of LacI and Lactose
-
[B]o : Initial concentration of Lactose<br>
+
|-
 +
|K3AB :
 +
|Rate constant of complex formation
 +
|-
 +
|[PA]o :
 +
|Protein concentration of LacI at the time of Lactose addition
 +
|-
 +
|[B]o :
 +
|Initial concentration of Lactose
 +
|}
-
#E7 production
+
===E7 production===
-
#* Transcription of E7 gene mRNA
+
* '''Transcription of E7 gene mRNA'''
[[Image:E71.JPG‎ |300px|E71]]<br>
[[Image:E71.JPG‎ |300px|E71]]<br>
-
[MC]: E7 mRNA Concentration<br>
+
{|border="0"|
-
k1C: kinetic constant of transcription<br>
+
|[MC]:
-
a : constitutive portion , 0<a<1<br>
+
|E7 mRNA Concentration
-
KC: Hill constant<br>
+
|-
-
nC: Hill coefficient<br>
+
|k1C:
-
d1C: degradation constant for mRNA<br>
+
|kinetic constant of transcription
 +
|-
 +
|a :
 +
|constitutive portion , 0<a<1
 +
|-
 +
|KC:
 +
|Hill constant
 +
|-
 +
|nC:
 +
|Hill coefficient
 +
|-
 +
|d1C:
 +
|degradation constant for mRNA
 +
|}
-
#* Translation of E7 Protein
+
* '''Translation of E7 Protein'''
[[Image:E72.JPG‎ |200px|E72]]<br>
[[Image:E72.JPG‎ |200px|E72]]<br>
-
[PC] : Protein concentration<br>
+
{|border="0"|
-
k2C : kinetic constant of translation<br>
+
|[PC] :
-
d2C : degradation constant for Protein<br>
+
|Protein concentration
 +
|-
 +
|k2C :
 +
|kinetic constant of translation
 +
|-
 +
|d2C :
 +
|degradation constant for Protein
 +
|}
 +
 
 +
==Variables of Iron and Ai2 controlled production of Lysis==
-
==Iron and Ai2 controlled production of Lysis==
+
'''Variables'''
-
===Variables===
+
{|border="0"|
-
Variables
+
|#Ai-2:
-
#Ai-2 : A
+
|A
-
#ai-2-phos : B
+
|-
-
#LsrR : C
+
|#ai-2-phos:
-
#SupD derivatives : D
+
|B
-
#T7ptag : E
+
|-
-
#Lysis : F
+
|#LsrR:
 +
|C
 +
|-
 +
|#SupD derivatives:
 +
|D
 +
|-
 +
|#T7ptag:
 +
|E
 +
|-
 +
|#Lysis:
 +
|F
 +
|}
-
#Phosphorylation of ai-2
+
===Phosphorylation of Ai-2===
[[Image:Phos.JPG‎ |200px|Phos]]<br>
[[Image:Phos.JPG‎ |200px|Phos]]<br>
-
A: Ai-2<br>
+
{|border="0"|
-
B: Ai-2-phos<br>
+
|A:
-
kPF: Forward Phosphorylation<br>
+
|Ai-2
-
S: Concentration of Protein Kinase<br>
+
|-
-
kPB: Backward dephosphorylation<br>
+
|B:
 +
|Ai-2-phos
 +
|-
 +
|kPF:
 +
|Forward Phosphorylation
 +
|-
 +
|S:
 +
|Concentration of Protein Kinase
 +
|-
 +
|kPB:
 +
|Backward dephosphorylation
 +
|}
-
# LsrR production
+
===LsrR production===
-
#* Transcription of LsrR gene mRNA
+
*'''Transcription of LsrR gene mRNA'''
[[Image:LsrR1.JPG‎ |200px|LsrR1]]<br>
[[Image:LsrR1.JPG‎ |200px|LsrR1]]<br>
-
[MC]: LacI mRNA concentration<br>
+
{|border="0"|
-
k1C : kinetic constant of transcription<br>
+
|[MC]:
-
U : system input<br>
+
|LacI mRNA concentration
-
d1C: degradation constant for mRNA,br>
+
|-
 +
|k1C:
 +
|kinetic constant of transcription
 +
|-
 +
|U:
 +
|system input
 +
|-
 +
|d1C:
 +
|degradation constant for mRNA
 +
|}
-
#* Translation of LsrR protein
+
* '''Translation of LsrR protein'''
[[Image:LsrR2.JPG‎ |200px|LsrR2]]<br>
[[Image:LsrR2.JPG‎ |200px|LsrR2]]<br>
-
[PC] : Protein concentration<br>
+
{|border="0"|
-
k2C : kinetic constant of translation<br>
+
|[PC] :
-
d2C : degradation constant for Protein<br>
+
|Protein concentration
 +
|-
 +
|k2C:
 +
|kinetic constant of translation
 +
|-
 +
|d2C:
 +
|degradation constant for Protein
 +
|}
-
#Complex formation between Ai-2-phos and LsrR
+
===Complex formation between Ai-2-phos and LsrR===
-
[[Image:Complex2.JPG‎ |200px|Complex2]]<br>
+
[[Image:Complex2.JPG‎ |350px|Complex2]]<br>
-
[ComAB] : Complex of LsrR and Ai-2-phos<br>
+
{|border="0"|
-
k3BC : Rate constant of complex formation<br>
+
|[ComAB] :
-
[PC]o: Protein concentration of LsrR at the time of start of complex formation <br>
+
|Complex of LsrR and Ai-2-phos
-
[B]o: Initial concentration of Ai-2-phos<br>
+
|-
 +
|k3BC :
 +
|Rate constant of complex formation
 +
|-
 +
|[PC]o:
 +
|Protein concentration of LsrR at the time of start of complex formation
 +
|-
 +
|[B]o:
 +
|Initial concentration of Ai-2-phos
 +
|}
-
The hard part here is to determine what is the initial concentration of the protein LsrR and Ai-2-phos. A way out of this mess is to simplify the equation even more.  
+
The hard part here is to determine what is the initial concentration of the protein LsrR and Ai-2-phos. A way out of this mess is to simplify the equation even more.
-
#AND GATE odes
+
===AND GATE odes===
-
#*Transcription of SupD gene mRNA
+
*'''Transcription of SupD gene mRNA'''
-
[[Image:SupD1.JPG‎ |200px|SupD]]<br>
+
[[Image:SupD1.JPG‎ |300px|SupD]]<br>
-
[MD]: t7 mRNA Concentration <br>
+
{|border="0"|
-
k1D: kinetic constant of transcription<br>
+
|[MD]:
-
a : constitutive portion , 0<a<1<br>
+
|t7 mRNA Concentration  
-
KD: Hill constant<br>
+
|-
-
nD: Hill coefficient<br>
+
|k1D:
-
d1D: degradation constant for mRNA<br>
+
|kinetic constant of transcription
 +
|a :
 +
|constitutive portion , 0<a<1
 +
|-
 +
|KD:
 +
|Hill constant
 +
|-
 +
|nD:
 +
|Hill coefficient
 +
|-
 +
|d1D:
 +
|degradation constant for mRNA
 +
|}
-
#*Transcription of T7ptag gene mRNA
+
*'''Transcription of T7ptag gene mRNA'''
-
[[Image:T71.JPG‎ |200px|T71]]<br>
+
[[Image:T71.JPG‎ |300px|T71]]<br>
-
[ME]: supD mRNA Concentration <br>
+
{|border="0"|
-
k1E: kinetic constant of transcription<br>
+
|[ME]:
-
a : constitutive portion , 0<a<1<br>
+
|supD mRNA Concentration
-
KE: Hill constant<br>
+
|-
-
nE: Hill coefficient<br>
+
|k1E:
-
d1E: degradation constant for mRNA<br>
+
|kinetic constant of transcription
 +
|-
 +
|a :
 +
|constitutive portion , 0<a<1
 +
|-
 +
|KE:
 +
|Hill constant
 +
|-
 +
|nE:
 +
|Hill coefficient
 +
|-
 +
|d1E:
 +
|degradation constant for mRNA
 +
|}
-
#*Complex formation between tRNA and mRNA
+
*'''Complex formation between tRNA and mRNA'''
-
[[Image:Complex3.JPG‎ |200px|Complex3]]<br>
+
[[Image:Complex3.JPG‎ |350px|Complex3]]<br>
-
[ComDE] : Complex of tRNA and mRNA t7<br>
+
{|border="0"|
-
k3DE : Rate constant of complex formation<br>
+
|[ComDE] :
-
[MD]o : concentration of tRNA at the time of start of complex formation<br>
+
|Complex of tRNA and mRNA t7
-
[ME]o : Initial concentration of mRNA t7<br>
+
|-
 +
|k3DE :
 +
|Rate constant of complex formation
 +
|-
 +
|[MD]o :
 +
|concentration of tRNA at the time of start of complex formation
 +
|-
 +
|[ME]o :
 +
|Initial concentration of mRNA t7
 +
|}
-
 
+
*'''Translation of t7 protein'''
-
#*Translation of t7 protein
+
[[Image:T72.JPG‎ |200px|Complex3]]<br>
[[Image:T72.JPG‎ |200px|Complex3]]<br>
-
[PF] : Protein concentration<br>
+
{|border="0"|
-
k2F : kinetic constant of translation<br>
+
|[PF] :
-
d2F : degradation constant for Protein<br>
+
|Protein concentration
 +
|-
 +
|k2F :
 +
|kinetic constant of translation
 +
|-
 +
|d2F :
 +
|degradation constant for Protein
 +
|}
-
#Lysis
+
===Lysis production===
-
#*Transcription of Lysis gene mRNA
+
*'''Transcription of Lysis gene mRNA'''
-
[[Image:Lysis1.JPG‎ |200px|Lysis]]<br>
+
[[Image:Lysis1.JPG‎ |300px|Lysis]]<br>
-
[MF]: Lysis mRNA Concentration <br>
+
{|border="0"|
-
k1F: kinetic constant of transcription<br>
+
|[MF]:
-
a : constitutive portion , 0<a<1<br>
+
|Lysis mRNA Concentration
-
KF: Hill constant<br>
+
|-
-
nF: Hill coefficient<br>
+
|k1F:
-
d1F: degradation constant for mRNA<br>
+
|kinetic constant of transcription
 +
|a :
 +
|constitutive portion , 0<a<1
 +
|-
 +
|KF:
 +
|Hill constant
 +
|-
 +
|nF:
 +
|Hill coefficient
 +
|-
 +
|d1F:
 +
|degradation constant for mRNA
 +
|}
-
 
+
*'''Translation of Lysis protein'''
-
#*Translation of Lysis protein
+
[[Image:Lysis2.JPG‎ |200px|Lysis2]]<br>
[[Image:Lysis2.JPG‎ |200px|Lysis2]]<br>
-
[PF] : Protein concentration<br>
+
{|border="0"|
-
k2F : kinetic constant of translation<br>
+
|[PF] :
-
d2F : degradation constant for Protein<br>
+
|Protein concentration
 +
|-
 +
|k2F :
 +
|kinetic constant of translation
 +
|-
 +
|d2F :
 +
|degradation constant for Protein
 +
|}
 +
-----------------------------------------------------------------------------------------------------------------
 +
<br><br>
 +
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Latest revision as of 05:42, 27 October 2008

Contents

ODEs used in modeling

In this section, we will have a short discussion on the ODEs used for Deterministic Modeling in our iGem project.

For the modeling exercise, we use mainly simple linear ODEs to describe the system we have at hand. These ODEs are used to describe processes such as constitutive transcription/ translation etc. Nonlinear equations are used for more complex processes such as regulated transcription. The choice of these equations are in hope that they can describe the system to a relative degree of accuracy and yet be simple enough to be solved. This is very much an engineering compromise between a rigorous description and getting an answer that is good enough which is required.

Our systems are based on the assumptions that Lumped parameter models are sufficient to describe them.
Lumped parameter models are those in which spatial variations are ignored and the dependent variables are uniform throughout the whole system. The only variations are those with respect to time leading to mathematical description in terms of the ODEs used.


The following equations show the break down of the different equations that will be used in this modeling exercise. By understanding this section, it would make the understanding of the system of ODEs used in our iGEm modeling exercise.

Modeling cartoon.gif

Constant synthesis & Linear Synthesis

Linear Synthesis

  • Simple ode to describe constant synthesis
  • Gives an explicit analytical solution
  • Unique solution once a IC is posed

Linear Degradation

Linear Degradation

  • Rate of degradation is proportional to how much of the molecule is present
  • Gives an explicit analytical solution
  • Constant half life

Simple Forward Reaction

Complex Formation

[C] : Complex
kc : Rate constant of complex formation

This equation ignores the fact that dissociation of the complex occurs. We can do so if the dissociation is much slower than the formation.

  • Single solvable equation for the unknown C
  • Simple, unique solution available with I.C

Phosphorylation and Dephosphorylation

Assumptions:

  • Linear kinetic rate laws apply only if XT is much less than the Michaelis constants of both kinase and phosphotase

Phosphorylation and Dephosphorylation

XT : total cost of X protein in phosphorylated and unphosphorylated form
S : protein kinase concentration
k2 : accounts for protein phosphotase
  • Modeled after simple linear kinetics
  • Gives a hyperbolic signal response curve when X plotted vs S

Regulated Transcription

Regulated1

[P]: Protein Formed
µ: Repression, µ=0;
Activation, µ=1
K: Hill Constant Value of input that gives 50% response
n: Hill coefficient Slope of signal-response curve at this input signal
d: degradation of protein
k1: basal gene expression
k: signal-dependent gene expression
a: correlation between k1 and k, 0<a<1

Regulated2
This ODE attempts to capture characteristics of the mRNA dynamics
For our modeling, all our detection systems activates some form of transcription. Therefore µ=1 in all cases for our modeling exercise. Regulated3

ODE system used in model

The system can be viewed as two parts. The first part comprises of lactose induced production of colicin E7 and the immunity protein. The second part comprises of a detection mechanism that produces the lysis protein upon the detection of both Iron ions and Ai-2 ( Autoinducer 2).

Lactose controlled production of E7 + Imm

Variables

LacI = A
Lactose = B
E7 = C

LacI production

  • Transcription of LacI gene mRNA

Lac1

[MA]: LacI mRNA concentration
k1A: kinetic constant of transcription
U: system input
d1A: degradation constant for mRNA
  • Translation of LacI Protein

Lac2

[PA]: Protein concentration
k2A: kinetic constant of translation
d2A: degradation constant for Protein

Complex formation between LacI and Lactose

Complex

[ComAB] : Complex of LacI and Lactose
K3AB : Rate constant of complex formation
[PA]o : Protein concentration of LacI at the time of Lactose addition
[B]o : Initial concentration of Lactose

E7 production

  • Transcription of E7 gene mRNA

E71

[MC]: E7 mRNA Concentration
k1C: kinetic constant of transcription
a : constitutive portion , 0<a<1
KC: Hill constant
nC: Hill coefficient
d1C: degradation constant for mRNA
  • Translation of E7 Protein

E72

[PC] : Protein concentration
k2C : kinetic constant of translation
d2C : degradation constant for Protein

Variables of Iron and Ai2 controlled production of Lysis

Variables

#Ai-2: A
#ai-2-phos: B
#LsrR: C
#SupD derivatives: D
#T7ptag: E
#Lysis: F

Phosphorylation of Ai-2

Phos

A: Ai-2
B: Ai-2-phos
kPF: Forward Phosphorylation
S: Concentration of Protein Kinase
kPB: Backward dephosphorylation

LsrR production

  • Transcription of LsrR gene mRNA

LsrR1

[MC]: LacI mRNA concentration
k1C: kinetic constant of transcription
U: system input
d1C: degradation constant for mRNA
  • Translation of LsrR protein

LsrR2

[PC] : Protein concentration
k2C: kinetic constant of translation
d2C: degradation constant for Protein

Complex formation between Ai-2-phos and LsrR

Complex2

[ComAB] : Complex of LsrR and Ai-2-phos
k3BC : Rate constant of complex formation
[PC]o: Protein concentration of LsrR at the time of start of complex formation
[B]o: Initial concentration of Ai-2-phos

The hard part here is to determine what is the initial concentration of the protein LsrR and Ai-2-phos. A way out of this mess is to simplify the equation even more.

AND GATE odes

  • Transcription of SupD gene mRNA

SupD

[MD]: t7 mRNA Concentration
k1D: kinetic constant of transcription a : constitutive portion , 0<a<1
KD: Hill constant
nD: Hill coefficient
d1D: degradation constant for mRNA
  • Transcription of T7ptag gene mRNA

T71

[ME]: supD mRNA Concentration
k1E: kinetic constant of transcription
a : constitutive portion , 0<a<1
KE: Hill constant
nE: Hill coefficient
d1E: degradation constant for mRNA
  • Complex formation between tRNA and mRNA

Complex3

[ComDE] : Complex of tRNA and mRNA t7
k3DE : Rate constant of complex formation
[MD]o : concentration of tRNA at the time of start of complex formation
[ME]o : Initial concentration of mRNA t7
  • Translation of t7 protein

Complex3

[PF] : Protein concentration
k2F : kinetic constant of translation
d2F : degradation constant for Protein

Lysis production

  • Transcription of Lysis gene mRNA

Lysis

[MF]: Lysis mRNA Concentration
k1F: kinetic constant of transcription a : constitutive portion , 0<a<1
KF: Hill constant
nF: Hill coefficient
d1F: degradation constant for mRNA
  • Translation of Lysis protein

Lysis2

[PF] : Protein concentration
k2F : kinetic constant of translation
d2F : degradation constant for Protein



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