Team:Kyoto/Modeling

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(Lactose and IPTG are inducer of lactose promoter)
(Modeling)
 
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'''Model1: R0011, a lactose promoter'''
'''Model1: R0011, a lactose promoter'''
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Before making the model for lysis cassette, we must characterize R0011, a lactose promtoer, because we use it to change the expression level of lysis cassette. We made some mathematical model for the lcatose promoter activity.
+
We used R0011, a lactose promoter, to express lysis cassette and to change its expression level. A Mathematical model for the activity of R0011 is established to characterize lysis cassette with RPU.  
-
 
+
'''Model2: Cell Lysis'''
'''Model2: Cell Lysis'''
 +
 +
We found that lysis cassette cannot kill cells entirely even if lysis cassette expressed enough. We made some model to explain thin phenomenon.
===Model1: R0011, a lactose promoter===
===Model1: R0011, a lactose promoter===
====Indroduction====
====Indroduction====
-
It is too laborious to measure the activity of R0011 with all IPTG concentrations, such as 0.01mM, 0.02mM,  , , ,  , 10mM. Measurement with some appropriate IPTG concentrations and making a mathematical model from the experimental data is beneficial to estimate the promoter activity with the other IPTG concentration and save the trouble.
+
It is too laborious to measure the activity of R0011 with all IPTG concentrations, such as 0.01mM, 0.02mM,  , , ,  , 10mM. Measurement with some appropriate IPTG concentrations and making a mathematical model from the experimental data is beneficial to estimate the promoter activity with the other IPTG concentration and to save the trouble.
 +
 
====LacI binds to lactose promoter and represses it====
====LacI binds to lactose promoter and represses it====
[[Image:KyotoFigM001.png|600px|center]]
[[Image:KyotoFigM001.png|600px|center]]
-
LacI, the repressor of lactose promoter, binds to DNA sequence of lactose promoter and represses it. R0011, a lactose repressor, has two operator regions where LacI binds <sup>[[#RefM1-1|[1]]]</sup>. The equilibrium reaction of binding and dissociation of LacI and the lactose promoter can be described as equation1.
+
LacI, the repressor of lactose promoter, binds to DNA sequence of lactose promoter and represses it. R0011, a lactose promoter, has two operator regions where LacI binds <sup>[[#RefM1-1|[1]]]</sup>. The equilibrium reaction of binding and dissociation of LacI and the lactose promoter can be described as equation1.
[[Image:KyotoModeling1-1.png|600px|center]]
[[Image:KyotoModeling1-1.png|600px|center]]
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* [X] means the concentration of LacI,  
* [X] means the concentration of LacI,  
* [D] means the concentration of the lactose promoter not binding LacI,  
* [D] means the concentration of the lactose promoter not binding LacI,  
-
* [DX] means the concentration of the lactose promoter binding a lacI
+
* [DX] means the concentration of the lactose promoter binding a lacI,
-
* [DX<sup>2</sup>] means the concentration of the lactose promoter binding two LacI.
+
* [DX<sub>2</sub>] means the concentration of the lactose promoter binding two LacI,
* K<sub>1</sub> means the equilibrium constant for the reaction between [D] and [X],
* K<sub>1</sub> means the equilibrium constant for the reaction between [D] and [X],
* K<sub>2</sub> means the equilibrium constant for the reaction between [DX] and [X].
* K<sub>2</sub> means the equilibrium constant for the reaction between [DX] and [X].
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Here,
Here,
-
* &alpha; is the promoter activity when all the lactose promoter bind two LacI respectively,
+
* &alpha; is the promoter activity when all the lactose promoter are bound by two LacIs,
-
* &beta; is the promoter activity when all the lactose promoter bind a LacI respectively,
+
* &beta; is the promoter activity when all the lactose promoter are bound by a LacI,
* [[Image:KyotoModeling1-a.png|100px]] is the ratio of [D] to the total concentration of lactose promoter,
* [[Image:KyotoModeling1-a.png|100px]] is the ratio of [D] to the total concentration of lactose promoter,
* [[Image:KyotoModeling1-b.png|100px]] is the ratio of [DX] to the total concentration of lactose promoter.
* [[Image:KyotoModeling1-b.png|100px]] is the ratio of [DX] to the total concentration of lactose promoter.
-
In this model, we hypothesized that the promoter activity is 0 when all the lactose promoter bind two LacI respectively.
+
In this model, we assumed that the promoter activity is completely repressed when two lacI bind to the operator regions on lactose promoter.
====LacI binds to lactose promoter and represses it====
====LacI binds to lactose promoter and represses it====
-
[[Image:KyotoFigM002.png|600px|center]]
+
[[Image:KyotoFigM002.png|300px|center]]
The inducer of lactose promoter, Lactose and IPTG, binds to LacI, and changes LacI conformation so that LacI dissociates from lactose promoter. The equilibrium reaction of the inducer and LacI is described as follows.
The inducer of lactose promoter, Lactose and IPTG, binds to LacI, and changes LacI conformation so that LacI dissociates from lactose promoter. The equilibrium reaction of the inducer and LacI is described as follows.
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* [X] is the concentration of LacI not binding with the inducer,  
* [X] is the concentration of LacI not binding with the inducer,  
* n is the number of the inducer molecule binds to one LacI molecule.
* n is the number of the inducer molecule binds to one LacI molecule.
-
The equbilium constant, K<sub>Xn</sub>, is described as equation6.
+
The equbilium constant, K<sub>X<sup>n</sup></sub>, is described as equation6.
[[Image:KyotoModeling1-6.png|600px|center]]
[[Image:KyotoModeling1-6.png|600px|center]]
-
We assume that the total concentration of LacI in a cell does not change in the log phase growth in which the cell growth does not change because the dilution effect for LacI due to the cell growth does not change and the expression and the degradation of LacI is constitutive.
+
We assumed that the total concentration of LacI in a cell did not change in the log phase growth in which the cell growth does not change because the rate of the cell growth was constant and it compensated the increasing of traslated lacI.
Therefore, let X<sub>T</sub> as the concentration of total LacI, and equation7 is applied.
Therefore, let X<sub>T</sub> as the concentration of total LacI, and equation7 is applied.
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====Lactose and IPTG are inducer of lactose promoter====
====Lactose and IPTG are inducer of lactose promoter====
-
[[Image:KyotoFigM003.png|600px|center]]
+
[[Image:KyotoFigM003.png|300px|center]]
From equation4 and equation8, the relationship between IPTG and lactose promoter activity is described as follows.
From equation4 and equation8, the relationship between IPTG and lactose promoter activity is described as follows.
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===Model2: Cell Lysis===
===Model2: Cell Lysis===
-
====Assumptions====
+
====Unexpected result of lysis cassette====
-
It is known that when lambda phage holin is expressed, they localized to plasma membrane.
+
-
Tough, holin in the plasma membrane is monomer because there is a few holins there firstly, holin becomes oligomer when enough holins exist in the plasma membrane. It is suggested that if the number of holin in the oligomer reaches about 62, holin oligomer makes a hole [[#RefM201|[1]]]. Once the hole is made, endolysin can go to the periplasm, and degrades peptideglycan and make cell lysis.
+
We cultivated E.coli transformed with the construction, Plac and lysis cassette on the low copy number plasmid, at various IPTG concentrations and A550 was measured at 16h, 18h and 20h. The result is shown below.
-
In order to make a model that can explain our experimental data for cell lysis, we made some assumptions.
+
[[Image:KyotoM004.png|500px|center|]]
-
The first is that there is limit of the number of holin in the plasma membrane. We think this is because the surface area of plasma membranes is limited. It is estimated that approximately 1000-3000 molecules of holin accumulates in the plasma membrane finally [[#RefM202|[2]]].
+
We predicted two things. One is that A550 is measured as the steady state at 16h, 18h and 20h. The other is that the more IPTG is introduced, the more A550 in the steady state decreases and completely reached zero. In the result, as you see Fig.1, OD550 was measured as the steady state at 16h, 18h and 20h in every IPTG concentration. In addition, we observed that OD550 decreased in the range from 0mM to 0.05mM of IPTG concentration. However, OD550 observed was a fixed value, (about 0.3 of OD550) at 0.05mMof IPTG. And E.coli was not completely killed by holin even in 1.0mM of IPTG!
-
Next one is that when only a few holin exist in the plasma membrane, the speed of binding between holin molecules is so low, but the speed is increased when the number of holin molecule becomes larger.
+
You may consider that this result occurred because of the mutation of our biobrick part. However, we denied the hypothesis by proposing the model the mutation happens in the next section.
-
The next is that the size of holin oligomer is limited and that there is 62 holin molecules in the holin oligomer at most.
+
====Mutation model====
 +
In this section, we explain the mutation model. We assume that our part works correctly in the beginning, accumulates the mutations on the way and loses its original function. As the result, OD550 does not reach zero even in IPTG 1.0 mM which is high enough. If this result really occurred by the mutant factor, OD550 at 16h, 18h and 20h must increase like an exponential function. However, in our result, OD550 at 16h, 18h and 20h in every IPTG concentration are measured as the steady state. Therefore the mutation model is not suited. Then, we will propose several stochastic models below.
-
We also hypothesized that the effect of endolysin is neglected while the number of holin molecules in the ologomer is under 62, but endolysin works when it reaches 62 and that the effect of endolysin becomes larager as the number of endolysin molecules increase, but there is limit for the effect of endolysin.
+
====Stochastic model====
 +
We propose four stochastic models.
-
====Equations====
+
1. First model is that the effect of endolysin for cell lysis is limited though the concentration of endolysin increases. We consider that when much endolysin is expressed, the size of holes made by holin restricts the transition rate of endolysin from cytosol to periplasm, where endolysin degrades peptidoglycan and makes cell lysis.
-
We made equations below.
+
-
[[Image:KyotoModeling2-1.png|800px|center]]
+
2. Second model is that the expression of holin and endolysin is not enough to kill every cell entirely. In this model, we assume that some cells are weak to the cell lysis and other cells are tough to that. If this is true, some proportional cells survive and the other proportional cells are died constantly (Fig.2).
-
[[Image:KyotoModeling2-2.png|800px|center]]
+
-
equation3
+
3. Third model is that quantity of expressed holin and endolysin is different from cells respectively. We assume that every cell has the threshold of lysis. In any concentration of IPTG, some proportional cells survive and the other cells are killed because of the difference of expression. (Fig.3)
-
equation4
+
-
====Equation1====
+
[[Image:KyotoFigM005.png|440px|thumb|left|Fig.2]] [[Image:KyotoFigM006.png|440px|thumb|right|Fig.3]]
-
equation1
+
-
In this equation,
+
-
[holin] means the concentration of holin in the plasma membrane,
+
-
Aholin is the constant including the transcriptionarate ,the translational rate and the rate of transport to the plasma membrane,
+
-
γholin means the dilution and degradation rate of holin.  
+
-
is the promoter activity of lactose promoter. [IPTG] is [Sx] in the eqaution9 of “Model for the activity of lactose promoter, R0011”.
+
-
To know more, go Model for the activity of lactose promoter, R0011
+
-
We also assume that although while [holin]<=3000, [holin] is increasing, [holin] is in steady state when [holin]>3000.
+
-
====Equation2====
+
{{clear}}
-
equation2
+
-
{holin size} means the number of holin molecules in holin oligomer,
+
-
Kholin_on is the binding speed of holin monomers or oligomers while [holin]<Bholin,
+
-
Kholin_size’ is the binding speed of holin molecules when [holin] >= Bholin.
+
-
Bholin is the thresh hold for the binding speed of holin molecules begin to increase.
+
-
Kholin_off is the dissociation speed of holin oligomers.
+
-
We also assume that once {holin size} reaches Bholin, holin oligomer make a hole and {holin size} does not change.
+
-
====Equation3====
+
4. Fourth model is that the transcription of holin and endolysin decreases because a part of the population of cells comes to be dormancy state <sup>[[#RefM2-1|[1]]]</sup> <sup>[[#RefM2-2|[2]]]</sup>. In any concentration of IPTG, a part of population of cells is always dormancy state and survive.
-
equation3
+
-
In this equation,
+
-
[endolysin] is the concentration of endolysin,
+
-
Aednolysin is the constant including the transcriptional rate and the translational rate of endolysin,
+
-
γendolysin is the dilution and degradation rate of endolysin.
+
-
  is the lactose promoter activity.
+
-
====Equation4====
+
[[Image:KyotoFigM007.png|350px|center|]]
-
equation4
+
-
In this equation
+
-
[cell] is the number of cells in the medium,
+
-
r is the cell growth rate,
+
-
Kcell is the carring capacity,
+
-
D is the constant for cell lysis
+
-
Kendolyisn is the threshold of the effect of endolysin for cell lysis.
+
-
We assume that while {holin size}<62, holin does not make a hole and endolysin does not degrade peptideglycan and the effect of endolysin is 0. We also assume that after holin molecules make holes, though, the effect of endolysin increases as the concentration of [endolysin] increases, there is limit for the effect of endolysin.
+
-
 
+
-
====Simulation====
+
 +
In the conclusion, it is not mutant but stochastic factor that brought this result. In addition, every four stochastic models can explain that OD550 was measured as the steady state at 16h, 18h and 20h and did not reach zero in every IPTG concentration.
====Reference====
====Reference====
-
# <html><a name="RefM201"></a></html>Ry Young et al. “The holin of bacteriophage lambda forms rings with large diameter” Molecular Microbiology (2008) 69(4), 784–793
+
# <html><a name="RefM2-1"></a></html>R Jayaraman. “Bacterial persistence: some new insights into an old phenomenon” J.Biosci.33(5),December 2008,795-805,
-
# <html><a name="RefM201"></a></html>Chang, C.Y., Nam, K., and Young, R. (1995) “S gene expression and the timing of lysis by bacteriophage l.” J Bacteriol 177: 3283–3294.
+
# <html><a name="RefM2-2"></a></html>Slava S.Epstein. “Microbial awakenings A theory of how microbes 'wake up' from dormancy could help to solve scientific mysteries and improve disease control” NATURE|Vol 457|26 February 2009
[[#top-section|^Top]]
[[#top-section|^Top]]
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Latest revision as of 03:54, 28 October 2010

Contents

Modeling

Models

Model1: R0011, a lactose promoter

We used R0011, a lactose promoter, to express lysis cassette and to change its expression level. A Mathematical model for the activity of R0011 is established to characterize lysis cassette with RPU.

Model2: Cell Lysis

We found that lysis cassette cannot kill cells entirely even if lysis cassette expressed enough. We made some model to explain thin phenomenon.

Model1: R0011, a lactose promoter

Indroduction

It is too laborious to measure the activity of R0011 with all IPTG concentrations, such as 0.01mM, 0.02mM, , , , , 10mM. Measurement with some appropriate IPTG concentrations and making a mathematical model from the experimental data is beneficial to estimate the promoter activity with the other IPTG concentration and to save the trouble.

LacI binds to lactose promoter and represses it

KyotoFigM001.png

LacI, the repressor of lactose promoter, binds to DNA sequence of lactose promoter and represses it. R0011, a lactose promoter, has two operator regions where LacI binds [1]. The equilibrium reaction of binding and dissociation of LacI and the lactose promoter can be described as equation1.

KyotoModeling1-1.png

Here,

  • [X] means the concentration of LacI,
  • [D] means the concentration of the lactose promoter not binding LacI,
  • [DX] means the concentration of the lactose promoter binding a lacI,
  • [DX2] means the concentration of the lactose promoter binding two LacI,
  • K1 means the equilibrium constant for the reaction between [D] and [X],
  • K2 means the equilibrium constant for the reaction between [DX] and [X].
  • The equilibrium constants K1 and K2 are described as equation2 and equation3 respectively.
KyotoModeling1-2.png
KyotoModeling1-3.png

From equation2 and equation3, equation4 is established.

KyotoModeling1-4.png

Here,

  • α is the promoter activity when all the lactose promoter are bound by two LacIs,
  • β is the promoter activity when all the lactose promoter are bound by a LacI,
  • KyotoModeling1-a.png is the ratio of [D] to the total concentration of lactose promoter,
  • KyotoModeling1-b.png is the ratio of [DX] to the total concentration of lactose promoter.

In this model, we assumed that the promoter activity is completely repressed when two lacI bind to the operator regions on lactose promoter.

LacI binds to lactose promoter and represses it

KyotoFigM002.png

The inducer of lactose promoter, Lactose and IPTG, binds to LacI, and changes LacI conformation so that LacI dissociates from lactose promoter. The equilibrium reaction of the inducer and LacI is described as follows.

KyotoModeling1-5.png

Here,

  • [SX] is the inducer concentration,
  • [X] is the concentration of LacI not binding with the inducer,
  • n is the number of the inducer molecule binds to one LacI molecule.

The equbilium constant, KXn, is described as equation6.

KyotoModeling1-6.png

We assumed that the total concentration of LacI in a cell did not change in the log phase growth in which the cell growth does not change because the rate of the cell growth was constant and it compensated the increasing of traslated lacI. Therefore, let XT as the concentration of total LacI, and equation7 is applied.

KyotoModeling1-7.png

From equation6 and equation7, equation8 is established [2].

KyotoModeling1-8.png

Lactose and IPTG are inducer of lactose promoter

KyotoFigM003.png

From equation4 and equation8, the relationship between IPTG and lactose promoter activity is described as follows.

KyotoModeling1-9.png

Simulation

We apply equation9 to the result of characterization of R0011 and parameters are decided by using MATLAB 7.10.0 (MathWorks).

KyotoGrpA003.png

The orange markers and error bars are experimental data and blue line is the expectation from this model. The data of 0.5mM IPTG does not be used in fitting the model.

Parameters are shown in table below.

symboldescriptionvalueunitnote
αthe transcription constant for [D]1.667RPUcalculated from the experimental result
βthe transcription constant for [DX]0.0000149RPUcalculated from the experimental result
K1the equilibrium constant for between [D] and [X]1.607mMcalculated from the experimental result
K2the equilibrium constant for between [DX] and [X]0.339mMcalculated from the experimental result
XTtotal concentration of lacI5.131mMcalculated from the experimental result
KXthe equilibrium constant for between [SX] and [X]0.05321mMcalculated from the experimental result
nthe number of IPTG molecules binding to a lacI1.285-calculated from the experimental result

Reference

  1. http://partsregistry.org/Part:BBa_R0011
  2. Uri Alon (2006) “An Introduction toSystems Biology” Chapman & Hall

^Top

Model2: Cell Lysis

Unexpected result of lysis cassette

We cultivated E.coli transformed with the construction, Plac and lysis cassette on the low copy number plasmid, at various IPTG concentrations and A550 was measured at 16h, 18h and 20h. The result is shown below.

KyotoM004.png

We predicted two things. One is that A550 is measured as the steady state at 16h, 18h and 20h. The other is that the more IPTG is introduced, the more A550 in the steady state decreases and completely reached zero. In the result, as you see Fig.1, OD550 was measured as the steady state at 16h, 18h and 20h in every IPTG concentration. In addition, we observed that OD550 decreased in the range from 0mM to 0.05mM of IPTG concentration. However, OD550 observed was a fixed value, (about 0.3 of OD550) at 0.05mMof IPTG. And E.coli was not completely killed by holin even in 1.0mM of IPTG!

You may consider that this result occurred because of the mutation of our biobrick part. However, we denied the hypothesis by proposing the model the mutation happens in the next section.

Mutation model

In this section, we explain the mutation model. We assume that our part works correctly in the beginning, accumulates the mutations on the way and loses its original function. As the result, OD550 does not reach zero even in IPTG 1.0 mM which is high enough. If this result really occurred by the mutant factor, OD550 at 16h, 18h and 20h must increase like an exponential function. However, in our result, OD550 at 16h, 18h and 20h in every IPTG concentration are measured as the steady state. Therefore the mutation model is not suited. Then, we will propose several stochastic models below.

Stochastic model

We propose four stochastic models.

1. First model is that the effect of endolysin for cell lysis is limited though the concentration of endolysin increases. We consider that when much endolysin is expressed, the size of holes made by holin restricts the transition rate of endolysin from cytosol to periplasm, where endolysin degrades peptidoglycan and makes cell lysis.

2. Second model is that the expression of holin and endolysin is not enough to kill every cell entirely. In this model, we assume that some cells are weak to the cell lysis and other cells are tough to that. If this is true, some proportional cells survive and the other proportional cells are died constantly (Fig.2).

3. Third model is that quantity of expressed holin and endolysin is different from cells respectively. We assume that every cell has the threshold of lysis. In any concentration of IPTG, some proportional cells survive and the other cells are killed because of the difference of expression. (Fig.3)

Fig.2
Fig.3

4. Fourth model is that the transcription of holin and endolysin decreases because a part of the population of cells comes to be dormancy state [1] [2]. In any concentration of IPTG, a part of population of cells is always dormancy state and survive.

KyotoFigM007.png

In the conclusion, it is not mutant but stochastic factor that brought this result. In addition, every four stochastic models can explain that OD550 was measured as the steady state at 16h, 18h and 20h and did not reach zero in every IPTG concentration.

Reference

  1. R Jayaraman. “Bacterial persistence: some new insights into an old phenomenon” J.Biosci.33(5),December 2008,795-805,
  2. Slava S.Epstein. “Microbial awakenings A theory of how microbes 'wake up' from dormancy could help to solve scientific mysteries and improve disease control” NATURE|Vol 457|26 February 2009

^Top