# Team:Dundee/Project/PP1Capacities

### From 2013.igem.org

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<h2>Theory</h2> | <h2>Theory</h2> | ||

<p>When considering the maximum number of PP1 molecules physically compacted into the periplasm of <i>E. coli</i> or onto the membrane surface of <i>B. subtilis</i>, considerations for wasted space must be taken into account. This wasted space arises due to the impossibility of using all of the available space due to the circular configuration of PP1.<br><br> | <p>When considering the maximum number of PP1 molecules physically compacted into the periplasm of <i>E. coli</i> or onto the membrane surface of <i>B. subtilis</i>, considerations for wasted space must be taken into account. This wasted space arises due to the impossibility of using all of the available space due to the circular configuration of PP1.<br><br> | ||

+ | |||

Before beginning such analysis, assumptions must be made regarding the shape and form which the PP1 molecule adopts. To best describe the PP1 protein shape, without unnecessary over complication, PP1 was approximated as a sphere with a radius equal to the average width of the unit cell in three dimensions. This approximation allows for the ability to assume random orientation of the PP1 molecules and gives an average result. The radius of PP1 is then calculated as ~9.1nm. .<br><br> | Before beginning such analysis, assumptions must be made regarding the shape and form which the PP1 molecule adopts. To best describe the PP1 protein shape, without unnecessary over complication, PP1 was approximated as a sphere with a radius equal to the average width of the unit cell in three dimensions. This approximation allows for the ability to assume random orientation of the PP1 molecules and gives an average result. The radius of PP1 is then calculated as ~9.1nm. .<br><br> | ||

In calculating the available volume of an <i>E. coli</i> or <i>B. subtilis</i> cell, the bacterial cells were approximated as cylindrical bodies with hemispherical ends. Furthermore, the dimensions of <i>E. coli</i> and <i>B. subtilis</i> cells were taken as being approximately equivalent and were based upon the known dimensions of <i>E. coli</i>. “Each bacterium measures approximately 0.5 μm in width by 2 μm in length.”[2] .<br><br> | In calculating the available volume of an <i>E. coli</i> or <i>B. subtilis</i> cell, the bacterial cells were approximated as cylindrical bodies with hemispherical ends. Furthermore, the dimensions of <i>E. coli</i> and <i>B. subtilis</i> cells were taken as being approximately equivalent and were based upon the known dimensions of <i>E. coli</i>. “Each bacterium measures approximately 0.5 μm in width by 2 μm in length.”[2] .<br><br> | ||

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<h2><i>E. coli</i></h2> | <h2><i>E. coli</i></h2> | ||

<p> | <p> | ||

- | For E. coli with a periplasmic width of 21nm, the periplasm volume can be calculated by the consideration of the cell as two parts.<br><br> | + | For <i>E. coli</i> with a periplasmic width of 21nm, the periplasm volume can be calculated by the consideration of the cell as two parts.<br><br> |

Calculating the periplasmic volume of <i>E. coli</i>, the inner form (representing the inner membrane and its cytoplasmically-enclosed contents) would have a cylindrical body of radius 0.25-0.021μm and a length of 1.5μm. The two hemispheres flanking the cylinder would be of radius 0.25-0.021μm. Calculating the total volume of the inner form gives:</p> | Calculating the periplasmic volume of <i>E. coli</i>, the inner form (representing the inner membrane and its cytoplasmically-enclosed contents) would have a cylindrical body of radius 0.25-0.021μm and a length of 1.5μm. The two hemispheres flanking the cylinder would be of radius 0.25-0.021μm. Calculating the total volume of the inner form gives:</p> | ||

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<ul style="padding-left:45px"> | <ul style="padding-left:45px"> | ||

<li>[1] What is the hydrodynamic radius (RH)? Accessed: 18/09/2013: <br><br><a href="http://ecoserver.imbb.forth.gr/pdf/hydrodynamic_radius.pdf">http://ecoserver.imbb.forth.gr/pdf/hydrodynamic_radius.pdf</a></li> | <li>[1] What is the hydrodynamic radius (RH)? Accessed: 18/09/2013: <br><br><a href="http://ecoserver.imbb.forth.gr/pdf/hydrodynamic_radius.pdf">http://ecoserver.imbb.forth.gr/pdf/hydrodynamic_radius.pdf</a></li> | ||

- | <li> [2] Escherichia coli, PortEco. Accessed: 18/09/2013: <br><br><a href="http://ecoliwiki.net/colipedia/index.php/Escherichia_coli">http://ecoliwiki.net/colipedia/index.php/Escherichia_coli</a></li> | + | <li> [2] <i>Escherichia coli</i>, PortEco. Accessed: 18/09/2013: <br><br><a href="http://ecoliwiki.net/colipedia/index.php/Escherichia_coli">http://ecoliwiki.net/colipedia/index.php/Escherichia_coli</a></li> |

<li> [3] Width of periplasm, BioNumbers. Accessed: 18/09/2013: <br><br><a href="http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=105387&ver=2">http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=105387&ver=2</a></li> | <li> [3] Width of periplasm, BioNumbers. Accessed: 18/09/2013: <br><br><a href="http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=105387&ver=2">http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=105387&ver=2</a></li> | ||

<li> [4] Protein serine/threonine phosphatase-1 (alpha isoform, type 1) complexed with microcystin-LR toxin, RCSB Protein Data Bank. Accessed: 18/09/2013: <br><br><a href="http://www.rcsb.org/pdb/explore/explore.do?structureId=1FJM#">http://www.rcsb.org/pdb/explore/explore.do?structureId=1FJM#</a></li> | <li> [4] Protein serine/threonine phosphatase-1 (alpha isoform, type 1) complexed with microcystin-LR toxin, RCSB Protein Data Bank. Accessed: 18/09/2013: <br><br><a href="http://www.rcsb.org/pdb/explore/explore.do?structureId=1FJM#">http://www.rcsb.org/pdb/explore/explore.do?structureId=1FJM#</a></li> |

## Revision as of 19:25, 2 October 2013

**PP1 Capacities**

## Introduction

Microcystin’s toxic action lies in its ability to bind to and deactivate the human Protein Phosphatase-1 (PP1). Microcystin covalently binds to PP1 in a one-to-one relation between a single Microcystin molecule and a single PP1 molecule. Therefore, higher binding potentials of a PP1 mop are achieved by producing larger amounts of PP1 in the bacterial chassis (*E. coli* and *B. subtilis*).

The two options explored for our mop bacteria were to export PP1 to the periplasm of *E. coli* and onto the membrane surface of *B. subtilis*. We investigated the PP1-binding capacity of both chassis options based upon geometrical packing, which allowed us to determine which chassis had a higher microcystin binding potential.

## Theory

When considering the maximum number of PP1 molecules physically compacted into the periplasm of *E. coli* or onto the membrane surface of *B. subtilis*, considerations for wasted space must be taken into account. This wasted space arises due to the impossibility of using all of the available space due to the circular configuration of PP1.

Before beginning such analysis, assumptions must be made regarding the shape and form which the PP1 molecule adopts. To best describe the PP1 protein shape, without unnecessary over complication, PP1 was approximated as a sphere with a radius equal to the average width of the unit cell in three dimensions. This approximation allows for the ability to assume random orientation of the PP1 molecules and gives an average result. The radius of PP1 is then calculated as ~9.1nm. .

In calculating the available volume of an *E. coli* or *B. subtilis* cell, the bacterial cells were approximated as cylindrical bodies with hemispherical ends. Furthermore, the dimensions of *E. coli* and *B. subtilis* cells were taken as being approximately equivalent and were based upon the known dimensions of *E. coli*. “Each bacterium measures approximately 0.5 μm in width by 2 μm in length.”[2] .

The equations describing the volumes of spheres of radius r and cylinders of radius r and height h are given:

Making use of this approximation and these equations, the volume of an *E. coli* cell can be calculated as the volume of a cylinder of radius 0.25μm and height 1.5μm and the volume of a sphere of radius 0.25μm (the two hemispheres can be brought together and calculated as a single sphere).

*E. coli*

For *E. coli* with a periplasmic width of 21nm, the periplasm volume can be calculated by the consideration of the cell as two parts.

Calculating the periplasmic volume of *E. coli*, the inner form (representing the inner membrane and its cytoplasmically-enclosed contents) would have a cylindrical body of radius 0.25-0.021μm and a length of 1.5μm. The two hemispheres flanking the cylinder would be of radius 0.25-0.021μm. Calculating the total volume of the inner form gives:

Therefore, the volume of the periplasm in *E. coli* is calculated to be:

To determine the number of PP1 molecules which could occupy the periplasm, we must determine the volume of space which a single PP1 molecule would occupy. This volume of a PP1 molecule can be taken as the volume of the unit sphere of diameter 9.1nm. However, when considering the amount of space that a single PP1 molecule will occupy the space in between molecules must also be considered. To take into account this wasted space, a cubic approximation was made:

By assumption, the PP1 molecules take up the excess room which would be included within a cube of length equal to their radius allowing the wasted space to be accounted for.

Thus, the volume of a PP1 molecule is given:

From this, the number of PP1 molecules which can suitably occupy the periplasm of *E. coli* can be calculated:

*B. subtilis*

When considering the packing of PP1 on the surface of *B. subtilis*, there is no defined volume for packing. *B. subtilis* is a Gram-positive cell and so the PP1 molecules are adhered in a monolayer to the surface of the cell membrane. Therefore, the approach to packing is different in this case.

In packing PP1 on the surface of the membrane, the number of PP1 molecules that can be placed “side-by-side” along the height of the membrane and around the circumference of the membrane was calculated.

The number of PP1 molecules able to be stacked along the length of the cylindrical centrepiece of the cell (excluding the hemispheres) and around the circumference of the cylinder top:

Multiplying the above results gives the total number of PP1 molecules which can occupy the surface of the cylindrical midsection of the bacteria.

The number of PP1 molecules which could occupy the hemispherical ends can be calculated by approximating the hemispherical ends as flat circles. This approximation is appropriate due to the smaller contribution at these ends.

Thus, the total number of PP1 which could occupy the surface of *B. subtilis* can be calculated as the sum of the above.

## Conclusion

In conclusion, we considered the volumes of the bacteria and PP1 and used a cube approximation that took into account volume which was wasted, in packing, by the spherical shape of the protein. For this model we assumed there were no other surface proteins and protein production was not limited by any factors.

Calculations show the maximum number of PP1 which can fit on the surface of *B. subtilis* is approximately 35,000. From this approximation, we can calculate that the number of bacterial mops required to clean a toxic level of microcystin in a litre of water as 5.3x10^{9} cells.

In *E. coli*, the PP1 which would bind microcystin is freely diffusing in three dimensions in the periplasm. The volume of the periplasm is much greater than the surface of *B. subtilis*. Therefore, *E. coli* has the capacitive potential to be a more efficient mop. The maximum number of PP1 which can be packed into the periplasm is approximately 83,000. Consequently, less bacterial mops are required to clean the same level of microcystin: 2.2x10^{9} cells.

Having shown that *E. coli* has the potential to be the more efficient mop, the geometric packing results were presented to the wet team in Week 1. When later in the project it was found that there were issues in exporting PP1 to the surface of *B. subtilis*, a focus on *E. coli* as the chassis for the bacterial mop was justified and appropriate both biologically and mathematically.

- [1] What is the hydrodynamic radius (RH)? Accessed: 18/09/2013:

http://ecoserver.imbb.forth.gr/pdf/hydrodynamic_radius.pdf - [2]
*Escherichia coli*, PortEco. Accessed: 18/09/2013:

http://ecoliwiki.net/colipedia/index.php/Escherichia_coli - [3] Width of periplasm, BioNumbers. Accessed: 18/09/2013:

http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=105387&ver=2 - [4] Protein serine/threonine phosphatase-1 (alpha isoform, type 1) complexed with microcystin-LR toxin, RCSB Protein Data Bank. Accessed: 18/09/2013:

http://www.rcsb.org/pdb/explore/explore.do?structureId=1FJM#