# Team:Dundee/Project/ProductionExport

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To solve the system of ODEs (5), we applied the appropriate initial conditions. The only non-zero initial conditions are the number of TatB-C complexes and TatA proteins. It is known that there are approximately 15 TatB-C complexes and 600 TatA proteins in a regular <i>E. coli</i> cell [5]. Hence, in our model we use 15 TatB-C complexes and 30 TatA assemblies. <br><br> | To solve the system of ODEs (5), we applied the appropriate initial conditions. The only non-zero initial conditions are the number of TatB-C complexes and TatA proteins. It is known that there are approximately 15 TatB-C complexes and 600 TatA proteins in a regular <i>E. coli</i> cell [5]. Hence, in our model we use 15 TatB-C complexes and 30 TatA assemblies. <br><br> | ||

- | The program v1_odes_solver_PP1_TatProduction_export solves this system. The deterministic model predicted that approximately 200 PP1 molecules would be exported to the periplasm. This was less than we anticipated. | + | The program v1_odes_solver_PP1_TatProduction_export solves this system. The deterministic model predicted that approximately 200 PP1 molecules would be exported to the periplasm. This was less than we anticipated. Indeed , based on these figures, 3.3g of cells would be required to clean up one litre of contaminated water as defined by WHO regulations. In order to check whether these deterministic results were reasonable, we next considered a stochastic modelling approach. |

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<H2>Stochastic Model</h2> | <H2>Stochastic Model</h2> | ||

- | + | Some of the components in the ODE model above are in low abundance. To examine whether low copy number had a significant effect on PP1 numbers, we formulated and analysed a stochastic model. <br><br> | |

- | The stochastic model, although representing the random nature associated with such processes, did not | + | The stochastic model, although better representing the random nature associated with such processes, did not produce significantly different results. Instead the model showed that because the high and low fluctuations cancel each other out, the deterministic solutions were a good approximation of the stochastic means for PP1cyto and PP1peri. |

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+ | Indeed , based on these figures, 3.3g of cells would be required to clean up one litre of contaminated water as defined by WHO regulations. In order to check whether these deterministic results were reasonable, we next considered a stochastic modelling approach. | ||

+ | Producing such large amounts would be unachievable for our Wet Team. Therefore, to ensure the construction of our ToxiMop was viable, we needed to investigate options that would increase the number of PP1 being exported into the periplasm.<br><br> | ||

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## Revision as of 11:43, 3 October 2013

**Production & Export**

## Introduction

The ToxiMop is an engineered *E. coli* bacterium that expresses PP1 and can be used as a molecular mop to remove microcystin from contaminated water. Central to successfully engineering this machine was to better understand PP1 production and export. This was crucial as we hypothesised that microcystin binding would predominantly take place with periplasmic PP1.

We explored both the Twin Arginine Translocase (Tat) and General Secretory (Sec) pathways as potential export routes for PP1. However, initial Western blot results indicated that PP1 was exported into the periplasm much more successfully via the Tat pathway than via Sec. Therefore, production and export based on Tat transport, was selected as a modelling focus to allow us to optimise the construction of our prototype ToxiMop.

## Building a Model for Tat Transport

The Tat machinery is a biological pathway that transports folded proteins from the cytoplasm into the periplasm. It consists of three small membrane proteins; TatA, TatB and TatC.

TatB and TatC together form a TatB-C complex. The protein destined for transport has a signal sequence at its N-terminus which is recognised by and binds to the TatB-C complex. This positions the protein ready for export. TatA proteins then polymerise and form a ring structure surrounding the protein allowing it to penetrate the membrane and pass into the periplasm. The signal peptide is cleaved off and this frees up the TatB-C complex and TatA proteins for further transport.

**Figure 1:** Processes involved in Tat transport.

## Tat-dependent Transport of PP1

PP1 has a molecular mass of 37kDa. Reasonably assuming that PP1 is spherical, it would require 20 TatA proteins to form a ring large enough to accommodate it and enable it to penetrate the membrane [1]. We define this structure as a TatA assembly.

For transport, PP1 in the cytoplasm (PP1_{cyto}) binds to TatB-C, forming a PP1 TatB-C complex (PP1B-C). The TatA assembly then surrounds the PP1 TatB-C complex. We define this product as PP1_{export}. PP1_{export} is then exported into the periplasm (PP1_{peri}), releasing the TatA assembly and TatB-C back into the membrane to assist in further rounds of transport.

We make the following further assumptions:

- TatA assemblies are pre-formed from TatA proteins
- PP1 exported to the periplasm remains in the periplasm
- All other processes are reversible

**Figure 2:** Ring structures formed by polymerisation of TatA proteins. Taken from Rodriguez *et al.* [2]

Hence we have the following framework for Tat transport of PP1:

## Production

Before transport, PP1 first needed to be produced. This involved inserting the PP1-encoding gene into a plasmid vector and transforming the plasmid into host cells. These cells then expressed the gene.

We now consider the transcription and translation required for this gene expression. This simple production scheme is derived by assuming that both mRNA and protein can degrade. Due to the heterologous nature of PP1, its degradation constant is particular significant.

Using the law of mass action and appropriate rate constants, we created a mathematical system that represents each reaction. These values and equations are shown below:

Reaction name | Constant | Value |
---|---|---|

Transcription | K_{Tc} |
0.03833 nM.s^{-1} |

mRNA degradation | K_{mdeg} |
0.0077 s^{-1} |

Translation | K_{Tl} |
0.75 s-^{1} |

PP1 degradation | K_{pdeg} |
0.0192 s^{-1} |

**Table 1:** PP1 production rate constants [3].

Our team developed a series of MATLAB programs to solve the models discussed here. The code for these programs along with further analysis is available at this repository [4]. The program v1_odes_solver_PP1Production solves this system numerically.

Our model predicts that we will produce approximately 1200 net PP1 in steady state in the cytoplasm of each *E. coli* chassis. This gives an initial indication of how many cells are required to mop up given concentrations of microcystin. We used this information to examine the practicality of the ToxiMop. For example, if we assume all the PP1 is exported to the periplasm and exploit the one-to-one binding of microcystin with PP1, then 0.6g of cells are required to clean up one litre of contaminated water that is classified as unsafe by World Health Organisation (WHO) regulations. The production of this mass of cells was easily achievable by our Wet Team.

**Figure 3:** PP1 production. Solutions of equation (3).

**Figure 4:** PP1 produced per cell division. Solutions of equation (3).

## Production & Export

Combining our separate schemes for protein production (2) and Tat transport (1), we built a model that describes PP1 Production & Export.Reaction name | Constant | Value |
---|---|---|

Transcription | K_{Tc} |
0.03833 nM.s^{-1} |

mRNA degradation | K_{mdeg} |
0.0077 s^{-1} |

Translation | K_{Tl} |
0.75 s^{-1} |

PP1 degradation | K_{pdeg} |
0.0192 s^{-1} |

Recognition binding | K_{1} |
8E3 M^{-1}s^{-1} |

Recognition unbinding | K_{2} |
8E3 M^{-1}.s^{-1} |

Assembly association | K_{3} |
200E4 M^{-1}s^{-1} |

Assembly disassociation | K_{4} |
0.00167 s^{-1} |

Export | K_{5} |
10 s^{-1} |

**Table 2:** PP1 production & export rate constants [3].

## Deterministic Model

To solve the system of ODEs (5), we applied the appropriate initial conditions. The only non-zero initial conditions are the number of TatB-C complexes and TatA proteins. It is known that there are approximately 15 TatB-C complexes and 600 TatA proteins in a regular*E. coli*cell [5]. Hence, in our model we use 15 TatB-C complexes and 30 TatA assemblies.

The program v1_odes_solver_PP1_TatProduction_export solves this system. The deterministic model predicted that approximately 200 PP1 molecules would be exported to the periplasm. This was less than we anticipated. Indeed , based on these figures, 3.3g of cells would be required to clean up one litre of contaminated water as defined by WHO regulations. In order to check whether these deterministic results were reasonable, we next considered a stochastic modelling approach.

**Figure 5:** PP1 production & export

**Figure 6:** 202 PP1 are exported to the periplasm and 1164 remain in the cytoplasm

## Stochastic Model

Some of the components in the ODE model above are in low abundance. To examine whether low copy number had a significant effect on PP1 numbers, we formulated and analysed a stochastic model.The stochastic model, although better representing the random nature associated with such processes, did not produce significantly different results. Instead the model showed that because the high and low fluctuations cancel each other out, the deterministic solutions were a good approximation of the stochastic means for PP1cyto and PP1peri.

**Figure 7:** PP1_{cyto} Stochastic mean & realisations [7]

**Figure 8:** PP1_{peri} Stochastic mean & realisations [7]

**Figure 9:** PP1_{cyto} Stochastic realisations & Deterministic Solution [8]

**Figure 10:** 2PP1_{peri} Stochastic realisations & Deterministic Solution [8]

With these cells the Wet Team started testing the ToxiMop. Their main experiment involved taking 20 µl from a 100 µg/ml microcystin solution and mixing it into a beaker with 200ml of TBS. 1g of ToxiMop cells were then added to the solution in a dialysis bag. Early results indicated that the ToxiMop was ineffective in mopping up the microcystin.

However, after some simple calculations that made use of the previous models, we tried to get an understanding of why the tests were unsuccessful.

_{peri}per cell, the experiment would require a minimum of 5.75g of cells to clean up the toxin. Due to this, 1g of cells was an insufficient amount to positively confirm the mopping of microcystin. We recommended the use of 5.75g of cells in a repetition of the same experiment. In addition, if we could increase the number of PP1 in the periplasm we would increase the probability of binding microcystin and improve the effectiveness of the ToxiMop. For use in realistic applications, such an increase in PP1 would reduce the total mass of cells required for clean-up, making our solution more feasible. Therefore, increasing the number of PP1 being exported into the periplasm continued to be our main focus.

## Increasing Tat machinery

As discussed, the second option we investigated was increasing the number of PP1 being exported into the periplasm. We attempted to achieve this by increasing the amount of Tat machinery per cell. Our models predicted that the cells retain 1200 PP1 in the cytoplasm which have no contribution in our ToxiMop application. By increasing the Tat machinery, we target transporting these excess proteins which results in greater PP1 numbers in the periplasm. The options we considered were; increasing TatA assemblies, increasing TatB-C complexes or simultaneously increasing both TatA assemblies & TatB-C complexes.All the options were simulated to examine their effects on the number of PP1 being exported. This allowed us to determine and inform the Wet Team which was the best option to pursue.

Theoretically our increases would be achieved by inserting selected genes into a plasmid with a copy number of 20. This would allow us to obtain 20 extra copies of the selected Tat machinery proteins that correspond to the particular genes. Therefore our cells could produce 20 times as much of the targeted proteins. For example; to increase TatA assemblies, we insert the

*tatA*gene into the plasmid, to increase TatB-C complexes, we insert the

*tatB*and

*tatC*genes, and to increase both TatA assemblies and TatB-C complexes we insert all three genes. The simulations are incremented from the current machinery amounts (30 TatA assemblies & 15 TatB-C complexes) to 20 times these values to reflect this potential increase.

## Increasing TatA assemblies

**Figure 11:** PP1_{cyto} Stochastic means for the corresponding number of TatA Assemblies [9]

**Figure 12:** PP1_{peri} Stochastic means for the corresponding number of TatA Assemblies [9]

**Figure 13:** Change in number of PP1_{peri} with increasing TatA Assemblies [9]

**Figure 14:** Mass of Cells required for ToxiMop experiment based on the number of TatA Assemblies [9]

Increasing the number of TatA assemblies, marginally increases the number of PP1 being exported. With such a small increase, the mass of cells required for our ToxiMop experiment remains fairly high. Therefore, we could not recommend this action.

## Increasing TatB-C complexes

**Figure 15:** PP1_{cyto} Stochastic means for the corresponding number of TatB-C Complexes [10]

**Figure 16:** PP1_{peri} Stochastic means for the corresponding number of TatB-C Complexes [10]

**Figure 17: ** Change in number of PP1peri with increasing TatB-C Complexes [10]

**Figure 18:** Mass of cells required for ToxiMop experiment based on the number of TatB-C Complexes [10]

As shown in the figures above, increasing the number of TatB-C complexes allows up to 3500 PP1 to be exported into the periplasm. This change is significant and the number of PP1 in the periplasm of our cells is sufficiently increased to improve our ToxiMop cells. With such an increase, the mass of cells required for clean-up is highly reduced allowing us to suggest this approach.

## Increasing TatA assemblies & TatB-C

**Figure 19:** PP1_{cyto} Increasing TatA assemblies & TatB-C

**Figure 20:** PP1_{peri} Stochastic means for the corresponding TatB-C Complex-TatA Assembly combinations [11]

From a numerical stance and with respect to PP1 number alone, this approach would produce the most efficient ToxiMop cells. However, when comparing the difference between the inclusion of TatB and TatC genes against their inclusion alongside TatA, considerations must also be made regarding the increased stress placed upon cells which must produce all three proteins.

In conclusion, increasing TatA assemblies alone does not provide a sufficiently significant change in the number of exported PP1. Combining this with an increased TatB-C complex number produces the highest potential for PP1 export. Therefore, for our ToxiMop experimental cells we recommended this option based upon its numeric merits. However, we instructed the Wet Team to consider the stress being placed upon the cells with these newly introduced protein productions.

## References

- 1. Leake, Mark C., et al. "Variable stoichiometry of the TatA component of the twin-arginine protein transport system observed by in vivo single-molecule imaging." Proceedings of the National Academy of Sciences 105.40 (2008): 15376-15381.
- 2. Rodriguez,Fernanda, et al. “Structural model for the protein-translocating element of the twin-arginine transport system.” Proceedings of the National Academy of Sciences 110( 2013). 1092-101.

- 3. Stamatakis, Michail, and Nikos V. Mantzaris. "Comparison of Deterministic and Stochastic Models of the
*lac*Operon Genetic Network." Biophysical journal 96.3 (2009): 887-906. - 4. MATLAB program repository: https://github.com/cdjohnston/CraigiGEM-MATLAB
- 5. Palmer, Tracy, and Ben C. Berks. "The twin-arginine translocation (Tat) protein export pathway." Nature Reviews Microbiology 10.7 (2012): 483-496.
- 6. Higham, Desmond J. "Modeling and simulating chemical reactions." SIAM review 50.2 (2008): 347-368.
- 7. https://github.com/cdjohnston/CraigiGEM-MATLAB/blob/master/Production%20%26%20Export/Stochastic%20Models/v1_ssa_PP1_TATproduction_export.m
- 8. https://github.com/cdjohnston/CraigiGEM-MATLAB/blob/master/Production%20%26%20Export/Stochastic%20Models/v2_ssa_PP1_TATproduction_export.m
- 9. https://github.com/cdjohnston/CraigiGEM-MATLAB/blob/master/Production%20%26%20Export/Stochastic%20Models/v6_ssa_PP1_TATproduction_export.m
- 10. https://github.com/cdjohnston/CraigiGEM-MATLAB/blob/master/Production%20%26%20Export/Stochastic%20Models/v7_ssa_PP1_TATproduction_export.m
- 11. https://github.com/cdjohnston/CraigiGEM-MATLAB/blob/master/Production%20%26%20Export/Stochastic%20Models/v8_ssa_PP1_TATproduction_export.m