The β-barrel Assembly Machinery Complex

Outer membrane proteins (OMPs) of Gram-negative bacteria are synthesized in the cytoplasm and transported across the inner membrane by SecYEG translocon into the periplasm. The survival factor A (SurA) chaperones carry the unfolded membrane proteins across the periplasm to the BAM complex, which is responsible for the insertion and assembly of OMPs into the outer membrane [1].

In E. coli BAM complex consists of a membrane protein BamA and four lipoprotein subunits BamBCDE. These subunits associate with BamA through periplasmic POTRA domains. In vitro reconstitution of the E. coli BAM complex and functional assays showed that all five subunits are required to achieve the maximum activity of BAM [1].

In theory, recruiting the BAM complex in a cell-free system could be extremely beneficial as it could integrate OmpA and lgA protease beta-domain into the membranes of liposomes from the inside without requiring any additional protein complexes. Also, this would make a cell-free system more ubiquitous, because BAM complex does not require any signal sequence for proper protein insertion. In order to ensure quick integration, BamA needs to be consistently present at high yields throughout the expression of OmpA and lgA. For this reason, it is essential to stimulate its expression by an initial addition of mRNA, ensuring rapid expression of BamA. For this reason, with the help of Edinburgh iGEM team (special thanks to Freddie Starkey), a mathematical model for BamA kinetics was created.

Mass Action Equations

First of all, to represent chemical reactions and to render a start for mechanistic modelling, Mass Action Equations were used. It is known that the laws of mass action state that the rate of a chemical reaction is directly proportional to the product of the activities or concentrations of the reactants. The mass action equations in Figure 1 can be used to represent basic protein expression:

Fig. 1

Fig. 1 Mass Action Equations for Protein Expression

Each of these equations is used in triplicate to represent the expression of BamA, OmpA and lgA respectively and from these mass action equations a system of ordinary differential equations can be derived.

Ordinary Differential Equations

The model uses a set of differential equations:

Fig. 2

Fig. 2 System of differential equations for BamA kinetics

Determination of the System Values

In order to solve this system it is first necessary to derive values for all the parameters used:

1. copies_BamA, copies_OmpA, copies_IgA - Relative number of plasmid copies.

It is important to consider the effect of different starting masses of DNA of BamA, OmpA, and lgA, therefore to calculate the number of plasmids from which each protein can be expressed. Assuming that we add 25-1000 ng of DNA to our system [1], a single base pair has mass of 650 Da, and the length of each plasmid is known, mass of each plasmid was calculated:

1. peT2Ab with BamA - 5089.5 kDa
2. pRSETb with OmpA - 2550.95 kDa
3. pRSETb with lgA - 2490.8 kDa

Knowledge that 1 ng equals to 6.022∗1017 kDa, allows to calculate the number of plasmids present for a particular number of ng of DNA added (Tab. 1).

Tab. 1 Number of plasmids present for a particular number of ng of DNA added

peT2AB with BamA
DNA added (ng)	Number of copies
25	2.985*1015
250	2.958*1016
1000	1.183*1017
pRSETb with OmpA
DNA added (ng)	Number of copies
25	5.902*1015
250	5.902*1016
1000	2.361*1017
pRSETb with IgA
DNA added (ng)	Number of copies
25	6.04415
250	6.04416
1000	2.41817

2. tr_BamA, tr_OmpA, tr_IgA - Transcription rate.

Transcription rate of T7 RNA polymerase is approximately 60 nucleotides per second [2]. Length of each gene was 2430 nucleotides for BamA, 1038 nucleotides for OmpA, and 945 nucleotides for lgA. Transcription rate per minute for each gene was calculated:

1. tr_BamA = (60/2430)∗60 = 1.48 mRNAs per minute
2. tr_OmpA = (60/1038)∗60 = 3.47 mRNAs per minute
3. tr_lgA = (60/945)∗60 = 3.81 mRNAs per minute

3. deg_{mRNA BamA}, deg_{mRNA OmpA}, deg_{mRNA IgA} - mRNA degradation rate.

Degradation rate is calculable from half-life using the formula: degX = ln(2)/halflife [3], where X shows the transcript of the target gene. Average mRNA half-life approximately is 5 minutes, however we screened half-lives of 1, 5, 10, and 15 minutes for each protein in order to more precisely evaluate the variability of the results. Degradation rates per minute were calculated:

1. 1 minute half-life - 0.69 mRNAs per minute
2. 5 minute half-life - 0.14 mRNAs per minute
3. 10 minute half-life - 0.07 mRNAs per minute
4. 15 minute half-life - 0.05 mRNAs per minute

4. trl_BamA, trl_OmpA, trl_IgA - Protein translation rate.

Translation rate is about 20 amino acids per second [2]. Lengths of target proteins are 2430 nucleotides for BamA, 1038 nucleotides for OmpA, and 945 nucleotides for lgA. Translation rates per minute were calculated:

1. trl_BamA = (20/810)∗60 = 1.48 proteins per minute
2. trl_OmpA = (20/346)∗60 = 3.47 proteins per minute
3. trl_lgA = (20/315)∗60 = 3.81 proteins per minute

5. deg_BamA, deg_OmpA, deg_IgA - Protein degradation rate.

Protein half-life was determined using ProtParam Tool, which uses the N-end rule [4] to determine protein half-life. The estimates given for each of BamA, OmpA, and lgA are >10 hrs in E. coli. In order to reflect the inexact nature of these computationally derived half-lives, we screened over possible half-lives of 10, 20, and 30 hours for each of BamA, OmpA, and lgA. Applying prior used degradation rate formula degX = ln(2)/halflife [3], this yielded degradation rates per minute of:

1. 10 hour half-life - 1.16∗10⁻³ proteins per minute
2. 20 hour half-life - 5.78∗10⁻⁴ proteins per minute
3. 30 hour half-life - 3.85∗10⁻⁴ proteins per minute

Starting Conditions

In order to examine the effects of higher initial mass of BamA RNA, 6 different values were screened over (Tab. 2). Assuming that addition of RNA in IVTT system is between 1 and 5 µg, the masses of sense and antisense strands of BamA in kDa [6] are 830.382 and 820.8, respectively, and conversion is 1 µg = 6.022∗10²⁰ kDa, the number of RNA molecules added can be calculated using the formula µgadded∗(6.022∗10²⁰/((830.382+820.8)/2)).

Tab. 2 Initial BamA mRNA

RNA added (µg)	RNA added (molecules)
0	0
1	7.29*1017
2	1.46*1018
3	2.19*1018
4	2.92*10
5	3.65*1018

The primary aim of this model was to identify parameters leading to rapidly-achieved and consistently high levels of BamA under conditions of co-expression of BamA, OmpA, and lgA. Prior to this it was important to identify particular parameters leading to these conditions and to examine some general trends. The number of molecules of mRNA and protein for each average, minimum and maximum plot and each of BamA, OmpA, and lgA after 2 hours were calculated (Fig. 3) and summarized (Tab. 3).

Fig. 3.1 Fig. 3.2

Fig. 3Minimum, maximum and average levels of mRNAs and protein for BamA, OmpA and lgA

Tab. 3Average, minimum and maximum number of protein molecules after 2 hours' Protein Number of molecules Average/td> BamA 2.8*10²⁰ OmpA 2.78*10²¹ IgA 3.44*10²¹ Minimum> BamA 1.95*10¹⁸ OmpA 2.13*10¹⁹ IgA 2.63*10¹⁹ Maximum> BamA 1.28*10²¹ OmpA 1.28*10²² IgA 1.58*10²²

In average, minimum, and maximum cases the protein expression of OmpA, lgA, and BamA follows the same trend which is unaffected by fluctuations of mRNA level. Each protein is expressed at a rate primarily proportional to its length and to a magnitude primarily dependent on the mass of available DNA.

Sensitivity Analysis

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Fourier Amplitude Sensitivity Testing (FAST) indices represent the proportion of the output variance of the model attributable to a particular variable and its interactions. Focusing on BamA expression as the protein of interest, total order FAST sensitivity indices were calculated using the BamA protein level each 20 minutes as the model output (Fig. 4).

Fig. 4

Fig. 4 FAST sensitivity analysis of BamA

As it can be seen from the graph, number of BamA plasmid copies contributes most to output variance over the whole time span. Also, BamA mRNA degradation rate is considerably faster than BamA degradation rate - with mRNA halflife of the order of minutes and protein halflife of the order of hours - hence the greater FAST index.

Conclusions

Prior to starting the wet lab experiments, we had hypothesized that the addition of mRNA into our system would ensure that BamA folded and inserted into liposome membrane more efficiently, thus enhancing the expression of OmpA and IgA. Therefore it was decided to purify BamA mRNA and add it to the reaction mixture as a template instead of DNA as we assumed that skipping the transcription step would increase protein synthesis rate. After creating a mathematical model that calculated the necessity of mRNA addition to IVTT system, we were able to generate a more efficient transcription-translation system in which we used both the purified BamA mRNA and DNA. This model clearly revealed that using both mRNA and plasmid DNA in our system was essential as BamA mRNA did increase the rate of protein expression while this effect was proportional to the mass of initial DNA. However, after experimenting in the wet lab, we chose to use purified BamA as it was desired to reach higher expression yields of membrane proteins and it proved to be more effective as BamA mRNA degradation rate was considerably faster than BamA protein degradation rate.

References

1. Gu, Y. et al. Structural basis of outer membrane protein insertion by the BAM complex. Nature 531, 64-69 (2016).
2. Biolabs, N. PURExpress® In Vitro Protein Synthesis Kit | NEB. International.neb.com (2018). at < https://international.neb.com/
3. Philips, R. What is faster, transcription or translation?. Book.bionumbers.org (2018). at
4. Exponential decay. En.wikipedia.org (2018). Accession: at
5. Bachmair, A., Finley, D. & Varshavsky, A. In vivo half-life of a protein is a function of its amino-terminal residue. Science 234, 179-186 (1986).
6. RNA Molecular Weight Calculator | AAT Bioquest. Aatbio.com (2018). Accession: at

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