Introduction

Team Vilnius-Lithuania aims to use the PURE cell free system [1] in order to integrate proteins into the membrane of the liposome from the inside. The BamA complex is responsible for integrating these proteins; OmpA and lgA hence in order to ensure quick integration BamA needs to be consistently present at high levels throughout the expression of OmpA and lgA. This mechanistic model aims to examine the simultaneous expression of these proteins and compare results primarily across different starting volumes of BamA RNA in order to quantify the effectiveness of an initial addition of RNA in ensuring fast expression of BamA.

Mass Action Equations

Mass Action Equations are commonly used to represent chemical reactions and provide a starting point for mechanistic modelling of a variety of phenomena. The laws of mass action state that the rate of any chemical reaction is proportional to the product of the masses of the reacting substances, with each mass raised to a power equal to the coefficient that occurs in the chemical equation [2]. The mass action equations in Figure 1 can be used to represent protein expression:

Each of these equations is used in triplicate to represent 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 simple set of ordinary differential equations (ODEs):

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

copies_BamA, copies_OmpA, copies_lgA - Number of plasmid copies - In order to take into account the effect of different starting masses of DNA for BamA, OmpA and lgA it is neccesary to calculate the number of plasmids present from which each protein may be expressed. DNA added to the PURE system should be between 25 and 1000ng per reaction [1] and hence it was decided to screen over a number of values in this range. A single base pair has mass of 650 Daltons [3] hence it is possible to calculate mass in kDa of each plasmid with its particular insert when each plasmid length is known [4] [5]:

The conversion 1ng = 6.022*10¹⁷ kDa allows the calculation of the number of plasmids present for a particular number of ng of DNA added.:

Although it would have been desirable to screen over a greater number of values within the 25-1000ng range the resulting increase in size of the already large parameter space made this intractable.

tr_BamA, tr_OmpA, tr_lgA - Transcription Rate - Transcription is known to occur at approximately 60 nucleotides per second [6]. Length of each gene was found to be 2430 nucleotides for BamA [7], 1038 nucleotides for OmpA [8] and 945 nucleotides for lgA [9]. Computing the transcription rate per minute for each of BamA, OmpA and lgA:

deg_{mRNA BamA}, deg_{mRNA OmpA}, deg_{mRNA lgA} - mRNA Degradation Rate - Average mRNA half-life is known to be approximately 5 minutes however to reflect variation in this number between different mRNAs we screen here over half-lives of 1, 5, 10 and 15 minutes for each protein [10]. Degradation rate is calculable from half-life using the formula; \(deg = ln(2)/halflife\) [11]. Using this formula degradation rates per minute are calculated as:

trl_BamA, trl_OmpA, trl_lgA - Translation Rate - Translation is known to occur at approximately 20 amino acids per second [12]. Length of each protein was found to be for BamA [7], for OmpA [8] and for lgA [9]. Computing the translation rate per minute for each of BamA, OmpA and lgA:

deg_BamA, deg_OmpA, deg_lgA - Protein Degradation Rate - Protein half-life was determined using ProtParam Tool [13]. ProtParam uses the N-end rule [14] to determine protein half-life, the estimates given for each of BamA, OmpA and lgA are \(>\)10hrs in E.coli and 30hrs in reticulocytes hence an average 20hr half-life. In order to reflect the inexact nature of these computationally derived half-lives we screen over possible half-lives of 10, 20 and 30 hours for each of BamA, OmpA and lgA. Applying the same degradation rate formula as previously (\(deg = ln(2)/halflife\)) [11] this yields degredation rates per minute of:

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