MODEL
We did two modelling: (1) Biofilm dispersion function by Proteinase-K and (2) H. pylori eradication by SUMO-AMP system. Here we present the mathematical model and solution based on literature searching parameters to answer principally two most important questions revolving around the system: (1) how many products and (2) how long does it take to achieve the determined goal.
Our team believes that the delivery of both Prot-K and AMP would be best done by Escherichia coli. In which they are expressed inside the Escherichia coli using certain promoters. We will use different Escherichia coli batches for 2 main reasons:
- Prot-K and AMP must be released in different times, Prot-K first in order to dismantle Helicobacter pylori’s biofilm, and then some time later AMP, to directly annihilate Helicobacter pylori. The timing of their deliveries shall not be reversed in order.
- We believe that it is better for both substances to be made in different batches as it will maximize the production of respective substances because we will use different promoters to achieve production for each of Prot-K and AMP. Also it won’t increase the complexity on the microbiological scale because there could be interference between the production of both if it is done in the same place and we believe it wouldn’t be as effective and precise as what we would have liked.
Therefore, on this page we provide 2 separate models. One that revolves around the production of AMP and one that revolves around the production of Prot-K. In our first batch of Escherichia coli, we will focus on the production of Prot-K and to promote Escherichia coli’s lysis we will use the E7 protein. In our second batch of Escherichia coli, we will focus on the production of AMP and to promote Escherichia coli’s lysis we will use the insertion of holin.
We will use arabinose in order to halt the mentioned progress. The main idea in this modelling is that arabinose, araC, and the promoters for araBAD (PBAD) and araC (PC) had certain effects together which our team suggest is the most effective to be used in our system of model. The araBAD and araC genes are transcribed in opposite directions. Within and around the promoters for araBAD (PBAD) and araC (PC) lie binding sites for the AraC protein, the cyclic AMP (cAMP) receptor protein (CRP), and RNA polymerase. Alone or in combination, proteins bound to these regions in both the presence or the absence of the inducer, l-arabinose, tightly regulate expression from both promoters. Transcription from PBAD is inducible with l-arabinose. In the presence of arabinose, AraC protein bound at the araI site immediately adjacent to the RNA polymerase binding site of the PBAD promoter stimulates transcription of the araBAD operon. In the absence of arabinose, however, the AraC protein represses mRNA synthesis from PBAD by a mechanism requiring the formation of a DNA loop. Without arabinose, most copies of the ara regulatory region contain a DNA loop between the araO2 and araI sites mediated by AraC protein bound to both of these sites. This loop constrains AraC protein bound at araI from entering the inducing state and holds the uninduced level of PBAD expression low. Upon the addition of arabinose, the araO2-araI loop opens, and arabinose bound to AraC protein on the araI site drives AraC into the inducing conformation, thereby inducing PBAD.^{1,2}
With the arabinose utilization system in Escherichia coli, with a concentration of 0.01% arabinose, it would take 34 ± 10 minutes for the arabinose to take an almost complete effect takeover on the group of Escherichia coli in which ~90% of Escherichia coli do not show any indication that its natural processes had been affected by the induction of arabinose for 70 minutes, and in 0.2% concentration of arabinose it would take 16 ± 2.5 minutes to do so. ^{3} Our team would like to test for different molarities of arabinose, which is 0,1 mM, 1mM, and 10mM of arabinose which is equal to approximately 0.00015%, 0.0015%, and 0.015% of arabinose concentration in water (mixed with a little amount of glucose). By using an exponential approximation to the experimental data given by Megerle et. al., we will derive the time needed on our project for each arabinose molarity so that the Escherichia coli will start producing Prot-K.
- For 10mM it would take an average of:
34 * ((16/34) ^ (0.015)/(0.01) / (0.2/0.01)) which is approximately 32.13 minutes - For 1mM it would take an average of:
34 * ((34/16) ^ (0.01)/(0.0015) / (0.2/0.01)) which is approximately 43.71 minutes - For 0.1mM it would take an average of:
43.71 * ((34/16) ^ (0.01)/(0.00015) / (0.2/0.01)) which is approximately 63.72 minutes
After the Escherichia coli start producing Prot-K, we will take interest in the amount of time needed for each Escherichia coli to produce the maximum number of Prot-K that they are able to produce. We believe that such quantities can be thoroughly scouted and represented in some sort of differential equations. Our most appropriate modelling would have been somewhat similar with this:^{4,5}
Prot-K_{m}: concentration of translated Prot-K
Prot-K: concentration of transcribed AMP
Parameters are determined through thorough search of correlated literatures and are shown as followed:^{6-9}
Parameter |
Description |
Value |
cpbad |
Transcription rate of Pbad |
2.54 [6] |
lpbad |
Leakage factor of Pbad |
0.002 |
n |
Activation constant of Pbad |
0 or 1 |
dmRNA |
Degradation rate of mRNA |
0.231 [7] |
a |
Translation rate per amino acid |
1020 [8] |
sProt-K |
Length of Prot-K in amino acids |
277 |
deProt-K |
Degradation rate of Prot-K |
0.0001 [9] |
We admit that certain functions and interference that revolve around the nature of each variable are not pointed out significantly in our modelling system. These things are hard to recognize and inform mathematically unless we go through some in-situ observations in which with the current situation our members weren’t able to practice.
Our solution towards this differential equation could then be approximated into a further better extent with an exponential approximation revolving around comparison of results regarding the time needed for TetR to reach its maximum value and its production promoter’s transcription rate (on our AMP production modelling) with the numbers revolving around Prot-K. We believe that it would take approximately 206 minutes and 25 seconds for our Escherichia coli batch to produce all the necessary Prot-K and therefore ready to be consumed. Even though arabinose may not be able to affect a partial part of the Escherichia coli batch in some time as stated before by Megerle et. al. the approximate extra time needed to produce Prot-K will still give some time for the arabinose to affect the remaining Escherichia coli, so we assume that by then nearly 100% of the Escherichia coli had already manage themselves to produce a significant amount of Prot-K.
As for the number of Prot-K molecules produced per Escherichia coli, to increase accuracy of our approximation to a certain extent, we will impose polynomial approximation to results taken from Guzman et. al.^{10} regarding production of protein molecules inside Escherichia coli promoted with PBad. We believe that it would take approximately a single Escherichia coli will eventually make an average of 8,300 molecules of Prot-K given the time stated.
Assimilation of NH4+ will then be inevitable once the Escherichia coli reaches the stomach linings, in which there will be an absurd amount of nitrogen (at least in Escherichia coli standards). Forced assimilation of alternative nitrogen sources will then provide enough motivation on the promoter systems of glnA-ntrBC to deactivate NtrC and then the occuring of autophosphorylation activity of NtrB will ensure that no more NtrC can be activated via phosphorylation. The amount of nitrogen in NH4+ that is produced naturally and in heavy amounts on the stomach linings would then halt immediately the promotion effect from the PgLnAp2 promoters.^{11}
The result would then be the immediate halting process of TetR production which will then induce a lysis process of our Escherichia coli which is heavily assisted by E7. The Escherichia coli lysis system using E7 has been studied significantly and results with similar previous inductions of arabinose had been clarified. After the immediate induction of E7, it would take 30 minutes in 37^{o}C (inner body temperature) for the Escherichia coli to perform lysis.^{12}
- M. Better, “Gene Expression Systems”, 1999.
- L. Huo, K. J. Martin, and R. Schleif, “Alternative DNA loops regulate the arabinose operon in Escherichia coli”, 1988.
- J. A. Megerle, G. Fritz, U. Gerland, K. Jung, and J. O. Radler, “Timing and Dynamics of Single Cell Gene Expression in the Arabinose Utilization System”, 2008.
- T. Chen, H. L. He, & G. M. Church, “Modelling Gene Expression With Differential Equations”, 2019.
- https://2008.igem.org/Team:UC_Berkeley/Modeling
- C. J. Davidson, A. Narang, and M. G. Surette, “Integration of transcriptional inputs at promoters of the arabinose catabolic pathway”, 2010.
- J. A. Bernstein, A. B. Khodursky, P. H. Lin, S. Lin-Chao, and S. N. Cohen, "Global analysis of mrna decay and abundance in escherichia coli at single-gene resolution using two-color fluorescent dna microarrays", 2002.
- R. Young and H. Bremer, "Polypeptide-chain-elongation rate in Escherichia coli B/r as a function of growth rate", 1976.
- U. Arnold, “Differences in the Denaturation Behavior of Ribonuclease A Induced by Temperature and Guanidine Hydrochloride”, 2000.
- L. Guzman, D. Belin, M. J. Carson, and J. Beckwith, “Tight Regulation, Modulation, and High-Level Expression by Vectors Containing the Arabinose PBAD Promoter”, 1995.
- A. Herrero, E. Flores, and J. Imperial, “Nitrogen Assimilation in Bacteria”, 2019.
- C. Zhao, C. Sun, X. Chang, S. Wu, and Z. Lin, “Construction and Application of Cell Lysis Systems in the Expression of Mycotoxin Degrading Enzyme in Escherichia coli”, 2019.
It has come into an understandable conclusion that during Escherichia coli’s lysis there are 2 conditions that should be met well:
- Helicobacter pylori is in very near proximity within Escherichia coli
- A suitable amount of AMP has been created inside Escherichia coli
To clarify and re-iterate, it is to be reminded again that the need to insert promoters and subsequent engineering as explained in our project preview are necessary; in order to:
- Promote AMP production (through the means of producing SUMO+AMP and cleaving it with Ulp)
- Promote holin production to trigger lysis of Escherichia coli in order to fully deploy the produced AMPs.
Here, we would like to create such a model that is a fusion between 2 models:
- The “Timer & AMP” model that models the production of AMP inside the Escherichia coli led by the copA promoter.
- The “Kill Switch” model that models the Escherichia coli’s lysis timing
With each variable corresponding to points of interest as described:
Variable |
Description |
TetRm |
Concentration of translated TetR |
SAMPm |
Concentration of translated SAMP |
cIm |
Concentration of translated cI |
Ulpm |
Concentration of translated Ulp |
TetR |
Concentration of transcribed TetR |
cI |
Concentration of transcribed cI |
Ulp |
Concentration of transcribed Ulp |
SAMP |
Concentration of transcribed SAMP |
Parameter |
Description |
Value |
cpcopA |
Transcription rate of PcopA |
4.16 [5] |
lpcopA |
Leakage factor of PcopA |
0.002 |
n |
Activation constant of PcopA |
0 or 1 |
dmRNA |
Degradation rate of mRNA |
0.231 [4] |
lpTet |
Leakage factor of PTet |
0.002 |
kpTet |
Dissociation constant of PTet |
6 [8] |
xpTet |
Hills coefficient for PTet |
3 [9] |
lpcI |
Leakage factor of PcI |
0.002 |
kpcI |
Dissociation constant of PcI |
20 [8] |
xpcI |
Hills coefficient for PcI |
3 [9] |
a |
Translation rate per amino acid |
1020 [6] |
sTetR |
Length of TetR in amino acids |
206 |
deTetR |
Degradation rate of TetR |
0.1386 [10] |
scI |
Length of cI in amino acids |
228 |
decI |
Degradation rate of cI |
0.042 [10] |
sUlp |
Length of Ulp in amino acids |
233 |
deUlp |
Degradation rate of Ulp |
0.01263 |
sSAMP |
Length of SAMP in amino acids |
128 |
deSAMP |
Degradation rate of SAMP |
0.0063 |
kcUlp |
Turnover rate of Ulp |
3 |
As reflected by our model, the SUMO+AMP starts being produced almost at the same time as when the T7 promoter is set on, reaching a linear rate of production as early as the timer reaches 5 minutes. Slower rate of production will then start approximately at 40 minutes and will significantly affect the production at the one-hour mark (as soon as the production of ulp-1 becomes more significant), resulting in an instantaneous equilibrium around the 74 minutes mark with 3400 molecules of SUMO+AMP existing in the Escherichia coli as its highest ever.
It can be understood that the cleaving of SUMO+AMP into AMPs had already started since the beginning of the recorded time (with a near linear rate). The rate of cleaving will then increase as ulp-1 started being produced significantly at the one-hour mark; right before suddenly going back to another linear rate (but now with lower rate of cleaving than the previous rate) approximately at 103 minutes as the number of SUMO+AMP had almost drop to 0.
With each variable corresponding to points of interest as described:
Variable |
Description |
H_{m} |
Concentration of translated holin |
H_{A}_{m} |
Concentration of translated antiholin |
H |
Concentration of transcribed holin |
HA |
Concentration of transcribed antiholin |
D |
Concentration of transcribed Dimer |
Parameters are determined through thorough search of correlated literatures and are shown as followed
Parameter |
Description |
Value |
cpcopA |
Transcription rate of PcopA |
4.16 [5] |
lpcopA |
Leakage factor of PcopA |
0.002 |
n |
Activation constant of PcopA |
0 or 1 |
dmRNA |
Degradation rate of mRNA |
0.231 [4] |
cpcon |
Transcription rate of Pcon |
0.5 |
a |
Translation rate per amino acid |
1020 [6] |
sH |
Length of holin in amino acids |
219 |
deH |
Degradation rate of holin |
0.0348 [3] |
kf |
Forward rate |
0.00117 [3] |
kb |
Backward rate |
0.00003 [3] |
sAH |
Length of antiholin in amino acids |
103 |
deAH |
Degradation rate of antiholin |
0.0348 [3] |
deD |
Degradation rate of Dimer |
0.0348 [3] |
It is to be noted that there is actually no set nor perfectly modelled lethal level of Holin for Escherichia coli. Some Escherichia coli even have staggering differences against each other for the number of holin concentration it can maintain before going it lyses. The difference is stated to be a result of different conditions of each unique Escherichia coli and further deviated by random mutations presented (or occuring) inside each unique Escherichia coli and the effectiveness of each holin towards creating “optimal” holes on the Escherichia coli topology which by any means both effects are totally random, unrecognizable at hindsight, and uncontrollable.The lethal level of Holin for most Escherichia coli is believed to be around 170 molecules per cell, yet for some Escherichia coli that may contain unrecognizable “special conditions” the lethal level of Holin may deviate up to 1000 molecules per cell, with 300 molecules per cell being subjected as the most occurring observation between the abnormal subjects.^{7}
The model is then simulated numerically with various other limitations that are being played upon to best simulate the realism of how the system will actually work in situ, cancelling its inherent oversimplification.
The graph implies a nearly-constant rate of holin growth up until the first hour of induction. During the first 20 minutes of induction it can be inferred that holin growth rate is increasing, and then it remain the most constant during the next 10 minutes, followed by a slight decline in holin growth rate afterwards, probably due to the fact that after some time reactions and transcriptions inside Escherichia coli will slow down as it adjust itself into equilibrium. From the graph, it can be deduced that for the level of holin to reach 170 molecules per cell, it would take approximately 11 minutes to do so since induction, and for reaching 300 and 1000 molecules per cell, it would take approximately 16 and 36 minutes to do so since induction respectively.
Due to the unknown spread and distribution between the categories of Escherichia coli factors that determine directly its holin threshold before approaching lysis, it is almost impossible for us to determine the average time of Escherichia coli lysis for our project unless we do a distribution modelling assumption for the results. We agree that the further the deviation of the time needed for lysis to happen, then the rarer the occurrence would be (because it depends on numerous alterations regarding the conditions in each Escherichia coli), therefore we would like to approximate the statistical distribution of the extra time taken for lysis to happen (compared to the usual 11 minutes taken) using a heavy-tailed distribution, such as the Weibull distribution with its shape parameter ranging from 0 to 1.
We assume that half of Escherichia coli will need less than 16 minutes to perform lysis on our model and we are sure that the event that an Escherichia coli will need more than 36 minutes on our model will be rare, hence using standard statistical confidence we set that 95% of Escherichia coli will need less than 36 minutes to perform lysis on our model. Therefore by applying the Weibull cumulative model, we can perform approximation of the shape (k) and scale (lambda) parameters of the Weibull distribution by solving these equations:
Which yield the the shape (k) and scale (lambda) parameters of the Weibull distribution with the exact answer and numerical approximation as followed:
Solving the equation provide us with the solution in which the time needed for Escherichia coli lysis (expressed as T) can be assumed with the following Weibull statistical distribution model; added by the constant 11 minutes inherited on normal lysis time.
And therefore the mean time for each Escherichia coli to achieve lysis in our model would be:
E[T] = 11+7.48158(T(1+1/0.909449)
E [T] = 18.827723
Or approximately 18 minutes and 50 seconds.
Complete H. pylori eradication model
With each variable corresponding to points of interest as already described before with the addition of AMP as concentration of transcribed AMP. While parameters follow as what has been given in the previous 2 models with the addition of de_{AMP} which is the degradation rate of AMP (0.0021).
The model is then simulated accordingly with similar alterations as mentioned in previous models, and thus resulted in the given graph:
As can be inferred from our model, the moment when ulp-1 has started its significant production also indicates the moment when holin has started its slight production. Using the results inferred from the previous model, this moment will approximately start at the one-hour mark.
Our mathematical formulation and simulation suggests that the nature of holin production in the fused model is indifferent compared with its production nature on its stand-alone model. On its stand-alone model it would take approximately 12.5 minutes to reach a holin level of 200 molecules per cell and on this model it would take approximately 100 minutes (after significant production at 60 minutes) to reach a holin level of 200 molecules. Therefore, holin production is faster approximately 8 times on its stand-alone model rather than the last fused one.
Hence, on our latest fused model, it would take in average approximately 60 + 8(18.827723) minutes for lysis to happen in an Escherichia coli: 210 minutes and 37 seconds to do so. It is obvious that at this time, almost no SUMO+AMP particles will be left uncleaved; showing incredible efficiency of its use to provide the maximum number of AMPs.
As for the number of AMPs produced we will use a similar method of approximation as what we have done to approximate the time of lysis. As our graph infer, assuming near-linearity of AMP production since the 110 minute mark (start of time when almost no SUMO+AMP particles will be left uncleaved) of 120 AMP molecule produced each minute, therefore at the time of lysis that will occur approximately 100 minutes after, there would approximately be 12000+100(120) AMP molecules set to be released. Approximately an arsenal of 24.000 AMP molecules ready to fight H. pylori.
Proteinase-K and AMP interaction
We find out that PGLa-AM1 has proteinase K cleavage site using Peptidecutter.^{10} Therefore, we need to adjust the amount of AMP that will be produced so it can achieve the concentration to kill H. pylori. To adjust the amount of AMP, we need to know the amount of AMP that will be digested by proteinase K. We find that proteinase K has specific activity of 13 U/mg for synthetic substrate Suc-(Ala)3-NH-Np at enzyme concentration 8 μg/ml (0.28 μM) in a volume of 1 ml.^{11} Using the specific activity of proteinase K, we can calculate the amount of product that is produced and therefore we can know the amount of AMP that is digested by proteinase K.
The MIC for our AMP is 1 μg/mL and our product is prepared for 75 mL. That means we need 75 μg of AMP in our product. Since our AMP has a molecular weight of 2 kDA, we need 0.0375 μmol. But because 265.2 μmol of AMP will be digested by proteinase K, we need a total of 265.2375 μmol of AMP in our product. Converting to gram, as low as 500 mg PGLa-AM1 is needed to give therapeutic effect.
However this data has to be interpreted carefully since we are lacking parameters related to the activity of Proteinase-K as an enzyme that work digesting antimicrobial peptide. The gastric environment is also complex and consist of enzyme that may also digest Proteinase-K. Some parameter such as the effect of low pH exposure, the residual activity after dispersing biofilm, and its substrate saturation are opened for exploration. Further studies are needed as this model is an oversimplification of the real situation.
- T. Chen, H. L. He, & G. M. Church, “Modelling Gene Expression With Differential Equations”, 2019.
- https://2013.igem.org/Team:TU-Delft/Timer-Sumo-KillSwitch
- https://2008.igem.org/Team:UC_Berkeley/Modeling
- J. A. Bernstein, A. B. Khodursky, P. H. Lin, S. Lin-Chao, and S. N. Cohen, "Global analysis of mrna decay and abundance in escherichia coli at single-gene resolution using two-color fluorescent dna microarrays", 2002.
- M. Barrio, K. Burrage, A. Leier, and T. Tian, "Oscillatory regulation of hes1: Discrete stochastic delay modelling and simulation", 2006.
- R. Young and H. Bremer, "Polypeptide-chain-elongation rate in Escherichia coli B/r as a function of growth rate", 1976.
- C. Y. Chang, K. Nam, and R. Young, "S gene expression and the timing of lysis by bacteriophage lambda", 1995.
- MCLab, "Sumo protease", 2013.
- Team Aberdeen iGEM team 2009, "Dissociation constants," 2009.
- ExPASy - PeptideCutter [Internet]. [cited 2021 Oct 18]. Available from: https://web.expasy.org/cgi-bin/peptide_cutter/peptidecutter.pl
- Bajorath J, Hinrichs W, Saenger W. The enzymatic activity of proteinase K is controlled by calcium. Eur J Biochem. 1988;176(2):441–7.