Team:YAU-China/Model
To understand, predict and ultimately control the behavior of our engineered bacteria quorum sensing, we have developed dynamic model of the system, based on differential equations which describe and integrate the individual processes. This model involves several entities going from the molecular level( genes, RNAs, proteins, and metabolites) up to the cellular and population levels, distinct intracellular and extracellular compartments, and a wide range of biological and physical processes( transcription, translation, signal transduction, growth, diffusion). Here we can show the concentrate of PelA and PslG produced by our engineered bacteria and the biofilm removing time through calculating.
So that our goal of this model is to create a generic quorum sensing model:
1. Through the passive solution of bacterial destruction of the wild Pseudomonas aeruginosa biological capsule time.
2. Through the active solution of engineering bacterial destruction of wild Pseudomonas aeruginosa biological capsule time.
Through the passive way of cracking engineering bacteria destruction organisms were touched
Pf4 phage is a mild phage that specifically recognizes Pseudomonas aeruginosa. Pf4 is one of the components of biofilms that can infect cells and lyse them. Where there are biofilms, there are phages. After knocking out the Pf4-related coding gene of the engineered bacteria, we made it more sensitive to Pf4, which was specifically attacked by the Pf4 phage secreted by the wild bacteria, so that the two enzymes could flow out to break the biofilm of the wild bacteria.
1.1 Production of Pf4 phages
In pseudomonas aeruginosa , the production of Pf4 phages is promoted by the shear enzyme XisF4 and inhibited by the inhibiting enzyme Pf4r. The concentration of xisF4 and Pf4r determines the release efficiency of Pf4 phages. A through the passive way of cracking engineering bacteria destruction organisms were touched.
1.1.1 The translation of xisF4
The xisF4 translation depends on the translation rate of the strain, the mRNA length and the quantity of mRNA. The translation velocity is expressed in molar concentration in one cell per time unit.
$$ V_{translation,x}=\frac{\left[{xisF4}_{mRNA}\right].K_{translation}.\left(Ribosomes/RNA\right)}{RNA length}$$
$K_{translation}$：Pseudomonas aeruginosa translation rate (nucleotides/s)
$\left[{xisF4}_{mRNA}\right]$ :xisF4 mRNA concentration in one Pseudomonas aeruginosa cell
$Ribosomes/RNA$: Number of ribosomes per mRNA
RNA length : Number of nucleotides on the xisF4 mRNA
For the convenience of mathematical operation, we merge the k${}_{translation}$ and Ribosomes/ RNA and to a constant.
1.1.2 The translation of pf4r
The pf4r translation depends on the translation rate of the strain, the mRNA length and the quantity of mRNA. The translation velocity is expressed in molar concentration in one cell per time unit.
$$ V_{translation,Pf4r}=\frac{\left[{Pf4r}_{mRNA}\right].K_{translation}.\left(Ribosomes/RNA\right)}{RNAlength} $$
$K_{translation}$: Pseudomonas aeruginosa translation rate (nucleotides/s)
$\left[Pf4r\right]$: pf4r mRNA concentration in one Pseudomonas aeruginosa cell
$Ribosomes/RNA$: Number of ribosomes per mRNA
$RNAlength$: Number of nucleotides on the pf4r mRNA
For the convenience of mathematical operation, we merge the k${}_{translation}$ and Ribosomes/ RNA and to a constant.
1.2 Extracellular polysaccharide hydrolase destroys biofilm
PelA and PslG are two enzymes encoded by Pseudomonas aeruginosa, and their function is to destroy the biofilm. Therefore, PelA and PslG can be considered as a kind of material enzyme X.
1.2.1 X transcription
The X transcription depends on the transcription rate of the strain and the length of the X gene. The Avogadro number is used to express the transcription velocity in molar concentration in one cell per time unit.
$$ V_{transcription,x\ mRNA}=\frac{K_{transcript}\left({RNA}_{{polymerase}/{gene}}\right)}{DNAlength.N_A.V_{intracell}} $$
$K_{transcript}$: Pseudomonas aeruginosa transcription rate (nucleotides/s)
${RNA}_{polymerase/gene}$: Number of RNA polymerase per gene
$DNAlength$: Number of nucleotides on the X gene
$V_{intracell}$: Volume of a bacterial cell (L)
For the convenience of mathematical operation, we merged the $K_{transcript}$ , ${RNA}_{polymerase/gene}$ and $V_{intracell}$ to a constant.
1.2.2 The translation of X
The X translation depends on the translation rate of the strain, the mRNA length and the quantity of mRNA. The translation velocity is expressed in molar concentration in one cell per time unit.
$$ V_{translation,x}=\frac{\left[x_{mRNA}\right].K_{translation}.\left(Ribosomes/RNA\right)}{RNAlength} $$
$K_{translation}$: Pseudomonas aeruginosa translation rate (nucleotides/ s)
$\left[x_{mRNA}\right]$: X mRNA concentration in one Pseudomonas aeruginosa cell
$Ribosomes/RNA$: Number of ribosomes per mRNA
$RNAlength$ : Number of nucleotides on the X mRNA
For the convenience of mathematical operation, we merge the $K_{translation}$ and $Ribosomes/RNA$ and to a constant.
1.2.3 Degradation
Some of the X protein and mRNA are degraded. A degradation constant is used to model the degradation velocity.
$$ V_{degradation,x}=K_{deg,x}.{[x]}_c $$
$K_{deg,x}$: X degradation constant（s-1）
$$ V_{degradation,xmRNA}=K_{deg,xmRNA}.{[xmRNA]}_c $$
$K_{deg,xmRNA}$: X mRNA degradation constant （s-1）
1.3 Biofilm removel
The biofilm is removed by the enzyme and the process is modeled assuming a Michaelis- Menten kinetics.
$$ V_{remo,biof}=K_x{[x]}_w\frac{[Biof]}{K_m+[Biof]}V_{intracell}[PAOI*] $$
$K_x$: catalytic constant of the X enzyme（s-1）
$K_m$: ( Michaelis constant of the X enzyme (mol/ l)
$\left[PAOI*\right]$: Concentration of engineered bacteria（mol/ l）
1.4 Solve
The system of ODEs was solved using Matlab R2016a. And we used the ode15s solver.
The complete set of ODEs is detailed here:
$$ \frac{\mathrm{d}\left[\mathrm{Pf4}\right]}{\mathrm{dt}}\mathrm{=}{\mathrm{V}}_{\mathrm{translation,xisF4}}\mathrm{-}{\mathrm{V}}_{\mathrm{translation,Pf4r}} $$
$$ \frac{{d[X-mRNA]}_c}{dt}\mathrm{=}V_{transcription,XmRNA}-V_{degradation,XmRNA} $$
$$ \frac{{d[X]}_c}{dt}\mathrm{=}V_{translation,X}-V_{degradation,X} $$
$$ \frac{d[Biof]}{dt}=-V_{remo,biof} $$
1.5 Result
This model can remove the biofilm in one hour
Pyrolytic engineering bacteria to destory the biofilm by an active mode
In addition to the passive pyrolysis of engineering bacteria by Pf4 phage, we also designed an active pyrolysis engineering bacteria. During the growth of Pseudomonas aeruginosa, quorum sensing system (QS) is used to regulate the production of some secondary metabolites. Naturally, when the BHL signal molecule in the environment reaches a certain threshold, BHL will bind to RhlR protein to form a complex, and combine with PA2069 promoter recognized by BHL signal molecule to initiate the production of downstream lysozyme-activated protein PrtN, thus inducing the autolysis of engineering bacteria and making the two enzymes flow out of the biofilm of wild-breaking bacteria.
2.1 Combine with RhlR-BHL
Both engineering bacteria and wild bacteria can produce BHL signal molecules and release them freely into the environment. When BHL signal molecules reach the threshold in the environment, they can combine with RhlR to form RhlR-BHL complex and play a role in cells. Using cell/L and Avogadro number, the concentration of BHL signal molecules can be approximately equal to the number of proteins in each cell.
$$ [RhlR]_c=(NumberofRhlR/cell)\frac{[PAO1*]}{NA} $$
This is how RhlR-BHL was formed
$$ RhlR-BHL\leftrightarrow RhlR-BHL $$
Assuming that the RhlR- BHL complexation kinetics is very fast compared with the rest of the system, we assume that the free and complexation forms are in equilibrium.
$$ Vcomplexation=Vdissociation $$
$$ k1\left[RhlR\right]C\left[BHL\right]C=k2\left[RhlR-BHL\right]C $$
$$ \left[RhlR-BHL\right]C=\frac{\left[RhlR\right]C\left[BHL\right]C}{Keq,RhlR-BHL} $$
$$ Keq,RhlR-BHL=\frac{k2}{k1} $$
Keq，RhlR-BHL: Equilibrium Constants of Complexes of the RhlR-BHL（mol / L）
2.2 Generation of PrtN
The production of lysin-activated protein PrtN includes activation of prtN transcription and translation by RhlR-BHL complex. In addition, we should consider its degradation.
2.2.1 Activation of PA2069 promoter
According to the concentration of its activator (RhlR-BHL complex), Michaelis equation was used to model it. The strength of the promoter is also considered.
$$ {PrtN}_{DNA/cell}={PrtN}_{DNA0/cell}\frac{{[RhlR-BHL]}_C}{K_{a,RhlR-BHL}+{[RhlR-BHL]}_C}\mathrm{/cdotKp.RhlR} $$
${PrtN}_{DNA0/cell}$: Total PrtN DNA per cell
${PrtN}_{DNA/cell}$: Number of PrtN DNA Activated per Cell
$K_{a,RhlR-BHL}$: Activation Constants of RhlR-BHL Complexes (mol/l)
$\mathrm{Kp.RhlR}$：The effect of promoter
2.2.2 Transcription of the prtN
The transcription of prtN depends on the transcription rate of the strain and the length of the prtN . The Avogadro constant is used to express the transcription rate of the molar concentration in the cell at each time unit.
$$ V_{transcription,PrtNmRNA}=\frac{{PrtN}_{DNA/cell}K_{transcript}({RNA}_{polymerase/gene})}{DNAlength.N_A.V_{intracell}} $$
$K_{transcript}$: Transcription rate of Pseudomonas aeruginosa (nucleotide/s)
${RNA}_{polymerase/gene}$: Number of RNA polymerases per gene
$DNAlength$: Number of Nucleotides on prtN
$V_{intracell}$: Bacterial cell volume (L)
2.2.3 prtN translation
The translation of prtN depends on the translation rate of the strain, the length of the RNA and the quantity of the RNA. The translation velocity is expressed in terms of the molar concentration in a cell per unit of time.
$$ V_{translation,PrtN}=\frac{\left[{PrtN}_{mRNA}\right].K_{translation}.(Ribosomes/RNA)}{RNAlength} $$
$K_{translation}$: Transcription rate of Pseudomonas aeruginosa (nucleotide/s)
$\left[{PrtN}_{mRNA}\right]$: PrtN mRNA Concentration in a Pseudomonas aeruginosa Cell
$Ribosomes/RNA$ ：Number of ribosomes per RNA
$RNAlength$: Number of Nucleotides on prtN mRNA
2.2.4 The degradation of the PrtN
Some PrtN proteins and RNA were degraded. The degradation constant is used to simulate the degradation rate.
$$ V_{degradation,x}=K_{deg,x}.{[PrtN]}_c $$
$K_{deg,x}$: Degradation Constants of PrtN (s-1)
$$ V_{degradation,xmRNA}=K_{deg,xmRNA}.{[PrtNmRNA]}_c $$
$K_{deg,xmRNA}$: Degradation Constants of prtN mRNA（s${}^{-1}$）
2.3 Extracellular polysaccharide hydrolase destroys biofilm
PelA and PslG are two kinds of enzymes coded by Pseudomonas aeruginosa. Their functions are to destroy biofilm. Therefore, PelA and PslG can be considered as a substance enzyme X.
2.3.1 The transcription of X
The transcription of X depends on the transcription rate of the strain and the length of the X gene. Avogadro number is used to express the transcription speed of a cell in molar concentration.
$$ V_{transcription,xmRNA}=\frac{K_{transcript}({RNA}_{polymerase/gene})}{DNAlength.N_A.V_{intracell}} $$
$K_{transcript}$: Transcription Rate of Pseudomonas aeruginosa (nucleotide/s)
${RNA}_{polymerase/gene}$: Number of RNA polymerases per gene
$DNAlength$: Number of nucleotides on X Gene
$V_{intracell}$: Bacterial cell volume (L)
In order to facilitate mathematical operations, we merge $K_{transcript}$, ${RNA}_{polymerase/gene}$ and $V_{intracell}$ into constants.
2.3.2 The Translation of X
The translation of X depends on the rate of translation, the length of the mRNA and the quantity of the mRNA. The translation velocity is expressed in terms of the molar concentration in a cell per unit of time.
$$ V_{translation,x}=\frac{\left[x_{mRNA}\right].K_{translation}.\left(Ribosomes/RNA\right)}{RNAlength} $$
$K_{translation}$: Transcription rate of Pseudomonas aeruginosa (Nucleotide/s)
$\left[x_{mRNA}\right]$: Concentration of X${}_{mRNA}$ in a Pseudomonas aeruginosa cell
$Ribosomes/RNA$: Number of ribosomes per mRNA
$RNAlength$: Number of Nucleotides on X mRNA
To facilitate mathematical operations, we merge $K_{translation}$ and $Ribosomes/RNA$ into constants.
2.3.3 Degradation
Some X proteins and mRNA are degraded. The degradation constant is used to simulate the degradation rate.
$K_{deg,x}$: Degradation Constants of X（s${}^{-1}$）
$$ V_{degradation,xmRNA}=K_{deg,xmRNA}.{[xmRNA]}_c $$
$K_{deg,xmRNA}$: Degradation Constants of X mRNA（s${}^{-1}$）
2.4 Biofilm removal
The biofilm was removed by enzymes and Michaedis- Menten kinetics was assumed to simulate the process.
$$ V_{remo,biof}=K_x{[x]}_w\frac{[Biof]}{K_m+[Biof]}V_{intracell}[PAO1] $$
$K_x$: Catalytic constants of X ( s${}^{-1}$）
$K_m$: Michaelis constant of X ( mol/ l)
2.5 Resolution
The ODEs system was solved by Matlab R2016a.
The complete ODEs are described below：
$$ \frac{{d\left[PrtN\right]}_c}{dt}\mathrm{=}V_{transcription,PrtN}-V_{degradation,PrtN} $$
$$ \frac{{d[X-mRNA]}_c}{dt}\mathrm{=}V_{transcription,XmRNA}-V_{degradation,XmRNA} $$
$$ \frac{{d[X]}_c}{dt}\mathrm{=}V_{translation,X}-V_{degradation,X} $$
$$ \frac{d[Biof]}{dt}=-V_{remo,biof} $$
2.6 Result
This model can remove the biofilm in 4.2 days.