Section 1 : Overview
We want to find a quantitative way to simulate our adsorption system. There are two mathematic models in all. We simulated the pmrCAB regulation system and the 3D model of Cell diffusion.
Section 2 : Gene Regulation System
Theory
Assumptions
We can roughly divide variables influence cell reactions into 5 classes, including physical environment,
cell physiology, correlated but non-overlap pathways, chemical reactions and gene
transcription/translation. Then we build the math model based on this classification.
For a single cell, we put our focus on the stochastic actions in the reaction period, where slight factors
like correlated reactions should be concerned. But the physical environment(like temperature) is hard to
influenced by the cell’s reaction and remains stable. So we build the SSA model(fig.3) without concerning
about the Tb3+ changes in pmrB activation.
It’s not the same situation that happens to the culture medium scale model. In this situation, we can simulate
the system by concentration. The metabolism of cells would influence the environment, which means the main
factor of the system is the main chemical reactions and physical parameters. So in the deterministic model,
we assume that the system is mainly affected by the chemical functions and we build a cell diffusion model
to describe the reaction behavior.
Reactions
We analyzed the influence process and come up with a standard function form:
$$
\sum_{X_i\in S'\subset S}X_i-^k\rightarrow\sum_{Y_i\in S''\subset S}Y_i \tag{1.1}
$$
S represents a set of reactants, while S’ and S’’ are subsets of S. The parameter k represents
the reaction rate constant. All of the chemistry reactions belong to this form. Biochemistry language is
normally regarded as biology language, while mathematics is regarded as physical language.
We can transform the gene expression regulation system into a chemical way.
$$
PmrB + Tb^{3+}<=^{k1}_{k2}=> PmrB_{activated}\tag{1.2}
$$
$$
PmrB_{activated}+PmrA\longrightarrow PmrA_{phosphoryl}\longrightarrow PmrC_{activated}\tag{1.3}
$$
There are 5 steps in the regulation system, including gene transcription, mRNA translation,
ion absorption(along with the protein function activation), transcription promoter activation and combination
with multiple ions. These reactions can be demonstrated by reaction(1.1). The central regulation reaction is
PmrB activated by Tb3+ , which can phosphorylate PmrA. The production of Fusion Protein relies on
PmrC promoter activated by phosphorylated PmrA protein.
Modeling
The rate constants come from the dataset[1] primarily. Then we built the model and optimized it with
our own experiment result. The experiment shows that when 150μM Tb3+ was added into cultures
when the OD600 reached 0.6-0.8, the recombinant protein is induced to overexpression. The different ratio
of Tb3+ reduction between whether adding Tb3+ is 0.63. As the recycle experiment of
cells with Si-tag shows 84% of the cells can be recycled, which means 84% of the cells were detected as
active after adding the overexpression agent.
The parameters we used to build the model are showed in the table below:
| Parameter | Description | Value |
|---|---|---|
| c(Tb3+) | Concentration of Tb3+ | 150 μM |
| c(E.coli) | Concentration of engineering bacterium | 0.015 μM |
| konA | km of PmrA transcription | 10 min-1 |
| konB | km of PmrB transcription | 10 min-1 |
| kmonC | km of pmrC promoter express Fusion Protein | 0.6 min-1 |
| ktlA | km of PmrA translation | 0.483 min-1 |
| ktlB | km of PmrB translation | 0.483 min-1 |
| ρactC | Regulation strength of phosphoryl-PmrA activate pmrC promoter | 0.84 |
| ρactA | Regulation strength of PmrA phosphorylation by PmrBact | 0.97 |
| KactA | EC50 of PmrA activated by PmrB(LBT) | 4.43 Cs |
| kmactA | Maximum PmrA Activation rate | 100 Cs-1min-1 |
| ρapsB | Regulation strength of PmrB(LBT) activation by Tb3+ | 0.63 |
| KapsB | EC50 of PmrB(LBT) activated by Tb3+ | 0.09 Cs |
| kmapsB | Maximum PmrB(LBT) activation rate | 0.95 Cs-1min-1 |
| kapsP1 | Apsorption rate of Fusion Protein with 1st Tb3+ | 0.0444 Cs-1min-1 |
| kapsP2 | Apsorption rate of Fusion Protein with 2nd Tb3+ | 0.0079 Cs-1min-1 |
| dPmrA | PmrA degradation rate | 0.015 min-1 |
| dPmrB | PmrB degradation rate | 0.03 min-1 |
| dP | Fusion Protein degradation rate | 0.015 min-1 |
| dmA | PmrA mRNA degradation rate | 0.034 min-1 |
| dmB | PmrB mRNA degradation rate | 0.034 min-1 |
model
The ODEs of the expression with pmrCAB system regulation model are as follows:
$$ \frac{d[Tb^{+3}]}{dt} = -k_{apsP1}[Tb^{+3}][P] -2k_{apsP2}[Tb^(+3)]^2[P] -k_{apsB}[PmrB][Tb^{+3}] +k_{actA}[PmrA][PmrB_{act}]\tag{2.1} $$ $$ \frac{d[mRNA_A]}{dt} = k_{on}n_{cell} -d_{mRNA_A}[mRNA_A]\tag{2.2} $$ $$ \frac{d[mRNA_B]}{dt} = k_{on}n_{cell} -d_{mRNA_B}[mRNA_B]\tag{2.3} $$ $$ \frac{d[PmrA]}{dt} = k_{tlA}[mRNA_A] -d_{PmrA}[PmrA] -v_{actA}([PmrA],[PmrB_{act}])\tag{2.4} $$ $$ \frac{d[PmrB]}{dt} = k_{tlB}[mRNA_B] -d_{PmrB}[PmrB] -v_{apsB}([PmrB],[Tb^{+3}]) +k_{actA}[PmrA][PmrB_{act}]\tag{2.5} $$ $$ \frac{d[P]}{dt} = v_{p_{product}}([PmrA_(act)]) -d_{P}[P] -k_{apsP1}[P][Tb^{+3}] -k_{apsP2}[Tb^{+3}]^2[P]\tag{2.6} $$ $$ \frac{d[PmrA_(act)]}{dt} = v_{actA}([PmrA],[PmrB_{act}]) -d_{PmrA}[PmrA_{act}] -k_{onC}n_{cell}[PmrA_{act}]\tag{2.7} $$ $$ \frac{d[PmrB_(act)]}{dt} = v_{apsB}([PmrB],[Tb^{+3}]) -k_{actA}[PmrA][PmrB_{act}]\tag{2.8} $$ $$ \frac{d[P_{1*Tb}]}{dt} = k_{apsP1}[P][Tb^{+3}]\tag{2.9} $$ $$ \frac{d[P_{2*Tb}]}{dt} = k_{apsP2}[Tb^{+3}]^2[P]\tag{2.10} $$
The expression reactions of PmrB and PmrA are promoted by T7 promoters which are isolated to the regulation system. So the expression of Fusion Protein is given by Michaelis-Menten equation. We also use Goldbeter-Koshand function to demonstrate the switch in this gene pathway. The degradation of bio-molecules involved in this system is given by the linear function. Some sub-functions are showed below:
$$ v_{actA}([PmrA],[PmrB_(act)]) = km_{actA}*[(1-\rho_{actA})+\rho_{actA}\frac{[PmrA]}{(K_{actA}+[PmrA])}]\tag{2.11} $$ $$ v_{apsB}([PmrB],[Tb^{+3}]) = km_{apsB}*[(1-\rho_{apsB})+\rho_{apsB}\frac{[Tb^{3+}]}{K_{aspB}+[Tb^{3+}]}]\tag{2.12} $$ $$ v_{p_{product}}([PmrA_{act}]) = km_{onC}*[(1-\rho_{actC})+\rho_{actC}\frac{[PmrA_(act)]}{K_{actC}+[PmrA_{act}]}]\tag{2.13} $$
The ODE function performs well as the figure below shows the overall process:
Fig. 1. Tb3+ concentration and Fusion Protein expression
In the Tb3+ existing environment, the figures below shows that the concentration of Tb3+ has a great impact on Fusion Protein expression regulation.
Fig. 2. Tb-expression
To exceed the continuous deterministic simulation, we simulate the behaviours of 1 cell with stochastic simulation algorithm (SSA) . Here’s the simulation result with time.
Fig. 3. stochastic simulation
Activation of PmrB
The main improvement of our system is the better Lanthanide binding activity, which means the optimized expression system is more sensitive to Lanthanide ions. We simulate the expression effect of the pmrCAB system different from Maximum PmrB(LBT) activation rate constant.
Fig. 4. 'Tb^{3+}' concentration after 1 min adsorbtion and km_{apsB}
Firstly, the figure shows that the system has no branch point, the changes of expression and adsorption remain continuous.
Fig. 5. FP expression in 0.5 min and in 30 min
Secondly, the parameter has a sigma curve change with the expression and Tb3+ absorption. The activity of regulation will become fairly low when kmapsB lower than 0.2 Cs-1min-1. According to the model result, we can find a proper binding affinity (0.6-0.8) for our protein design.
Fig. 6. specific score simulation
Finally, from the figure above, we find the branch occurs at the point of kmapsB lower than 0.05 Cs-1min-1. So we simulated a more specific scope and find an activating point for both function.
Binding Sites
Another improvement of our system is the adding of Lanthanide binding sites. In the figure below shows the differences between the number of different binding sites.
Fig. 7. FP expression and Tb adsorbtion with time for different number of binding sites
The results conflict with the hypothesis that the increasing number of binding sites improves the regulation system's working inefficiency. But the result shows a great grads data curves.
Section 3 : Cell Diffusion Model
We simulated the diffusion of cells and Tb3+ molecules using MATLAB. The results are displayed in the plots below.
Assumption
The changes of temperature has no obvious effect on our REE Miner.
The liquid in the container is stationary.
There is nothing else in the container which can influence REE Miner.
Theoretical Model
Firstly, we assume there is a one-dimensional reaction diffusion process. The basic diffusion function is the base of the following models' establishment.
$$ \frac{\partial n_{cell}(x,t)}{\partial t} = D_{cell}\frac{\partial^2*n_{cell}(x,t)}{\partial x^2}\tag{1.1} $$
Then the boundary condition can be defined as follows, with the partial of particle concentration is 0 at site of x=0, and the number of particles in and out decide the dynamic rate of change where x=L.
$$ \frac{\partial n_{cell}(x,t)}{(\partial x)}|_{x=0}=0\tag{1.2} $$ $$ \frac{\partial n_{cell}(x,t)}{(\partial x)}|_{x=L} = -k_{on}*n_{cell}(L,t)+k_{off}*\sigma_{cell}(t)\tag{1.3} $$
The dynamic function are as follows
$$ \frac{d\sigma_{cell}(t)}{dt} = k_{on}*n_{cell}(L,t) - k_{off}*\sigma_{cell}(t)\tag{1.4} $$
With the analysis of stable state, we can get the relationship of \( k_(on)/k_(off) = σ_(cell)(t^*)/(n_(cell)(L,t^*)) \), therefore, we can have the following algorithm:
$$ n^{i,j+1}=n^{i,j}+\frac{D_{cell}*\Delta t}{(\Delta x)^2}*[n^{i+1,j}-2n^{i,j}+n^{i,j+1}]\tag{2.1} $$
Based on the dynamics algorithm(2.1), we derived this model to 3D model, and build a diffusion model. The main part of this model is the algorithm(2.2).
$$ \sigma^{i,j,m}=n^{i,j,m}+ \Delta t*D_{cell}(i,j,m)*(\frac{n^{i+1,j,m}-2n^{i,j,m}+n^{i-1,j,m}}{w^2}+\frac{n^{i,j+1,m}-2n^{i,j,m}+n^{i,j-1,m}}{d^2}+\frac{n^{i,j,m+1}-2n^{i,j,m}+n^{i,j,m-1}}{h^2})\tag{2.2} $$
The result are as follows.
Fig. 8. Dynamic 3D model of cell diffusion
