导航栏
Overview

The simulation of biological systems is of great significance in modern biological research, which can help us to deeply understand complex biological processes and reveal the interactions between different biological factors. By constructing mathematical models, we are able to predict the gene expression of a system and optimize the experimental design, thus significantly reducing the consumption of experimental resources and time cost.


In this study, we built a model focusing on the salicylic acid (SA) sensing mechanism, shRNA expression and bacteria lysis process. The model use ordinary differential equations to describe the dynamic interactions between these biological processes. This modeling strategy enables us to analyze the response under different stimulus conditions, and thus answer the questions of "how much shRNA can be expressed by our engineered bacterium, is it enough?" and "does the engineered bacterium express sufficient amount of shRNA before lysis?" ,Overall, our model was built to address some of our concerns.

Model 1: Salicylic Acid Biosensor

Why we built the model?

Our engineered bacterium senses SA, which binds to NahR, and their complex activates the expression of downstream genes, as shown in Figure 1. In the presence of SA, Psal promoter is activated to drive the expression of shRNA. We aim to estimate the expression level of shRNA through modeling to evaluate whether it can effectively mediate gene silencing. Additionally, we want to simulate the effects of different concentrations of SA on shRNA expression to understand how shRNA levels change with varying SA concentrations. This will help us gain a deeper understanding of our system. 待放回路图 Figure 1. The mechanism of the SA sensor.

What questions we want to answer?

  • How much shRNA will be expressed?
  • What effect do various SA levels have?

    • Assumptions

    1. The initial value of SA is fixed because SA metabolism in plants is so complex that it is impractical to simulate.

    2. The SA concentration in the engineered bacteria is the same as in the plant.

    3. Protein-ligand complex NahR_SA is not degraded.


    We use ordinary differential equations to construct our model and simulating gene expression. The influence of transcription factors on the gene transcription rate is described using a Hill function.


    Reaction formula

    $$\text {Gene} \stackrel{K_{\text {mNahR}}}{\longrightarrow} mRNA_{\text {NahR}}\quad\quad(r1.1)$$

    $$\text mRNA_{NahR} \stackrel{K_{\text {NahR}}}{\longrightarrow} NahR\quad\quad(r1.2)$$

    $$NahR + SA \stackrel{K_{\text {on1}}}{\underset{K_{\text {off1}}}{\rightleftharpoons}} NahR\_SA\quad\quad(r1.3)$$

    $$\text {Gene} \stackrel{Hill}{\underset{K_{\text {SA}}}{\longrightarrow}} shRNA\quad\quad(r1.4)$$

    $$\text SA \stackrel{k_{\text {SA}}}{\longrightarrow} \varnothing\quad\quad(r1.5)$$

    $$\text mRNA_{NahR} \stackrel{k_{\text {mNahR}}}{\longrightarrow} \varnothing\quad\quad(r1.6)$$

    $$NahR \stackrel{k_{\text {NahR}}}{\longrightarrow} \varnothing\quad\quad(r1.7)$$

    $$\text shRNA \stackrel{k_{\text {shRNA}}}{\longrightarrow} \varnothing\quad\quad(r1.8)$$


    Ordinary differential equations involved in biosensor model

    $$\frac{d[SA]}{dt}=-k_{SA}[SA]\quad\quad(f1.1)$$

    $$\frac{d[mRNA_{NahR}]}{dt}=K_{mNahR}-k_{mNahR}[mRNA_{NahR}]\quad\quad(f1.2)$$

    $$\frac{d[NahR]}{dt}=K_{NahR}[mRNA_{NahR}]-k_{NahR}[NahR]-k_{on1}[NahR][SA]+k_{off1}[NahR\_SA]\quad\quad(f1.3)$$

    $$\frac{d[NahR\_SA]}{dt}=k_{on1}[NahR][SA]-k_{off1}[NahR\_SA]\quad\quad(f1.4)$$

    $$\frac{d[shRNA]}{dt}=\beta_{sal}\frac{[NahR\_SA]^{n_1}}{K_1^{n_1}+[NahR\_SA]^{n_1}}-k_{shRNA}[shRNA]\quad\quad(f1.5)$$

    Considering that our NahR is constitutively expressed, the initial concentration of NahR is not zero,but rather reaches a steady state. So, before simulating nematode infestation, we need to obtain the initial value of NahR.

    $$ \begin{cases} \ [SA] = 0 &\\ \\ \ \frac{d[mRNA_{NahR}]}{dt} = 0 \\ \\ \ \frac{d[NahR]}{dt} = 0 \end{cases} $$

    Then we can get the initial values of mRNA and NahR, which are 104.99 nM and 0.568 mM, respectively. And we set the initial concentration of salicylic acid to 7.2 μM.

    Figure 2. The expression level of shRNA under SA degradation conditions.

    shRNA levels declined after reaching a peak, attributed to the degradation of SA. However, since our model simulates bacterial changes after nematode infection, we assume a constant level of SA. Consequently, we adjusted the model by removing the SA degradation term (f1.1).

    Figure 3. The expression level of shRNA under constant SA.

    We observed that the concentration of shRNA stabilized at approximately 53.47 nM,this is an adequate concentration of shRNA. However, during our simulation, we learned from our IHP (click here learn more) that the concentration of SA may be vary among different plants. Therefore, we adjusted the initial concentration of SA to investigate the impact of different SA levels on our project.

    Figure 4. The activation of the system for shRNA expression under different concentrations of SA.

    At concentrations below 0.1 µM, the system exhibits little to no noticeable response. However, at concentrations of 1 µM and above, shRNA expression increases significantly. After 10 µM, the activation effect approaches saturation, and further increases in concentration have a limited impact on expression levels.This is also confirmed in our result .


    Conclusion

  • The steady-state concentration of shRNA is approximately 50 nM and this concentration is generally considered effective for achieving gene silencing.
  • Our SA sensor can definitely work in most plants, except for those with extremely high SA, such as rice.
  • Excessive SA concentrations only have a slight effect on the final shRNA expression level.


    • The parameter values were got from literature experimental data and other iGEM teams.

    Table 1. Parameters and values in SA sensor model

    Parameter Description Value Unit Source
    βsal Maximum transcription rate of the Psal 1.5625 nM·min-1 [1]
    KmNahR Transcription rate of NahR mRNA 4.8505 nM·min-1 [2]
    kmNahR Degradation rate of NahR mRNA 4.62×10-2 min-1 [3]
    K1 Dissociation constant of NahR for the promoter Psal. 2.44×103 nM [4]
    KNahR Translation rate of NahR 3.1229 nM·min-1 [2]
    kNahR Degradation rate of NahR 5.775×10-4 min-1 [5]
    kon1 The binding rate of NahR and SA 2.4×10-2 nM-1·min-1 [-]
    koff1 The dissociation rate of NahR and SA. 6×102 min-1 [-]
    kshRNA Degradation rate of shRNA 2.772×102 min-1 [3]
    n1 Hill coefficient of NahR to Psal 1.55 -- [1]
    Model 2: Bacteria Lysis

    Why we built the model?

    When NahR senses SA, the engineered bacteria begin to express shRNA. However, at the same time, the lysis module also starts to express as the Figure 5 shows . If lysis occurs at an early stage of bacterial growth, it may lead to insufficient accumulation of shRNA, which is undesirable. Therefore, we have constructed a lysis model to gain a deeper understanding of the lysis process.

    Figure 5. The mechanism of the bacteria lysis.

    What questions we want to answer?

  • Can sufficient shRNA be produced before the lysis of the engineered bacteria?

    • Assumptions

    1. The Natural degradation rate of HrpR_HrpS complex is 0.

    2. The bacteria are lysed when the Holin and PGHs protein reache steady state.


    Reaction formula

    $$\text {Gene} \stackrel{K_{\text {SA}}}{\underset{Hill}{\longrightarrow}} mRNA_{\text {HrpR}}\quad\quad(r2.1)$$

    $$\text mRNA_{HrpR} \stackrel{K_{\text {HrpR}}}{\longrightarrow} HrpR \quad\quad(r2.2)$$

    $$\text {Gene} \stackrel{K_{\text {mHrpS}}}{\longrightarrow} mRNA_{\text {HrpS}}\quad\quad(r2.3)$$

    $$\text mRNA_{HrpS} \stackrel{K_{\text {HrpS}}}{\longrightarrow} HrpS \quad\quad(r2.4)$$

    $$\text HrpR + HrpS \stackrel{k_{\text {on2}}}{\underset{k_{\text {off2}}}{\rightleftharpoons}} HrpR_HrpS \quad\quad(r2.5)$$

    $$\text {Gene} \stackrel{K_{\text {HrpRS}}}{\underset{Hill}{\longrightarrow}} mRNA_{\text {Holin}}\quad\quad(r2.6)$$

    $$\text mRNA_{Holin} \stackrel{K_{\text {Holin}}}{\longrightarrow} Holin \quad\quad(r2.7)$$

    $$\text {Gene} \stackrel{K_{\text {HrpRS}}}{\underset{Hill}{\longrightarrow}} mRNA_{\text {PGHs}}\quad\quad(r2.8)$$

    $$\text mRNA_{PGHs} \stackrel{K_{\text {PGHs}}}{\longrightarrow} PGHs \quad\quad(r2.9)$$

    $$\text where, K_{SA}=\beta_{sal}\frac{[NahR\_SA]^{n_1}}{K_1^{n_1}+[NahR\_SA]^{n_1}} , K_{HrpRS}=\beta_{HrpL}\frac{[HrpS\_HrpR]^{n_2}}{K_2^{n_2}+[HrpS\_HrpR]^{n_2}}$$

    Degradation processes as follows

    $$\text mRNA_{HrpR} \stackrel{k_{\text {mHrpR}}}{\longrightarrow} \varnothing\quad\quad(r2.10)$$

    $$\text HrpR \stackrel{k_{\text {HrpR}}}{\longrightarrow} \varnothing\quad\quad(r2.11)$$

    $$\text mRNA_{HrpS} \stackrel{k_{\text {mHrpS}}}{\longrightarrow} \varnothing\quad\quad(r2.12)$$

    $$\text HrpS \stackrel{k_{\text {HrpS}}}{\longrightarrow} \varnothing\quad\quad(r2.13)$$

    $$\text mRNA_{Holin} \stackrel{k_{\text {mHolin}}}{\longrightarrow} \varnothing\quad\quad(r2.14)$$

    $$\text Holin \stackrel{k_{\text {Holin}}}{\longrightarrow} \varnothing\quad\quad(r2.15)$$

    $$\text mRNA_{PGHs} \stackrel{k_{\text {PGHs}}}{\longrightarrow} \varnothing\quad\quad(r2.16)$$

    $$\text PGHs \stackrel{k_{\text {PGHs}}}{\longrightarrow} \varnothing\quad\quad(r2.17)$$


    Ordinary differential equations involved in lysis model

    $$\frac{d[mRNA_{HrpR}]}{dt}=\beta_{sal}\frac{[NahR\_SA]^{n_1}}{K_1^{n_1}+[NahR\_SA]^{n_1}}-k_{mHrpR}[mRNA_{HrpR}]\quad\quad(f2.1)$$

    $$\frac{d[HrpR]}{dt}=K_{HrpR}[mRNA_{HrpR}]-k_{HrpR}[HrpR]-k_{on2}[HrpS][HrpR]+k_{off2}[HrpS\_HrpR]\quad\quad(f2.2)$$

    $$\frac{d[mRNA_{HrpS}]}{dt}=K_{mHrpS}-k_{mHrpS}[mRNA_{HrpS}]\quad\quad(f2.3)$$

    $$\frac{d[HrpS]}{dt}=K_{HrpS}[mRNA_{HrpS}]-k_{HrpS}[HrpS]+k_{off2}[HrpS\_HrpR]-k_{on2}[HrpS][HrpR]\quad\quad(f2.4)$$

    $$\frac{d[HrpS\_HrpR]}{dt}=k_{on2}[HrpS][HrpR]-k_{off2}[HrpS\_HrpR]\quad\quad(f2.5)$$

    $$\frac{d[mRNA_{Holin}]}{dt}=\beta_{HrpL}\frac{[HrpS\_HrpR]^{n_2}}{K_2^{n_2}+[HrpS\_HrpR]^{n_2}}-k_{mHolin}[mRNA_{Holin}]\quad\quad(f2.6)$$

    $$\frac{d[Holin]}{dt}=K_{Holin}[mRNA_{Holin}]-k_{Holin}[Holin]\quad\quad(f2.7)$$

    $$\frac{d[mRNA_{PGHs}]}{dt}=\beta_{HrpL}\frac{[HrpS\_HrpR]^{n_2}}{K_2^{n_2}+[HrpS\_HrpR]^{n_2}}-k_{mPGHs}[mRNA_{PGHs}]\quad\quad(f2.8)$$

    $$\frac{d[PGHs]}{dt}=K_{PGHs}[mRNA_{PGHs}]-k_{PGHs}[PGHs]\quad\quad(f2.9)$$

    Figure 6. The relationship between lysis time and shRNA concentration.

    We can clearly see that shRNA reaches its steady state long before the bacterial lysis factors, Holin and PGHs, do. This shows that our delay mechanism is working as expected, giving enough time for shRNA to accumulate before lysis starts. The delay is doing its job, making sure that shRNA reaches the needed levels, and preventing early lysis from happening too soon.


    Conclusion

  • An adequate amount of shRNA can be produced before the engineered bacteria undergo lysis.


    • In addition to parameters of the previous model , these are the values of the additional parameters.

    Table 2. Parameters and values in bacteria lysis model

    Parameter Description Value Unit Source
    kmHrpS Degradation rate of HrpS mRNA 4.62×10-2 min-1 [3]
    KHrpS Translation rate of HrpS 2.98 nM·min-1 [2]
    kHrpS Degradation rate of HrpS 5.775×10-4 min-1 [5]
    kon The binding rate of HrpR and HrpS 0.3 nM-1·min-1 [6]
    koff2 The dissociation rate of HrpR_HrpS. 1.8×102 min-1 [6]
    KmHrpS Transcription rate of HrpS mRNA 4.3026 nM·min-1 [2]
    kmHrpR Degradation rate of HrpR mRNA 4.62×10-2 min-1 [3]
    KHrpR Translation rate of HrpR 2.87 nM·min-1 [2]
    kHrpR Degradation rate of HrpR 5.775×10-4 min-1 [5]
    βHrpL Maximum transcription rate of the PHrpL 5 nM·min-1 [-]
    K2 Dissociation constant of HrpRS for the promoter PHrpL. 3.78×102 nM [Calculated]
    n2 Hill coefficient of HrpRS to P HrpL 2 -- [6]
    kmHolin Degradation rate of Holin mRNA 4.62×10-2 min-1 [3]
    KHolin Translation rate of Holin 7.0676 nM·min-1 [2]
    kHolin Degradation rate of Holin 5.775×10-4 min-1 [5]
    KPGHs Translation rate of PGHs 3.5741 nM·min-1 [2]
    kPGHs Degradation rate of PGHs 5.775×10-4 min-1 [5]

    [1] https://2009.igem.org/Team:PKU_Beijing/Modeling

    [2] Johnson GE, Lalanne JB, Peters ML, Li GW. Functionally uncoupled transcription-translation in Bacillus subtilis. Nature. 2020 Sep;585(7823):124-128.

    [3] Hambraeus G, von Wachenfeldt C, Hederstedt L. Genome-wide survey of mRNA half-lives in Bacillus subtilis identifies extremely stable mRNAs. Mol Genet Genomics. 2003 Aug;269(5):706-14.

    [4] Park HH, Lim WK, Shin HJ. In vitro binding of purified NahR regulatory protein with promoter Psal. Biochim Biophys Acta. 2005 Sep 15;1725(2):247-55.

    [5] BNID 111930, Milo R, Jorgensen P, Moran U, Weber G, Springer M. BioNumbers--the database of key numbers in molecular and cell biology. Nucleic Acids Res. 2010 Jan;38(Database issue):D750-3.

    [6] https://2013.igem.org/Team:HIT-Harbin/Modeling

    Go back to Model