Overview

Fig 1: The detection kit model: Engineered E. coli cell for detection of DSF by fluorescence

This model aims to explain theoretically the functioning of the bio detection kit for early detection of bacterial blight in rice. The system is a genetically modified E. coli cell. The receptor kinase "RpfC" on the surface acts as the receptor for the ligand "DSF". Once DSF binds to RpfC, it induces an allosteric conformational change that causes autophosphorylation of the DSF-RpfC complex. This activated complex then phosphorylates downstream target "RpfG". The activated RpfG in turn acts as an enzyme to convert cyclic di-GMP to 2 GMP.

Meanwhile, at another site in the cell, there is a vc2-riboswitch upstream of a transcribed mRNA from an infused plasmid. This riboswitch can be in one of two conformations, A or B. Conformation B binds to its ligand cyclic-di-GMP and blocks translation of downstream targets. Conformation A (after unbinding from ligand) allows translation to happen. Thus, for the blocking of translation to happen, binding of cyclic di-GMP to the riboswitch is necessary.

Therefore, if DSF binds, cyclic-di-GMP is degraded creating a chemical potential in the cell and the cyclic-di-GMP from the vc2-riboswitch detaches into the cytoplasm and the mRNA becomes available for translation, forming the fluorescence-tagged protein gfaspurple, which can be then detected as purple fluorescence.

Hypothesis

To mathematically model the detection kit system such that it replicates the experimental results, and ultimately, helps predict features of the system under varying parametric conditions.

Components

1. DSF binding to RpfC
2. Autophosphorylation of DSF-RpfC complex
3. Phosphorylation of RpfG by activated DSF-RpfC complex
4. Degradation of cyclic di-GMP by activated RpfG
5. Detachment of cyclic di-GMP from VC2-riboswitch
6. Change in conformation of VC2-riboswitch to promote/inhibit translation (and subsequent expression) of gfaspurple

System

The following reactions are used to model the system:

$${ \begin{align*} \ce{DSF + RpfC <=>[k_1][k_2] DSF\cdot RpfC}\\ \ce{DSF\cdot RpfC ->[k_3] DSF\cdot RpfC\cdot P}\\ \ce{DSF\cdot RpfC\cdot P + RpfG <=>[k_4][k_5] RpfG\cdot P + DSF\cdot RpfC}\\ \ce{RpfG\cdot P + cdiGMP ->[k_6] RpfG\cdot P} \end{align*} } $$

Applying the Law of Mass Action, the rates of change of each reaction component are written as a system of differential equations:

$${ \begin{align*} &\frac{d}{dt}(\mathrm{DSF}) = -k_1*(DSF)*(RpfC) + k_2*(DSF\cdot RpfC)\\ &\frac{d}{dt}(RpfC) = -k_1*(DSF)*(RpfC) + k_2*(DSF\cdot RpfC)\\ &\frac{d}{dt}(DSF\cdot RpfC) = k_1*(DSF)*(RpfC) - k_2*(DSF\cdot RpfC) - k_3*(DSF\cdot RpfC) + k_4*(DSF\cdot RpfC\cdot P)*(RpfG) - k_5*(DSF\cdot RpfC)*(RpfG\cdot P)\\ &\frac{d}{dt}(DSF\cdot RpfC\cdot P) = k_3*(DSF\cdot RpfC) - k_4*(DSF\cdot RpfC\cdot P)*(RpfG) + k_5*(DSF\cdot RpfC)*(RpfG\cdot P)\\ &\frac{d}{dt}(RpfG) = -k_4*(DSF\cdot RpfC\cdot P)*(RpfG) + k_5*(DSF\cdot RpfC)*(RpfG\cdot P) + k_6*(RpfG\cdot P)*(cdiGMP)\\ &\frac{d}{dt}(RpfG\cdot P) = k_4*(DSF\cdot RpfC\cdot P)*(RpfG) - k_5*(DSF\cdot RpfC)*(RpfG\cdot P) - k_6*(RpfG\cdot P)*(cdiGMP)\\ &\frac{d}{dt}(cdiGMP) = -k_6*(RpfG\cdot P)*(cdiGMP) \end{align*} } $$

Fig 2: Kinetic model for vc2-riboswitch functioning through three regulatory mechanisms: The ribosome can switch reversibly between the two conformations A and B, with conformation A allowing higher translation of downstream mRNA (ON state). The ligand L (cyclic di-GMP) binds to conformation B of the riboswitch, slowing down translation (OFF state).

Upon solving the above system of ordinary differential equations for a given set of initial conditions and varying initial DSF concentrations, we obtain the ligand (cyclic di-GMP) concentrations as a function of time and DSF concentrations. Using this, we calculate the final protein (gfaspurple) concentration as a function of the ligand concentration as: $${ \begin{align*} P = \frac{k_f}{k_{dP}}\cdot \frac{k_{pA}k_7 + k_{pB}(1 + k_8L)}{k_{dA}k_7 + k_{dB}(1 + k_8L)} \end{align*} } $$

Assumptions

1. DSF binding to RpfC is a typical ligand-receptor binding equilibrium reaction
2. Autophosphorylation of the DSF-RpfC complex is a unidirectional reaction, independent of inorganic phosphate concentration.
3. There is a transfer of phosphate in the activation of RpfG by the DSF-RpfC-P complex, following an equilibrium reaction.
4. Activated (phosphorylated) RpfG degrades cyclic di-GMP
5. There are three possible states for the riboswitch - A, B, BL - with A as the one that allows most translation to happen (open conformation), B (closed conformation) and BL (closed conformation bound to ligand L) allowing less translation
6. All the different states for the riboswitch degrade with the mRNA, so does gfaspurple.
7. We don’t consider intrinsic degradation of DSF, RpfG, cyclic di-GMP.
8. There is an effective transcription rate for the synthesis of the mRNA of gfaspurple.

Plots

The model is run using the above parameters and the following initial conditions:

The initial concentration for DSF is varied from 0.01 μM to 10 μM. The model is run for a tmax of 15s.
The following plots are obtained for [DSF], [cyclic di-GMP], [gfaspurple] vs time for varying [DSF0]; and [gfaspurple] vs [DSF0]:

Fig 3: As time increases, the remaining free DSF decreases as it binds to RpfC and aids in the subsequent reactions, as is shown in the plot.The rate of decrease in DSF level is almost the same for all initial concentrations of DSF.

Fig 4: As time progresses, more RpfG is phosphorylated, which leads to an increased degradation of cyclic di-GMP, as is seen in the plot. For increasing initial DSF concentrations, the rate of decrease in cyclic di-GMP concentration also increases

Fig 5: As time increases, final protein level also increases with the decrease in cyclic di-GMP concentration. This follows the principle that cyclic di-GMP binding to vc2-riboswitch reduces translation of gfaspurple mRNA. It is to be noted here that till an initial DSF concentration of 2.0 μM, there is hardly any protein synthesis, but from [DSF0] onward, the protein level quickly reaches saturation. Thus, the required threshold value for [DSF0] is somewhere between 2.0 and 4.0 μM

Fig 6: Final gfaspurple levels with increasing [DSF0]. At around [DSF0] = 2.8 μM, the protein concentration rapidly increases, reaching a saturation point of around 5 μM

Results

Thus, we find that as we keep increasing DSF from 0.01 to around 2.5 μM, the observed fluorescence (measured directly from gfaspurple concentration) does not increase by much. However, when the DSF concentration crosses a certain threshold level in the interval 2.5-3 μM, the fluorescence skyrockets - indicating the detection point. After that, the protein level again does not increase much.

Conclusion

The results prove the hypothesis, at least qualitatively. We see a clear phase transition point of the model where the kit transitions from switch OFF state to switch ON state for a certain initial DSF concentration, which in turn can be adjusted to the actual value found for Xanthomonas infection by suitably modifying some of the parameters in the model.

Limitations

Some of the parameter values were set arbitrarily, to roughly fall in the biologically acceptable range. By performing more advanced experiments, we can set the parameter values more precisely.

Future scopes

As a natural extension to this model, one can introduce randomness in the interactions and implement something similar to a Watts-Strogatz Random graph to incorporate more real life scenarios.