Motivation
A biosensor system is often used in biological systems to detect an analyte of interest. Our intention behind utilising a biosensor system in our project was to have a controlled release of the protein of interest, that is the invertase inhibitor. Since fructose is a product of hydrolysis of sucrose, its concentration is directly proportional to the activity of invertase. Our construct therefore produces an anti-invertase enzyme whose quantity is regulated by the concentration of fructose in the cellular environment.
We used the pFruB-Cra system to modulate the output protein. The expression of the FruR gene is regulated by an IPTG-induced pLac promoter, which is induced before packaging the bacteria in our polymer inoculant. FruR is a transcription factor with an affinity for fructose-1-phosphate (F1P) — an important component of most metabolic pathways. It is formed from D-fructose to ensure that fructose does not leave the cell. The FruR protein prevents transcription of the regulated promoters. pFruB is the promoter region following FruR and is repressed by the FruR transcription factor.
We planned on simulating the dynamics of the pFruB-Cra system in silico but the lack of lab access due to COVID lockdowns coupled with an unavailability of any literature on the kinetic studies of the FruR system limited us to a theoretical model. We have however planned a few experiments that we would have conducted to determine the kinetics of the model. You can find them on our experiments page here.
Given below is a figure showing the dynamics governing the working of the construct.
Figure 1: The dynamics governing the pFruB-Cra construct
Molecular Modelling
Since FruR and its interactions are not extensively characterised, we had to conduct molecular docking studies in order to elucidate the interactions between FruR, pFruB and fructose. We followed the steps below to conduct this study
We first conducted homology modelling of FruR using SWISS-MODEL in order to obtain its predicted 3D structure (We understand that a more accurate model could have been generated by using Modeller and a suite of other softwares but we unfortunately did not have access to the computational facilities required to conduct an extensive analysis).
Once we obtained the three-dimensional structures, we used AutoDock to dock fructose-1-phosphate to FruR and obtained the interaction data.
We then tried predicting the region/electrostatic patch that FruR uses to bind to the DNA-binding region (pFruB).
1. Homology Modelling
SWISS-MODEL 91011121314 is a web-based integrated service dedicated to protein structure homology modelling. Homology modelling comprises of (i) identification of structural template(s); (ii) alignment of target sequence and template structure(s); (iii) model-building; and (iv) model quality evaluation.
We have used GMQE and QMEAN as parameters to gauge the reliability of our models.
GMQE (Global Model Quality Estimation) is a quality estimation procedure which combines properties from the target template alignment and the template structure. They are combined using a multilayer perceptron. The resulting GMQE score is expressed as a number between 0 and 1, reflecting the expected accuracy of a model built with that alignment and template, normalized by the coverage of the target sequence. Higher numbers indicate higher reliability.
QMEAN is a composite estimator based on different geometrical properties and provides both global (i.e. for the entire structure) and local (i.e. per residue) absolute quality estimates on the basis of one single model. The QMEAN Z-score provides an estimate of the "degree of nativeness" of the structural features observed in the model on a global scale Benkert et al., 2009. It indicates whether the QMEAN score of the model is comparable to what one would expect from experimental structures of similar size. QMEAN Z-scores around zero indicate good agreement between the model structure and experimental structures of similar size. Scores of -4.0 or below are an indication of models with low quality.
FruR
As illustrated in the Virtual Screening Model, the Ramachandran Plot provids us an initial estimate on the structural assessment of the protein. We constructed a Ramachnadran plot of the homology-modelled FruR, which is shown below.
Figure 2: The Ramachandran Plot for FruR
We found that 97.07% of the protein is Ramachandran favoured, however 1.88% are rotamer outliers and 31 out of the 6175 amino acids contain bad angles.
We then assessed the quality of our protein by plotting its predicted similarity to the target protein versus the corresponding residue number. It can be seen that there is a significant overlap between the template and the target, proving the reliability of our homology model.
Figure 3: Local quality estimate for FruR
2. AutoDock Models
FruR - F1P
We docked FruR and Fructose-1-Phosphate using AutoDock. AutoDock offers a variety of search algorithms to explore a given docking problem. These include Monte Carlo Simulated Annealing (SA); a Genetic Algorithm (GA); and a hybrid local search GA, also known as the Lamarckian Genetic Algorithm (LGA). In general, the LGA performs the best out of SA, GA, and LGA, in finding the lowest energy of the system.
We decided to use the Lamarckian GA and we conducted 100 GA runs to obtain the statistically significant docked conformations. The results of the docking are illustrated in the image below along with the interactions.
Figure 4: Visualisation of interactions between FruR and F1P on PyMol
From the results we derive that the binding of F1P happens to an active binding pocket of FruR surrounded by Gly, Gln and Arg residues.
3. BindUP Predictions
BindUP Paz et al., 2016 is an automatic server to predict DNA and RNA binding proteins given the three dimensional structure of the protein (or a structural model). The DNA and RNA binding prediction is based on the electrostatic patches on the protein surface and does not rely on either sequence or structural homology. In addition to providing functional prediction (i.e. whether the protein binds nucleic acids or not), the server displays the largest positive and/or negative electrostatic patches on the protein.
The largest positive patch we found on FruR was ARG60 THR61 ARG62 GLN86 GLN89 ARG90 GLY91 ARG294 LEU301 GLU302 LEU305 LEU308, which is highlighted in the animation below.
Legend for the animation below:
- Red: Electrostatic patch
- Blue: Chain A FruR
- Peach: Chain B FruR
Figure 5: Largest positive patch on FruR
We observed that the DNA-binding region of FruR is very close to the F1P ligand-binding pocket.
Mathematical Model
The assumptions we have made for the model are listed below:
As cells would be IPTG-induced before being added to the inoculant, we have assumed that the total number of FruR molecules remains constant during the course of the action of the cells.
Phosphorylation and dephosphorylation of fructose is not considered to simplify the model. We assume that all the fructose that is taken up by the cell is immediately phosphorylated. Hence, the amount of fructose is roughly equal to the amount of fructose-1-phosphate. Subsequently, the metabolic pathway from F1P leading to FBP, F6P, G6P is also ignored Chavarría et al., 2014.
The FruR-F1P complex degrades slowly enough such that the FruR in the complex is not recycled back to bind to pFruB again.
The binding rate for F1P and FruR is unaffected by pFruB.
F1P binds to FruR and helps it unbind from pFruB as a concerted non-reversible displacement reaction. Our molecular model confirms this assumption that F1P binding to FruR precludes FruR binding to pFruB. This observation, therefore, serves as a validation of our assumption that F1P binds to FruR and helps it unbind from pFruB as a concerted non-reversible displacement reaction.
There is a constant external source of FruR which maintains its concentration at a steady, constant level throughout the timescale of the model.
Figure 6: The metabolic pathway of glucose and fructose
Reactions Involved
The reactions involved in the working of the biosensor constructs are
$$ \text{Fructose} \xrightarrow{\text{Fructokinase}} \text{Fructose-1-phosphate}\\ \text{FruR} + \text{pFruB} \xrightarrow{} \text{[pFruB-FruR]}\\ \text{F1P} + \text{[pFruB-FruR]} \xrightarrow{} \text{[F1P-FruR]} + \text{pFruB}\\ \text{[F1P-FruR]} \xrightarrow{} \text{FruR} + \text{F1P}\\ \\ \text{Plasmid} \xrightarrow{} \text{Invertase Inhibitor} \xrightarrow{} \emptyset\\ \text{Plasmid} \xrightarrow{} \text{mCherry} \xrightarrow{} \emptyset\\ $$
Differential Equations
The set of ODEs ~ Ordinary Differential Equations that model the FruR-Cra kinetics are illustrated below:
$$ \begin{aligned} \frac{d[A]}{dt} &= \frac{e_A C_N}{\left( 1 + \frac{[C_2]}{k_f} \right)} - \beta_A[A]\\ \frac{d[M]}{dt} &= \frac{e_M C_N}{\left( 1 + \frac{[C_2]}{k_f} \right)} - \beta_M[M]\\ \frac{d[F]}{dt} &= - k_1[F][C_2] + k_2[C_1]\\ \frac{d[R]}{dt} &= - k_3[R]C_B + k_2[C_1]\\ \frac{d[C_1]}{dt} &= k_1[F][C_2] - k_2[C_1]\\ \frac{d[C_2]}{dt} &= k_3[R]C_B - k_1[F][C_2]\\ \end{aligned} $$
Here,
- \( [R] \) is the FruR concentration;
- \( [M] \) is the mCherry concentration;
- \( [A] \) is the anti-invertase concentration;
- \( [C_1] \) is the concentration of the [F1P-FruR] complex;
- \( [C_2] \) is the concentration of the [pFruB-FruR] complex (in the binding domain); and
- \([C_B] = \frac{C_N}{V}\) is the approximate concentration of binding domains per cell where \(C_N\) is the plasmid copy number and \(V \) is the cell volume.
The parameters used in the above set of equations along with their descriptions are tabulated below.
Note: We were aware of probability-based techniques to perform parameter estimations (such as Bayesian parameter estimation), but as there were a large number of undetermined parameters in our model, it would not make any stastical sense to estimate such a large number. The parameter space would be too large for us to derive any significant or meaningful results. Therefore, we chose to limit ourselves to a theroretical description which we hope teams in the future can use and improve upon.
| Parameter | Description |
|---|---|
| \( e_A \) | Expression rate for anti-invertase\(^\dagger\) |
| \( e_M\) | Expression rate for mCherry\(^\dagger\) |
| \( \beta_A \) | Degradation rate of anti-invertase |
| \( \beta_M \) | Degradation rate of mCherry |
| \( k_1 \) | Binding rate of F1P to FruR |
| \( k_2 \) | Unbinding rate of F1P from FruR |
| \( k_f (=\frac{k_2}{k_1}) \) | Dissociation constant for F1P and FruR |
| \( k_3 \) | Binding rate of FruR to pFruB |
| \( k_4 \) | Unbinding rate of FruR from pFruB |
| \( k_R (=\frac{k_4}{k_3}) \) | Dissociation constant for FruR and pFruB |
\(^\dagger\) - Expression rate is a modification made to decrease the number of parameters in the model. Since the degradation and translation rates of mRNA are in the order of seconds while those for proteins are of the order of minutes, we combine the two equations for expression of protein into one compact equation. The expression rate for a protein \(B\) is given by \( e_B = \frac{\alpha_B \alpha_{mB}}{\beta_{mB}} \) where \( \alpha_B \) is the translation rate of the protein; \( \alpha_{mB} \) is the trancription rate of the protein mRNA; and \( \beta_{mB} \) is the degradation rate of the protein mRNA.
The experimental output (concentration of mCherry) can be corroborated to the model (pFruB activity) using the following equation Leveau, J.H.J. & Lindow, S.E., 2001. $$ L = \frac{\partial F}{\partial OD}\mu\left( 1 + \frac{\mu}{m}\right) $$
where
- \(L\) is the promoter activity;
- \(F\) is the fluoroscence of mCherry;
- \(\mu\) is the growth rate;
- \(m\) is a maturation constant of the fluoroscent protein (mCherry) (This is obtained from literature); and
- \( \frac{\partial F}{\partial OD} \) is the slope of the plot of the fluoroscence output versus the optical density of the culture.
In phase 2, we plan to predict the overall efficiency of the pFruB-Cra system and the amount of inoculant required per sugarcane using information from experiments. We would obtain some of the contants from experiments as well as various parametric estimation methods, thus improving the model.
Insights
We validated our assumption using homology modelling that F1P binding to FruR precludes FruR binding to pFruB. This helped us frame our mathematical model.
We realised that we would have to design a FruR specificity experiment to characterise pFruB-FruR interactions and validate our docking studies.
For experimental estimation of parameters, we would have to first perform Bayesian Parameter Estimation and then validate our results by the design of a Gel Shift Assay and Site-directed mutagenesis.