Differential equations are widely used in biology, especially in mathematical modelling to describe the dynamics and behaviour of biological systems. In this paper, differential equations are explored to control the growth and dormancy of Escherichia coli in the toxin-antitoxin system and the Flp-FRT system. The following includes differential equation modelling, numerical simulation and parameter tuning, as well as the final biological interpretation.
Consider the toxin-antitoxin system and the Flp-FRT system for controlling growth and dormancy in E. coli. The systems contain two major proteins, the toxin HicA and the antitoxin HicB, as well as the Flp recombinase and the FRT site. The toxin HicA inhibits the growth of E. coli, while the antitoxin HicB binds to HicA and neutralises its toxicity.Flp recombinase is used to induce the switching system. This paper models for two scenarios:
(1) In the absence of Flp induction, the changes in the concentrations of HicA and HicB over time are described.
(2) In the case of Flp induction, calculate the steady-state concentration of Flp after induction and explain its biological significance. For case I, in the absence of Flp induction, describing the changes in the concentrations of HicA and HicB over time can be achieved by differential equations. The following are some equations for this system:
Based on the data obtained from the experiments, we set the initial conditions as HicA(0) = 0.3 HicB(0) = 0.2 Next, we can use numerical methods, and in this paper, we use Euler's method to solve these differential equations in order to obtain the changes in the concentrations of HicA and HicB with time. A specific example is shown below:
The Euler method was then used to simulate the change in concentration of HicA and HicB over time. First, a time step Δt was chosen and then iteratively calculated according to the following equation:
The above iterations were repeated until the desired time point and steady state were reached. For Case I, in the case of Flp induction, we focus on the behaviour of Flp and its effect on bacterial growth and dormancy. Firstly, in this paper, the differential equation of Flp is established:
Next, we will use the Euler method to simulate the steady-state concentration of Flp after induction. First, a time step Δt is chosen and then it is calculated iteratively according to the following equation:
We will continue the simulation until the concentration of Flp no longer changes significantly or a steady state condition is reached.Steady state conditions can be defined as the concentration of Flp changing less than a certain threshold in two consecutive time steps.
From the figure, it can be found that HicA and HicB reach steady state at around 30h, when dynamic equilibrium is reached. Biological significance: Flp recombinase plays a key role in the Flp-FRT system. When Flp reaches steady-state concentrations after induction, the system may be dormant. This is due to the ability of Flp to recombine the FRT site, thereby inhibiting the growth of E. coli. In the dormant state, the bacteria may be in a state of hypoactivity, not growing and dividing. This may have a survival advantage for the bacteria to survive and cope with unfavourable conditions in the external environment. Therefore, by controlling the rate of Flp production (k_{Flp_production}) and degradation (k_{Flp_degradation}), the control of dormancy and arousal of the bacteria can be realised, which is of great significance for applications in the fields of bioengineering and synthetic biology. By adjusting these parameters, the growth and dormancy of bacteria can be precisely controlled, which helps to achieve microbial control and bioproduction in specific application scenarios.
A population of microorganisms in which the increase in population density leads to changes in their physiological and biochemical properties during their growth, displaying characteristics not found in small numbers of organisms or in individual organisms. Data fitting, a way of representing existing data by mathematically substituting it into a mathematical equation. Scientific and engineering problems can be obtained through such methods as sampling, experimentation and other methods of a number of discrete data, based on these data, we often want to get a continuous function (curve) or more dense discrete equations with the known data, this process is called fitting. This paper focuses on the help of data fitting in group sensing experiments.
Due to time reasons, we were unable to measure the values of FI/OD600 for more groups of [3O C6 HSL] at different concentrations, so we chose to use data fitting to generalize the effect of different concentrations of [3O C6 HSL] on E. coli FI/OD600. Based on the theory of population induction modeling, we chose to use Logistic5 mathematical model for data fitting
where FI/OD600 is used as the dependent variable y and [3O C6 HSL] is used as the independent variable x
Convergence of the fit was finally achieved after 19 iterations, reaching the Chi-sqr tolerance value of IE-9.
parameter:
statistics:
Summary:
Fitting renderings:
Residual plots:
Through data fitting, we overcame the difficulty of not being able to measure more groups of data due to time reasons, which reduced the burden of synthetic biology experiments; at the same time, the visualization of the plotted curves data achieved through data fitting allowed a clearer understanding of the mathematical relationship between [3OC6 HSL] and FI/OD600, which assisted in the planning of biological experiments
The aim of this experiment was to investigate the structural dockability of DMBT1 with Pseudomonas aeruginosa flagellin FilC and type IV bacteriophage protein PilA by computational methods. We modeled the docking of DMBT1 with Pseudomonas aeruginosa flagellin FilC and Pseudomonas aeruginosa type IV bacteriorhodopsin PilA using AlphaFold2 and ClusPro tools to explore the spatial conformation secondary structure and docking effect between them.
AlphaFold2 is a protein structure prediction algorithm developed by DeepMind, Inc. whose core principles include multiple sequence alignment, inter-residue distance prediction, deep learning, and iterative optimization.AlphaFold2 uses a large amount of biological information, especially protein evolutionary information. This information is obtained by performing multiple sequence alignment on a large number of protein sequences. It mainly predicts the distances between individual residues in proteins and the associated angular information. The model is based on a deeply learned Transformer structure and uses knowledge of physics and chemistry to optimize the structure after predicting the initial protein structure. Through these methods, AlphaFold2 is able to predict the 3D structure of a protein from its amino acid sequence.
ClusPro is a tool specifically designed for protein-protein docking. It mainly uses the FFT method for docking search and is able to systematically evaluate all possible protein combinations in a short time. After docking, all possible protein combinations receive an energy score representing the stability of such combinations. Combinations with lower scores are more likely to be real biological combinations.ClusPro also performs a greedy clustering process after docking, clustering the 1,000 docking locations based on their similarity. The largest clustered groups are usually considered the most likely biologically active combinations because the most stable structures usually occur the most often in simulations, so the most likely docking locations from these dockings are identified through clustering.
Using the AlphaFold2 model, we input the amino acid sequences of DMBT1, FilC and PilA. The predicted 3D structure of each protein was obtained by calculation and the structure results were saved in pdb format. The model obtained several possible 3D structures for each amino acid sequence, which were ranked 0, ranked 1, ranked 2, etc. in order of confidence.
Using the ClusPro tool, we entered the predicted structures of DMBT1 with FilC and DMBT1 with PilA. After setting the docking position, the ClusPro model started to calculate the structural docking between the two pairs of proteins, and several clusters consisting of the docking results were obtained.
AlphaFold2 successfully predicted the 3D structures of DMBT1, FilC and PilA. We used the online Mol* View tool for visualization and obtained their structures as shown below, respectively:
FilC:
PilA:
Among multiple possible structure predictions, we chose the Rank 0 structures of DMBT1, FilC and PilA with the highest confidence level for docking experiments, and the results of ClusPro are shown in the table below:
Cluster scores of docking between DMBT1 and FiLC:
Cluster scores of docking between DMBT1 and PilA:
Each row of the above table represents a cluster, i.e., a collection of specific protein-protein interaction structures. Each cluster has a central structure, called "Representative", an energy score (i.e., activation energy) for that structure, called "Weighted Score", and a lowest energy score, called "Lowest Energy". The lower the score for these structures, the better the stability of the structure. For the docking of DMBT1 with FilC, the largest cluster has 39 members, with a central activation energy of -1477.6kcal/mol and a minimum activation energy of -1786.0kcal/mol. for DMBT1 with PilA, the largest cluster has 41 members, with a central activation energy of -1132.0kcal/mol and a minimum activation energy of -1192.2kcal/mol. 1192.2kcal/mol.
The result of the docking is shown below:
Docking between DMBT1 and FiLC visualized:
Docking between DMBT1 and PilA visualized:
Please look at the "Software" Page.