Team:Lund/Model
Significance
The inhibition model was engineered by answering increasingly complex questions. Based on a rigid understanding of mathematics and molecular chemistry, it was expanded to manage single bacteria, then inhibitors, and finally multi-bacteria problems. This workflow highlights the power of numerical models in molecular biology. Simple models can easily be used to answer very complex questions.
The current version provides a unique way of evaluating various inhibitors. Using the model provided clear evidence against producing the inhibitor QFGGGNPP, and reinforced our decision to produce the inhibitor CsgC. In doing so, we saved several weeks of work in the lab, as well as increasing the possible fitness of our end product GMO. The model is also designed to be easy to use and can tackle multi-bacteria and multi-scale simulations. This makes it useful for answering questions such as probiotic bacteria delivery method and dosage. To our knowledge, the model is the most complex of its kind to be presented at iGEM.
Introduction
The Inhibition Model aimed to provide valuable insight into the molecular dynamics underlying the formation of Curli fibrils and how different inhibitors affect these dynamics. The model was then used to evaluate inhibitors to express in our probiotic bacteria.
Amyloid formation is an extensively investigated topic in research. The molecular dynamics of Curli have been successfully probed both in-vitro and in-vivo. There are mainly two subunits making up Curli fibrils. CsgA, which makes up the fibril, and CsgB, which anchors the fibril to the cell membrane. The rate-limiting steps are nucleation and elongation. Nucleation is mostly referring to a CsgB peptide attaching to the cell membrane, enabling the formation of a new fibril, but can also refer to two CsgA monomers coming together to form a freely floating fibril. CsgB nucleation is several orders of magnitude faster than CsgA nucleation and we will therefore neglect the CsgA nucleation. Elongation is the process of CsgA monomers attaching to fibril endpoints. Several research groups have investigated the production rate of CsgA and CsgB in E.Coli, and the nucleation, and elongation rates 1. In-vitro, the rate of CsgB production is on the order of .1 pM/s, CsgB production on the order of 100 pM/s, and the elongation rate is on the order of 104-105 (mole*s)-1. Interestingly, the elongation rates found in-vitro are not large enough to fully describe the time it takes for E.Coli to form a dense biofilm 2. Indeed, when the Curli formation was investigated in-vivo the rate of formation was far greater 3. In this model, we will use the value 21 000 (mole*s)-1 reported in 4.
Several iGEM teams (for example Team TU Delft 2014) have come up with models for describing the curli production of a bacteria. We have worked on implementing the newest research into the models as well as extending the models to enable multi-bacterial, and multi-scale, systems where diffusion becomes a critical factor. However, for our purpose, we never needed to use these more complex models.
This report is divided into three subparts. First, we will explain how we model uninhibited Curli growth using our Simple inhibition model and our Complex inhibition model. We will then move on to discuss how we use these models to simulate published results to fit inhibitor rate parameters. Lastly, we will show how we can use simulations of probiotic - E.coli interactions to yield a numeric value for the inhibitor effectiveness.
Modeling Curli Formation
Curli is formed extracellularly in three steps. First, a CsgB protein is secreted into the extracellular environment and attached to the cell membrane. A CsgA protein then has to be secreted and attached to the CsgB molecule, acting as the first step on a then repeated nucleation process. Importantly, new CsgA molecules always have to diffuse from the cell membrane to the tip of the Curli fibril. Previous iGEM teams have decided against taking this diffusion process into account to greatly simplify the model. One can simply assume that the CsgA concentration is uniformly distributed across the entire volume. This holds for most smaller peptides and proteins (<50kDa) at smaller length scales (<1 mm). When modeling larger inhibitors, or when simulating macro systems in vivo, however, we would need a more advanced model.
To model the systems, we initially made the following assumptions:
The system is closed.
Bacteria are perfect spheres.
Bacteria do not form colonies.
Production of protein is homogenous and protein synthesis is not a limiting factor.
CsgA monomers diffuse as perfect spheres in a quiescent solution.
CsgB monomers attach to the cell membrane immediately and do not diffuse away.
Curli fibres grow evenly distributed across the entire bacteria cell membrane.
Curli fibres grow independently of each other.
No modelled agents interact with any other substances in the solution, and protein does not denature.
We then designed two models. The Simple Inhibitor Model (SIM), and the Complex Inhibitor Model (CIM). Both models can simulate the formation of Curli at various bacteria concentrations, time scales, and length scales. We also implemented a simple way to include advanced inhibitors in the models. How the models are derived and implemented can be found in our jupyter notebooks.
PDF on CIM
Notebook on the SIM
Figure 1: Simulation of 15 h Curli formation using CIM. In order to prove the usefulness of CIM, we have decreased the curli diffusion coefficient 1000-fold to the point when diffusion becomes a limiting factor. Top left: The mass distribution of curli at different distances from the bacteria. We can see that the curli fibrils reach 1 μm away from the bacteria cell membrane. Top right: CsgA concentration profile at different distances from the bacteria cell membrane. We can see that the concentration swiftly decreases after only a few 100 nm from the bacteria cell membrane. Bottom left: The curli fibril endpoint distribution at different distances from the bacteria cell membrane. We see that fibrils mainly grow “in a front”, and are not evenly distributed across the entire volume. Bottom right: Fibril size distribution. We can see that most fibrils are only made up of a few hundred CsgA subunits.
Figure 2: Simulation of 10 h Curli formation using SIM. The simulation is for 1012 bacteria/dm3 in suspension. Top left: The mass distribution of curli at different distances from the bacteria. We can see that the curli fibrils reach 1 μm away from the bacteria cell membrane. Bottom left: The curli fibril endpoint distribution at different distances from the bacteria cell membrane. We see that fibrils grow evenly distributed across the entire volume. Bottom right: Fibril size distribution. We can see that most fibrils are only made up of several hundred CsgA subunits.
We can see that diffusion indeed can affect the growth of amyloids such as Curli drastically if the solvent viscosity or the subunit size is large enough. As the diffusion coefficient of CsgA (13 kDa) is 120 μm2 / s, and the size of a Curli fibre never exceeds 1 μm, we can assume that SIM will be a good enough approximation for our purpose. From these simulations we can also see that only less than 20 % of all CsgA secreted by a bacteria attaches to the fibres of that same bacteria, the rest attach to neighbouring bacteria. This highlights the importance of colony formation when modeling Curli growth. Similar mathematical models by Georg Meisl et al.[Electrostatically-guided inhibition of Curli amyloid nucleation by the CsgC-like family of chaperones - PubMed (nih.gov)] are only able to accurately describe the process for the first 10 h of growth in vivo, after which the colony behaviour starts to influence the process.
Fitting Inhibitors
Now that we have a model that accurately predicts the formation of Curli on bacteria, we could move on to modeling the effects of different inhibitors on the process. We used SIM to simulate experiments reported in the literature and then fitted our inhibitors against the reported results. For this to work, we needed to find studies that quantified the inhibitors’ effect on specifically Curli. Most of our inhibitors had reported effects on most amyloids or biofilms, but few came with actual measurements on Curli. We were able to investigate two inhibitors: QFGGGNPP (QFG), and CsgC. A thorough description of how we derived and fitted their respective rate parameters can be found in our github.
QFG notebook
CsgC notebook
Evaluating Inhibitors
As the two inhibitors target vastly different parts of the aggregation reaction, it is difficult to evaluate which of them is more effective to produce simply from their reaction constants. Instead, we asked the question:
“In a solution with 50% E coli, and 50% probiotic bacteria, what inhibitor production rate would be required in the probiotic bacteria for it to be able to reduce the Curli formation by 50%?”
Answering the question gives us a numeric value for each inhibitor that is independent of the different reaction mechanisms, and it would also give us insight into the efficiency of our probiotic bacteria in-vivo.
A detailed description of how the production rates were computed can be found in our Github in the file
"Fitting Production Rate"
Figure 3: Required inhibitor production rate for CsgC and QFG. CsgC 0.24 proteins/bacteria/s, 45160 proteins/bacteria/s.
Figure 3 clearly shows how CsgC is a much more effective inhibitor than QFG. For every CsgC protein produced, the probiotic bacteria would need to secrete 105 QFG protein. One could also argue that while 0.2 molecules/bacteria/s is quite a reasonable production rate, > 10 000 is not. Based on these results, we decided against expressing the inhibitor QFG in our probiotic bacteria.
Outlook
The approach presented in this section is to our knowledge the first of its kind in iGEM. Although the assumptions made are many and far-reaching, and the computed required production rates probably are far from the truth, the model can compare inhibitors that affect very different reaction mechanisms. SIM and CIM are easy to use and easy to adapt for other iGEM teams in need of performing similar simulations. They are made to be extended to multi-bacterial, and multi-scale systems, although that was not necessary here. If we were able to measure the actual production rates of our probiotic bacteria, as well as growth rates of E.coli and our probiotic bacteria, we would be able to extend our model to tackle the question: “How close do our bacteria need to be to the E.coli colony to be able to repress their Curli growth?”. Answering this question would give crucial insight into which delivery method would be required when producing a probiotic pill.
You can find the iGEM Judging release of our model here.