Basically a hydrogel is a giant network consisting of a lot of polymer chains, which locks a lot of water inside. It is similar to a sponge, but the texture is much more delicate. Here is a simplified drawing of a hydrogel. You might actually have seen hydrogel-like structures in your life! For example, tofu is an excellent example of hydrogel.
The hydrogel is a matrix that contains our bacteria. Just a recap, our bacteria needs to detect the communication signal between pathogenic bacteria, and secretes a protein (called Dispersin B) that helps break down the biofilm formed in the urinary tract. Biofilm protects the pathogenic bacteria from being killed or flushed away. For details regarding our design please visit https://2021.igem.org/Team:Manchester/Design .
For the hydrogel to function well with our design, the secreted protein and signaling molecules need to move reasonably fast in the hydrogel to reach the biofilm. One way to figure this out is to investigate the diffusion behaviour of the protein in the hydrogel.
For those who are not familiar with, diffusion is basically molecules moving from a region of high concentration to a region with very low concentration. Diffusivity is essentially a measure of how fast this process is. An example of this is the smell of perfume pervading the room. However, the medium of diffusion often influences how fast the particles move in it. For example, a drop of ink diffuses differently in water and honey, as the diffusivity of ink is higher in water.
The model for diffusivity is important because we need to ensure that the hydrogel will not be the limiting factor of the function of our design. In the other word, we need to validate whether we can assume the concentration of DspB protein reaching biofilm is only limited by the secretion rate of our engineered bacteria.
We modelled the influence of the hydrogel network on the diffusion behaviour of our protein . The way we quantified this is to model the relative diffusivity of the hydrogel. The relative diffusivity is basically how fast the protein moves in hydrogel versus how fast it moves in pure water. The model we adopted uses a few parameters to calculate this relative diffusivity. For example, we need to know the dimensions of the polymer chain, the size of the diffusing molecules, the size of the network mesh, and how much polymer there is compared with water.
Here is the equation for the model:
$$ \frac{D}{D_{0}}=\operatorname{erf}\left(\frac{r_{FV}}{r_{s}}\right)\cdot\exp\left(-\left(\left(\frac{r_{s}}{r_{FV}}\right)^{3}\cdot\frac{\phi_{p}}{\left(1-\phi_{p}\right)}\right)\right)+\left(1-\operatorname{erf}\left(\frac{r_{FV}}{r_{s}}\right)\right)\exp\left(-\pi\cdot\left(\frac{r_{s}+r_{f}}{ξ+2r_{f}}\right)^{2}\right) $$It looks very complicated, but here is what these parameters mean. Don't worry about the terms "erf" and "exp" for now.
Here are the parameters needed:
Here are the assumptions we made when employing this model:
Here is a brief explanation of what this model means:
We were able to find most of the parameters in the literature, but we could not find the reported values for mesh size of the polymer. It turned out to be quite challenging to estimate, and we had to adapt additional models to calculate the unknown parameters such as mesh size.
For example, this is the model (Peppas et al., 2000) we used to model mesh size.
As there are no reported values of Cn specific to Sodium Carboxymethylcellulose (NaCMC), we calculated this using the polymer “end-to-end” distance (<r0>, reported by Lopez, et al., 2015), the backbone bond length (which is variable l, 0.515nm) and the number of monomers per polymer chain (n, chain molar weight divided by monomer molar weight).
Here is the equation we used:
In addition, we needed to find out the average molar mass between crosslinks to eventually calculate mesh size. We managed to find a model that calculates this value, as shown here:
Here are the parameters needed for this model:
We could not find values on the Flory interaction parameter unfortunately, so we ended up using another formula to find that one out. We adopted a model proposed in 1987 ( https://doi.org/10.1016/0142-9612(88)90006-3 > ) that presented a way to estimate the value of Flory interaction parameter.
As described by Mikos and Peppas in the literature, the assumptions for this model are:
Here are the parameters in this paper that we needed to find:
When we combined these with the parameters that are reported, we were finally able to calculate the value for the mesh size and the diffusivity.
We were aware of the amount of uncertainties we were dealing with. To make sure our conclusion is useful, we used an ensemble modelling approach. We found alternative values for the parameters that were reported, and created log-normal distributions using the protocol proposed by Tsigkinopoulou, et al., 2018.
The log-normal distributions basically tell us how likely it is to have the parameter at a certain value. For example, if the molar mass of carboxymethyl cellulose is around 250kg/mol, it is impossible for all chains to have this exact number. However, the chains have molar mass that are likely to fall within a certain range, as shown here:
After we created the distributions, we randomly sampled those parameters from their distributions, used the randomly sampled parameters to perform all the calculations, and kept a record of the values. We repeat this process one hundred thousand times and then we are able to visualise the distribution of our results - the mesh size and diffusivity.
Here are the results of our initial design:
Firstly, this is the histogram for relative diffusivity:
The first thing you can see here is that most of our calculations resulted in very small diffusivity values. This is a problem and we needed to find out why this happened.
We first plotted the variation of relative diffusivity with mesh size. From this plot, the relative diffusivity only started to increase when mesh size was about 3nm. We used this value as a threshold and divided the results into the "good" and "bad" groups. By “good” it means the parameter combinations produced relative diffusivity values that are high enough.
Most of the parameters do not have a strong impact of the “good” or “bad” nature of one specific set of results, as the PDF of the overall distribution of these parameters follows closely to the histogram of the normalised PDF of the random parameters in the “good” range. This means the distribution of the parameters that resulted in “good” or “satisfactory” results are similar to its overall distribution.
The distribution of the “good” values for mass swelling ratio is significantly deviated from the overall distribution. This shows that the mass swelling ratio in the swollen state (q) has a strong influence on the final result. This makes sense from the materials science perspective, as the mass swelling ratio is related to how much the polymer network of the hydrogel has “expanded” during the water absorption process, which is an indication of the mesh size.
Since the mass swelling ratio in the swollen state is largely controlled by the degree of crosslinking, the simulations indicate that the crosslinker concentration in the current hydrogel formulation needs to be altered.
Since we know the mass swelling ratio in the swollen state (q) was the cause of the problem, we need to change it. Luckily, it can be increased simply by reducing the concentration of the crosslinker. So, instead of using the 20% w/w citric acid as the crosslinker, we reduced the concentration in our design to 10%. We the performed the calculations again with the updated design. The model results predicted a sufficiently-large mesh size and adequate diffusivity with the new design. This prediction confirms from a technical perspective that the cellulose-based biopolymer hydrogel can satisfy the pivotal design goals of our device.
Nevertheless, there are other functional properties such as stiffness or fracture toughness that also need to be modelled and experimentally characteristised. Furthermore, the hydrogel-catheter interface interaction needs to be modelled to investigate the adhesion of the hydrogel onto the catheter surface. We hope future teams can build upon what we have achieved here and utilise these results to make further advancements in this area.
Here's a list of the parameters we used in the calculation if you are interested
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