Team:Manchester/Model/Hydrogel
Background Information
What is a hydrogel?
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 .
Reasons for Modelling
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.
Theoretical framework
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:
- Radius of solute (rs)
- Radius of free volume of void (rFV)
- Radius of polymer chain (rf)
- Volume fraction of polymer (greek letter Φp)
Here are the assumptions we made when employing this model:
- Solutes can only diffuse through hydrogel via either a. free volume voids or b. through the polymer network alongside the aqueous solution
- The free volume of void is basically the gap between water molecules that are trapped in the hydrogel.
- Solute particles behave like rigid spheres in the hydrogel network.
Here is a brief explanation of what this model means:
- The multi scale diffusion model calculates the probability of the solute diffusing through these two mechanisms. (basically the "erf" and "exp" are mathematical functions that help computing the probabilities). The first term in the equation describes the probability of solute diffusing via the volume of free voids of space , and the second term describes the probability of the solute diffusing with water through the polymer network.
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.
$$ \xi=v_{2,s}^{\frac{1}{3}}l\sqrt{\frac{2C_{n}\overline{M_{c}}}{M_{r}\ }}\ $$
- v2,s: volume fraction of polymer in swollen state
- polymer bond length
- Cn: Polymer Characteristic ratio, a measure of the “flexibility” of polymer chains
- (Mc): average molar mass between crosslinks
- Mr: molar mass of repeat unit
- Most of the parameters were reasonably easy to calculate, as we had the mass-swelling ratio data from a published paper. However, we needed to determine the characteristic ratio of our polymer, which is a measure of the flexibility of the polymer chains.
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:
$$ C_{n}\ = \lim_{x \to +\infty} \frac{\left(r^{2}_{0}\right)}{nl^{2}} $$
- This model assumes the polymer chains are unperturbed (no external forces or solvent effects) and are freely rotatable. This assumption is valid in our design as in the urethra, since the carboxyl groups remain charge neutral.
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:
- We adopted model proposed by Peppas, et al. ,2000 , which uses the swelling ratio of the hydrogel in relaxed and swollen states to predict the average molar mass between crosslinks.
- Assumption: polymer chains do not contain ionic charges. Although our hydrogel contains carboxyl groups, this assumption can be made because the pH level in the urinary tract prevents carboxyl groups on carboxymethylcellulose from becoming negatively charged.
$$ \frac{1}{\overline{M_{c}}}=\frac{2}{\overline{M_{n}}}\ -\ \ \frac{\frac{\overline{v}}{V_{1}}\ \left[\ln\left(1-v_{2,s}\right)+v_{2,s}+\chi_{1}v_{2,s}^{2}\right]}{v_{2,r}\left[\left(\frac{v_{2,s}}{v_{2,r}}\right)^{\frac{1}{3}}-\left(\frac{v_{2,s}}{2v_{2,r}}\right)\right]} $$
Here are the parameters needed for this model:
- Mn: molar mass of polymer chain
- 𝜒: Flory interaction parameter (greek letter "chi")
- This basically describes the interaction between polymer chain molecules and the water molecules.
- v: specific volume of polymer (reciprocal of density)
- V2,r: volume fraction of polymer in relaxed state
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.
$$ \chi\ =\ 0.34\ +\ \frac{V_{1}}{RT}\left(δ_{1}\ -\ δ_{2}\right)^{2}\ +\ 2I_{12}δ_{1}δ_{2} $$
As described by Mikos and Peppas in the literature, the assumptions for this model are:
- the volumechange of mixing is zero.
- the mixing process is random.
- the segment interactions are binary.
Here are the parameters in this paper that we needed to find:
- V1: specific volume of water
- δ 1: the solubility parameter of water
- δ2: the solubility parameter of polymer
- I12: binary parameter.
- For the binary parameter, we adopted the Wilson equation that is simplified to one parameter form. (This was retrieved from section 8.18, Properties of Gases and Liquids by Bruce Poling et al., 2001) The assumption in this simplification is that the energy parameter between the water molecules and polymer molecules is constant.
- ΔHv1: enthalpy change of vaporisation of species 1 (in this case water)
- Z is the coordination number. We estimated the coordination number by multiplying the number of monomers per chain by 6.7. The condition here is that we assume 6.7 water molecules per monomer when we have a degree of subsitution of 0.7.
$$ l_{ij}= -\ β\ (ΔH_{v1}-RT) $$
$$ \beta = \frac{2}{z} $$
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.
Initial Results
What has been done to ensure the results are reliable?
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.
- The probability density function (PDF) describes the overall distribution of each parameter. For each parameter, the PDF was computed for the range of the randomly sampled data. The “good” parameter set was then normalised and plotted on top of the overall PDF to compare the distribution of the “good” parameters and all the parameters sampled.
- Here are the plots of the parameter distributions on top of their histograms:
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.
Design Improvement
What does this mean for the design and project overall?
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 are the results when crosslinker concentration is reduced to 10%:
- As shown below, a lower crosslinker concentration significantly increases the mesh size of the hydrogel network, and therefore increases the relative diffusivity of the Dispersin B protein.
- Since the crosslinker concentration is inversely proportional to the structural integrity of the hydrogel, final crosslinker concentration was decided to be between 10 and 15% w/w.
Here's a list of the parameters we used in the calculation if you are interested
Supporting Information
- Parameters used:
- Radius of solute (rs): For Dispersin B we used 6nm as the hydrodynamic radius, as reported by Pavlukhina, et al. in 2012.
- Radius of free volume of void (rFV): A value of 2.69Å was used to be the radius of free volume of void in water, as reported by Axpe, et al .
- Radius of polymer chain (rf ): For the deterministic calculation, the average polymer chain radius was considered to be 0.5938 nm, which is an average value of the radius for carboxymethyl (1.3nm, Zhivkov, A. M., 2013 ) and hydroxypropylmethyl cellulose chains (0.85nm, Arai, K, et al, 2020 )
- Mass swelling ratio of the polymer in swollen state (q): 1.6 ( Dharmalingam, K, Anandalakshmi, R, 2019
- Mass swelling ratio of polymer in relaxed state(q’):
- Erosion ratio (percentage weight loss when hydrogel is dried). Value = 3.5% (error = 0.66%) ---> mass swelling ratio in relaxed state q' = swelling ratio in swollen state * (1- erosion rate) = 1.6*(1-0.035) = 1.544
- Source: https://www.sciencedirect.com/science/article/pii/S0141813019312371#f0015
- Hydrodynamic radius of Autoinducer 2 was calculated to be 3.0Å using bond lengths reported (source: doi: 10.1021/jp800125p
- Density of NaCMC: 1.6gcm -3 (source: https://www.chemsrc.com/en/cas/9004-32-4_304486.html
- Density of HPMC: 1.39 gcm -3 (source: https://www.chemsrc.com/en/cas/9004-65-3_121194.html )
- Hildebrand Solubility parameter of NaCMC (deltaCMC) and HPMC (deltaHPMC): 28.9 and 23.6 respectively (source: Brady, J., Dürig, T., Lee, P. I., & Li, J.-X., 2017
- Flory characteristic ratio of polymers:
- 50 and 54 (reference: https://doi.org/10.1002/bip.1970.360090704
- 45.0 (calculated using polymer length)
- Polymer molar weight: NaCMC: 250 kDa (Sigma), HPMC: 88kDa (sigma)
- Average repeat unit molar mass: 199.095 g/mol (calculated as weighted average (3:1 ratio) between NaCMC (Degree of substitution = 0.7, Sigma) and HPMC (22% Methoxyl substituent and 8.1 % Hydroxypropyl substituent. Source: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4370968/
- Polymer bond length (l): 0.515nm, Source: https://cdn.intechopen.com/pdfs/45490/InTech-Electric_properties_of_carboxymethyl_cellulose.pdf
References
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Arai, K., Horikawa, Y., Shikata, T., & Iwase, H. (2020). Reconsideration of the conformation of methyl cellulose and hydroxypropyl methyl cellulose ethers in aqueous solution. RSC Advances, 10(32), 19059-19066.
Brady, J., Dürig, T., Lee, P. I., & Li, J.-X. (2017). Chapter 7 - Polymer Properties and Characterization (Y. Qiu, Y. Chen, G. G. Z. Zhang, L. Yu, & R. V. B. T.-D. S. O. D. F. (Second E. Mantri (eds.); pp. 181–223). Academic Press. https://doi.org/https://doi.org/10.1016/B978-0-12-802447-8.00007-8
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G. Mikos, A., & A. Peppas, N. (1988). Flory interaction parameter χ for hydrophilic copolymers with water. Biomaterials , 419–423. https://doi.org/https://doi.org/10.1016/0142-9612(88)90006-3
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Pavlukhina, S. V, Kaplan, J. B., Xu, L., Chang, W., Yu, X., Madhyastha, S., Yakandawala, N., Mentbayeva, A., Khan, B., & Sukhishvili, S. A. (2012). Noneluting Enzymatic Antibiofilm Coatings. ACS Applied Materials & Interfaces, 4(9), 4708–4716. https://doi.org/10.1021/am3010847
Zhivkov, A. M. (2013). Electric Properties of Carboxymethyl Cellulose (p. Ch. 8). IntechOpen.