Introduction

This simulation was designed to investigate the amount of lysostaphin and nutrients present at the line where the hydrogel ends, and wound begins for different designs of patches. In order to effectively eradicate the Staphylococcus aureus culture that is present in the wound, it is important that the amount of lysostaphin exceeds a certain concentration at the wound. Furthermore, there should not diffuse nutrients into the wound since not only our engineered E. coli likes nutrients of course, but also harmful bacteria in the wound will thrive well on these nutrients.

For convenience’s sake, we decided it to be the easiest if we feed the E. coli by adding a nutrient rich layer on top of the hydrogel patch. This way the nutrient rich layer can be easily replaced once depleted. Because replacement requires the help of a specialist like a nurse, we aim to maximize the time a patch can stay on the skin without anyone touching it. We will try to find an optimal composition of the hydrogel patch, so no nutrients diffuse entirely through the hydrogel to reach the malificent bacteria. However, this should not inhibit the delivery of lysostaphin to the infected wound.

Model

COMSOL Multiphysics 5.3a was used to simulate the transport of lysostaphin and nutrient particles within a porous medium. COMSOL uses the finite element method (FEM) to calculate the value of variables within an element for a given timestep. COMSOL allows the user to easily select physical processes, such as diffusion. Also, one can simply construct a grid, set boundary conditions and select simulation times. Its simplicity made the choice for us to use this program very easy.

The physical properties included in the simulation are diffusion following Ficks law (eq:1), and an equation describing the effective diffusivity of particles within a porous medium (eq:2). In this equation the diffusivity (D_e/D_o) scales with the inverse of the porosity, while we used a quadratic inverse relation with the porosity in the first model, as was argued to be correct in (1). In order to avoid confusion between the tortuosity used in COMSOL and the tortuosity determined in the first model, the tortuosity in COMSOL will be referred to as τ_c where equation 3 holds.

In the first model, where we experimented with the diffusivity of GFP, we got a tortuosity (τ) value between 1 and √1.5. For this model we will use the average of these values, which leads to τ = √1.237 and τ_c = 1.237.

Patch composition

First the shape of the hydrogel was chosen to be a square 30x30 mm block. We thought about it to be used for an infected wound on the arm. Its total thickness was chosen to have a value of 4 mm. Our decision fell on this thickness, because as a patient you do not want a very thick layer of hydrogel on your skin, since it is inconvenient and more prone to being damaged by shear stresses. We compose the patch of three different layers as can be seen in figure 1. The top layer is the one containing nutrients that supply the cells with food and allow growth and lysostaphin production. However, the most important layer, namely the layer that houses the lysostaphin secreting E. coli, can be found on a buffer layer. A buffer layer of hydrogel sits between the cell layer and the wound. This buffer layer should be small enough to ensure a sufficient amount of lysostaphin flowing from E. coli towards the wound, but the buffer layer should be large enough, along with the gel where the cells reside, to ensure that as little nutrients as possible end up in the wound by diffusion.

In physics displacement of molecules can be influenced by diffusivity as said above, but flux can also have influence on this displacement. Through movement of the patch or forces on the gel, while it is on the skin of a patient, flux can be generated. However, these external influences are inpredictable and for this reason very difficult to put in a model. So for simplicity we assume an ideal situation where there are not any external forces or movements that cause flux through the gel.

Skin

Since we want to simulate our patch on top of the skin, we also had to simulate a piece of skin in COMSOL. However, we assumed that wound is not just a superficial wound, but an open wound because superficial wounds are generally able to heal by themselves and are not commonly infected. So for simplicity we assumed that the wound was open and covered with exudate. Moreover, as we are working with an open wound, the blood vessels are extremely close to the surface if not damaged and open. This is why we did not use a permeability constant for the skin, we assumed it negligible. In conclusion, the diffusivity through both exudate and blood were assumed to be equal to water due to the high water contents (approximately 92% for blood plasma (2)) and our target, S. aureus, thrives in exudate with high water content. (3) Furthermore, for the reason that blood flows to the wound and away from the wound we set a boundary where the blood is. This boundary works as a sink in the sense that nutrient and lysostaphin concentration are zero here, because there is flux away from the wound additional to diffusion.

Figure 1: Structure of the 3D model in COMSOL Multiphysics. The hydrogel parts are shown in grey, from top to bottom: nutrient layer, cell layer, buffer layer. The layers fulfilling the goals of exudate and blood are the top and bottom blue layers respectively.

Cell layer

As can be seen in the second model on lysostaphin production, we assumed a completely saturated hydrogel for cells to get a lysostaphin production rate per cell. For this model we do not assume growth of cells, actually, we assume that the cells’ growth and death rate are in equilibrium, establishing a steady state where there is a constant number of cells. Since we obtained a lysostaphin production rate per cell and nutrient uptake rate, along with diffusion constants, we can vary the amount of cells in the gel and the thickness of the gel to get results on possibly fitting values.

As can be seen further in this model, due to the fact that we vary the thickness of the buffer layer and not the cell layer, we argue that for convenience in handling a minimum thickness of 1 mm of the patch is needed (excluding the nutrient layer). This means that the minimum thickness is equal to the thickness of the cell layer, because the buffer layer’s thickness can be zero. Since using 1 mL as the cell layer’s volume eases calculations a lot, we fixed the thickness of the cell layer to 1.111 mm causing the cell layer to have an exact volume of 1 mL.

Nutrient layer

As one can find in Table 1 with the parameters that will be used in the COMSOL Multiphysics model, we will be using the concentration of nutrients of standard LB Broth (Miller) which we also use in our laboratory. This nutrient-rich layer has to be able to be replaced by a new nutrient layer. Just like we argued for the cell layer in the patch, handling of this layer should for this reason be made easy by its thickness to enable replacement. Therefore we chose to use a fixed thickness of 1 mm for this layer.

Parameters

Using literature, calculations and results from previous models and experiments, we came up with parameters and variable ranges for our simulation in COMSOL, which can be found in table 1 and 2 respectively.

Table 1: Parameters used for the 3D model in COMSOL Multiphysics.

Calculation 1

To decide what kind of source of nutrients we would use, we decided to look at commonly used nutrient sources for the bacteria we use in our hydrogel: E. coli. For general laboratorial growth of E. coli often Lysogeny broth (LB) medium is used, so we decided to take this nutrient rich medium as a first source to feed our bacteria.

The LB we use in the lab of our university is called ‘LB Broth (Miller)’ from Sigma-Aldrich. As stated on their website, this type of LB consists of 10 g/L tryptone, 10 g/L NaCl and 5 g/L yeast extract. (8) We neglected the mass of NaCl, since NaCl is not that important to the total uptake of nutrients that is used to produce lysostaphin. Tryptone is a mixture composed of amino acids (AAs) and longer peptides. (9) Yeast extract is not much different since it exists out of various growth factors, sources of sulfur and vitamin B complex, but mainly peptides and AAs since yeast extract is mainly produced by digestion of proteins by proteases. (10) Because both tryptone and yeast extract are largely amino acids and peptides we treated them as if they were only AAs and peptides. We then assumed that the mass ratio single AAs versus oligopeptides was 50/50. The average molecular weight for a single amino acid was calculated by averaging molecular weights of the twenty most common AAs in nature as shown in table 2. Values for molecular weight of amino acids were obtained through the database of the National Centre of Biotechnology Information (NCBI) called PubChem Compound.

Table 2: Amino acids with their respective molecular weights in gram per mol.

For the oligopeptides we determined the average oligopeptide MW by multiplying the average oligopeptide length of 13 (2 to 25 AAs) by the calculated average MW of a single amino acid, resulting in MW_oligo= 1779.72 g/mol. (11,12) Since we have 10 g/L tryptone and 5 g/L yeast extract, this results in a 15 g/L mixture of AAs and oligopeptides. By assuming a 50/50 mass ratio of AAs and oligopeptides, so 7.5 grams of both, we could conclude the following concentrations with the average MW values: c_AAs = 0.05478 mol/L, c_oligo = 0.004214 mol/L.

We then took the average of these concentrations to estimate the concentration nutrients in the LB we used and thus in the nutrient rich layer we put on top of our patch: c_nut = 0.029499 mol/L

Calculation 2

After calculating the concentration of nutrients in LB Broth (Miller) we of course want to put this into the model, however, we will need a diffusivity value for that. We came across an article by Polson, A. where he experimentally determined the diffusivity of several amino acids in water. (13) We will not elaborate on these values, but we took the averages for AAs that Polson, A. determined and took the average of these values to get an averaged diffusivity for AAs: D_0,AAs = 7E-10 m²/s

Since we said that the mass ratio within the nutrients is 50% amino acids and 50% oligopeptides (see Calculation 1), we also have to determine a separate diffusivity value for oligopeptides. Since we calculated a plausible molecular weight for oligopeptides earlier (in Calculation 1), and with the knowledge that MW scales approximately with the diffusivity, we assumed that a molecule with similar MW as MW_oligo had a similar diffusivity.(7) We found a diffusivity value for the neuropeptide Cortistatin of rats with MW_corstatin = 1721.0 g/mol. So with MW_oligo = 1779.72 g/mol, we determined the following: D_0,oligo ≈ D_0,corstatin = 3.70E-10 m²/s (14)

Just like the concentration of nutrients, we want the diffusivity of the nutrients to have one value and not be split up. So again we took the average, although we discussed the fact this would favour small values, but we came across a website of the FCSXpert Team (Fluorescence Correlation Spectroscopy) of Sensor Technologies LLC. They stated that, whatever average calculation you used, it is not a bad way to calculate it as long as it is noted how it is calculated. (15) So for our calculation we took D_0,AAs and D0,oligo, summed them and divided them by 2 to get: D_0,nut = 5.35E-10 m²/s

For the diffusivity of NaCl through water we found the value of D_0,NaCl = 1.607 * 10^-9 m²/s. (16) However, since NaCl is a common salt present in all kinds of solutions, we did not think that taking the average would actually realistically influence the total diffusion constant, because the medium in the gel itself will already contain salts like NaCl. Moreover, as said before, it does not contribute to the production of lysostaphin. Going back to the nutrient diffusivity, thus without taking NaCl into account, we do not want the diffusivity through water (D₀) but through the hydrogel, the effective diffusivity D_e. We use equation 2 to calculate D_e from D_0,nut, ε_t and τ_c. For the nutrients we then get: D_e,nut = 3.98E-10 m²/s

Calculation 3

To determine the uptake of E. coli cells of nutrients, we first assumed that the E. coli bacteria in our gel would always use the maximum amount of nutrients available, whether it is used for growth, protein production, division or other processes. Selvarasu et al. conducted experiments to determine the uptake of amino acids by E. coli that are programmed to maximize growth rate during the growth phase using linear programing (LP). (17) They showed AA uptake during three growth phases in which the uptake was maximized in the first growth phase. So to match our assumption we chose to use their data acquired during the first growth phase in column three, Table III of their article, the biosynthetic demand per amino acid in mmol/gDCW/h (DCW is short for Dry Cell Weight).

We divided the biosynthetic demand per AA by 3600 to get the demand per AA in mg/gDCW/s. Next, we divided it by 3600 to get the values in mg/gDCW/s, since COMSOL Multiphysics commonly works with seconds instead of hours. To get the average demand of per AA we summed all the synthetic demand values per AA and divided them by the amount of amino acids registered (twenty). After that, to get from gDCW to cells divided the synthetic demand by the average dry cell weight per cell of 280 fg (18). This gets us the average AA demand per cell in mmol/cell/s. We multiply this by 1000 to get the value in mol/cell/s, but to get the total amount of AAs needed per cell we still have to multiply the current value by 20, since our current value is for a single average AA, giving:

Uptake of nutrients per cell, U_nut,cell =4.22E-19 mol/cell/s

Table 3: Variables used to obtain various combinations of models to review the contribution of patch composition and seeding density to the total hydrogel patch.

To get results out of the model we take the measure that the MBEC for lysostaphin should be reached at the latest moment at 12 hours after the bacteria are induced to produce lysostaphin.

We will determine the moment of reaching MBEC at the skin for lysostaphin for several thicknesses of hydrogels with varying seeding densities. Three thicknesses of hydrogel will be modelled, starting with the hydrogel with a nutrient layer of 1 mm, cell layer of 1,111 mm and a buffer layer of 1.8889 mm for a total of 4 mm hydrogel. Secondly, a hydrogel with nutrient layer of 1 mm, cell layer of 1.111 mm and buffer layer of 0.9445 (half of the original) for a total thickness of 3.056 mm. Lastly, the buffer layer will be completely removed. Only the cell layer and nutrient layer will be left for a thickness of 2.111 mm.

Moreover, for the three mentioned variants of hydrogel four seeding densities will be tested in steps of x10. The start will be at the amount of cells in OD = 0.01 which is 3 million per millilitre and we will end at 3 * 10¹⁰ cells per millilitre for which we definitely know we reach MBEC. The experiments used in model 2 show that with 7.5 * 10⁹ cells enough lysostaphin is produced to eradicate the biofilm of S. aureus. For the reason we wanted to scale up consistently, we chose to end at 3 * 10¹⁰ to make sure we establish MBEC: 2.963 * 10^-6 mol/L = 2.963 * 10^-3 mol m^-3

Results

Starting for the hydrogel with a 1 mm nutrient layer, a 1.111 mm cell layer and 1.889 mm of buffer layer with a cell count of 3 million. The model shows a high concentration of nutrients at t = 0h in the nutrient layer just like expected. (figure 2)

Figure 2: Nutrient layer of 1 mm, cell layer of 1.111 mm and buffer layer of 1.889 mm. t=0h. Nutrient concentration is shown. Axis shown in mol/m_-3.

Over time we see diffusion of the nutrients through the gel, but not only diffusion, the uptake of nutrients by cells is also taken into account. However, after twelve hours we still see a large concentration of nutrients of around 5 * 10^-4 mol/m^-3 or 5 *10^-7 mol/L all throughout the gel as can be seen in figure 3.

Figure 3: Nutrient layer of 1 mm, cell layer of 1.111 mm and buffer layer of 1.889 mm. t=12h. Nutrient concentration is shown. Axis shown in mol/m_-3.

For the lysostaphin production, at t = 0h in figure 4 an extremely small concentration of lysostaphin can be seen in the cell layer.

Figure 4: Nutrient layer of 1 mm, cell layer of 1.111 mm and buffer layer of 1.889 mm. t=0h. Lysostaphin concentration is shown. Axis shown in mol/m_-3.

At t = 12h, as can be seen in figure 5, a low concentration of lysostaphin has been established.

Figure 5: Nutrient layer of 1 mm, cell layer of 1.111 mm and buffer layer of 1.889 mm. t=12h. Lysostaphin concentration is shown. Axis shown in mol/m_-3.

Since we want to see if the MBEC for S. aureus is reached, we determine the concentration of lysostaphin along the x,y-plane between gel and exudate, the exudate-plane. We also determine the concentration of nutrients on this plane to see what amounts of nutrients are flowing through the gel into the exudate with malignant bacteria. Figure 6 shows the mentioned plane with an approximate average nutrient concentration of 7 * 10^-5 mol/m³ across the plane. Figure 7 depicts the approximate average lysostaphin concentration across the same plane, showing a concentration of 10-15 * 10^-13 mol/m^-3.

Figure 6: Nutrient layer of 1 mm, cell layer of 1.111 mm and buffer layer of 1.889 mm. t=12h. Nutrient concentration is shown. Axis shown in mol/m_-3.

Figure 7: Nutrient layer of 1 mm, cell layer of 1.111 mm and buffer layer of 1.889 mm. t=12h. Lysostaphin concentration is shown. Axis shown in mol/m_-3.

Now that we have shown what results we obtain and how we obtain them with COMSOL, we will not show all the pictures for the other hydrogel composition and seeding density combination, however, we put the obtained values in Table 4.

Table 4: Results

Discussion

As we established the 3D in COMSOL, the complete hydrogel patch looked promising since there are nice gradients that can be seen for diffusion of particles. We kept encountering these gradients, but the surfaces we wanted to use to determine concentrations have varying values all over as can been seen in figure 6 and 7. We kept trying to make educated guesses, but sometimes the axis gave such values that it was almost impossible to get decent results. However, it is not as bad as it seems, because the lysostaphin concentration values increased by tenfold, making reading errors not that influencing. The trend stays the same. This also accounts for the concentration of nutrients, although are reading errors much more influential here because of the small differences.

Moreover, we also see the logical decreasing trend of nutrient concentration at the exudate surface, because more bacteria use more nutrients. For some reason this does not account for nutrient concentrations with high cell counts. We think this might be due to the fact that at certain moment the nutrients are depleted, the model does not take this account and calculates values that are not realistic.

For future modelling there should be done more experiments to increase the accuracy of certain constants used in this model. Other models we established, were also partly used in this model, but their individual inaccuracy was already separately discussed. Improving the first model will already improve the accuracy of the properties of the gel and this way indirectly diffusion constants.

Due to the fact that many parameters and values the model put out are inaccurate because literature errors, human errors and small errors in formulas, we cannot conclude anything quantitatively from this model. However, since trends in obtained data is visible we still think we can carefully draw some minor conclusions.

Conclusion

For the nutrients we saw that at all times, in comparison to lysostaphin, large amounts of nutrients entered the exudate and will feed the S. aureus infection. The lysostaphin showing up in the exudate had relatively low rates, not reaching the MBEC we calculated. As can be seen in figure 5 already, and as we have seen in other modelled pictures with COMSOL (not shown) there is severe accumulation of lysostaphin in the layer it is produced. We suspect that the lysostaphin has trouble getting out of the cells, because even at cell densities of 3 * 10¹⁶ m^-3 (which is very high) not enough lysostaphin reaches the exudate while this amount of cells is definitely able to produce enough.

Future insight

We think, concluding from our model that with a larger pore size lysostaphin will have less trouble diffusing through the hydrogel and reaching the exudate. Also, because of this low diffusivity and the bacteria producing enough lysostaphin, we definitely think it is not necessary to have hydrogel completely filled up lysostaphin if the hydrogel’s properties are properly modified. Moreover, we think it is not necessary to have a thick hydrogel but rather a thin one. This will lead to more convenience and it will not lead to problems with lysostaphin production.

Lastly, nutrients will keep being a problem because of their high diffusivity. They will easily diffuse through the gel where lysostaphin is having trouble. Therefore nutrients are definitely a problem that has to be faced in the future because an infection cannot be treated if you feed it at the same time with the same hydrogel patch.

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