Team:Tec-Chihuahua/Model

Erwinions








From the very moment we made this our project, we were repeatedly asked the same question: “how can you make sure enough AMPs get to the bee larvae?” This is a valid and crucial concern. After all, one of the key elements to our project’s success is that the antimicrobial peptides reach the bee larvae with a high enough peptide concentration to inhibit pathogenic bacteria. The method we chose was PLGA nanoencapsulation.

Therefore, in order to measure the final amount of AMPs that will be delivered to each larva, we model the amount of peptide released from the nanoparticles in their path through the insides of the nurse bee and into the jelly fed to larvae. The model we developed will help us to determine the initial dosage of AMPs necessary to reach their final destination in desirable concentrations.


Theoretical Foundation

Given previous research that confirms antibiotics supplemented to nurses in food get to the brood1, the nanoparticles must travel through the nurse bee body along with the artificial food. Therefore, is important to examine the alimentary canal as well as the hemolymph system, since this is the path we postulate the nanoparticles will travel.

The nanoparticles must travel through the nurse bee body firstly along with the artificial food, therefore, is important to examine the alimentary canal, the blood and the organs of circulation of the bee since these are the main systems in the pathway from PLGA nanoparticles to the larvae.2

The cellular layer segregates juices and enzymes and the resulting products of digestion will be absorbed by the peritrophic membrane and travel through the stomach walls to the hemolymph. Therefore nanoparticles will go to the blood or hemolymph system. Once they are in the hemolymph they will circulate through the body by the single blood vessel which is a long slender tube that extends from the back abdomen through thorax and ends at the head.2

The abdomen segment of the vessel tube is the heart and by rhythmic pulsations, it makes the blood circulate. The blood with the nanoparticles will travel to the aorta and will irrigate all the structures of the head, including the hypopharyngeal and mandibular glands. The hypopharyngeal synthesize the royal jelly that will mix with the secretions of the mandibular glands and the PLGA nanoparticles. The mix will then be delivered to the larvae through the excretory ducts that are the normal pathway for which the royal jelly is delivered to the larvae.2


The polylactic-co-glycolic acid (PLGA) is a biodegradable and biocompatible polymer that is used for the fabrication of drug delivery devices. PLGA could be used to transport proteins, peptides and macromolecules such as DNA and RNA. PLGA is ideal for drug delivery because its degradation can be used to control the drug’s release profile.

While zero order kinetics would be the most desirable case, this is currently unachivable. There are three possible non-zero ordered behaviours for polymer degradation. Polymer degradation/erosion (A) is dependent on the degradation of the polymer coating. Diffusion (B) is currently the most common release mechanism, wherein release is dependent on the dissolved drug’s concentration. Finally, solvent penetration or device swelling (C) is dependent on the permeability of the device.

The release behavior of PLGA is linked to the degradation which occurs by hydrolysis. The degradation profile is through bulk degradation as a consequence of the water penetration into the particle and by the polymer degradation. The process can be divided into two phases:

  1. The initial burst where there is a quantity of drug released at time zero related to the drug type, drug concentration, and polymer hydrophobicity.
  2. The progressive drug liberation through the weakened PLGA layer. The solvent hydrolyzes the polymer and creates a channel for where the drug can be released by diffusion and erosion until polymer degradation is complete.

The factors that influence the PLGA degradation are the following:3

  1. Polymer composition determines the hydrophilicity and the rate of degradation of the particles. A high percentage of glycolic acid in the composition increases weight loss.
  2. Polymer coating, which can result in a different reaction, e.g. Chitosan coating makes for sturdier nanoparticles that begin degradation in more acidic environments.
  3. The average molecular weight is related to the polymer chain size. Heavier molecular weights correspond to lower degradation rates.
  4. The drug or molecule type in the PLGA matrix modifies the degradation mechanism and the rate of degradation. It is clear that the chemical properties of the drug affect the release profile. However, there is not a clear correlation between the drug chemistry and the behavior.
  5. The size and shape also impacts matrix degradation. A higher surface area in comparison to volume shows higher matrix degradation making drug liberation faster.
  6. Environment pH may also affect the rate degradation of PLGA. Strongly alkaline and acidic media leads to a faster degradation whilst less acidic and neutral media exhibits slower acceleration on the rate degradation.
  7. Also, another important parameter is the amount of drug load. PLGA loaded with high drug concentrations increase the initial burst release.
  8. Finally, in biological systems, the degradation may be influenced by enzymatic activity. However, there are not conclusive results in how the enzymes play a role in the degradation process.

Modeling

The model is based on the diffusion from the antimicrobial peptide in the interior of the PLGA nanoparticle to the exterior of it. Since our peptides will be delivered through PLGA 50:50 nanoencapsulation, we decided to use a simple Fickian diffusion model suggested by previous research4, 5, 6.
The diffusion equation can be seen here:

Where:


Symbol Variable Unit
Mt/M∞ Drug released in time t in relation to the total amount of nanoencapsulated drug Unitless, from 0 to 1
T Time Seconds
D Diffusion coefficient, dependent on the drug-polymer interations Centimeter squared per second
R Radius Centimeters
⍺ The initial burst, which is a normal behavior of PLGA-drug release systems where quantity of initial drug is released independently of time. A fraction, in relation to the total amount of drug


We adapted this equation to our needs, adding a new variable ẟ that represents the remainder of dosage still within the nanocapsule such that ẟ at a given time is equal to the subtraction of the amount of peptide released at that time from the full load.

Since our capsules will travel through two stages, inside an adult bee and within royal jelly, we also established a delivery time (td) and expressed the diffusion equations in relation to this travel path.

Since programming to infinity would imply an infinite calculation time, it was also necessary to define the appropriate boundary conditions for the summation. Since this summation converges, we can find an n at which every new summation term is insignificant (adds an amount smaller than 0.00001).

Before receiving the data for the radii range in our nanoparticules, we picked boundary conditions based on a wide range of previously reported4 diffusion coefficients and radii, as well as time (from 0 to 35 days, far beyond what we need).


Time (s) Diffusion coefficient (cm2/s) Radius (cm) Highest significant n
t=3,024,000s D=5x10-19 R=2.2x10-5 1
R=0.00013 4
D=4x10-16 R=2.2x10-5 15
R=0.00013 62
t=0s Any value 317


As predicted, a lower exponent value requires more summation terms. At 0, the lowest possible value, n was significant (as defined by us) up to 317. This value was used in all of our codes (presented at the bottom of the page).

Experiment

Our td was discovered experimentally with the help of team UAlberta. They timed the hours it takes for a bee to get rid of said peptide.

The limitation that calculating D provides is clear. This coefficient is an interaction between the drug and its polymer coating, along with the solvent. To overcome this, we have to experimentally evaluate the drug’s release mechanism and pick the best-fit line.

We must do this under two conditions: one simulating hemolymph and another for jelly. For this, we have to prepare two buffer solution at these pH ranges, and kept them at the desired temperature, as established below.


Conditions Simulated pH Temperature
Bee Hemolymph 8.35,6 32 - 35 ºC 9
Worker/Royal Jelly 3.5 - 4.57, 3.98 32 - 35 ºC 9, 30 - 33 ºC 10

Our friends at Tec-Monterrey kindly wrote a protocol for encapsulating our peptides in PLGA 50:50 (51 kDa) coated with Chitosan. We could periodically test the amount released over a period of time in the buffer solutions.

The best fit line for each case lets us figure out the diffusion coefficient for these conditions. (A code for this purpose is presented at the end of this page).

Once we have the diffusion coefficients, we substitute them in the equation system we developed. We input them into modeling software to find the behavior of diffusion through both environments (find this code at the bottom of the page).


Limitations to our approach

There are several challenges regarding this experimental approach. Here we present them in an upfront manner, and explain what is to be done:

  1. Since the initial burst is also determined by drug concentration, once we find the diffusion coefficient, this will not be enough to successfully predict another concentration’s behavior but rather provide an approximation. Further experimentation is required.
  2. This model for PLGA diffusion has been shown to lose prediction accuracy beginning on the 10th day. This limitation is insignificant to the scope of our project; as shown by our collaboration with UAlberta, bee metabolism takes only X days and larvae are also most susceptible 12-36 hours after hatching. Not only this, but the brood need only 9-11 days to have their cells capped and start becoming pupae, which rarely die from Foulbrood (since it can no longer infect them).13
  3. The viscosity of jelly might result in a reduced diffusion capacity, which must be taken into account for the amount of remaining peptide.
  4. Bees have other pH values in their guts, head, and glands. However, these all fall in between the pH values chosen for hemolymph and royal jelly.14
  5. We underwent experimentation only in apidaecin. We’d need to repeat this process for the remaining AMPs or their combination.

Results & Discussion

With the modelling tools provided and given experimental release profiles, we can simulate an average release within the path nanocapsules are meant to follow. Here, we can see a simulated profile with examples of radii and diffusion rates taken from other polymer-encapsulated drugs. 4

What’s important is that we have to be able to see two characteristics in our graph: (1) concentration at Td is at least enough to inhibit bacterial growth in the larval cell and (2) that the minimum inhibitory concentration gets released over the next 4 to 5 days, before larvae are encapsulated within their cell.

Minimum inhibitory concentrations for honeybee peptides and their combinations have been reported on extensively.15 In this example, we suppose % of an abaecin load gets initially released. By delivery, the nanocapsule has % and releases % during the next 4 days. Taking into account our limitations, we can know that if % is what gets released during these days, and MIC is X, the initial load must be X times that.

The simulation and codes provided here will be of use in later stages of this project, and perhaps for similar pharmacokinetic analyses. It could be further refined by adding the transition within each part of the nurse bee, but that implies tracking the nanocapsule at several times through the nurse bee’s body. This experimentation would feeding using and killing bees, an ethical consideration you can read more on it in our Human Practices.

Matlab Coding

Without further ado, here we present the codes used for our modeling, to be used as confirmation or reference (values provided are examples):


clear all;
close all;
clc;

D=5*10^-19;
t=0;
R=2.2*10^-5;
x=exp(-(D*t*(pi^2))/(R^2));
z=0;

for n=1:1:inf
            zi=(x^(n^2))/(n^2);
            if zi<0.00001
                        disp(n);
                        break;
            end;
z=z+zi
end;




clear all;
close all;
clc;

format longEng;

sampnumb=input(['How many times was concentration meassured? '])
valoresexp=zeros(1,sampnumb);
R=2.2*10^-5; %en cm

for i=0:1:sampnumb-1
            valoresexp(:,i+1)=input(['What is the value at ' num2str(i*24) ' hours? ']);
end

a=valoresexp(1,1);
j=0;
sses=zeros(1,2);

for D=5*10^-19:10*10^-19:4*10^-16
            j=j+1;
            for k=0:1:sampnumb-1
            t=k*86400
            Z=0;
            errorssqs=zeros(sampnumb,1);
            for n=1:1:317
            Z=Z+(1/(n^2))*exp(-((n^2)*(pi^2)*D.*t/(R^2)))
            end;
            y=1-(6/(pi^2)).*Z+a;
            error=(valoresexp(1,k+1))-y;
            errorsq=error*error;
            errorsqs(k+1,1)=[errorsq];
            end;
            sse=sum(errorsqs)
            sses(j,:)=[D sse]
end;

[val,idx] = min(sses(:,2))
sse=val;
D = sses(idx,1);
disp('The diffusion coefficient that best explains this behaviour is');
disp(D);



clear all;
close all;
clc;

R=4.5*10^-5; %in cm
t=0:3600:864000 %in s
a=0.3;

D1=3*10^-18; %in cm2 por s
D2=4*10^-17; %in cm2 por s

t1=0:3600:345600;
t2=345600:3600:864000;

Z=0;
for n=1:1:317
            Z=Z+(1/(n^2))*exp(-((n^2)*(pi^2)*D1.*t1/(R^2)))
end;
sum1=(6/(pi^2)).*Z;

Z=0;
for n=1:1:317
            Z=Z+(1/(n^2))*exp(-((n^2)*(pi^2)*D2.*(t2-(t1(end)))/(R^2)))
end;
sum2=(6/(pi^2)).*Z;
sum2(1,1)=1;

u=sum1-a;
v=(u(end)).*(sum2);

plot(t1,u);
hold on;
plot(t2,v);
hold off;

xticks([0 86400 172800 259200 345600 432000 518400 604800 691200 777600 864000]);
xticklabels({'0','1','2','3','4','5','6','7','8', '9', '10'});
xlabel('Time (days)');
ylabel('Fraction released');





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