Our mathematical model aims to predict the capacity of our probiotic to kill helminths in the human intestine based on
literature parameters. By assessing variables that go from our probiotic's delivery to
the chitinase's expression and action, we can anticipate the real-world effectiveness of our product. It
can also support decision-making at all stages of drug development [1].
It is essential to determine the adjusted posology of the biopharmaceutical product before in vivo testing, to guarantee its safety and efficiency, and to save time and resources.
Getting parameters
To test the capacity of a bacterial bio-drug that expresses a heterologous enzyme in the human
intestine to kill a parasite we need to discuss which criteria can influence its action. So, first,
it's necessary to know how much viable bacteria can actually reach the intestine after ingestion the variation of the number of cells there. Then,
bearing this loss along the way in mind, how much of both chitinases can the Lactobacillus
acidophilus produce per cell, time and volume? With that variable set, we needed to know the
amount of endo and exochitinase that can enable a reduction in the parasitic load. But we still need to
consider an estimated range of intestinal helminths per infected person, and that number will vary from
species to species. With all those parameters settled, we can assemble them into mathematical equations
and generate graphs showing us if the project is feasible and at which points we can improve it. Given
the dynamicity of living organisms and nature itself, our strategy was to adopt minimums and maximums in
each variable.
Viability up till intestine
In order to reach their functional site, produce and secrete the final product, our probiotic bacteria has
to overcome some harsh conditions, such as gastric acid and bile salts. With that in mind, the probiotic needs
to be ingested in adequate quantities that can provide the survival of viable numbers of microorganisms in the
intestine. Several reports determined the survival rates of probiotics in the gastrointestinal tract (GIT).
Here we present our assessment of those data and the conclusions taken.
First, Naissinger, et al. [2] tested commercial probiotics' survival in the GI tract. They counted viable
cells in two distinct moments: after being prepared for consumption (activated, diluted, or removed from the
package) and after succeeding in the simulated digestion, counting each step of the passage. According to the
study, for a probiotic to offer benefits to the individual, it needs to get to the small intestine with a
minimum concentration of 6 log CFU/g. Considering an expected decrease of 2 log CFU/g along the digestion, it
is recommended that the supplement has at least 8 log CFU/g. The researchers observed a decrease between 1-4
log CFU/g in all 11 samples tested in the GI simulation. Therefore, the approximate loss was about 6 to 55%.
Marteau, et al. [3] assessed the survival of some lactic acid bacteria using a dynamic model of the stomach
and small intestine that can simulate the successive passages in the GIT in vivo. The species evaluated were
L. bulgaricus, L. acidophilus, Streptococcus thermophilus and Bifidobacterium bifidum.
According to their results, the L. acidophilus and L. bulgaricus strains were the most resistant
ones, with a great fraction of ingested bacteria reaching the duodenum alive. This occurrence was mainly
observed during the 20 to 30 minutes after a meal, when the stomach pH was still relatively high.
A study developed by Bezkorovainy in 2001 [4] stated that the survival rates of probiotics are about 20 to
40%. The half-life of B. bifidum and L. acidophilus was 140 minutes, with the stomach pH
differing from 5.0 to 1.8 in 80 minutes after ingestion. The author also evaluated determinants of growth in
the gut, criteria that we initially established as a component of our model. But in this cited article, we
realized some inconclusive data related to this factor. Can probiotics actually adhere to intestinal cells? In
vitro studies using tissue cultures suggested that they do adhere to the mucosa. But, on the other hand, in
vivo tests proposes otherwise. However, even with that unsettled information, it is undeniable that orally
administered probiotics can change the intestinal microbiota in a positive way, presenting uncountable
benefits.
Finally, Yao, et al. [5] evaluated the microencapsulation of a Lactobacillus species for storage and
delivery viability. They simulated GIT conditions for the free probiotic and the ones encapsulated with
gelatin and alginate. The results for free probiotics showed that almost all of them died after 40 minutes of
contact with the stomach and small intestine conditions. On the other hand, the microencapsulation with
alginate-gelatin was the most efficient, showing small percentages of reduction. It also proved to be
resistant to aerobic storage and heating. This article was important not only for gathering data for the
modelling but to think about the shelf life of our bio-drug. The encapsulation with gelatin and alginate seems
like a good idea since the material is already used to encapsulate, it is cheap and can form microgels with
relatively simple injection processes.
By discussing the presented data, we decided to consider a range of decreased viability of our probiotic in
the mathematical model. Being the minimum loss settled at 10% and the maximum loss of 50%. Since we
would not be able to test our L. acidophilus strain under the aforementioned conditions, stipulating a
range of values can make our simulation closer to the real world.
Probiotic factor
Having the viability loss of L. acidophilus settled, we need to estimate the amout of CFU present in the intestine according to time, that is, the bacteria population size according to time. We settled this factor as the probiotic factor. The data was gathered from an article that evaluated the survival ability of L. casei in vivo before administration and after 3, 6, 9, 24, 30, 48, and 72h after administration to the mice [6].
Fig 1. Probiotic factor of L. casei in vivo. Each test represents average viable count from mice. Reprinted from Su, P., et al. [6].
Before administration, the probiotic bacteria wasn't detected, although, 3 hours after the administration, L. casei was encountered reaching a level of 1–1,12×109 CFU/g in feces at 6 and 9h. Then a gradual reduction is observed until 72h, when no bacteria is detected.
Expression rate of chitinase
To reach an estimated value for this variable, we collected data from heterologous expression and secretion
of chitinase in Escherichia coli , and Lactobacillus. Boer, et al. [7] expressed two different
chitinases from Trichoderma harzianum, a fungus, in E. coli. They also tested both enzymes on
the bacteria's surface and in different cell compartments. Inside the cytoplasm, the yields were around 10
mg/L for one and 2 mg/L for the other chitinase. On the periplasmatic region, however, they reached a yield of
about 1 mg of active enzyme per liter of culture. Since we aim to export the chitinase outside the cell, we
need to consider the decrease usually observed in the expression rate.
Additionally, Nguyen, et al. [8] expressed a chitinase from a Gram-positive bacterium into Lactobacillus
plantarum. In this case, the gene was expressed intracellularly because they did not successfully
secrete the enzyme. The yield obtained was 5 mg of soluble chitinase per liter of fermentation medium in
larger-scale cultivations. On the other hand, a related chitinase gene was expressed using E. coli as
an expression system and the authors reached 20 mg per liter [9, 10].
All the cited references express the production rate in mg per liter, but Savijoki, et al. demonstrated a
high level of heterologous protein secretion using a more precise unit [11]. They reached the secretion rate
of 5x10⁵ molecules of the protein per cell per hour in Lactobacillus and Lactococcus using a
secretion system based on the Lactobacillus brevis S-layer signal.
Therefore, considering that we aim to express, extracellularly, a fungus-derived chitinase in Lactobacillus
acidophilus with the highest secretion rate possible, we adopted the value of 5x10⁵ Chitinase
molecules/cell/h. Using this unit instead of mg/L is more favorable considering that we can associate
it with the CFU data of bacteria arriving at the intestinal tract.
But to get closer to a real prediction, another scope needs to be taken into account, the volume of
intestinal fluids. An article released in 2005 dosed the variability of the gastrointestinal fluids,
evaluating meal-induced changes and the transit of solid indigestible capsules [12]. Using this as a
reference, our mathematical model adopted the values: minimal small intestine volume = 45 mL
and maximum small intestine volume = 319 mL.
[Chitinase] that destroys the parasite
Since there is no research evaluating the effect of chitinases against human intestinal worms, to refine
this parameter, we applied an article that tests a fungal chitinase against C. elegans and two other horse
parasitic nematodes [13]. The authors employed 150 uL of 50 mg/mL enzyme solution in about 1.500 nematodes and
incubated at optimum temperature for 12, 24 and 36h. Using chitinase to treat Strongylus equinus,
Caenorhabditis elegans and Haemonchus contortus separately the killing rates were:
Our team’s objective is to first test the chitinases against C. elegans, so in the mathematical model
only these killing rates (highlighted in bold) were considered. We also consider the incubation periods used
by the researchers. They also found that chitinase effectively kills the eggs of nematodes, showing its
potential to act as biological control.
Amount of helminths/human intestine
The final variable was to determine the average abundance of the targeted worms. This data is usually
measured by coprological methods and the average abundance is determined by the total number of individuals of
a particular parasite in a sample of a host species, divided by the total number of hosts examined for that
species. The helminth population in humans tends to have an aggregated distribution among hosts, with a few
infected individuals having a large number of parasites and a larger number of infected individuals with a
lower parasite load [14, 15].
Since the infection from Ascaris lumbricoides is one of the most frequent geo-helminthiasis
distributed in Brazil [16], we searched for the prevalence of eggs per gram (epg) of feces in different age groups
in underdeveloped regions. Supali, et al. [17] assessed the frequency of soil-transmitted helminths infections
in eastern Indonesia, gathering the following data:
On the other hand, Hall, et al. [18] targeted the distribution of Ascaris among 1765 people in Bangladesh.
The results obtained by them show a worm burden in different age groups. Accordingly, we selected the
maximum and minimum loads that corresponded to 6,9 and 23,4 worms (adults) per intestine .
Assembling equations
So, the variables considered in the assembly of mathematical equations were:
Fig 2. Maximun and minimun of each selected variable. Probiotic factor not displayed because it wasn't defined in ranges but in hours after ingestion, as showed in Fig 1.
We used the software RStudio to create our model.
The first equations that we assembled were to calculate the max and min viable amount of probiotic cells in
the intestine in relation to time:
Then, we created a function to represent the relation between time and probiotic factor via numeric fitting
to double exponential decay, since we did not have enough points to fit a polynomial curve. We fitted the
function in the points obtained from the graph. The resultant equation was:
Then, we performed indefinite integration and obtained the function:
where f(x) is the probiotic factor and x is time.
To calculate the chitinase production we joined the previous function with the other parameters:
Then, we divided the resultant value by the intestinal volume:
Caption:
viable_lac_max_dose = Maximum viable probiotic cells that arrive in the intestine according to the dose.
viable_lac_min_dose = Minimum viable probiotic cells that arrive in the intestine according to the dose.
probiotic_dose: Probiotic dose in CFU.
viability_min: Minimum viability (10%).
viability_max: Maximum viability (50%).
probiotic_factor: Factor that represents the bacteria population size according to time(h).
chitinase_expression: Chitinase production by the probiotic cells minus chitinase degradation.
recombinant_production_chitinase: Amount of chitinase in mg produced per cell per hour.
protein_degradation_rate: Degradation of protein in mg per hour.
intestinal_volume: Intestinal volume in ml.
chitinase_concentration: Concentration of chitinase in the human intestine, in mg/ml.
We performed these calculations with the variable's maximum and minimum values, obtaining the resultant
graphs shown below.
Results
Relation between probiotic factor and time
We successfully obtained a function that represents the points of the probiotic factor graph, the function was represented in red.
Then, we integrated the function, as represented in blue.
Viability up till intestine
In the graphs below we show the results of the modelling of the probiotic cells viability in the small
intestine, according to the administered dose.
Exochitinase expression
We plotted graphs that show the exochitinase production, according to the administered dose:
Endochitinase expression
We plotted graphs that show the endochitinase production, according to the administered dose:
Exochitinase concentration in the small intestine
Additionaly, we calculated the concentration of exochitinase in the small intestine, according to the
administered dose, and plotted the graphs:
Endochitinase concentration in the small intestine
Additionaly, we calculated the concentration of endochitinase in the small intestine, according to the
administered dose, and plotted the graphs:
Improvements and next steps
The next step of our model is the calculation of the relation between the chitinase concentration and
the anthelmintic activity. Unfortunately due to the lack of experience and time, we could not calculate
the anthelmintic effect of the resultant chitinase concentrations. However, we invite future iGEMers to
continue this challenge and improve our model.
To join this challenge, check out our code and RStudio tutorial. With these documents, you can recreate our model and adapt the variables to the ones you need.
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