Introduction into kinetic modeling
What is kinetic modeling?
Kinetic modeling is used to describe dynamic (biological) systems and to simulate the output of different reactions within this system. The dynamic system can be divided into several compartments. The reactions between these compartments are described by rate laws, which are incorporated using ordinary differential equations (ODEs). These ODEs thereby describe the dynamics of compartmentalized reaction networks. Traditional reaction rate laws describe the changes in amounts or concentrations of reactants or products per unit time.
Mass action vs. Michaelis-Menten kinetics
One of the simplest forms of kinetics is the law of mass action. This law states that the velocity of a chemical reaction is proportional to the products of the mass or concentrations of the reactants.
v1= velocity 1
k1 = proportionality constant
v ...reaction rate
[S] ...substracte concentrate
Vmax ...maximum rate
KM ... [S] at Vmax/2
Another type of kinetics is Michaelis-Menten kinetics, which is commonly used as a rate law for enzymatic reactions in biochemical reaction networks. This type of kinetics is a good approximation for first-order reactions in which the rate is proportional to one concentration. The Michaelis-Menten equation is based on the assumption of the substrate and the enzyme-substrate complex being in an instantaneous equilibrium.
Deterministic vs. stochastic models
In deterministic models the values of parameters such as molecular concentrations are assumed to be continuous. As a result, there is only one outcome of reactions with these parameters. However, for most biological systems a deterministic approach may not be sufficient to model the complex biological networks. Therefore, a stochastic model can be selected. In contrast to deterministic models, stochastic models include the natural fluctuations in the concentrations of molecular species. Thereby, the probability for different outcomes of reactions in a biological system can be simulated by using a stochastic model. Further, it is possible to design hybrid models implementing a deterministic and stochastic approach.
Kinetic modeling in our project
For the analysis of the kinetics of our Friendzymes in the small intestine, we implemented a deterministic model to answer the following questions:
- What is the optimal time for taking the Friendzyme pill prior to eating a meal with critical fructan concentration?
- At what rate must our Lactobacilli secrete Friendzymes to sufficiently relieve the symptoms in patients for a meal high in fructans?
- How would modified enzymes (e.g. higher turnover constant) change the dynamics of the overall system?
A more detailed insight into our model, can be found here.
Hofmeyr, J.S. (2020). Kinetic modelling of compartmentalised reaction networks. Bio Systems, 197, 104203. https://doi.org/10.1016/j.biosystems.2020.104203
Resat, H., Petzold, L., & Pettigrew, M.F. (2009). Kinetic modeling of biological systems. Methods in molecular biology (Clifton, N.J.), 541, 311–335. https://doi.org/10.1007/978-1-59745-243-4_14
Srinivasan, S., Cluett, W.R., & Mahadevan, R. (2015). Constructing kinetic models of metabolism at genome-scales: A review. Biotechnology journal, 10(9), 1345–1359. https://doi.org/10.1002/biot.201400522
Tummler, K., Lubitz, T., Schelker, M., & Klipp, E. (2014). New types of experimental data shape the use of enzyme kinetics for dynamic network modeling. The FEBS journal, 281(2), 549–571. https://doi.org/10.1111/febs.12525
Several people suffer from fructan sensitivity, where the ingestion of elevated levels of inulin and levan lead to symptoms such as pain and discomfort in the abdomen, bloating and diarrhea. In the following tutorial we want to illustrate the first steps of creating a model for inulin and levan degradation by our therapeutic pill Friendzyme with the software MATLAB SimBiology. We will display how to set up degradation reactions and how to simulate a meal intake. Furthermore, we will demonstrate how to use the model analyzer and plot results. This tutorial gives a short introduction into some of the many possibilities for modeling in MATLAB SimBiology.
The start of our model
Open MATLAB and select the SimBiology Model Builder App from the app bar. A new window appears. Do not close the first window, otherwise the whole application of SimBiology closes as well.
The new window contains a browser, diagram and property editor window.
To create a new model, click on the model icon and give it a name, for example “bokuigem21_1”.
For getting started, the diagram window gives you the best overview and you can start building your model from that point. There are different modeling blocks that can be used to build a model. The specific blocks are in a bar above the diagram window. You can put them into your workspace by drag and drop.
For the creation of your first model, you need a compartment where the species are placed, and reactions are happening. Compartments can be defined as the volume of a cell or biological systems where the species are interacting with each other. Species and reactions need to be in a defined compartment otherwise they are invalid. The next step is to define the species that are used in the model. These are the educts or the products which are linked by reactions. By a double click on the items they can be renamed. You can connect building blocks via ctrl + holding the mouse. Be sure to connect the blocks in the right direction, reactants must be on the left side, whereas products must be on the right side of the arrow.
Blocks Symbol Explanation + Examples Compartment Isolated region (volume, room): bacteria, human cell, reaction room, … Species (Bio)chemical compounds: receptors, ligands, enzymes, antibodies, … Reaction Chemical reactions: transport of substrates, enzyme-catalyzed reaction, mass action reaction, … Parameters Constants and/or variable units describing the kinetics of the system: rate constants, Michaelis-Menten constant, … Events Describe the change of a value of a compartment, species or parameter caused by a specific point in time or a time-independent condition. (e.g., temperature) No. Name (+ name in the property editor) Block 1. Small intestine Compartment 2. Inulin Species 3. Levan Species 4. Fructose Species 5. Glucose Species 6. Inulin degradation Reaction 7. Inulin flowthrough Reaction 8. Levan degradation Reaction 9. Levan flowthrough Reaction 10. Fructose conversion Reaction 11. Fructose flowthrough Reaction 12. Glucose uptake Reaction 13. Glucose flowthrough Reaction
Add the building blocks to the small intestine compartment by drag and drop, rename them and connect the species and reactions in the right direction as described in 4.
To check for errors, the verify button can be pressed. This is a nice tool to check if there is any kind of mistake in the model. Thereby, in the browser window a small message window appears which contains a description of the errors.
In our model, the small intestine is chosen as the compartment, in which the reactions are taking place. Click on the compartment and open the property editor to set the volume of the small intestine. Since the volume is not changing select “Constant = true”.
The blue reaction describes the irreversible degradation of inulin and levan to fructose and further to glucose. The property editor is already set to false (irreversible) for the reaction. If the reaction was reversible, the command would need to be changed to true.
As the law of this reaction is Henri-Michaelis-Menten, select this kinetic law in the property editor for the degradation reaction .
The orange reactions describe the diffusion of substances by the natural flow through the small intestine. The reaction is then written as [small intestine].inulin -> null. The kinetic law is set to unknown as none of the options properly describes this reaction.
Parameters are the rate constants that define the reactions. They can be set to constant or be adjusted by using repeated or initial assignments.
As a next step, you can define the reaction rate constant k, reaction rate Vm and Michaelis-constant Km since we use a Henri-Michaelis-Menten equation for each reaction.
As can be seen in the figure beneath, the units can be chosen which are necessary to define the amount of the species and the reaction rate constant Vm and Km. The units for the species are gram/liter and for Km and Vm gram/seconds. The units of the parameters depend on the specific kinetic law that is used. Be careful to use the right units for the parameters.
To have a better overview you can choose the Browser window to see all parts of the model with their names, values, units etc.
Further applications such as doses and plots
Doses describe the temporal increase of a species within the model, e.g., by oral or intravenous administration. We can use this for mimicking the ingestion of a meal.
|Repeat dose||Same dose amount is given at equal time intervals e.g., 10 mg every 5 hours|
|Schedule dose||Different dose amounts at each administration which are given at desired time intervals e.g., 10 mg at time 0, 15 mg after 20 min, 20 mg after 35 min|
|Target name||The species related to the dose|
|Start time||Time of the first dose e.g., administrated after 5 min|
|Amount||Amount of each dose|
|Rate||How quickly the dose is administered, e.g., 2 milligram per hour|
|Interval||The time between the doses: 5 with the unit “day” = 5 times a day|
|Repeat count||Gives the total number of doses to be administered|
To open the doses window, click on “show doses” icon (marked red in the screenshot beneath) in the browser window.
This command opens a new tab for dose settings. There, it is possible to assign individual doses to species in the model. First, choose the type of dose in the first table, click on the dose sign and choose “schedule dose” for the intake of inulin and levan. In the field "target name", the species that should depend on this dose can be entered and will be connected automatically. The dose needs to be set to active.
The species should now depict the dose icon, as can be observed in the figure below.
How to visualize your data
The next step is to visualize and validate your data. This can be done in the model analyzer.
In the tab named "program" different parameters for the plot can be changed.
In the section "doses", inulin and levan need to be selected with the checkmark at the column "use", otherwise the doses will not be considered for creating the plot. The stop time for the plot is set to 300 minutes. In the section “states to log”, the different species which should be plotted can be selected.
The plot can be created by clicking on the “Run”-Icon.
In the Property Editor, the colors of the lines can be chosen by a double click on the color. By clicking the tick mark in the property editor, the chosen species is not shown in the plot anymore. In the plot, the steady increase of inulin and levan can be observed, indicating that the dosing event worked properly.
The section explorer on the left can be used to adapt parameters and to observe the resulting change in the plots. To open the panel, click on "model" in the section browser on the left side. To change the values of species or parameters, drag them to the explorer section. If the box “overlay results” in the explorer is checked, the new values will be saved. In the following plot, the starting concentration of inulin is changed from 0 to 0.5 gram/liter. Values can be tuned in the area red marked on the left side. It can be observed that the second plot shows a modified curve for inulin. Using this explorer tab, you can test different settings of your model to investigate the influence of certain parameters on the model performance.
This tutorial should give you a feeling on how to build a model in SimBiology with an exemplary part of our iGEM 2021 model. A detailed description of our whole model and our results can be found here.
We would like to take this opportunity to thank MATLAB for providing us with a license for the software and would like to recommend using this software to any scientist who is planning on getting started with kinetic modeling.
We modeled the kinetics of fructan degradation in the small intestine of a patient that took a Friendzyme pill prior to their meal. The starting point of our kinetic model is a scaffold that adheres to the mucosa of the small intestine through their mucoadhesive properties.(1) Following, the enclosed Lactobacillus plantarum secretes three different enzymes: endoinulinase, levanase and xylose isomerase. Endoinulinase stepwise degrades inulin to fructose and levanase degrades levan to fructose.(2)(3) Since many people are also sensitive to fructose and suffer from fructose malabsorption, we additionally included the enzyme xylose isomerase in our model, which catalyzes the isomerization of fructose to glucose. (4)(5)
Approximately 3 hours after the pill has been swallowed, the patient eats a pizza margherita, a meal high in fructans. With our model we would like to analyze the breakdown of inulin and levan, and determine the efficacy of our therapy using kinetic values from literature. Furthermore, we would like to identify important parameters for optimizing the degradation reactions.
The outline of our model can be seen in figure 1.
Our enzymes are produced by lactobacilli that are enclosed in a scaffold, which is attached to the mucosa of the small intestine. The enzymes are released by the Lactobacillus and diffuse through the pores of the scaffold into the small intestine. In the small intestine, the enzymes are degraded by proteolytic enzymes.(5) Additionally, a flow through the small intestine continuously decreases the enzyme concentration in the small intestine. This flow was modeled based on the average residence time in the small intestine.
Enzyme production by Lactobacillus and diffusion into the small intestine
In our model, lactobacilli embedded in the scaffold continuously produce an enzyme mix consisting of equal amounts of endoinulinase, levanase and xylose isomerase. The enzymes are secreted into the scaffold with a secretion rate of ksecretion and subsequently released with a diffusion rate of krelease into the small intestine. The diffusion for every enzyme is modeled individually, as indicated by the respective abbreviations for each enzyme: I for inulinase, L for levanase and XI for xylose isomerase.
With the diffusion out of the scaffold, the enzyme concentration in the small intestine increases.
- The amount of scaffold and Lactobacillus does not change over the observed time. This approximation is reasonable as the scaffold is attached to the intestinal mucosa for approximately 24 hours. Due to this adhesion, the scaffold is also not affected by the simulated flow through the small intestine.
- To simplify the model, the lactobacilli produce and secrete all three enzymes with the same production rate.
- We estimated the diffusion rate to be very high and to be not hindered by the scaffold. However, the diffusion rate is heavily dependent on the pore size and the size of the enzymes.(6)
Decrease of enzyme in the small intestine
The enzymes in the small intestine are subject to proteolytic degradation mostly due to pancreatic enzymes. This leads to a steady decline of the enzyme concentration in the small intestine, which is also modeled by mass action law with the degradation rates kdegr_I, kdegr_L and kdegr_XI.
An additional flow term models the flow outside of the small intestine. After the average residence time is over, the enzyme concentration decreases in the small intestine with the average flux k_flow according to mass action law, since the flow is proportional to the enzyme concentration in the compartment. This is modeled by using the event function of SimBiology, triggering the flow through and thereby decrease of the enzymes after the average residence time of 3.5 hours.
- The small intestine is considered to be a continuously stirred bioreactor and the changes in local concentrations are not considered. As the scaffold adheres to the mucosae directly at the beginning of the small intestine, the enzymes stay in the small intestine for the duration of the average residence time.
Overall ordinary differential equations
Inulin and levan degradation to fructose and glucose
The degradation of the incoming fructans inulin and levan is mediated by the secreted enzymes endoinulinase and levanase to the intermediate fructose. This intermediate is further converted to glucose by the enzyme xylose isomerase. The reactions were modeled using Michaelis-Menten-kinetics.
Inulin and levan degradation to fructose
After food consumption, inulin and levan reach the small intestine, where they interact with the steadily secreted enzyme mix. The influx of inulin and levan were modeled as scheduled doses after a time frame of 180 minutes to imitate the ingestion of the Friendzyme pill prior to the meal. The rates of the influx were approximated based on gastric emptying times found in the literature. Inulin is only degraded by the enzyme endoinulinase, whereas levan is degraded by the enzyme levanase. Both reactions yield the intermediate fructose.
Due to degradation and diffusion reactions of the enzymes and the dependency of Vm on the enzyme concentration, the maximum reaction rates Vm were defined as variable and assigned with a continuous assessment with the following reactions.
- The polymer degradation mechanism was greatly simplified in our model. Polymers are chains, built of monomeric units. The enzyme reaction leads to the cleavage of one monomer and a polymer with (n-1) subunits. In our model, the reaction was modeled as single step mechanism and all polymers with different chain lengths were treated as uniform substrates. Additionally, inulin and levan both consist of fructose monomeric units and one glucose terminal unit. Due to the small amount of glucose, the reaction was simplified to only yield fructose as product. Our approximations are based on a publication, where the kinetics of the degradation of inulin was modeled and where it was shown that despite these great simplifications a valid model could be derived.(2)
- The reaction was modeled with a complete product turnover. This is often not the case for enzymatic reactions, depending on the substrate affinity and amount of enzyme.
- Inulin is also a substrate of the enzyme levanase, however with substantially lower affinity than levan.(3) Therefore, the interaction of levanase with inulin was not considered.
- Glucose and fructose contents in the meal were not considered for simplification of the model.
Fructose isomerization to glucose
The intermediate fructose is further converted to glucose catalyzed by the enzyme xylose isomerase. The maximum reaction rate was set to variable as stated in the inulin and levan reaction.
The reaction catalyzed by the enzyme xylose isomerase is reversible and glucose can also be converted into fructose. This reaction was not considered in our model as we assumed the product glucose to be transported across the membrane immediately, thereby decreasing the product concentration in the reaction space of the small intestine.(7)
Glucose is being taken up by the intestinal epithelial cells by the transporters SGLT-1 and GLUT-2.(8) The uptake was modeled in a single step reaction.
- The passive absorption of glucose was neglected.
- The model is only valid for glucose concentration below 25 mM, where the transport capacity of SGLT-1 is not exceeded yet. With a steady uptake of glucose this threshold is not reached in our model.
- The number of transporters in the epithelial cells was modeled to be constant for the observed time frame. Therefore, constant Vm and Km values were used.
Flow through of the sugars
As the sugars enter the small intestine with a time shift of 180 minutes compared to the enzymes, the flow through is modeled by a separate event function and flow rate k_flowF, with the event being triggered at a later timepoint.
Overall ordinary differential equations
Results and discussion
All parameters were derived from literature and approximated to fit the conditions in the small intestine. In general, published values were finetuned up to a factor of 10 to consider e.g., pH-dependent changed activity of enzymes. In table 1, the calculated values of our model with the according references are summarized. The detailed calculations to derive these parameters can be found below:
|Enzyme mix||0||gram liter-1||modeled|
|Small intestine||0.764||liter||Karthikeyan, Salvi, and Karwe (2021) (9)|
|Xylose isomerase||0||gram liter-1||modeled|
|Secretion rate enzymes||0.0067||min-1||estimated|
|Release rate enzymes||5.6∙10-3||min-1||estimated|
|Vm inulinase||0||gram liter-1 min-1||modeled|
|Km inulinase||1.64||gram liter-1||Ricca et al. (2009) (2)|
|Vm levanase||0||gram liter-1 min-1||modeled|
|Km levanase||12||gram liter-1||Wanker, Huber, and Schwab (1995) (3)/td>|
|Vm xylose isomerase||0||gram liter-1 min-1||modeled|
|Km xylose isomerase||1.80||gram liter-1||Van Bastelaere, Vangrysperre, and Kersters-Hilderson (1991) (7)|
|Degradation rate enzymes||1.4∙10-3||min-1||Pongracz, Pfisterer, and Missbichler (2010) (5)|
|Vm glucose uptake||2.7||gram min-1||Zheng et al. (2012) (8)|
|Km glucose uptake||3.37||gram liter-1||Zheng et al. (2012) (8)|
|Flow through intestine||4.8∙10-3||min-1||Karthikeyan, Salvi, and Karwe (2021) (9)|
|Dose||Scheduled time (min)||Amount (g)||Rate (g min-1)||Reference|
|Inulin||180||6.00||0.05||Moshfegh et al. (1999) (10) |
Lavigne et al. (1978) (11)
|Levan||180||2.50||0.0208||Ua-Arak et al. (2017) (12) |
Lavigne et al. 1978 (11)
For the analysis of our model, we firstly simulated the system with parameters derived from literature to get an overview on the kinetics of the system. The ODEs were solved using the SolverType ode15s (stiff/NDF). In a next step, we modified parameters and finetuned the system to answer important questions on the efficacy and feasibility of our therapeutic approach. For comparisons, the original model is indicated by a star (*).
Figure 2: Concentrations of fructans and enzymes in the small intestine over time
The diagram shows a steep rise in inulin and levan concentration after 180 minutes as the fructan components of the simulated ingested pizza reach the small intestine. It can be observed that the degradation reaction starts immediately, leading to a rise in fructose content. Additionally, the isomerization of fructose to glucose can be observed. After approximately 120 minutes the symptom triggering fructans are degraded, fructose is isomerized to glucose. Glucose is continuously taken up by SGLT-1 and GLUT-2, leading to a total clearance of all sugars within 130 minutes.
The enzyme mix is steadily produced and excreted into the small intestine after taking the Friendzyme pill and attachment of the scaffold to the enterocytes. After 210 minutes, the flow through the small intestine results in the flattening of the concentration curve. However, as the encapsulated lactobacilli are not affected by the flow, enzymes keep getting excreted and the overall concentration in the small intestine rises.
With the literature-derived parameters, the breakdown of fructans in the small intestine works efficiently. The main goal is to degrade the compounds as fast as possible to avoid reaching concentrations higher than the clinical threshold (0.26 gram liter-1).(13) Furthermore, the breakdown should be efficiently carried out before inulin and levan can reach the large intestine, where they are known to trigger symptoms. Therefore, the overall residence time should not exceed the average residence time of 3.5 hours in the small intestine. With the calculated and approximated parameters, the overall fructan concentration reaches 0.22 gram liter-1 and all sugars are cleared within 2.25 hours.
What is the optimal time for taking the Friendzyme pill prior to eating a meal with critical fructan concentration?
To determine the importance and influence of the time between taking the Friendzyme pill and eating a meal high in fructans, we simulated the model with a ∆t of 120, 180* and 240 minutes. ∆t refers to the time between taking the pill and eating a meal.
Figure 3: Concentrations over time with a ∆t of 120 minutes
Figure 4: Concentrations over time with a ∆t of 180 minutes
Figure 5: Concentrations over time with a ∆t of 240 minutes
Figure 2 shows that a ∆t of 120 minutes does not lead to a sufficient breakdown of the incoming fructans as the enzymes have not yet reached sufficient concentrations. As inulin and levan amounts exceed the clinical threshold for more than 50 minutes, it is highly recommended to wait at least 180 minutes prior to food consumption. Figure 4 shows that a ∆t of 240 minutes results in improved breakdown of the compounds, as more enzymes have already been excreted to the small intestine. However, this result should not be extrapolated to even longer time shifts. The enzyme concentration will decrease eventually due to the flow through the small intestine, reduced output of the lactobacilli, bacterial cell death and degradation by proteolytic enzymes. Reduced degradation rates can already be observed with the first flow through event at 210 minutes for the ∆t 120 and ∆t 180 models. Overall, the results indicate that with the used parameters the Friendzyme pill should be taken 3 to 4 hours prior to eating a critical meal. We consider this timing as convenient, as the patient could easily take the pill prior to going out for lunch or dinner.
At what rate must our lactobacilli secrete Friendzymes to sufficiently relieve the symptoms in patients for a meal high in fructans?
To address this question, we varied the secretion rate of our lactobacilli to 0.003, 0.006 and 0.008 min-1 and compared the results to the original value of 0.007 min-1.
Figure 6: Different concentrations of fructans and enzymes depending on the secretion rate of lactobacilli
Figure 5 shows that the efficacy of the therapy is highly dependent on the secretion rate of the encapsulated lactobacilli. Decreasing the secretion rate by a factor of 2 results in fructan concentrations nearly twice the clinical threshold. Therefore, the undisrupted production and functioning excretion of the enzymes is of high importance for our therapeutic approach. The degradation of fructans can certainly be improved by increasing the secretion rate, however, it is questionable whether rates higher than 0.007 min-1 can be reached. The estimated parameter of 0.007 min-1 is already considered very high when comparing to the growth rates of lactobacilli reported in the literature (see calculation of parameters). Our next experiment addresses the influence of the enzymatic performance on the model performance.
How would modified enzymes change the dynamics of the overall system?
We were interested, whether the high dependency of the therapeutic success on the secretion rate of our lactobacilli could be minimized by using optimized enzymes with a higher turnover or Michaelis-Menten constant. Therefore, we used the model with the insufficient secretion rate of 0.003 min-1 and changed kcat and Km of inulin, as it makes up the majority of the fructan content.
Figure 7: Secretion rate of 0.003 min-1 with original Km (1.6428 gram liter-1) and kcat value (0.1056 s-1)
Figure 8: Secretion rate of 0.003 min-1 with optimized Km (0.821 gram liter-1) and original kcat value (0.1056 s-1)
Figure 9: Secretion rate of 0.003 min-1 with original Km (1.6428 gram liter-1) and optimized kcat (kcat = 0.2112 s-1) value
Figure 10: Secretion rate of 0.003 min-1 with optimized Km (0.821 gram -1) and kcat (0.2112 s-1) value
In figure 7 and 8, the influence of optimized Km and kcat parameters without changing the respectively other parameter can be observed. Improving the Michaelis-Menten constant Km (Km = 0.821 gram liter-1) results in sufficient degradation, with fructan amounts exceeding the clinical threshold only for a negligible time frame and amount. For the optimized kcat (kcat = 0.2112 s-1) model, critical fructan concentrations are not exceeded. The simulated model with both parameters optimized results in an even better degradation efficiency than achieved with the original model. Therefore, using optimized enzymes can regulate the high dependency of sufficient breakdown of fructans on the secretion rate and amount of enzyme in the small bowel and leads to an overall improvement of the degradation rate. We suggest screening the enzymes for high performance under physiological conditions in the small intestine. Additionally, site specific or random mutagenesis could be used to optimize the enzymes. Nevertheless, the time and effort to obtain enzymes with higher binding affinity and/or catalytic performance must be kept in mind.
Summary and outlook
Overall, the simulations of our model gave us first results on the efficacy of our therapeutic approach as well as potential properties to be adjusted for optimization. As the rate of produced enzyme and the diffusion into the small intestine were estimated to rather high values, the actual therapeutic effect might not be as sufficient as observed in our simulations. However, considering the results of the experiment with altered Km and kcat values, other parameters could be optimized to address this issue. If there are less lactobacilli in the scaffold or the secretion rate is not as efficient as approximated, the introduction of high-performing enzymes would still result in efficient degradation rates.
In general, it must be stated that all kinetic parameters as well as species and compartment amounts were derived from literature or estimated yielding only a hypothetical model. Our next step would be to validate and optimize our model by implementing experimental data from the lab. Nevertheless, we gained important insights into kinetic modeling itself and obtained a deeper understanding of the involved reactions of our Friendzyme therapy.
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