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Modeling

Introduction

The main objective of our modeling section was to create a mathematical system that could explain the relationship between blood glucose levels, the size of the bacterial population, and the resulting insulin production, for different conditions related to state of energy homeostasis in the human body. The relevance of our model is such that it provides a theoretical framework that can be applied to the development of our device for people that depend on insulin, while predicting the behaviour of their blood glucose when using it, ensuring a proper insulin administration.

Bacteria

We obtained experimental data from laboratory section of bacterial population growth over time. After observing the growth curves, we assumed a logistic growth based in a qualitative approach, and what has been reported in previous works [1] Using linear regression and applying limits to estimate the variables to generate the corresponding values for the minimum and maximum regions of the equations, it was possible to develop a function that was similar to the experimental results.

Glucose

The glucose that is being considered is, primordially, correspondent to the levels that are present on the human blood, the main reason is that the sensor that is present at the bacteria membrane is thought to detect the levels on the surroundings, which should be at an equilibrium with the glucose levels on blood. Our system is thought to detect the glucose that the user has assimilated in his/her blood, in order to produce an adequate transcriptional response resulting in insulin that might be beneficial for the user’s metabolism. Glucose is considered as being negatively regulated by the amount of insulin in bloodstream and is consumed at a basal level at a rate proportional to the glucose already present.

Figure 1: Experimental values on bacteria population over time, it is possible to appreciate that the development seems to be logistic, shuffle bacteria is used as an example of this method, but similar models on EnvZ, Rosseta and DH5a were developed.

The process of ingesting is considered as very chaotic and specific to each person , we decided to elaborate a system of pulses with an aleatory amplitude and periodicity. This aleatory amplitude was considered as following a Poisson distribution. The decision to implement a Poisson distribution was made under the considerations that we had to generate pulses that represented a general person from a certain population, the Poisson allows to have values related to the population average without generating values that would be unrealistic such as negative or extremely high that would surge from other distributions like the normal distribution. These pulses were taken as the following:

We also take into account that there is an amount of time between food eating and sugar level changes associated with them, so we are planning to build a system that uses delay equations to represent this period. Other important instances that could influence the glucose levels are the starving response and the glucose-consuming activity, those factors are dependent on the glucose levels given for a certain time and vary along the day, these equations were developed using a delay equations system, which might be helpful because these changes are not instantaneously taking place after its application.. Activity was thought to a function that causes a reduction on the glucose levels according to a certain person’s physiology. There is a constant consumption of glucose in the body that is not related to sport activity, like the consumption used to sustain the brain and other metabolic pathways [2]. We expected a differential response according to the glucose levels already present, when those levels are low, an

specific person has low energy levels, which could be correlated to a deficit on energy supplies like glucose. On the other hand, when glucose levels are too high, people tend to feel a lethargy sensation, it is commonly known as “el mal del cerdo” (the pig’s disease) in our region. Under that theoretical basis, we assumed that our activity function would have an optimum glucose burn at a medium glucose rate, which might be different inter and intra population because of the metabolic variances that could arise when changing our study subject. The two tails of the function should be asymmetric since both have different origins, considering these circumstances, we assumed that the graphic that represents this behaviour should be like the Figure 3.

The resulting function of the variables obtained by our methodology is shown in this figure, when compared to the experimental values, little difference in the values is seen. By generating the corresponding calculation, it was also possible to obtain the equilibrium values for our system.

The starving function followed a different approach, the response for each glucose level scenario were also analyzed and compiled at a single function with its respective graphic. When a certain person reaches extremely low levels of glucose, the body uses the glucose stocked in the body as a emergency resource, resulting in an increase in blood glucose. This response is dependent in the amount of glucose, which reduces considerably with medium levels of glucose [3]

The conjunction of these factors emulates the influence of lifestyle on glucose dynamics, these must be considered with the dynamics that result from the metabolism, which also involve the protein regulation of the insulin. The glucose regulation was considered as being the result of an exponential equation with a steady state at an specific vale xb, which should be considered as the value

Figure 3: A simulation of how the glucose consuming activity should be represented in the body, considering Xb as the optimum point for those kinds of activities, which also is considered to be the maximum value in the graphic. The present graphic was obtained using the program Geogebra with an input of an proximate function that represents the changes at every blood glucose value

that optimizes the body functions. The system that could represent this later development should be represented as follows: Ax(e^(x−xb) − 1)

It is remarkable that when we equal this function to zero, we get a steady state when x = 0, xb, where zero is a repulsive state and xb is an attractive point, causing that our glucose levels tend to xb. The final section to cover with our model on glucose involves the influence generated by the insulin in our body over glucose changes, x∞(x,y), it relates to the changes on the glucose level in association with the insulin present in the organism. When generating a function that involves all previous modules, the next equation was developed:

Insulin

The insulin quantity depends on two principal factors to consider, as the insulin is produced as a response of high levels of glucose, one important factor is the glucose already present. On the other hand, there must be a limit for insulin production, which might be named as y∞ , when the insulin present in the body reaches this quantity, we can see an important decrease on the insulin production rate. When estimating the value in which y reaches an equilibrium, we developed the following.

Figure 4: A similar methodology as the one followed in the previous graphic is used to represent the expected response when starving, it is noticeable that when reaching the optimum value for glucose in blood, the starving response is almost unnoticeable. The starving response is an emergency alternative that requires the usage of the main reserves and activates when glucose is dangerously low

it is remarkable that when xy = x, the resulting exponential would be 1, which translates on a y∞ that equals ym 2, an half of the maximum quantity of insulin. The values of this function are contained in the range [0, ym] which makes sense as both are logical limits to the insulin in the organism. The relation between the quantity of glucose present in the body and the insulin that is produced by the sensing system should be dependent on the glucose levels at a certain time.

We have two principal scenarios, when there’s no glucose, we can see that the expression x xM gets a zero value, resulting in no production of insulin, on the other hand, if x equals its maximum value xM, the same fraction would result in a value of 1 resulting in a production rate of aM multiplied by rb + (x − xb).

Union

Once the different parts that composed the model were developed and studied separately, simulations on the whole model were run using python programming, based on previous studies [7] [6] [4] that showed the relation between glucose and insulin, our results were similar to the expected. Some parts of our model could not be considered as there are some technical issues that need to be attended before generating the pulses on feeding, it is expected that the complete model should work as a generalization of the one presented in these simulations, but we require to build a complete simulation to emphasize this point. We developed a mathematical model that regulates glucose levels of diabetic people to 80mg/dL, an optimum value according to Normoglycemia definitions. Normoglycemia is defined as blood glucose concentration in the range of 70–100 mg/dl. [5].

In order to test our glucose sensor, we measured fluorescence intensity of RFP in our bacteria populations. Experimental results showed that under a 154 mg/dL glucose concentration, the system reaches its highest expression peak of the gene promoter. We included experimental data in our computational simulations and provided the theoretical framework of Glucose-Insulin dynamics in our system. Previous iGEM teams found similar results, Missouri Miners iGEM Team developed a glucose sensor with specific dose responses with different glucose concentrations than the concentrations we obtained, they found a positive and down regulation dynamic of the system by measuring fluorescence intensity due to the production of eYfp [8]. Our mathematical approach help us to understand the dynamic of the system and the parameters involved using experimental data.

Figure 5: A comparison is made between the glucose levels (up) which begins at a level of 170 mg/dL and the insulin levels (down), it is possible to appreciate that the glucose levels are expected to be regulated to a 80 mg/dL level, generating a feedback on the insulin levels, which tends to zero when glucose is optimized. It was assumed that an insulin unit reduces 30 mg/dL which was indicated as a safe assumption by our medical section

Once we were sure the model had a behaviour similar to what we expected when developing the corresponding equations, the comparison between both kinds of equations was required to ensure that there was a relation in the key points that established how our model interacted with itself, as it is a continuous model that varies according to the conditions at the preceding moment. We also had to ensure that our modeling results were not a mere result of an appropriate selection of initial values, as the main function of insulin decrease the glucose values, we evaluated different scenarios with a glucose level over the

Figure 6: Theoretical simulation of Glucose-insulin dynamics with an optimum glucose value Xm of 154 mg/dL. Glucose and insulin are integrated in a single table in order to demonstrate how the regulation system works, it is noticeable that both share similar turning points and reach an equilibrium at similar times. In order to test our glucose sensor, we measured our reporter gene’s expression by measuring its absorbency. Our reporter gene (OmpR controlled mRFP) behaves in different conditions of glucose concentration. According to experimental results, under a glucose concentration of 154 mg/dL we have the highest peak of expression of our reporter gene. By taking into consideration these results, the simulation considers the highest peak of expression of our reporter to be also the highest peak of expression of insulin gene. The simulation shows that under a glucose concentration of 154 mg/dL, insulin reaches its highest peak and then starts to decrease while glucose levels also decrease.

recommended levels, showing the effect of glucose. In order to consider levels under this line, it would be needed to integrate the feeding pulses to reestablish the correct balance on the system.

In the whole, our simulation could be considered as a system that evaluates many important factors on the insulin/glucose regulation, with certain predictive capacity and a well balanced equilibrium that could be reached at a relatively short amount of time.

Conclusion

The whole integration of this model explains the dynamics that regard insulin and glucose in the organism, which could be generalized to a system that regulates the insulin production synthetically like the system that we developed.

Figure 7: Different glucose levels are configured as beginning points (up), every scenario has a corresponding insulin generation response. The intensity and duration of every response increases along the initial quantity of glucose to be regulated. One important factor to consider is that this simulation integrates a single deviation from the equilibrium while a biological system reunites a series of different pulses that might be analysed as a series of pulses analogues to the ones presented here.

The implementation of our mathematical model could facilitate the configuration of the system according to a biological basis, using experimental data to accomplish the correct regulation of glucose. The system might be modified to regulate the values of the bacteria in order to get optimum results and the model should be modified according to these changes to avoid disparities between experimental results and mathematical framework. This model could be implemented in the area of biosensors while being of utility in other areas of knowledge as some of these equations could be of use for dieticians, predicting the behavior of their patient’s organisms or for endocrinology and other diabetes related fields as it predicts the generation of insulin according to a wide variety of parameters. In order to achieve this goal, we are planning to share the possible implications of this work with written resources comparing different people metabolism and lifestyles

References

[1] Li, X. Pietschke, C. Fraune, S. Altrock, P.M. Bosch, T.C.G. Traulsen, A. Which games are growing bacterial populations playing?. The Royal Society Publishing. 2015 8

[2] Cuendet GS, Loten EG, Jeanrenaud B, Renold AE. Decreased basal, noninsulin-stimulated glucose uptake and metabolism by skeletal soleus muscle isolated from obese-hyperglycemic (ob/ob) mice. Journal of Clinical Investigation. 1976;58(5):1078-1088.

[3] Berg JM, Tymoczko JL, Stryer L. Biochemistry. 5th edition. New York: W H Freeman; 2002. Section 30.3, Food Intake and Starvation Induce Metabolic Changes. Available from: http://www.ncbi.nlm.nih.gov/books/NBK22414/

[4] Ellingsen C, Dassau E, Zisser H, Grosman B, Percival MW, Jovanovic L, Doyle FJ. Safety Constraints in an Artificial Pancreatic Cell: An Implementation of Model Predictive Control with Insulin on Board. Journal of Diabetes Science and Technology. 2009;3(3): pp, 536–544.

[5] American Diabetes Association. Standards of medical care in diabetes—2006. Diabetes Care. 2006;29 Suppl 1:S4–42. [6] Huang M, Li J, Song X, Guo H. Modeling impulsive injections of insulin: Towards artificial pancreas. Society for Industrial and Applied Mathematics. 2012, pp.1524–1548

[7] Wang Q, Molenaar P, Harsh S, Freeman K, Xie J, Gold C, Rovine M, Ulbrecht J. Personalized State-space Modeling of Glucose Dynamics for Type 1 Diabetes Using Continuously Monitored Glucose, Insulin Dose, and Meal Intake: An Extended Kalman Filter Approach. Journal of Diabetes Science and Technology. 2014, pp.331-345

[8] Missouri Miners iGEM Team. Glucose sensor. 2011. url:https://2011.igem.org/Team:Missouri Miners/Data