Team:NUDT CHINA/Model
Model
Introduction
In our project, we employ a designer cell approach to control hyperglycemia by degrading hepatic Glucagon Receptors (GCGR) in a glycemic dependent manner. To predict and demonstrate the in-vivo performance of our circuit and avoid animal uses, we constructed a set of mathematical models on molecular, cellular and whole-body level. To be specific, a cellular glucose sensing model, a protein degradation model, and a whole-body glucose-insulin-glucagon model were built based on fundamental biochemical and physiological principles.
The glucose sensing model simulates the transcriptional activation of the designed circuit under the regulation of glucose; the protein degradation model concentrates on the ubiquitination and degradation of GCGR; the glucose-insulin-glucagon model represents how the impaired cellular glucagon sensitivity caused by the disruption of GCGR affects glycemia. These three models constitute a closed-loop system, that is, the output of the former model also serves as the input of the latter. Computer simulation suggested that our circuit established the glycemic homeostasis of type 2 diabetic objects into a physiological level; it also suggested an improved performance of closed-loop system on preventing hypoglycemia comparing to the glucose-insensitive circuit. Complementary to our wet lab experiments, this model further demonstrates our design for in vivo use and the significant importance of the glucose sensing device in achieving intelligent regulation of glycemia.
Figure 1. Architecture of the mathematical model
Modeling
General Assumption
1 Since the biochemical reactions in cells are too complex to be completely simulated, the interactions of intermediates throughout the whole process were ignored.
2 Considering that differentiated cells in vivo are under homeostatic condition, where the cell proliferation rate can be generally ignored, and the intercellular microenvironment remains stable.
3 We ignore the effect of the endogenous trim21 enzyme on the degradation of target protein in our study object.
4 We assume that the degradation of GCGR directly interacts with whole body glucose homeostasis by affecting glucagon sensitivity.
The glucose sensing model
The glucose sensing model is a simplified dynamic representation of the glucose-responsive process, which mainly refers to the signaling pathway of the transcription factor ChREBP (Figure 2.). The mechanism of the ChREBP regulation has been studied extensively while a detailed mathematical model of ChREBP pathway is lacking so far. The previous work from Kabashima et al. and Tara Jois et al. have given us a biochemical and physiological insight into ChREBP pathway1,2, and the model that describes the pentose phosphate pathway in rat liver cells presented by Sabate have provided parameter guidance for our work3.
Figure 2. ChREBP pathway
In the ChREBP pathway, extracellular glucose transports into cells and produces xylose-5-phosphate (X5P) via pentose phosphate pathway. Respectively, v1_1, v1_2, v1_3, v1_4, v1_5 account for the enzymatic reaction rate of each step in the glucose metabolism. Intracellular X5P increases correspondingly and it is immediately used as a glucose signaling compound that recruits and activates a specific protein serine threonine phosphatase (PPase)1 (in our case, PP2A) at the rate v2. Activated PPase then catalyzes the dephosphorylation of ChREBP with rate v3 that is dependent on the concentration of PPase. Then ChREBP translocates to the nucleus and binds to the glucose sensing promoter (ChoRE), activating the transcription of downstream genes, at the rate v4 that is proportional to the concentration of ChREBP. In order to keep our model as accurate as possible, we also consider that the intermediate metabolites decrease at the rate ki*[A] (i=1,2,3……, A refers to the metabolite) in other biochemical reactions. Based on mass-action law and Michaelis-Menten-type kinetic mechanism, the dynamic equations for the metabolites are defined as follow:
Utilizing this model, we can obtain the relationship between the initial concentration of glucose and the concentration of mRNA. This model is connected to the protein degradation model via the concentration of mRNA as outputs.
The protein degradation model
Adapted from a previous model developed for our 2018 project, the protein degradation model simulates the process from mRNA translation to the degradation of GCGR via ubiquitin-proteasome pathway (Figure 3.). This model links the concentration of mRNA to the final concentration of GCGR.
Figure 3. Ubiquitination and degradation of GCGR
Glucagon-IgG-Fc and E3 ligase Trim21 can be expressed in cells after the translation of mRNA. Glucagon-IgG-Fc fusion protein binds with GCGR at the rate v0f, forming the substrate (S). Such substrate binds with E3 ligase at the rate v3f. Simultaneously, ubiquitin (Ub) is activated by the ubiquitin-activating enzyme E1 and transferred from E1 to ubiquitin-conjucating enzyme E2. Since the process of binding is reversible, we use v0f, v1f, v2f, v3f to represent the forward reaction rates and v0b, v1b, v2b, v3b for the reverse reaction rates. The ubiquitination of a substrate protein occurs primarily by sequential transfers of single ubiquitin molecules to the substrate4,5, yielding a polyubiquitin chain. Here we assume that eight ubiquitin molecules are linked to form a polyubiquitin chain, and we model the rate that Ub is transferred from E2 to substrate as v4_n (n=1,…,8). Then ubiquitinated substrates ubnSE3 dissociate with E3 at the rate v5_n (n=1,…，8). The ubnS decrease at the rate v6_n (n=1,…，8) for the effect of deubiquitinating enzymes that remove ubiquitin from substrate proteins. Since at least four ubiquitin molecules must be attached to a lysine residue on the condemned protein in order for it to be recognized by the 26S proteasome6, only ubiquitinated substrates with Ub chains longer than 4 can bind with 26S proteasome for degradation at the rate v7_n (n=4,…，8), and finally are degraded by 26S at the rate v8_n (n=4,…，8). The model equations were formulated following the law of mass action：
The glucose-insulin-glucagon model
The glucose-insulin-glucagon model describes the relationship among blood glucose concentration(B), insulin concentration(I) and glucagon concentration(G) (Figure 4.).
Figure 4. The overall framework of the glucose-insulin-gulcagon model
In this model, the following assumptions are made7:
- the secretion of the two hormones, insulin and glucagon, is dependent only on the blood glucose concentration
- the rate of insulin secretion depends only on the blood glucose concentration with increasing blood glucose concentration leading to an increased rate of secretion
- the secretion of glucagon depends only on the blood glucose concentration with decreasing blood glucose concentration leading to increased secretion of glucagon
- there exists a maximum rate of insulin secretion regardless of the blood glucose level due to the fixed capacity of pancreas
- there exists a maximum rate of glucagon secretion
- the rate of glucose utilization in the body is directly proportional to both the blood glucose concentration and the insulin concentration in a one-compartment model for insulin
- insulin sensitivity is a constant
- the level of glucagon decays at a rate proportional to the amount of glucagon present
- the rate of hepatic glucose production is assumed to be directly proportional to the glucagon concentration
Respectively, the equations of these three variables are shown below:
The meaning of each parameters are as follows:
Utilizing this model, we can obtain the relationship between the concentration of glucagon and blood glucose. Thus, the whole closed-loop model is established.
Results and Discussion
Part1
In order to characterize the relationship between the glucose concentration，which is the input of the glucose response module, and the GCGR predator mRNA production, we performed a computational analysis by the glucose sensing model.
The overall framework of our glucose sensing model is shown in Figure 5.a, with glucose concentration as the input and GCGR Predator mRNA production as the output. The dynamic changes of all parameters in our simulation are shown in Figure 5.e. To further determine the relationship between the input glucose concentration and GCGR Predator mRNA production, we challenged the model with different input glucose level and predict the production of GCGR Predator mRNA. As is shown in Figure 5.b, quadruple glucose input level resulted in a doubled mRNA concentration output. Meanwhile, Sensitivity analysis also revealed the time-dependent sensitivity of mRNA production on input glucose level (Figure5.c). Such result matches well with our wet-lab data, which showed similar trend on glucose sensitivity (Figure 5.d), indicating that our model might have correctly reflected the cellular process in our case.
Figure5.(a) The overall framework of the glucose sensing model ; (b) (1) line1 initial Glucose mRNA (2) line2 4* initial Glucose mRNA ; (c) Sensitivity curve ; (d) Experimental diagram ; (e) The dynamic changes of all parameters in Figure 5.a.
Part2
Protein degradation model is used to simulate the GCGR Predator mediated GCGR degradation. To validate if our model fits wet lab experiment, we compared the final GCGR abundance using 0 mRNA or the stable value of model #1 as input.
Figurg 6. (a) Model diagram ; (b) Experimental diagram.
Results showed that without GCGR Predator mRNA, the level of the GCGR stabled at 0.57. In comparison, the introduction of GCGR Predator mRNA caused immediate drop on the GCGR level and the cellular concentration gradually stabled at around 0.37, a ~50% change could be observed (Figure 6.a).
In order to further verify the accuracy of the protein degradation model, we revisited the experimental data. Western blotting showed that GCGR decreased for ~50% 48 h post transfection (Figure 6.b), which is similar to the modeling results.
Part3
In order to verify the contribution of GCGR Predator on whole body glucose homeostasis, we established a glucose-insulin-glucagon model to provide a testing platform by simulating the homeostasis of blood glucose, insulin and glucagon of both healthy people(Figure 1.a) and T2D patients(Figure 1.b).
Figure 7. (a) Steady state curve of normal human body ; (b) Steady state curve of T2D patients.
Simulation showed that under common conditions, a healthy object maintained a glucose level of around 96 mg/dL. When changing the insulin sensitivity for 50%, we could observe a significant increase of both blood glucose and insulin (Figure 7.b) level, which is similar to the hyperglycemia and hyperinsulinemia symptom of T2D.
Integrated Analysis
By integrating the three parts into a whole, we can simulate the effect of the whole regulation mechanism on blood glucose homeostasis.
The circuit starts from the change of blood glucose concentration and transmits the signal of blood glucose concentration change through the glucose response module, which leads to the change of GCGR Predator mRNA expression level. Furthermore, it regulates the degradation level of GCGR Predator to GCGR, so as to regulate the response of the body to GCG. The change of the sensitivity of the body to the response of GCG will further lead to the change of blood glucose.
By integrating three models, we obtained glycemia simulation results under different conditions. Healthy object showed stabled glycemia level at around 96 mg/dL. By decreasing insulin sensitivity by 50%, object showed significant hyperglycemia at around 133 mg/dL. When GCGR-Predator is introduced into the model with a strong constitutive promoter, object showed mild hypoglycemia (~53 mg/mL), which is consistent with the suggestions provided by HP expert interview as well as previous literatures. However, when GCGR Predator system is controlled by glucose sensing device, which provides feed-back control over the degradation of GCGR, simulation showed that the glycemia level would be stabled at around 86.7 mg/dL, indicating that the close-loop controlling system may provide a more robust control over the glycemia level (Figure 8.).
Figure 8. (a) Integration of three models; (b) Glycemia simulation under different conditions.
Reference
1 Kabashima et al. Xylulose 5-phosphate mediates glucose-induced lipogenesis by xylulose 5-phosphate-activated protein phosphatase in rat liver. National Acad Sciences vol. 100, no. 9, 5107–5112, doi:10.1073/pnas.0730817100(2003).
2 Tara Jois & Mark W. Sleeman. The regulation and role of carbohydrate response element binding protein in metabolic homeostasis and disease. Journal of Neuroendocrinology Volume 29, Issue10 e12473, doi:10.1111/jne.12473(2017).
3 Llorenc et al. A model of the pentose phosphate pathway in rat liver cells. Molecular and Cellular Biochemistry 142, doi: 10.1007/BF00928908 (1995).
4 Pierce NW, Kleiger G, Shan SO & Deshaies RJ. Detection of sequential polyubiquitylation on a millisecond timescale. Nature 462: 615–619, doi:10.1038/nature08595 (2009).
5 Lida Xu & Zhilin Qu. Roles of Protein Ubiquitination and Degradation Kinetics in Biological Oscillations. PLoS ONE Volume 7 , Issue 4 , e34616, doi:10.1371/journal.pone.0034616 (2012).
6 Hicke L . Protein regulation by monoubiquitin. Nature Reviews Molecular Cell Biology. 2 (3): 195–201, doi:10.1038/35056583 (2001).
7 Caleb L. Adams & D. Glenn Lasseigne. An extensible mathematical model of glucose metabolism. Part I: the basic glucose-insulin-glucagon model, basal conditions and basic dynamics. LETTERS IN BIOMATHEMATICS, vol.5, NO.1,70-90, doi:10.1080/23737867.2018.1429332(2018).
The glucose sensing model
The protein degradation model
The glucose-insulin-glucagon model
