Features of KPD that a mathematical model should explain

A model of KPD incorporating the hypothesis of reversible β-cell deactivation.

Here we illustrate schematically the model described by Eqs. (1–4) (Materials and methods). A: Glucose promotes insulin secretion, and insulin lowers glucose, in a negative feedback loop. The effect of insulin on glucose disposal is mediated by the insulin sensitivity SI , and the amount of insulin secreted is proportional to the mass of active beta cells β. This couples these fast dynamics to the slower dynamics in panel B. B: Longer timescale dynamics incorporate two distinct types of glucotoxicity. An intermediate scale, reversible glucotoxicity with rate kINg(G) produces an inactive pheonotype βIN, which recovers at rate kRE. Simultaneously, a slower process causes permanent β-cell deactivation or death at rate kDh(G). The dependence of these rates on G couples these processes to those in panel A.

Simulations showing the onset and remission of KPD in our model.

Solid black lines show daily blood glucose averages. Red dashed lines demarcate a period of increased sugar consumption, while blue dashed lines demarcate a period of insulin treatment, if present. Horizontal dotted lines indicate 80 mg / dL (black, typical normoglycemia) and 130 mg / dL (gray, diabetes control threshold per American Diabetes Association). A: With a low rate kIN of reversible β-cell inactivation, blood glucose returns to normal after the period of high sugar consumption. B: With a high rate kIN of reversible β-cell inactivation, the same period of high sugar consumption produces a sharp rise in the blood glucose level, associated with a sharp drop in β-cell function, which persists after the period of sugar consumption ends. A sufficiently long period of insulin treatment can produce an insulin-free remission. C: A lower rate kRE of β-cell reactivation, compared to B, increases the time required to produce remission with the same insulin treatment. D: A higher rate kD of permanent β-cell death, compared to C, results in a failure to achieve insulin-free remission.

The reversible β-cell dynamics have two fixed points for parameters which produce KPD-like behavior.

After assuming that the fast-timescale glucose/insulin dynamics relax to their fixed-point at a given value of β, we compute dβ/dt for the intermediate-timescale, reversible dynamics as a function of β for the parameter values shown in Fig. 2(A,B). Fixed points are given by values of β where the curve crosses the dashed line an thus dβ/dt = 0. A: For model parameters where simulations do not show KPD-like behavior, there is a single stable fixed point of the intermediate-timescale dynamics. B: For model parameters where simulations show KPD-like behavior, there are two stable fixed points, corresponding to a high β / moderate G state and a low β / high G state.

Glucose or insulin intake can push the β-cell dynamics between the fixed points.

We compute dβ/dt for the intermediate-timescale, reversible dynamics, averaged over a day. We use the parameter values shown in Fig 2A,B. Purple curves correspond to the low daily sugar intake (region outside the blue and red dashed lines in Fig. 2; note that the curves are only slightly different from the black curves in Fig. 3), red curves to the high sugar consumption between the red dashed lines in Fig. 2, and blue to a gradual release insulin treatment, as between the blue dashed lines in Fig. 2). A: For model parameters where simulations do not show KPD-like behavior, there is a single stable fixed point of the intermediate-timescale dynamics, which is qualitatively unchanged by the high-glucose-consumption condition. Thus, as in Fig. 2A, β-cell function remains healthy. B: For model parameters where simulations show KPD-like behavior, there are two stable fixed points. The high-β fixed point is not present in the high-sugar-consumption condition (red), and thus this condition produces a flow from the high-β fixed point to the low-β fixed point, as seen between the red lines in Fig. 2B–D. The insulin-treatment condition (blue) instead destroys the low-β fixed point, and thus produces a flow from the low-β fixed point to the high-β fixed point, allowing for remission as seen in Fig. 2B,C.

Phase diagram of the model

Fixed-point structure of the model (fixed points of black curves in Fig. 3) as a function of total β-cell population and insulin sensitivity SI , relative to the values used in Fig 2B. Color indicates fasting glucose (with all values of G above 130 mg/dL the same red), while hatching indicates the number of fixed points. No hatching indicates existence of just one stable fixed point at high β and intermediate or normal fasting glucose. Single-hatched regions have two fixed points, allowing for transient loss of β-cell function and remission, while double-hatched regions have only the low-β fixed point (i. e., total insulin dependence). A: For the β-cell inactivation rate as in Fig. 2A, the bistability which produces KPD-like presentation and remission is not seen, even for β-cell populations and SI values which produce fasting hyperglycemia consistent with T2D. B: For the β-cell inactivation rate as in Fig 2B, mild fasting hyperglycemia tends to correspond with a susceptibility to KPD. Reduction of β-cell populations (e.g., due to the slower, permanent glucotoxicity as in 2D) can produce greater hyperglycemia and even total insulin dependence. Improvements in insulin sensitivity, in contrast, contribute to maintenance of remission.