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Insulin granule trafficking in -cells: mathematical model of glucose-induced insulin secretion

¹Institute of Systems Analysis and Computer Science, Consiglio Nazionale delle Ricerche, Rome; ²Department of Systems Analysis and Informatics, University of Rome "La Sapienza," Rome; and ³Institute of Internal Medicine, Catholic University School of Medicine, Rome, Italy

Submitted 27 November 2006 ; accepted in final form 16 April 2007

	ABSTRACT

TOP ABSTRACT Glossary DEVELOPMENT OF THE MATHEMATICAL... RESULTS DISCUSSION APPENDIX A: ANALYSIS OF... APPENDIX B: ANALYSIS OF... APPENDIX C: ESTIMATION OF... REFERENCES

 
A mathematical model that represents the dynamics of intracellular insulin granules in -cells is proposed. Granule translocation and exocytosis are controlled by signals assumed to be essentially related to ATP-to-ADP ratio and cytosolic Ca²⁺ concentration. The model provides an interpretation of the roles of the triggering and amplifying pathways of glucose-stimulated insulin secretion. Values of most of the model parameters were inferred from available experimental data. The numerical simulations represent a variety of experimental conditions, such as the stimulation by high K⁺ and by different time courses of extracellular glucose, and the predicted responses agree with published experimental data. Model capacity to represent data measured in a hyperglycemic clamp was also tested. Model parameter changes that may reflect alterations of -cell function present in type 2 diabetes are investigated, and the action of pharmacological agents that bind to sulfonylurea receptors is simulated.

-cell; insulin secretion; insulin granule dynamics; type 2 diabetes

Since the early observations of the first and second phase of insulin secretion in response to a step in extracellular glucose (11, 13), pathways of stimulus-secretion coupling have been identified. In the triggering pathway, the glucose stimulus causes an increase in the ATP-to-ADP ratio. This leads to closure of ATP-dependent K⁺ (K_ATP) channels, cell membrane depolarization, and Ca²⁺ influx through voltage-sensitive Ca²⁺ channels. The consequent increase in cytosolic Ca²⁺ concentration ([Ca²⁺]_i) stimulates exocytosis of insulin granules. Still not completely characterized are other signaling pathways, and several mediators may be involved in the K_ATP-independent Ca²⁺-dependent (amplifying) pathway (43, 21, 31, 30). ATP and Ca²⁺ cover a major role in the machinery that regulates granule translocation from the trans-Golgi network to the cell membrane, granule priming, fusion with the cell membrane, and release of contents in the extracellular space (41). Other important aspects of islet behavior are as follows: 1) the heterogeneity in the responsiveness to glucose of the -cell population (26) and 2) the presence of spontaneous sustained oscillations in the concentration of signaling molecules and in the insulin release, revealed both in isolated cells and islets, and in animal models as well as in human patients (19, 40).

Complex mathematical models of glucose-stimulated insulin secretion were proposed previously by Grodsky and coworkers (20, 37) (storage-limited model) and by Cerasi and coworkers (12, 36) (signal-limited model). Subsequently, models of insulin secretion were proposed that provide estimates of -cell function by using C-peptide data (49, 8, 33) and that are more easily utilizable in clinical practice. Complex models of the Hodgkin-Huxley type that represent the bursting behavior of the -cell electrical activity and the ATP and Ca²⁺ oscillations were also presented (46, 4, 38).

The mathematical model proposed in the present paper focuses on the dynamics of formation, translocation to cell membrane, and exocytosis of insulin granules in -cells. We have not attempted to model in detail the biochemical events that lead to changes in ATP concentration ([ATP]) and cytoplasmic Ca²⁺ concentration (or other signaling molecules) in response to stimulation by extracellular glucose or to the experimental manipulations that cause cell membrane depolarization. Instead, these processes are represented as time changes of rate coefficients regarded as "control" signals that regulate granule trafficking. The model provides a preliminary framework in which a number of experimental observations can be accommodated and analyzed in quantitative terms. The capacity of the model to analyze experimental data obtained in a hyperglycemic clamp was ascertained. In addition, to our knowledge, this is the first model that tries to represent the pharmacological action of drugs that stimulate insulin secretion.

	Glossary

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State Variables (Equation Number Where Quantities First Appear)

I

Pool of proinsulin aggregates (Eq. 1)

V

Pool of granule membrane material (Eq. 1)

R

Reserve pool (Eq. 3)

D

Pool of docked granules (Eq. 4)

D_IR

Pool of immediately releasable granules (Eq. 4)

F

Pool of granules fused with plasma membrane (Eq. 6)

Rate coefficient of granule externalization and priming, related to the ATP-to-ADP ratio (min^–1) (Eq. 3)

Rate coefficient of granule fusion with cell membrane, related to [Ca²⁺]_i (min^–1) (Eq. 5)

Additional Variables

C

Pool of unbound L-type Ca²⁺ channels

G

Extracellular glucose concentration (input variable) (mmol/l) (Eq. 7)

ISR

Insulin secretion rate (pmol/min) (Eq. 11)

IS

Insulin amount secreted in a time T during the first-phase response (pmol) (Eq. A4)

G_m

Measured plasma glucose concentration (mmol/l) (Eq. C1)

I_p

Measured plasma insulin concentration (pmol/l) (Eq. C1)

G_a

Estimated arterial glucose concentration (mmol/l) (Eq. C1)

Parameters in the Equations of Granule Dynamics

k

Rate constant of formation of proinsulin-containing granules in trans-Golgi network (min^–1) (Eq. 1)

b_I

Biosynthesis rate of proinsulin aggregates at a given glucose concentration (min^–1) (Eq. 1)

_I

Rate constant of degradation of proinsulin aggregates (min^–1) (Eq. 1)

b_V

Rate of biosynthesis of granule membrane material (min^–1) (Eq. 2)

_V

Rate constant of degradation of granule membrane material (min^–1) (Eq. 2)

_V

Time delay related to recycling of granule membrane material (min) (Eq. 2)

C_T

Pool of total Ca²⁺ channels (Eq. 4)

k₁⁺

Rate constant of association for the binding between granule and Ca²⁺ channel (min^–1) (Eq. 4)

k₁^–

Rate constant of dissociation for the binding between granule and Ca²⁺ channel (min^–1) (Eq. 4)

Rate constant of insulin release from granules fused with cell membrane (min^–1) (Eq. 2)

Parameters and Functions in the Equations Representing Stimulus-secretion Coupling

Rate constant in the equation for (min^–1) (Eq. 7)

_b

Basal value of (min^–1) (Eq. 7)

Oscillatory function that represents events inducing oscillations in (min^–1) (Eq. 7)

h

Function of G representing the activatory action of glucose on (min^–1) (Eq. 7)

_G

Time delay related to time required by glucose metabolism for activation of (min) (Eq. 7)

G*

Glucose concentration threshold for the activation of (mmol/l) (Eq. 8)

Maximal value of h (min^–1) (Eq. 8)

Glucose concentration over which h remains constant and equal to (mmol/l) (Eq. 8)

Rate constant in the equation for (min^–1) (Eq. 9)

_b

Basal value of (min^–1) (Eq. 9)

h

Function of representing the activatory action of on (min^–1) (Eq. 9)

k

Parameter representing the sensitivity of on the activatory action of (Eq. 10)

Additional Parameters and Functions

I₀

Insulin amount contained in a granule (amol) (Eq. 11)

N_c

Average number of -cells in an islet (Eq. 11)

N_i

Number of islets in pancreas (Eq. 11)

N

Total number of -cells in pancreas (Eq. 17)

f

Fraction of -cells responding to glucose stimulus (function of glucose concentration G) (Eq. 11)

f_b

Basal value of the fraction f (Eq. A5)

K_f

Parameter in the function f(G) (mmol/l) (Eq. A5)

V_G

Glucose distribution volume (liters) (Eq. C1)

V_C

Volume of C-peptide accessible compartment (liters) (Eq. 17)

Q

Cardiac output (l/min) (Eq. C1)

S_I

Insulin sensitivity index [l·min^–1(pmol/l)^–1] (Eq. C1)

S_d

Dynamic responsivity index (Eq. 17)

S_s

Static responsivity index (min^–1) (Eq. 18)

a and b

Parameters in the approximate expression of the first-phase ISR (min^–1) (Eq. A1)

x_b

Denotes the basal value of the generic variable x

Denotes the steady-state value of the variable x

	DEVELOPMENT OF THE MATHEMATICAL MODEL

TOP ABSTRACT Glossary DEVELOPMENT OF THE MATHEMATICAL... RESULTS DISCUSSION APPENDIX A: ANALYSIS OF... APPENDIX B: ANALYSIS OF... APPENDIX C: ESTIMATION OF... REFERENCES

 
Granule Dynamics in a -Cell

Let us consider for the moment a single -cell. A schematic diagram of the formation and trafficking of insulin-containing granules is given in Fig. 1. Essentially, the model describes the kinetics of four intracellular pools of insulin granules: the reserve pool, the pool of docked granules, the pool of immediately releasable granules, and the pool of granules fused with cell membrane. In the following, the model of Fig. 1 will be illustrated and formulated as a mathematical model.

View larger version (5K): [in this window] [in a new window]   Fig. 1. Diagram of the kinetics of granule pools in the -cell. I, pool of proinsulin aggregates; V, pool of granule membrane material; R, reserve pool; D, pool of docked granules; C, pool of Ca2+ channels; DIR, pool of immediately releasable granules; F, pool of granules fused with plasma membrane. Other symbols explained in the text. The quantity F represents the number of insulin granules that, in the unit time, release insulin in the extracellular space.

(1)

where the overdot indicates derivative with respect to time, _I is a degradation rate constant, and b_I is a production rate that describes proinsulin biosynthesis and aggregation. Changes in proinsulin biosynthesis rate do not appear to be involved in the first and second phase of insulin secretion elicited by a step increase in ambient glucose concentration (42, 22). However, proinsulin synthesis has been reported to increase with glucose concentration (45), so b_I is actually a function of the extracellular glucose concentration, G. The governing equation for V is

(2)

where _V is a degradation rate constant and b_V is the rate of production of the granule membranes. The last term in the right-hand side of Eq. 2 accounts, in simplified way, for the contribution to granule membrane formation resulting from the (complete) recycling of membrane material. This material becomes part of the plasma membrane at exocytosis and is successively removed and returned to the trans-Golgi network. We have denoted here by F the number of granules that are fused with cell membrane, by the rate constant of the fusion process, and by _V the constant time interval required for recycling. Incomplete recycling may easily be accounted for by means of a multiplicative coefficient smaller than 1.

To write the governing equation for the reserve pool, R, we assume that granules formed at the trans-Golgi network immediately enter the reserve pool and that granules leave this pool to become docked granules with a rate constant . If granules are substantially stable, that is, they cannot be disaggregated or degraded, we can write:

(3)

The mechanism by which granules are translocated from trans-Golgi to plasma membrane in -cells has not yet been completely elucidated, although it appears to involve unidirectional movement along microtubules by means of motor proteins (52). Experimental data from mouse and rat -cells show that only 1–2 granules/min per -cell are released at extracellular glucose concentration around 3 mmol/l, and around 20–30 granules/min per cell are released at the peak of the first phase of insulin secretion after a step increase in glucose (2, 7, 48). Thus granule transport must involve only a small fraction of the reserve pool that has been estimated to contain 9,000–13,000 granules (48). To keep the model simple, we include in the translocation also granule priming, so a unique rate coefficient, , will represent the processes that produce docked granules that are competent for release. As discussed in the next section, has to be considered a rate coefficient that is actually dependent on the glucose concentration G and represents a main rate-limiting step in the response to glucose.

According to a well-established view (2, 48), within the pool of docked and primed granules here denoted by D, a pool of immediately releasable granules, D_IR, must be distinguished. The latter granules are likely to be tightly associated with L-type Ca²⁺ channels (2). Thus we assume (see Fig. 1) that the pool D_IR is formed through the binding of docked granules with unbound L-type Ca²⁺ channels, C. For simplicity, we will also assume that the Ca²⁺ channels that become unbound at granule fusion are immediately available to bind new granules. Denoting by C_T the constant pool of total Ca²⁺ channels (bound plus unbound), we have C = C_T – D_IR and we may write:

(4)

(5)

where k₁⁺ and k₁^– are the rate constants of association and dissociation, respectively, for the binding between docked granule and Ca²⁺ channel, and is a rate coefficient that accounts for the factors that promote the fusion of granules with cell membrane. Actually, granule fusion is strictly related to changes in [Ca²⁺]_i. As seen in the following sections, we consider a rate coefficient that is affected by changes in the ATP-to-ADP ratio or by experimental manipulations, such as the increase in extracellular K⁺ concentration ([K⁺]), that cause depolarization of the cell membrane and thus an increase in [Ca²⁺]_i.

The rate of granule membrane fusion with cell membrane is given by the last term in the right-hand side of Eq. 5, i.e., D_IR(t), so the equation for the pool of granules undergoing fusion with the cell membrane, F(t), is given by

(6)

where is the rate constant that regulates the duration of the fusion event. Although insulin cargo release follows the formation of the fusion pore with some delay (41), for simplicity, we lump the two events together by considering a unique rate constant. So F(t) is the number of granules releasing insulin in the unit time at time t in a single -cell. When granules have completed the fusion process and released their content in the extracellular space, the material that forms granule membrane is recycled to the trans-Golgi network. This flow of internalized material, delayed by a time _V, appears thus in Eq. 2.

Stimulation by Changes in Extracellular Glucose

The above model of granule dynamics must be complemented by equations that relate the glucose stimulus to the quantities that govern the machinery of granule trafficking. This has been done on the basis of the scheme in Fig. 2, modified from Fig. 1 in Ref. 21. The ATP-to-ADP ratio, and the other glucose-dependent factors that enhance granule supply from the reserve pool to the docked pool (21, 31, 30), are assumed to be globally represented by the phenomenological parameter that appears in Eqs. 3–4. [Ca²⁺]_i is represented by the rate constant in Eqs. 5–6. The fast oscillations in [Ca²⁺], related to changes in cell membrane potential, have been neglected in the present study. More comprehensive models of ATP and Ca²⁺ kinetics have been proposed (4, 38, 5).

View larger version (15K): [in this window] [in a new window]   Fig. 2. Schematic representation of the triggering and amplifying pathways that couple glucose stimulus to insulin secretion. Solid arrows, material fluxes; dotted arrows, control signals; +, stimulation; –, inhibition. The rate constants and are the main signals that regulate granule trafficking and insulin secretion. Modified from Ref. 21.

(7)

where is a rate constant, _b is the basal value at low glucose, and is an oscillatory forcing function that represents the events inducing [ATP] oscillations. Glucose activation is modeled by the function h, and the time delay _G is the time required by glucose metabolism (41, 48). Data on isolated -cells and cell clusters stimulated by glucose (26) show that cell response is maximal and almost constant for G >10–12 mmol/l and much lower below 6 mmol/l. So h has been chosen equal to zero for G smaller than a threshold G*, linearly increasing for G in the glucose concentration range (G*, ), and then constant. We have:

(8)

where is the maximal value of h and is reached at G = .

The increase in the ATP-to-ADP ratio leads to the closure of K_ATP channels and increase in [Ca²⁺]_i (see Fig. 2). So, the rate coefficient will depend on . In writing the governing equation for , 1) we account for the existence of a basal value _b, and 2) we assume linearity to model the action of [ATP] on [Ca²⁺]_i. We obtain the following equation:

(9)

where is a rate constant, and the function h, which embodies the action of ATP on Ca²⁺, is simply defined as

(10)

We remark that, for G below the threshold G* and = 0, the rate coefficients and stay at their basal values, and the ISR may be thought to be determined by the "constitutive pathway" of insulin secretion. When G exceeds the threshold, h is no longer equal to zero, and the ISR becomes determined by the "regulated pathway."

In summary, as shown in Fig. 2, extracellular glucose modulates the mitochondrial signal according to Eqs. 7–8, and regulates granule translocation and priming. In turn, affects , and thus the fusion of D_IR granules with cell membrane, according to Eqs. 9–10. So, in the present model, it is the parameter that has a major role both in the triggering pathway and in the amplifying pathway (43, 21). Equations 7–10 represent in a very rough way the coupling of glucose metabolism to the signals that activate insulin secretion. In particular, we neglected the dependence of production and degradation rates of ATP on [Ca²⁺]_i, as well as the kinetics of Ca²⁺ stores in the endoplasmic reticulum, which may cause oscillations in the concentrations with a phase shift between and (1, 15).

The -Cell Population

According to Eq. 6, the insulin amount released in the unit time by a single -cell in the extracellular space is given by I₀F, where I₀ is the insulin amount contained in a granule, if complete emptying of granules that undergo fusion is assumed. Actually, by targeting fluorescent probes to granule membrane or granule lumen (50), it has been found that membrane fusion is not always associated with insulin release. Thus only a fraction (15–40%) of granule insulin amount, with this fraction being possibly modulated by the stimulus intensity, may be released.

To compute the insulin amount released in the unit time by the whole -cell population (ISR), we must account for both the size of the population (total number of -cells) and the fraction of cells that actually release insulin in the given experimental situation. Denoting by N_c the average number of -cells per islet and by N_i the number of islets, the total number N of -cells may be written as N = N_c x N_i. In the experiments with isolated islets, if a single islet is considered, we have N_i = 1.

When the intact pancreas in vivo is considered, the number of active -cells is regulated by several factors. The -cell number is known to undergo time changes in relation to the balance among cell replication, cell death by apoptosis, and neogenesis by differentiation from ductal epithelium. Regulation of these processes appears to be altered in obese and type 2 diabetic subjects (9, 14). Estimates of the total number N of pancreatic -cells have been reported for mouse (1.06 x 10⁶) and rat (2.76 x 10⁶) (6). In studies on human pancreata (14), the number of islet equivalents (IEq, islet with average size of 150 µm diameter) per kilogram of body weight was found to be 6,640 IEq/kg (decreased to 2,641 IEq/kg in type 2 diabetes).

Another important factor is the heterogeneous responsiveness of the cell population to glucose (neglecting other secretagogues). The fraction of responsive islet cells and clusters increases with the extracellular glucose (26), and cell subpopulations with different intrinsic sensitivity to glucose are likely to exist. In addition, because glucose diffuses in the islet volume and is consumed by -cells, even in isolated islets, the cells will be exposed to different glucose concentrations both in the basal state and during changes in glucose concentration (3). In spite of the rich vascularization of pancreas, extracellular glucose concentration will decrease as the distance from vessels increases, and a wide range of glucose concentrations is likely to be found. Moreover, it is known that insulin itself, acting as a paracrine signal, sensitizes -cells to glucose stimuli and may thus participate in cell recruitment (47).

To keep the model as simple as possible, we have adopted a rough description of -cell recruitment by glucose stimulus, by assuming that only a fraction, f, of the total cell population responds to glucose. This fraction will increase as the glucose concentration G increases, with a time delay for recruitment equal to _G, so we have f(t) = f[G(t – _G)]. Under the further assumption that -cells respond to stimulation in a synchronous manner, which may be realistic on a not-too-fast time scale thanks to cell-to-cell electrical and chemical interactions (38), we may finally write for the insulin secretion rate of the whole cell population at time t the following expression:

(11)

Given the time course of glucose concentration G, Eqs. 1–11 predict the time evolution of the control rate coefficients and , the evolution of the quantities that describe granule dynamics (I, V, R, D, D_IR, F), and the evolution of the recruited cell fraction f and hence of the ISR. The following parameters may be considered constant, independent of the degree of stimulation of the -cell: k, _I, _V,_V, C_T, k₁⁺, k₁^–, , _G, ,_b, G*, , , , _b, k, and I₀ (parameters of granule dynamics and signaling model); N_c and N_i (parameters of the -cell population). By contrast, other quantities in the model, namely b_I, b_V, , and f, may be affected by changes in the concentration of glucose (and other secretagogues) or as a consequence of changes in the experimental conditions. An equation for f as a function of G is given in APPENDIX A, and b_V will be taken time and glucose independent.

Steady-state Conditions

To represent the basal condition (glucose concentration G_b), or a situation in which glucose concentration is kept constant for a long time at a basal or suprabasal value, we may consider the steady-state solution of the model in which b_I, , , and f take values corresponding to the assigned value of G, = 0, and the pools I, V, R, D, D_IR, and F are constant. These constant values are obtained by setting to zero the time derivatives in model equations. We observe that, in the steady state, we have the following chain of equalities:

(12)

Taking Eq. 12 into account, from Eqs. 1–3 we see that I, V, and R are given in the steady state by the following expressions:

(13)

Thus we have

(14)

Using Eqs. 4–6, D, D_IR, and F can be conveniently expressed in terms of the steady-state value of R:

(15)

To guarantee the positivity of D in the steady state, the constraint C_T – R > 0 has to be satisfied.

For the ISR in the steady state, Eq. 14 gives

showing that, as intuitive, the rate of insulin secretion in the long time is essentially determined by the proinsulin biosynthesis rate, b_I. Because b_I and f depend on G, the above equation is the "dose-response relationship" predicted by the model when glucose levels (and all other quantities) are kept constant for a long time. As we will see in the following, this relationship may not reflect the ISR data measured from an early part of the second phase of insulin secretion.

Assessment of Model Parameter Values

For some of the parameters of the model of Eqs. 1–11, we may use experimental measurements or estimates, e.g., I₀ = 1.6 amol in rat granules (41) and N_c = 1,000–2,000 cells for an islet (2). The granule fusion event plus the insulin cargo release has been reported to require a time of the order of a few seconds (41), so we may take an estimate for in the range 6–30 min^–1. Moreover, C_T = 500 was reported as an upper estimate for the number of Ca²⁺ channels (2). We will take _G = 1 min (2).

Other indications about the values of model parameters can be obtained in view of the available experimental data concerning the basal state in isolated islets. At the glucose concentration of 3 mmol/l, only 1–2 granules·min^–1·cell^–1 are released in the mouse or rat (48). We will assume that, at the basal state, F (number of granules released per minute by a -cell) and the other quantities equal to F that appear in Eq. 12 are equal to 1. With this choice, assuming N_c = 10³ active cells/islet and f = 1 (that is, 100% of islet cells release insulin), the ISR of a single islet predicted by Eq. 11 at the basal state is equal to 1.6 x 10^–3 pmol/min (or 9.28 pg/min), which is in the range of the values experimentally observed (22, 44, 47).

To determine the baseline values of model parameters, we proceed by taking F = 1. The pool of immediately releasable granules, D_IR, is reported to contain 50 granules in the basal state (2, 7, 48). By taking D_IR = 50, from Eq. 12 we estimate that the value of in the basal state, denoted as _b, is _b = 0.02 min^–1. A plausible estimate is that the reserve pool, R, contains 10,000 granules, and that 1,000 granules are docked with the plasma membrane: thus we take D + D_IR =1,000. The chosen R value leads to the estimate for in the basal state (denoted by _b) as _b =10^–4 min^–1. To proceed further, we take as plausible the value of 10 for both I and V at the basal state (it may be seen that this assumption does not affect the general pattern of the first and second phase of insulin secretion). So kIV = 1 (from Eq. 12) implies k = 10^–2 min^–1. Also, from Eq. 13 we have b_Ikb_V/(kb_V + _IV) = 1. The quantity kb_V/(kb_V + _IV) ranges from 0 to 1 and increases with b_V, so we may look at it as a factor that takes into account the variability in the vesicle membrane production rate related, for instance, to variability in lipid metabolism within the -cell. Taking for this factor the value 0.25 at the basal state, we deduce for the basal b_I, denoted as b_I,b, the estimate b_I,b = 4 min^–1 and, using Eq. 1 written in the basal state, we also obtain _I = 0.3 min^–1. Let us now assume that the degradation rate constant for V is larger than the degradation rate constant for I, e.g., _V = 2_I, and use a similar reasoning as above. This leads to the following estimates: _V = 0.6 min^–1 and b_V,b = 6 min^–1.

Concerning _V, k₁⁺, k₁^–, and C_T, we have tentatively assumed, on the basis of simulation results, the following baseline values: _V = 5 min, k₁⁺ = 1.447 x 10^–5 min^–1, k₁^– = 0.10375 min^–1, C_T = 500. With these parameter values, the (basal) steady-state values of model variables are thus as follows: I = 10, V = 10, R = 10,000, D = 950, D_IR = 50, F = 3.33 x 10^–2, and ISR = 9.28 pg·min^–1·islet^–1 (f = 1). The values of the other parameters were chosen as discussed in APPENDIX A.

	RESULTS

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Stimulation by High K⁺

Data of islet response to stimulation by high extracellular [K⁺] in low glucose and in the presence of diazoxide have been given by various authors (see Ref. 21). Henquin et al. (22) report time courses in mouse islets of [Ca²⁺]_i and of the insulin secretion rate in response to a step increase in extracellular K⁺ and to a sequence of impulsive K⁺ increments with pulses lasting 6 min and interpulse intervals of 6 min. The continuous increase in K⁺ caused a sustained [Ca²⁺]_i rise, whereas the insulin secretion rate exhibited a large initial peak followed by a slow return to the initial value. The impulsive increments in K⁺ caused an intermittent rise of [Ca²⁺]_i and a sequence of decreasing peaks of the insulin secretion rate. The pattern of the response, but not its basal value, was influenced by the glucose level (0 or 3 mmol/l).

This type of stimulation is modeled here by assuming that the time course of reflects the time course of [Ca²⁺]_i reported in Ref. 22, whereas retains its basal value _b. Thus, starting from the basal value _b, the time course of in the case of continuous stimulation has been obtained as the response of a linear first-order system with rate constant to a step function (the step increase in K⁺). A component increasing linearly with slope s has been added to mimic more closely the measured [Ca²⁺]_i in the time interval 0–50 min. So, if the step increase in K⁺ occurs at t = t₁, we have (t) = _b for t < t₁ and

(16)

see the plot in Fig. 3A. The values of and s were chosen on the basis of simulation results, for a K⁺ increase from 4.8 to 30 mmol/l (22).

View larger version (20K): [in this window] [in a new window]   Fig. 3. Simulated islet response to K+ stimulation. A (continuous stimulation) and C (intermittent stimulation): time course of (dotted line), D (dashed-dotted line), and DIR (solid line). B and D: insulin secretion rate (ISR) time course with baseline parameters (solid line) and alternative parameters (dotted line). Baseline parameters: k = 10–2 min–1, bI,b = 4 min–1, I = 0.3 min–1, V = 0.6 min–1, bV,b = 6 min–1, b = 10–4 min–1, b = 0.02 min–1, = 1 min–1, s = 4 x 10–3 min–2, = 1 min–1, V = 5 min, k = 1.447 x 10–5 min–1, k= 0.10375 min–1, CT = 500, = 30 min–1, I0 = 1.6 amol, Nc = 1,000, and f = 1. Alternative values: k= 5.788 x 10–5 min–1, k= 0.255 min–1, CT = 300, and other parameters as in baseline.

For the intermittent stimulation, Fig. 3C shows the time course of (a sequence of 4 pulses lasting 6 min with values of , , and s being the same as for the continuous stimulation). The predicted response is illustrated in Fig. 3, C and D. Note in Fig. 3C the refilling of pool D_IR during the interpulse interval. When C_T, k, and k were set to alternative values that produce the same amount of granules in the various pools at the steady state, the pattern of the response was qualitatively similar. The ISR is reported in Fig. 3, B and D, by the dotted line (83 granules released in the first 5 min). The simulated ISR of the islet in Fig. 3D reproduces the declining amplitude of the peaks experimentally observed at zero glucose (22). The data show a slight broadening of the peaks, possibly related to the incomplete synchrony of -cells in an islet.

Stimulation by Extracellular Glucose

As seen in the previous section, insulin secretion induced by K⁺ rise can be simulated in the model by an increase in the rate coefficient and shows a first-phase response only. In that case, the ISR pattern is monophasic, and, after the peak, the secretion rate declines toward the basal value. The secretory response to a step increase in extracellular glucose is, instead, biphasic. The second phase of glucose-stimulated insulin secretion had been initially considered to be sustained by proinsulin biosynthesis. Subsequently, this interpretation was refuted (18, 42). Confirmed by model simulations (data not shown), an increase in the rate of proinsulin biosynthesis b_I, concomitant with the increase in , does not produce a secondary response if is kept at _b. As previously discussed, a critical role is played by the increase in the rate coefficient modulated by extracellular glucose, because intervenes in both the triggering and the amplifying pathway (21, 43).

To compute the model response to glucose stimuli, the functions h(G) and f(G) have to be specified. To this aim, we rely on data of insulin secretion by the perfused rat pancreas, reported by Grodsky (20). In particular, we used the data of insulin amount released during the first phase of the response to different values of a step change in glucose concentration (IS, see Fig. 4). An approximate expression of the first-phase ISR and of the insulin amount secreted, as predicted by the present model, is derived in APPENDIX A. APPENDIX A also gives the form chosen for the function f (Eq. A5) and describes the procedure followed to estimate the parameters of h(G) and f(G) from Grodsky's data. Figure 4 shows the IS data (20) with the fitting curve of Eq. A4 (parameter values reported in APPENDIX A). The number of -cells used in the simulations in this and the following section is equal to the estimate reported for the rat.

View larger version (9K): [in this window] [in a new window]   Fig. 4. Insulin amount secreted in the first-phase response to different values of a step glucose stimulation (IS). Filled squares, data (perfused rat pancreas, means ± SE) replotted from Ref. 20; continuous line, model fitting (parameter values reported in APPENDIX A).

View larger version (16K): [in this window] [in a new window]   Fig. 5. Pancreatic response to step increase in glucose concentration. A: time course of G (from 1 to 16.7 mmol/l, dashed line), D (dotted line), and DIR (solid line). B: ISR time course for a step in G to 300 mg/dl (solid line), 500 mg/dl (dotted line), and 150 mg/dl (dashed line). Dashed-dotted line: time course of the ISR when the G step does not induce granule translocation and priming. C: ISR time course when glucose stimulation (16.7 mmol/l) is interrupted in a rest period from 60 to 65 min. D: ISR response to glucose rise as a fast ramp (slope 50 mg·dl–1·min–1, solid line) and a slow ramp (slope 5 mg·dl–1·min–1, dotted line). Baseline parameters (rat pancreas): = = 4 min–1, G = 1 min, G* = 4.58 mmol/l, = 10 mmol/l, = 3.93 x 10–3 min–1, k = 350, fb = 0.05, Kf = 3.43 mmol/l, N = 2.76 x 106. Other parameters are same as the baseline parameters of Fig. 3.

View larger version (11K): [in this window] [in a new window]   Fig. 6. Glucose stimulation. A: response to a step in G from 1 to 16.7 mmol/l, given at t = 10 min and lasting 12 h, bI kept at bI,b (ISR, solid line; R, dashed line), and bI increased to 8 x bI,b (ISR, dotted line; R, dashed-dotted line). B: response to a G step from 1 to 16.7 mmol/l, given at t = 1 h in the presence of the endogenous oscillation (t) in . ISR, solid line; , dotted line. After t= 4 h, is set equal to 2 min–1 to simulate extracellular K+ rise. Other parameters are as in A (bI increased to 8 x bI,b).

Figure 6B shows the effect on and ISR of a sinusoidal oscillation possibly arising in the glycolytic pathway, represented by the function in Eq. 7. Because the oscillations of -cell signaling molecules appear to be enhanced by extracellular glucose (19), we tentatively assumed the amplitude of equal to (1/2)h(G) (G changed stepwise from 1 to 16.7 mmol/l). The oscillation period was set to 10 min. When G exceeds the threshold G*, the oscillation becomes clearly visible in both and ISR. Depending on the amplitude, frequency, and site (oscillations may also be present in the fraction f of active -cells), the oscillatory component may represent a substantial or a major part of the ISR (40). In Fig. 6B, we also simulate the effect of the experimental manipulation consisting in the increase of extracellular K⁺, in the presence of diazoxide, when glucose is at a stimulatory concentration (19). As in the preceding section, K⁺ rise is modeled by an increase in (we set = 2 at t = 4 h). When is kept at a high value, the second-phase response is preserved since remains at a high level and oscillates, but the oscillations in the ISR disappear. This behavior is consistent with the observations (19).

Simulation of Derangements of Physiological Mechanisms and Effect of Drugs

As seen in APPENDIX A, the product k, but not the values of the two factors, was estimated from Grodsky's data. Figure 7A shows that, when is increased and k decreased with k kept constant, the first-phase response to a step increase in G is unchanged, whereas the second phase has a larger amplitude and a more prompt decay. Pool R is more completely depleted as increases. This behavior agrees with the analysis in APPENDIX B and may be seen as the result of an increase in the sensitivity of second-phase ISR with respect to glucose. The pattern of the response at large , where the second phase may largely exceed the peak of the first phase, is more reminiscent of the ISR observed in the rat than of that observed in the mouse (22). For a given G step, a larger causes a larger rise in , and this agrees with the idea that the amplifying action may be stronger in the rat than in the mouse (22). Figure 7A also shows the ISR obtained with the alternative values of C_T, k₁⁺, and k₁^–, given in the legend of Fig. 3. Because the product k₁⁺C_T is larger with the alternative parameters than with baseline parameters, the second-phase ISR has a faster increase, as predicted by the initial value of the derivative (see APPENDIX B).

View larger version (16K): [in this window] [in a new window]   Fig. 7. ISR responses after glucose step from 1 to 16.7 mmol/l (bI increased to 8 x bI,b). A: changed with k kept constant and = 0. Solid line, baseline parameters; dotted line, x 5; dashed line, /5; dashed-dotted line, baseline parameters with alternative values of CT, k1+, and k1–. B: solid line, ISR response to the step in G given at 1.5 h and (t) as in Fig. 6B, with /2 min–1, k/5 min–1 and = 3 min–1. After t = 4 h, to simulate the effect of a drug that blocks ATP-dependent K+ (KATP) channels, k is exponentially increased from of the baseline value to the baseline value (time constant of the exponential, 10 min). Dotted-line plot, ISR response to the step in G given at 1 h, with bV = 1.5 bV,b and the other parameters as in Fig. 6B.

In the solid-line plot of Fig. 7B, we also simulate, within the extremely simplified representation of the stimulus-secretion coupling adopted in the present model, the action of pharmacological agents that bind to sulfonylurea receptors (27, 24) and cause closure of K_ATP channels, leading to increased conductance of voltage-sensitive Ca²⁺ channels and increased Ca²⁺ influx. An increase in the model parameter k translates into an increase in , that is, in the [Ca²⁺]_i. Thus, the drug action may be simulated by an increase in k. The solid-line plot of Fig. 7B shows, up to t = 4 h, the response to a glucose step of a -cell with impaired stimulus-secretion coupling. At t = 4 h, to simulate the drug effect, k is exponentially increased to the baseline value. This change causes a rise in and thus a transient increase in the ISR, as it would occur with a rise in extracellular K⁺. The amplitude of the oscillations is restored.

In general, the effect of antidiabetic drugs should guarantee that the signal is large enough to allow exocytosis of the docked granules that are fed by the reserve pool at rate R, as remarked in Ref. 21. If the quantity C_T – R in Eq. 15 for D becomes small (and positive), the pool D may in fact become exceedingly large.

C-peptide Secretion During Hyperglycemic Clamp in Humans

As a preliminary validation of the present model, we simulated the C-peptide response to a hyperglycemic clamp in healthy humans. Previously obtained data of glucose, insulin, and C-peptide were used (32). The C-peptide secretion rate was computed according to the model of Eqs. 1–11, and the whole body C-peptide kinetics were represented by the usual two-compartment model (17, 39) with the standard kinetic parameters computed for each subject (51).

To determine the time course of glucose concentration that actually stimulates the in vivo pancreatic response, we estimated the time course of arterial glucose concentration, G_a, as shown in APPENDIX C. In the insulin secretion model, the number N of -cells in human pancreas was set according to the estimate in Ref. 14. A parameter identification was not attempted; instead, a few parameters were modified with respect to the baseline values previously reported. In particular, we used the alternative values for k₁⁺, k₁^–, and C_T, and we set = 1.38 x 10^–3 min^–1, a value smaller than that used in the simulations of rat pancreatic response. Other parameters were chosen in the following ranges: _b = 0.01–0.02 min^–1, k was two to four times the value in the rat, and b_V = 6–8 min^–1. We found that C-peptide data were adequately approximated by assuming f constant at a low value (f = 0.06–0.07). This may be related to a very slow recruitment of the -cells during the course of the experiment (fasting subjects) or to inhibitory signals acting on the islets in vivo. To obtain the observed value of C-peptide preceding the clamp, we added to the ISR a "constitutive" insulin secretion rate given by I₀FfN with F = 1 and f = 0.05.

Figure 8 shows, for one of the subjects, the experimental data of glucose and C-peptide together with the time course of G_a, reconstructed according to Eq. C1. Figure 8B. reports the time courses of the ISR and of the C-peptide concentration as predicted by the model when the glucose stimulus in Eq. 7 is G_a (solid lines) or G_m (dotted lines). The initial more rapid increase of G_a with respect to G_m may be a factor that accounts for the occurrence of an appreciable first-phase ISR and the observed rapid increase of the C-peptide concentration. No significant differences in the C-peptide concentration calculated by the model with G_a or G_m were observed, likely because of the long time constant of plasma C-peptide kinetics (51).

View larger version (11K): [in this window] [in a new window]   Fig. 8. C-peptide response to hyperglycemic clamp in human subjects (data from Ref. 32). A: measured glucose concentration (triangles) with the spline approximation (dotted line) and time course of reconstructed arterial glucose concentration (solid line). B: measured C-peptide (CP) concentration (filled squares) with the time course predicted by the model and the predicted ISR (solid lines). The dotted-line plots represent the model response when the glucose stimulus is the measured glucose concentration, Gm.

Interpretation of -Cell Responsivity Indexes

Using the approximate expressions of the first- and second-phase ISR given in APPENDIXES A and B, we may derive the expressions for the dynamic and the static responsivity index of the -cell, denoted as _d and _s in Refs 8, 49, and 10, that are predicted by the present model for a step increase in glucose concentration.

According to Ref. 10, we define a "dynamic responsivity index," S_d, as the ratio

where IS is the insulin amount secreted during the first-phase response, given by Eq. A4, and V_C is the volume of the C-peptide-accessible compartment. Note that G* has a meaning similar to the threshold parameter h in Ref. 10. The index S_d is dimensionless, like the index _d. As seen from Eq. A4, the index S_d depends on G, so we consider the case of a small G step (a different expression may be derived for large G). By approximating exp(–aT) in Eq. A3 as 1 – aT, we get Z(G) = D_IR,bT. Furthermore, neglecting _b in , we obtain from Eq. A4 the equation for S_d:

(17)

The above expression shows that S_d essentially depends on the size of the pool of immediately releasable granules and on the product of the [ATP] sensitivity to glucose, /( – G*), times the [Ca²⁺]_i sensitivity to [ATP], k. These parameters are defined in Eqs. 8 and 10.

The "static responsivity index," in the spirit of the index _s in Ref. 10, takes into account the dynamics of the second-phase ISR. Thus we have obtained a dynamic index by computing the derivative at t = 0 of the second-phase ISR and dividing by the rate constant, k C_T, of ISR increase. From Eqs. B3 and B4, we get for this ratio the quantity I₀R_bfN. Proceeding as for the dynamic index, the static responsivity index S_s (with dimension of time^–1) has the expression

(18)

that depends on the [ATP] sensitivity to glucose and the size of the reserve pool. As noted in APPENDIX B, Eq. 18 is obtained under the assumption of large G (free [Ca²⁺]_i at its maximum level).

With the values of model parameters used in the present study to simulate the response to the hyperglycemic clamp in a normal human subject, we obtain S_d 670 x 10^–9 and S_s 26 x 10^–9 min^–1. These values are in the range of the _d and _s values estimated in Refs. 8, 49, and 10 under different experimental conditions and by means of a different model.

	DISCUSSION

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In this study, a model of insulin granule trafficking in pancreatic -cells is proposed. Essentially, the model describes the formation of insulin granules and the dynamics of four intracellular granule pools: the reserve pool, the pool of docked granules, the pool of immediately releasable granules, and the pool of granules fused with cell membrane. Granule translocation from the trans-Golgi network to the cell membrane and exocytosis of granules fused with the cell membrane are regulated by two rate constants, and , assumed to be related to the ATP-to-ADP ratio and other mitochondrial signals, and to the free [Ca²⁺]_i, respectively. As anticipated by Grodsky (20), the "labile" insulin may be grossly identified with the insulin contained in the D_IR pool. The "stable" insulin is the R pool, with the D pool, that feeds the D_IR pool as insulin is secreted, being in an intermediate position. The model links in a quantitative, although simplistic, way the dynamics of granule release by the single -cell, with the recruitment of a subpopulation of glucose-responsive cells within the whole -cell population.

The different roles of the triggering pathway and of the amplifying pathway in the glucose-stimulated insulin secretion are modeled according to Ref. 21. The rate coefficient , which represents events leading from glucose influx into the cell to ATP production, has an essential role in promoting the granule externalization and priming that supports the second-phase ISR. This rate coefficient, in particular the parameters and in its governing equation, is a rate-limiting step in the glucose-stimulated insulin secretion. The representative parameters of the part of the triggering pathway from the ATP-to-ADP ratio to [Ca²⁺]_i are and k, which thus represent another rate-limiting step in the insulin secretion (48). We stress that the model does not include the regulatory role of Ca²⁺ in granule translocation and thus in the second-phase insulin release (16, 25), nor Ca²⁺ implication in ATP metabolism (1, 15). Moreover, the events on the fast time scale of the rapid fluctuations in -cell membrane potential (38, 46) are not accounted for.

The pattern of the pancreatic response to glucose is shown in the present study by the expression of the ISR in the first and the second phase (see APPENDIXES A and B), by the simulation tests, and by the responsivity indexes of Eqs. 17–18. The first phase mainly depends on the insulin amount in the D_IR pool, and on the product of [ATP] sensitivity to glucose, /( – G*), times the [Ca²⁺]_i sensitivity to [ATP], k. Accordingly, the dynamic responsivity index S_d accounts for the fusion of D_IR granules (release of labile insulin), essentially regulated by the increase in [Ca²⁺]_i. Notably, agents that produce membrane depolarization, e.g., through closure of K_ATP channels, or potentiate the triggering pathway at some point, such as gastric inhibitory peptide (GIP) and glucagon-like peptide (GLP)-1, may result in an increased S_d. This action is, however, transient if it is not accompanied by adequate increase in the proinsulin synthesis rate and provision of granules competent for release. In contrast, the static responsivity index S_s mainly accounts for granule provision, which is regulated by the size of the reserve pool and the increase in the ATP-to-ADP ratio. The rate constant is a synthetic descriptor of the signals that regulate these events, and its increase, together with the increase in b_I, is able to produce a durable release of insulin without degranulation of the -cell.

The secretory response is also determined by the number, fN, of responsive -cells. Given the total number N of -cells in pancreas, possibly subject to slow but considerable changes in pathological conditions (14), the fraction f of recruited cells will change with the time in response to a variety of stimuli (glucose, insulin, GIP, GLP-1, etc.). The complex phenomena and interactions among the cells distributed in pancreas volume [for instance, the diffusion and consumption of glucose in the islets with consequent heterogeneity of glucose concentration (3) and electrical and chemical interactions (38)] are not truly represented in the present model. These phenomena are described by a dependence of the fraction f of active cells on G, at most with a time delay. It may be speculated that cell recruitment has a slower dynamics, also related to the concentration of insulin and other substances in the interstitial fluid that surrounds the -cells. In vivo, because of inhibitory signals, the maximal value attained by f is likely to be substantially smaller than the unity, and was estimated to be of the order of 6–7%.

Although the present model does not allow to discriminate among the various pathways involved in the pathophysiology of type 2 diabetes, it might provide some insight into the coupling between energy changes and insulin secretion. Therefore, should the mitochondrial derangement described in type 2 diabetes for skeletal muscle (29) occur also in -cell, then the parameters regulating the kinetics of the signals and would be altered with a subsequent reduction in insulin secretion. The problem also arises of understanding how the various events involved in the trafficking of intracellular granules are coordinated and how the physiological amount of granules in the various pools is maintained over time. There may be control mechanisms devoted to "sensing" the size of granule pools to avoid granule depletion or excessive crowding and devoted to check the status of the network of microtubules that vehicle granule translocation.

The model proposed allows a good fitting of experimental data obtained during a hyperglycemic clamp, suggesting that it may be applied to in vivo experimental conditions where glucose is stably increased over time. However, the experimental pattern of the ISR and the value of the responsivity indexes, determined for instance in an oral glucose tolerance test (8, 10), will also be determined by factors other than glucose, such as the signals arising in the enteroinsular axis. The present model can represent the effect of these signals through changes in the equations for the control variables and . For instance, the incretin effect may be translated into an increase in k. The k increase may also be related to an increase in the Ca²⁺ efficacy on exocytosis.

A similar consideration is also true for the effect of pharmacological agents on insulin secretion. The mechanism of action of different drugs that stimulate insulin secretion can be modeled, at least partially, by taking into account the drug effect at different levels, for instance on the K_ATP channels or on the metabolic energy level as it happens for acetylcarnitine. According to the observations in Ref. 21, it is important that drugs stimulating insulin secretion act not only at the level of the amplifying pathway but also of the triggering pathway; otherwise, the final effect would be that of an imbalance in the equilibrium between these pathways. Therefore, tailor-made drugs in diabetes should be designed to obtain the maximum effect.

A full description of these events cannot be given by the present model because the representation of the signaling pathways involved in the insulin secretion is still oversimplified. For instance, the role of long-chain acyl-CoAs and protein kinases (43, 53) is not taken into account. A more detailed description of the kinetics of signaling molecules should improve the capability of the model to investigate the events that lead to -cell dysfunction and predict the effects of agents stimulating insulin secretion.

	APPENDIX A: ANALYSIS OF FIRST-PHASE RESPONSE

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We derive here an approximate expression for the insulin amount released in the first-phase response following a step in glucose concentration. The parameters appearing in this expression will be estimated from data reported by Grodsky (20) (see Fig. 4). We assume, for the sake of simplicity, that the dynamics of and are much faster than granule dynamics (large and values). Although the measured [ATP] and free [Ca²⁺] in -cell cytosol after a glucose step show rise times of a few minutes (1), glucose appears to provoke the formation of subplasma membrane ATP microdomains, where ATP rapidly responds to a glucose increase with a half-time = 20 s (28). The state of nearby K_ATP and L-type Ca²⁺ channels is likely to rapidly change, leading to fast Ca²⁺ influx at sites close to the granules of the D_IR pool. The transient increase in [Ca²⁺]_i at these sites, and thus in , may be completed in 1 min. We will take as baseline values = = 4 min^–1, so the transient of and will require 1 min. Simulations of a recently proposed model for mitochondrial ATP production (5) show that mitochondrial ATP and [Ca²⁺] rapidly respond to a step increase in fructose bisphosphate, with a transient that is completed in <1 min.

Following the step change in glucose concentration to a suprabasal value G, and after the time delay _G, and will rapidly attain their steady-state values, denoted as and , respectively. From Eqs. 7 [with (t) = 0] and 9, we have = _b + h(G) and = _b + kh(G). Let us take the time origin so that (t) = and (t) = for t > 0. Assuming that pool D does not change appreciably during the first-phase response, we solve Eq. 5 with D(t) held constant at its initial (basal) value D_b, equal to , and initial condition D_IR(0) = D_IR,b. The pool D_IR will be depleted according to

(A1)

where a = k₁⁺ D_b +k₁^– + , with being a function of G, and b = k₁⁺D_bC_T. Then we solve Eq. 6 for F, with initial condition at the basal value F_b, = , and D_IR(t) given by Eq. A1, obtaining

(A2)

To compare the prediction of the present model with the data of insulin amount released by the perfused rat pancreas during the first-phase response, we must compute the integral of the ISR over a time interval, say of length T, that just contains the first ISR peak. Recalling Eq. 11, we compute the total number of insulin granules released by a single -cell in the interval T, that is, the integral of F(t) from t = 0 to t = T. Assuming a << and F_b small, this integral, denoted as Z(G), is given by

(A3)

Thus the insulin amount, IS(G), released in the first-phase response to a glucose step G can be written as

(A4)

where N here is the number of -cells in rat pancreas. The function f was chosen of the form

(A5)

where f_b is the fraction of responding cells at glucose concentration smaller than the threshold G* (for simplicity assumed equal to that in Eq. 8), and K_f is a constant related to the effectiveness of recruitment. According to Eq. A5, 100% of -cells may be recruited for a large enough G.

Eq. A4 was used to fit Grodsky's data. Taking the baseline values of model parameters, we have still to estimate the quantities G*, , the product k, f_b, and K_f. To simplify the estimation problem, we set = 10 mmol/l according to data in Ref. 26 and f_b = 0.05. Least-squares fitting, used to estimate the remaining parameters, provided the following values: G* = 4.58 mmol/l, k = 1.38 min^–1, and K_f = 3.43 mmol/l (see Fig. 4). By model simulation we chose, as baseline parameters, = 3.93 x 10^–3 min^–1 and k = 350. Finally, we note that a simple expression for the first-phase ISR can no longer be obtained if the dynamics of and are not much faster than granule dynamics. Model simulations show that the peak ISR becomes smaller and is delayed in time when and decrease, but the peak values in Fig. 5B can be recovered by increasing the product k. If, for instance, and are decreased from the baseline value of 4 min^–1 to = = 1 min^–1, the baseline value of k must be multiplied by 2.5.

	APPENDIX B: ANALYSIS OF SECOND-PHASE RESPONSE

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By a similar analysis as in APPENDIX A, an approximate expression for the second-phase ISR can be obtained as follows. Let us assume that, during this phase, the pool R decays exponentially from the basal value R_b to a final value , where is expressed as in Eq. 13 with the final values of and b_I, and ₁, respectively, and the other parameters unchanged. The rate constant of the exponential is , so we have (initial time t = 0)

(B1)

Using the quasistationary approximation _IR = 0 and summing the equations for D and D_IR, we may write

(B2)

For >> k D + k(valid when G is large and the concentration of free Ca²⁺ is at its maximum level), the differential equation in B2 may be written as a linear differential equation for D that can be solved using Eq. B1 and with D(0) = D_b, obtaining

(B3)

By the further quasistationary approximation = 0, we finally get

(B4)

that may be used with Eq. B3 to compute the approximate expression of the second-phase ISR. The derivative of F(t) at t= 0 is given by k C_T (R_b – k C_TD_b).

If k C_T = 7.24 x 10^–3 min^–1, and _b + = 3.93 x 10^–3 min^–1 (baseline parameters), the descending front of second-phase ISR has a large time constant (1/ > 4 h). The time constant becomes smaller if is larger, as shown in Fig. 7A. With the baseline parameters, the negative term in the above derivative may be neglected for G >10 mmol/l.

	APPENDIX C: ESTIMATION OF ARTERIAL GLUCOSE CONCENTRATION

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An estimate of arterial glucose concentration was obtained by using the measured plasma glucose concentration, G_m, and by computing the net tissue glucose uptake from the insulin sensitivity index and the plasma insulin concentration. Neglecting non-insulin-dependent uptake, the following equation can be written:

(C1)

where V_G is the glucose distribution volume, Q is the cardiac output, S_I is the insulin sensitivity, and I_p and I_p,b denote the current and the basal insulin concentration in plasma. The time course of G_a was thus obtained by computing the derivative _m from a smooth approximation of G_m. The parameters were assessed for each subject as follows: V_G was obtained by using the estimate of 0.16 l/kg body wt (23), and cardiac output was given by Q = 0.2 x kg body wt^0.75 (Brody's formula); S_I was available from Ref. 32; I_p,b was computed as the mean I_p value in the last 30 min preceding the clamp. An improved estimate of G_a might be obtained by means of a more complex circulatory model (34).

	FOOTNOTES

 

Address for reprint requests and other correspondence: G. Mingrone, Istituto di Medicina Interna, Università Cattolica del Sacro Cuore, Largo A. Gemelli, 8, 00168 Rome, Italy (e-mail: gmingrone{at}rm.unicatt.it)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

	REFERENCES

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