# Beneficial effects of intercellular interactions between pancreatic islet
cells in blood glucose regulation

Junghyo Jo
Laboratory of Biological Modeling,
National Institute of Diabetes and Digestive and Kidney Diseases,
National Institutes of Health, Bethesda, MD 20892, U.S.A.
Moo Young Choi
Department of Physics and Astronomy and Center for Theoretical Physics,
Seoul National University, Seoul 151-747, Korea
Corresponding author. Address: Department of Physics and Astronomy, Seoul
National University, Seoul 151-747, Korea. E-mail<EMAIL_ADDRESS>Duk-Su
Koh
Department of Physiology and Biophysics,
University of Washington, Seattle, WA 98195, U.S.A

###### Abstract

Glucose homeostasis is controlled by the islets of Langerhans which are
equipped with $\alpha$-cells increasing the blood glucose level, $\beta$-cells
decreasing it, and $\delta$-cells the precise role of which still needs
identifying. Although intercellular communications between these endocrine
cells have recently been observed, their roles in glucose homeostasis have not
been clearly understood. In this study, we construct a mathematical model for
an islet consisting of two-state $\alpha$-, $\beta$-, and $\delta$-cells, and
analyze effects of known chemical interactions between them with emphasis on
the combined effects of those interactions. In particular, such features as
paracrine signals of neighboring cells and cell-to-cell variations in response
to external glucose concentrations as well as glucose dynamics, depending on
insulin and glucagon hormone, are considered explicitly. Our model predicts
three possible benefits of the cell-to-cell interactions: First, the
asymmetric interaction between $\alpha$\- and $\beta$-cells contributes to the
dynamic stability while the perturbed glucose level recovers to the normal
level. Second, the inhibitory interactions of $\delta$-cells for glucagon and
insulin secretion prevent the wasteful co-secretion of them at the normal
glucose level. Finally, the glucose dose-responses of insulin secretion is
modified to become more pronounced at high glucose levels due to the
inhibition by $\delta$-cells. It is thus concluded that the intercellular
communications in islets of Langerhans should contribute to the effective
control of glucose homeostasis.

Key words: glucose homeostasis, islets of Langerhans, feedback, diabetes

##  1 Introduction

Homeostasis, maintenance of the constant physiological state, is one of the
main characteristics of life. In particular, glucose homeostasis is critical
because glucose is the energy source for our bodies; the malfunctioning of
this process causes several disease states including diabetes mellitus and
brain coma.

In order to understand glucose homeostasis, we first need to examine the
tissue controlling the blood glucose level, the islet of Langerhans in the
pancreas. It consists mainly of three types of endocrine cells: $\alpha$-cells
which secrete glucagon hormone increasing the glucose level, $\beta$-cells
which secrete insulin decreasing the glucose level, and $\delta$-cells which
secrete somatostatin, known to inhibit activities of $\alpha$\- and
$\beta$-cells. The hormone secretion of a cell influences the behavior of
neighboring cells, and is thus tightly correlated with the islet structure
[Hopcroft et al., 1985, Pipeleers et al., 1982]. In rodents, an islet contains
about 1,000 endocrine cells on average: $\beta$-cells, occupying the most
volume (70 to 80%) of an islet, populate largely in its core, whereas
non-$\beta$-cells are located on the mantle [Brissova et al., 2005].

To the first approximation, $\alpha$\- and $\beta$-cells should be sufficient
for glucose control because $\alpha$-cells can increase the glucose level
whereas $\beta$-cells can decrease the level. The importance of this bi-
hormonal mechanism for glucose homeostasis has been well recognized
[Cherrington et al., 1976]. However, it should be noted that endocrine cells
in the islet interact with each other rather than act independently. For
example, the electrical coupling between $\beta$-cells through gap-junctions
is known to enhance insulin secretion of coupled $\beta$-cells [Jo et al.,
2005, Pipeleers et al., 1982, Sherman et al., 1988]. In addition, it has been
recently reported that chemical interactions between neighboring cells through
hormones [Cherrington et al., 1976, Franklin et al., 2005, Orci & Unger, 1975,
Ravier & Rutter, 2005, Samols et al., 1965, Samols & Harrison, 1976, Soria et
al., 2000] and neurotransmitters [Brice et al., 2002, Franklin & Wollheim,
2004, Gilon et al., 1991, Moriyama & Hayashi, 2003, Rorsman et al., 1989,
Wendt et al., 2004], termed “paracrine interaction,” affect glucose
regulation.

Among these intercellular communications, enhancement of insulin secretion by
glucagon [Brereton et al., 2007, Samols et al., 1965, Soria et al., 2000]
seems to be paradoxical because $\alpha$-cells, playing the reciprocal role to
$\beta$-cells in glucose regulation, promote the activity of $\beta$-cells. In
contrast, insulin, secreted by $\beta$-cells, inhibits glucagon secretion of
$\alpha$-cells [Cherrington et al., 1976, Franklin et al., 2005, Ravier &
Rutter, 2005, Samols & Harrison, 1976, Soria et al., 2000], which appears
natural. Furthermore, the role of the third cell-type, $\delta$-cells, is
still not completely known although there have been reports that somatostatin
hormone, secreted by $\delta$-cells, suppresses the hormone secretion of both
$\alpha$\- and $\beta$-cells [Cherrington et al., 1976, Daunt et al., 2006,
Orci & Unger, 1975, Soria et al., 2000].

What is then the raison d’etre of the paradoxical interactions between
$\alpha$\- and $\beta$-cells and the inhibitory action by $\delta$-cells?
Despite previous studies as to these questions over the last thirty years
[Orci & Unger, 1975, Pipeleers, 1987, Soria et al., 2000, Unger & Orci, 1977],
there still lacks concrete understanding of the role of these interactions in
terms of glucose homeostasis. The primary difficulty in understanding these
interactions lies in the complexity of the islet system which includes many
interactions between different coexisting cell-types working in different
conditions.

In this paper, we analyze the interactions between $\alpha$-, $\beta$-, and
$\delta$-cells, which contribute to the precise control of the glucose level,
by means of a mathematical model incorporating experimentally known
interactions between islet cells (see above). As a result, our model predicts
that the intracellular interactions modify insulin and glucagon secretion in a
way to control the blood glucose level more efficiently.

##  2 Islet model

###  2.1 Activity of islet cells

We begin with a simplified model in which cells of each type can take one of
two states (active and silent). The state of a cell, represented by “Ising
spin” $\sigma$, is defined to be active ($\sigma=+1$) when the cell secretes
islet hormone; otherwise the state is defined as silent ($\sigma=-1$).
Accordingly, the state of an islet consisting of $\alpha$-, $\beta$-, and
$\delta$-cells can be represented by
($\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta}$). There is a total of
$2^{3}$ possible states of the islet, among which ($+1,-1,-1$) and
($-1,+1,+1$) describe the islet state at low and high glucose levels,
respectively. The main source of changing cell states is the blood glucose
level $\tilde{G}$, which globally influences all cells. In addition, the
paracrine interaction $\tilde{J}$ from neighboring cells locally affects cell
states. In this manner we obtain a simple Ising-type model, generally
characterizing two-state dynamics in statistical physics: The glucose level
$\tilde{G}$ corresponds to the external magnetic field and the paracrine
interaction $\tilde{J}$ to the local interaction between spins.

¿From the known cellular interactions illustrated in Fig. 1, one may determine
local stimuli $G_{\alpha}$, $G_{\beta}$, and $G_{\delta}$, which change the
states of $\alpha$-, $\beta$-, and $\delta$-cells, respectively, in the forms:

$\displaystyle G_{\alpha}$ $\displaystyle=$
$\displaystyle-G-\frac{1+\sigma_{\beta}}{2}J_{\alpha\beta}-\frac{1+\sigma_{\delta}}{2}J_{\alpha\delta}$
$\displaystyle G_{\beta}$ $\displaystyle=$ $\displaystyle
G+\frac{1+\sigma_{\alpha}}{2}J_{\beta\alpha}-\frac{1+\sigma_{\delta}}{2}J_{\beta\delta}$
$\displaystyle G_{\delta}$ $\displaystyle=$ $\displaystyle mG,$ (1)

where $G\equiv\tilde{G}-\tilde{G}_{0}$ measures the excess glucose level from
the basal glucose level $\tilde{G}_{0}$ during the fasting period. The
reciprocal nature of $\alpha$\- and $\beta$-cells in the responses to glucose
is manifested by the opposite signs in front of $G$ in the first equation (for
$G_{\alpha}$) and the second one (for $G_{\beta}$) of Eq. 2.1. In addition,
the asymmetric interaction between these two cell types is also reflected in
the second terms involving $J_{\alpha\beta}$ and $J_{\beta\alpha}$ of the
equations. For simplicity, we assume that the interaction strength
$J_{\beta\alpha}$ from $\alpha$\- to $\beta$-cells is the same as
$J_{\alpha\beta}$ from $\beta$\- to $\alpha$-cells and given by $J_{1}$, i.e.,
$J_{\alpha\beta}=J_{\beta\alpha}=J_{1}$. The last terms involving
$J_{\alpha\delta}$ and $J_{\beta\delta}$ describe the inhibition effects of
$\delta$-cells on $\alpha$\- and $\beta$-cells, both with negative signs.
Although the endogenous strengths of the interactions from $\delta$-cells to
$\alpha$\- and $\beta$-cells are not known, the exogenous stimulus of
somatostatin has been reported to inhibit both insulin and glucagon secretion
to a similar degree [Cherrington et al., 1976]. As a first approximation, it
is thus assumed that both interactions have the same strength:
$J_{\alpha\delta}=J_{\beta\delta}=J_{2}$. Here the interaction strengths
$J_{1}$ and $J_{2}$ are expressed in terms of the relative effects to glucose
stimulation, and therefore have the unit of mM corresponding to the hormonal
stimulus $\tilde{J}$, namely, a given amount of stimulus $\tilde{J}$ by
hormone is considered to produce the same effects on a cell as a certain
amount $J$ of glucose stimulation. In our simplified model, $\delta$-cells are
not influenced by neighboring $\alpha$\- and $\beta$-cells but stimulated
solely by glucose; therefore, $G_{\delta}$ depends only on $G$ in Eq. 2.1.
Like $\beta$-cells, $\delta$-cells become active, and secrete somatostatin
above a threshold level of glucose. The glucose sensitivity of $\delta$-cells
is expected to have a value between zero and unity, i.e., $0<m<1$ because the
threshold level for the activation of $\delta$-cells is lower than that of
$\beta$-cells [Efendić et al., 1979, Nadal et al., 1999]. We thus choose the
value $m=0.5$ in this study; the overall behavior does not depend
qualitatively on the value of $m$.

For given local stimulus $G_{\alpha}$ considering glucose stimulus $G$ and
effects of insulin and somatostatin, the transition rate of an $\alpha$-cell
from state $\sigma_{\alpha}$ to state $-\sigma_{\alpha}$ depends on the states
of other cell types as well as its own state, and is denoted as
$w_{\alpha}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$. This transition
rate should satisfy the detailed balance condition between two $\alpha$-cell
states $\sigma_{\alpha}$ and $-\sigma_{\alpha}$ at equilibrium:

$w_{\alpha}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})P(\sigma_{\alpha})=w_{\alpha}(-\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})P(-\sigma_{\alpha}),$
(2)

where the probability $P(\sigma_{\alpha})$ for state $\sigma_{\alpha}$ follows
the Boltzmann distribution $\exp[-G_{\alpha}(1+\sigma_{\alpha})/2\Theta]$ with
respect to the quantity $G_{\alpha}(1+\sigma_{\alpha})/2$ for the local
stimulus $G_{\alpha}$. Namely, $\alpha$-cells favor the state minimizing the
quantity $G_{\alpha}(1+\sigma_{\alpha})/2$. Here $\Theta$ measures the amount
of uncertainty, which is inevitable in biological systems. The origin may be
the heterogeneous glucose sensitivity of cells and/or the environmental noise
including thermal fluctuations. It is obvious that
$G_{\alpha}(1+\sigma_{\alpha})/2$ and $\Theta$ correspond to the energy and
the temperature, respectively, in statistical physics. In this study, the
“temperature” is taken to be unity ($\Theta=1$) in units of the “energy”,
which is biologically tantamount to the fluctuations caused by 1 mM change of
glucose stimulation.

The ratio between the reciprocal transition rates thus reads

$\displaystyle\frac{w_{\alpha}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})}{w_{\alpha}(-\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})}$
$\displaystyle=$
$\displaystyle\exp\Bigg{[}-\frac{1}{\Theta}G_{\alpha}\sigma_{\alpha}\Bigg{]}$
(3) $\displaystyle=$
$\displaystyle\exp\Bigg{[}\frac{1}{\Theta}\Bigg{(}G_{\alpha}^{\text{eff}}\sigma_{\alpha}+\frac{J_{1}}{2}\sigma_{\alpha}\sigma_{\beta}+\frac{J_{2}}{2}\sigma_{\delta}\sigma_{\alpha}\Bigg{)}\Bigg{]}$

with $G_{\alpha}^{\text{eff}}\equiv G+J_{1}/2+J_{2}/2$, where Eq. 2.1 has been
used to obtain the second line. There the three stimulation terms represent
effective glucose stimulation, paracrine interaction from $\beta$-cells, and
another from $\delta$-cells, respectively. Assuming that these stimuli affect
independently the $\alpha$-cell state, we write the transition rate in the
form

$\displaystyle w_{\alpha}$ $\displaystyle=$
$\displaystyle\frac{1}{2\tau}\left[1+\tanh\left(\frac{G_{\alpha}^{\text{eff}}}{2\Theta}\right)\sigma_{\alpha}\right]\left[1+\tanh\left(\frac{J_{1}}{4\Theta}\right)\sigma_{\alpha}\sigma_{\beta}\right]$
(4)
$\displaystyle\times\left[1+\tanh\left(\frac{J_{2}}{4\Theta}\right)\sigma_{\delta}\sigma_{\alpha}\right],$

where $\tau$ measures the characteristic time of the transition and it has
been noted that $\tanh(y\sigma)=\sigma\tanh y$ for $\sigma=\pm 1$. Note that
among possible transition rates satisfying Eq. 2, we adopt the Glauber
dynamics [Glauber, 1963] to choose the specific form of Eq. 4, which exhibits
the sigmoidal form ubiquitously describing response functions in biological
systems. However, the behavior of the system in general does not depend
qualitatively on the specific form of the transition rate satisfying Eq. 2.
Similarly, we obtain the transition rates $w_{\beta}$ and $w_{\delta}$ of
$\beta$\- and $\delta$-cells.

The transition rates of three cell types can be summarized as

$\displaystyle w_{x}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$
$\displaystyle=$
$\displaystyle\frac{1}{2\tau}\left[w^{x}+w^{x}_{\alpha}\sigma_{\alpha}+w^{x}_{\beta}\sigma_{\beta}+w^{x}_{\delta}\sigma_{\delta}+w^{x}_{\alpha\beta}\sigma_{\alpha}\sigma_{\beta}\right.$
(5)
$\displaystyle\left.+w^{x}_{\beta\delta}\sigma_{\beta}\sigma_{\delta}+w^{x}_{\delta\alpha}\sigma_{\delta}\sigma_{\alpha}+w^{x}_{\alpha\beta\delta}\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}\right],$

with $x=\alpha,\beta,$ and $\delta$, where the coefficients are given in
Tables 1 and 2. The master equation, describing the evolution of the
probability $P(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$ for the islet
in state $(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$, reads

$\displaystyle\frac{d}{dt}P(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$
$\displaystyle~{}~{}=w_{\alpha}(-\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})P(-\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})+w_{\beta}(\sigma_{\alpha},-\sigma_{\beta},\sigma_{\delta})P(\sigma_{\alpha},-\sigma_{\beta},\sigma_{\delta})$
$\displaystyle~{}~{}~{}~{}~{}+w_{\delta}(\sigma_{\alpha},\sigma_{\beta},-\sigma_{\delta})P(\sigma_{\alpha},\sigma_{\beta},-\sigma_{\delta})-w_{\alpha}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})P(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$
$\displaystyle~{}~{}~{}~{}~{}-w_{\beta}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})P(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})-w_{\delta}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})P(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$
(6)

with the transition rates $w_{\alpha}$, $w_{\beta}$, and $w_{\delta}$ in Eq.
5. Note that Eq. 2.1 describes the net flux to state
$(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$ simply given by the
difference between the in-flux to state
$(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$ from other states
$(-\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$,
$(\sigma_{\alpha},-\sigma_{\beta},\sigma_{\delta})$, and
$(\sigma_{\alpha},\sigma_{\beta},-\sigma_{\delta})$ and the out-flux from
state $(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$ to others.

¿From this master equation, it is straightforward to obtain the time evolution
of the ensemble averages of the cell states and their correlations. For
example, multiplying both sides of Eq. 2.1 by $\sigma_{\alpha}$ and summing
over all configurations, we obtain the evolution equation for the average
$\langle\sigma_{\alpha}\rangle\equiv\sum_{\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta}}\sigma_{\alpha}P(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$
of the state of $\alpha$-cells:

$\frac{d}{dt}\langle\sigma_{\alpha}\rangle=-2\langle\sigma_{\alpha}w_{\alpha}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})\rangle$
(7)

and similarly,

$\displaystyle\frac{d}{dt}\langle\sigma_{\beta}\rangle$ $\displaystyle=$
$\displaystyle-2\langle\sigma_{\beta}w_{\beta}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})\rangle$
(8) $\displaystyle\frac{d}{dt}\langle\sigma_{\delta}\rangle$ $\displaystyle=$
$\displaystyle-2\langle\sigma_{\delta}w_{\alpha}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})\rangle.$
(9)

Note that $1+\langle\sigma_{\alpha}\rangle$ gives twice the average activity
of $\alpha$-cells, etc.

Among the eight equations for the probability
$P(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$ corresponding to the eight
possible states of the islet, only seven are independent, due to the
normalization condition
$\sum_{\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta}}p(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})=1$.
Therefore, there exist four more equations in addition to the above three
describing the average of cell states. Those are evolution equations for
correlations of two cell states and of three cell states. The equation for the
correlation function $\langle\sigma_{\alpha}\sigma_{\beta}\rangle$ of the
$\alpha$-cell and $\beta$-cell states can again be derived from Eq. 2.1,
multiplied by $\sigma_{\alpha}\sigma_{\beta}$ and summed over all
configurations:

$\frac{d}{dt}\langle\sigma_{\alpha}\sigma_{\beta}\rangle=-2\langle\sigma_{\alpha}\sigma_{\beta}w_{\alpha}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})\rangle-2\langle\sigma_{\alpha}\sigma_{\beta}w_{\beta}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})\rangle.$
(10)

The equations for $\langle\sigma_{\beta}\sigma_{\delta}\rangle$ and
$\langle\sigma_{\delta}\sigma_{\alpha}\rangle$ are also obtained in the same
way:

$\displaystyle\frac{d}{dt}\langle\sigma_{\beta}\sigma_{\delta}\rangle$
$\displaystyle=$
$\displaystyle-2\langle\sigma_{\beta}\sigma_{\delta}w_{\beta}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})\rangle-2\langle\sigma_{\beta}\sigma_{\delta}w_{\delta}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})\rangle$
$\displaystyle\frac{d}{dt}\langle\sigma_{\delta}\sigma_{\alpha}\rangle$
$\displaystyle=$
$\displaystyle-2\langle\sigma_{\delta}\sigma_{\alpha}w_{\delta}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})\rangle-2\langle\sigma_{\delta}\sigma_{\alpha}w_{\alpha}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})\rangle.$
(11)

Note that correlations of two cell states represent the relative activity of
the two cells. Accordingly, it makes a good measure of the different responses
between two cells. Similarly, the equation for correlations of three cell
states is given by

$\displaystyle\frac{d}{dt}\langle\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}\rangle$
$\displaystyle=$
$\displaystyle-2\langle\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}w_{\alpha}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})\rangle-2\langle\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}w_{\beta}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})\rangle$
(12)
$\displaystyle-2\langle\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}w_{\delta}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})\rangle.$

Substituting the transition rates in Eq. 5 into Eqs. 7 to 12, we finally
obtain equations for the states of the three cell types and their
correlations:

$\displaystyle\tau\frac{d}{dt}\langle\sigma_{\alpha}\rangle$ $\displaystyle=$
$\displaystyle-w^{\alpha}_{\alpha}-w^{\alpha}\langle\sigma_{\alpha}\rangle-w^{\alpha}_{\alpha\beta}\langle\sigma_{\beta}\rangle-w^{\alpha}_{\delta\alpha}\langle\sigma_{\delta}\rangle-w^{\alpha}_{\beta}\langle\sigma_{\alpha}\sigma_{\beta}\rangle$
$\displaystyle-w^{\alpha}_{\alpha\beta\delta}\langle\sigma_{\beta}\sigma_{\delta}\rangle-w^{\alpha}_{\delta}\langle\sigma_{\delta}\sigma_{\alpha}\rangle+w^{\alpha}_{\beta\delta}\langle\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}\rangle$
$\displaystyle\tau\frac{d}{dt}\langle\sigma_{\beta}\rangle$ $\displaystyle=$
$\displaystyle-w^{\beta}_{\beta}-w^{\beta}_{\alpha\beta}\langle\sigma_{\alpha}\rangle-w^{\beta}\langle\sigma_{\beta}\rangle-w^{\beta}_{\beta\delta}\langle\sigma_{\delta}\rangle-w^{\beta}_{\alpha}\langle\sigma_{\alpha}\sigma_{\beta}\rangle$
$\displaystyle-w^{\beta}_{\delta}\langle\sigma_{\beta}\sigma_{\delta}\rangle-w^{\beta}_{\alpha\beta\delta}\langle\sigma_{\delta}\sigma_{\alpha}\rangle-w^{\beta}_{\delta\alpha}\langle\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}\rangle$
$\displaystyle\tau\frac{d}{dt}\langle\sigma_{\delta}\rangle$ $\displaystyle=$
$\displaystyle-w^{\delta}_{\delta}-w^{\delta}_{\delta\alpha}\langle\sigma_{\alpha}\rangle-w^{\delta}_{\beta\delta}\langle\sigma_{\beta}\rangle-w^{\delta}\langle\sigma_{\delta}\rangle-w^{\delta}_{\alpha\beta\delta}\langle\sigma_{\alpha}\sigma_{\beta}\rangle$
$\displaystyle-w^{\delta}_{\beta}\langle\sigma_{\beta}\sigma_{\delta}\rangle-w^{\delta}_{\alpha}\langle\sigma_{\delta}\sigma_{\alpha}\rangle-w^{\delta}_{\alpha\beta}\langle\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}\rangle$
$\displaystyle\tau\frac{d}{dt}\langle\sigma_{\alpha}\sigma_{\beta}\rangle$
$\displaystyle=$
$\displaystyle-(w^{\alpha}_{\alpha\beta}+w^{\beta}_{\alpha\beta})-(w^{\alpha}_{\beta}+w^{\beta}_{\beta})\langle\sigma_{\alpha}\rangle-(w^{\alpha}_{\alpha}+w^{\beta}_{\alpha})\langle\sigma_{\beta}\rangle$
$\displaystyle-(w^{\alpha}_{\alpha\beta\delta}+w^{\beta}_{\alpha\beta\delta})\langle\sigma_{\delta}\rangle-(w^{\alpha}+w^{\beta})\langle\sigma_{\alpha}\sigma_{\beta}\rangle$
$\displaystyle-(w^{\alpha}_{\delta\alpha}+w^{\beta}_{\delta\alpha})\langle\sigma_{\beta}\sigma_{\delta}\rangle-(w^{\alpha}_{\beta\delta}+w^{\beta}_{\beta\delta})\langle\sigma_{\delta}\sigma_{\alpha}\rangle$
$\displaystyle-(w^{\alpha}_{\delta}+w^{\beta}_{\delta})\langle\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}\rangle$
$\displaystyle\tau\frac{d}{dt}\langle\sigma_{\beta}\sigma_{\delta}\rangle$
$\displaystyle=$
$\displaystyle-(w^{\beta}_{\beta\delta}+w^{\delta}_{\beta\delta})-(w^{\beta}_{\alpha\beta\delta}+w^{\delta}_{\alpha\beta\delta})\langle\sigma_{\alpha}\rangle-(w^{\beta}_{\delta}+w^{\delta}_{\delta})\langle\sigma_{\beta}\rangle$
$\displaystyle-(w^{\beta}_{\beta}+w^{\delta}_{\beta})\langle\sigma_{\delta}\rangle-(w^{\beta}_{\delta\alpha}+w^{\delta}_{\delta\alpha})\langle\sigma_{\alpha}\sigma_{\beta}\rangle$
$\displaystyle-(w^{\beta}+w^{\delta})\langle\sigma_{\beta}\sigma_{\delta}\rangle-(w^{\beta}_{\alpha\beta}+w^{\delta}_{\alpha\beta})\langle\sigma_{\delta}\sigma_{\alpha}\rangle$
$\displaystyle-(w^{\beta}_{\alpha}+w^{\delta}_{\alpha})\langle\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}\rangle$
$\displaystyle\tau\frac{d}{dt}\langle\sigma_{\delta}\sigma_{\alpha}\rangle$
$\displaystyle=$
$\displaystyle-(w^{\delta}_{\delta\alpha}+w^{\alpha}_{\delta\alpha})-(w^{\delta}_{\delta}+w^{\alpha}_{\delta})\langle\sigma_{\alpha}\rangle-(w^{\delta}_{\alpha\beta\delta}+w^{\alpha}_{\alpha\beta\delta})\langle\sigma_{\beta}\rangle$
$\displaystyle-(w^{\delta}_{\alpha}+w^{\alpha}_{\alpha})\langle\sigma_{\delta}\rangle-(w^{\delta}_{\beta\delta}+w^{\alpha}_{\beta\delta})\langle\sigma_{\alpha}\sigma_{\beta}\rangle$
$\displaystyle-(w^{\delta}_{\alpha\beta}+w^{\alpha}_{\alpha\beta})\langle\sigma_{\beta}\sigma_{\delta}\rangle-(w^{\delta}+w^{\alpha})\langle\sigma_{\delta}\sigma_{\alpha}\rangle$
$\displaystyle-(w^{\delta}_{\beta}+w^{\alpha}_{\beta})\langle\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}\rangle$
$\displaystyle\tau\frac{d}{dt}\langle\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}\rangle$
$\displaystyle=$
$\displaystyle-(w^{\alpha}_{\alpha\beta\delta}+w^{\beta}_{\alpha\beta\delta}+w^{\delta}_{\alpha\beta\delta})-(w^{\alpha}_{\beta\delta}+w^{\beta}_{\beta\delta}+w^{\delta}_{\beta\delta})\langle\sigma_{\alpha}\rangle$
(13)
$\displaystyle-(w^{\alpha}_{\delta\alpha}+w^{\beta}_{\delta\alpha}+w^{\delta}_{\delta\alpha})\langle\sigma_{\beta}\rangle-(w^{\alpha}_{\alpha\beta}+w^{\beta}_{\alpha\beta}+w^{\delta}_{\alpha\beta})\langle\sigma_{\delta}\rangle$
$\displaystyle-(w^{\alpha}_{\delta}+w^{\beta}_{\delta}+w^{\delta}_{\delta})\langle\sigma_{\alpha}\sigma_{\beta}\rangle-(w^{\alpha}_{\alpha}+w^{\beta}_{\alpha}+w^{\delta}_{\alpha})\langle\sigma_{\beta}\sigma_{\delta}\rangle$
$\displaystyle-(w^{\alpha}_{\beta}+w^{\beta}_{\beta}+w^{\delta}_{\beta})\langle\sigma_{\delta}\sigma_{\alpha}\rangle-(w^{\alpha}+w^{\beta}+w^{\delta})\langle\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}\rangle.$

###  2.2 Glucose homeostasis

Heretofore we have focused on the cellular interactions at a given glucose
level. To study dynamics of glucose homeostasis, however, we should also take
into account the change of the glucose level and incorporate another equation
for glucose regulation into the model. Based on the fact that $\alpha$\- and
$\beta$-cells secrete glucagon and insulin, respectively, raising and reducing
the glucose level, the equation for the glucose level $G$ is taken to be

$\tau_{G}\frac{dG}{dt}=\frac{1+\langle\sigma_{\alpha}\rangle}{2}-\frac{1+\langle\sigma_{\beta}\rangle}{2},$
(14)

where $\tau_{G}$ is the characteristic time for the hormones to regulate the
glucose level. It is expected that $\tau_{G}$ is larger than the
characteristic time $\tau$ of the change in cell states. Equation 14 describes
the decrease or increase of the glucose level when $\alpha$-cells or
$\beta$-cells are active ($\sigma_{\alpha}=1$ or $\sigma_{\beta}=1$). Here,
for simplicity, we have used the same characteristic time $\tau_{G}$ for
glucagon and insulin to regulate glucose levels. Having different time
constants turns out merely to shift the stationary level of blood glucose. To
sum, we have a total of eight differential equations given by Eqs. 13 and 14,
which describe the process of glucose homeostasis.

##  3 Results

###  3.1 Asymmetric interactions between $\alpha$\- and $\beta$-cells

In our model, activities of $\alpha$-, $\beta$-, and $\delta$-cells are
determined by the external glucose level together with feedback loops of
intercellular interactions. A given cell, subject to a glucose stimulus,
secretes hormone which influences the behavior of neighboring cells. In
response, the neighboring cells reversely influence the given cell. These
mutual interactions through hormones constitute the feedback loop which is
widely employed for advanced system control in engineering [Bechhoefer, 2005].

The interactions between $\alpha$\- and $\beta$-cells are asymmetric: While
glucagon secreted from $\alpha$-cells enhances insulin secretion of
$\beta$-cells [Brereton et al., 2007, Samols et al., 1965, Soria et al.,
2000], insulin inhibits glucagon secretion [Cherrington et al., 1976, Franklin
et al., 2005, Ravier & Rutter, 2005, Samols & Harrison, 1976, Soria et al.,
2000]. The former positive interaction to the counterpart cells may seem
strange, but it eventually contributes to the construction of a negative
feedback loop for both cells. At low glucose levels, $\alpha$-cells secrete
glucagon, which enhances insulin secretion. In turn, insulin inhibits the
glucagon secretion of $\alpha$-cells. Therefore, their interactions as a whole
tend to suppress the glucagon secretion from $\alpha$-cells. Similar negative
feedback operates when $\beta$-cells are activated by high glucose
concentration. It is noteworthy that this feedback works more efficiently in
case that the glucose level varies. At a static glucose level, it should be
difficult for the mutual interactions between $\alpha$\- and $\beta$-cells to
arise simultaneously because insulin and glucagon are secreted at different
glucose levels.

In general, a negative feedback favors stability of a system because it
attenuates overaction of the system such as overshoot or undershoot. The
negative feedbacks in an islet system contribute to the stable recovery to the
normal glucose level $G_{\infty}$, when the system is externally perturbed by
stimuli such as a glucose dose. The normal glucose level $G_{\infty}$, reached
by $G\,(\equiv\tilde{G}-\tilde{G}_{0})$ at stationarity, depends on the
cellular interactions shown in Fig. 2. The asymmetric interaction $J_{1}$
lowers the basal glucose level because $\alpha$-cells activate $\beta$-cells
which secret insulin and thus reduces the glucose level. In addition, the
inhibitory interaction $J_{2}$ of $\delta$-cells, albeit the same for
$\alpha$\- and $\beta$-cells, suppresses the activity of $\beta$-cells more
than that of $\alpha$-cells at the normal glucose level, because the activity
of $\beta$-cells is higher than that of $\alpha$-cells resulting from the
asymmetric interaction between $\alpha$\- and $\beta$-cells. Accordingly, the
basal glucose level tends to increase as the strength $J_{2}$ of the
inhibitory interaction is increased.

Figure 3 demonstrates the smooth recovery of the glucose level in the presence
of cellular interactions (solid line), compared with the somewhat erratic
recovery, once reaching low glucose levels, in the absence of the interactions
(dashed line). For comparison, we also consider the behavior in the case of
symmetric interactions between $\alpha$\- and $\beta$-cells, i.e., where
glucagon inhibits insulin secretion and vice versa, only to find even more
erratic recovery (see the dotted line). Shown here is the recovery from the
high glucose state [$G=1$ mM (or $\tilde{G}=\tilde{G}_{0}$ \+ 1 mM),
$\langle\sigma_{\alpha}\rangle=-1$, and $\langle\sigma_{\beta}\rangle=1$]. The
recovery from a low glucose state gives the same results (data not shown)
although such erratic recovery is more pronounced for the glucose level
starting from a higher value.

To examine the stability in approaching the normal glucose level, we define
the balance function

$b(G)\equiv\tau_{G}\frac{dG}{dt}=\frac{1+\langle\sigma_{\alpha}\rangle}{2}-\frac{1+\langle\sigma_{\beta}\rangle}{2},$
(15)

which describes the glucose level change during the characteristic time. Since
the activity of cells represents their hormone secretion, $b(G)$ appropriately
describes the effectiveness of the glucose regulation by $\alpha$\- and
$\beta$-cells. If the characteristic time $\tau_{G}$ of glucose regulation is
much larger than the characteristic time $\tau$ of cell responses in Eq. 13,
i.e., $\tau\ll\tau_{G}$, the glucose level should be in a quasi-stationary
state at time $t$ shorter than $\tau_{G}$. Then the fast dynamics of cell
states in Eq. 13 saturates rapidly at a given glucose level and the seven
variables, activities and correlations, reach their fixed points depending on
the glucose level $G$. In particular $\langle\sigma_{\alpha}\rangle$ and
$\langle\sigma_{\beta}\rangle$ depend on $G$, giving the balance function in
Eq. 15 as a function of $G$, with a fixed point at $G=G_{\infty}$ (see Fig.
4). At low glucose levels ($G<G_{\infty}$), we have the balance function
greater than zero ($b>0$), or $dG/dt>0$, thus the glucose level grows with
time. At high glucose levels ($G>G_{\infty}$), the opposite behavior arises.
The resulting flow of the balance function is illustrated by the arrows in
Fig. 4 and it is concluded that the balance function correctly describes
glucose homeostasis. Further, the slope of $b(G)$ near the fixed point
$G=G_{\infty}$ represents how smoothly the glucose level approaches the normal
level: The slope of the balance function for the asymmetric interaction is
small at $G=G_{\infty}$, which results in the smooth recovery of the normal
glucose level shown in Fig. 3. This result is more evident with the
interaction strength $J_{1}$ larger and the characteristic time $\tau_{G}$
shorter.

If $\alpha$\- and $\beta$-cells would inhibit each other, how should the
result change? As suggested already [Saunders et al., 1998], the bidirectional
inhibitory interactions seem to be optimal in view of that $\alpha$\- and
$\beta$-cells play opposite roles in glucose regulation. Remarkably, however,
such symmetric interactions turn out to result in dynamically unstable
responses, as shown by the dotted line in Fig. 3. If this were the case,
glucagon secreted by $\alpha$-cells at low glucose levels would suppress
$\beta$-cells from secreting insulin. As the secretion of insulin decreases,
so would the inhibitory effects of insulin on the glucagon secretion diminish.
It should thus follow that glucagon secretion is not negatively controlled,
implying more glucagon secretion. Such an apparent positive feedback loop,
enhancing hormone secretion, gives rise to an instability in the islet system
(see Fig. 3).

###  3.2 Inhibitory interactions of $\delta$-cells

####  3.2.1 Suppression of co-secretion from $\alpha$\- and $\beta$-cells

There is basal hormone secretion from $\alpha$\- and $\beta$-cells even at the
normal glucose level [Cherrington et al., 1976], where it is not necessary to
change the blood glucose concentration with the help of glucagon or insulin.
Obviously, the simultaneous secretion of glucagon and insulin at the normal
level should be minimized because the opposite effects of the two would cancel
out, nullifying the net effects on the glucose level. Such wasteful co-
secretion of counteracting hormones can be prevented by $\delta$-cells
secreting somatostatin, which inhibits secretion of both glucagon and insulin.

In our model, the average activity of cells is given by
$(1+\langle\sigma\rangle)/2$. Accordingly, the average cell state
$\langle\sigma\rangle=\pm 1$ means that all cells are active/silent; in
particular $\langle\sigma\rangle=0$ corresponds to half of the cells being
active. In the absence of the inhibitory interaction of $\delta$-cells, Fig.
5(a) shows that both $\langle\sigma_{\alpha}\rangle$ and
$\langle\sigma_{\beta}\rangle$ take values greater than $-1$ even at the
normal glucose level. Namely, fluctuations associated with the biological
uncertainty $\Theta$ have some fraction of cells still active, leading to
basal hormone secretion. Here the presence of inhibitory interactions of
$\delta$-cells lowers the basal activity of $\alpha$\- and $\beta$-cells, as
shown in Fig. 5(b), which reduces co-secretion of the counteracting hormones,
glucagon and insulin.

Figure 6 displays the relation between $\langle\sigma_{\alpha}\rangle$ and
$\langle\sigma_{\beta}\rangle$, in the absence ($J_{2}=0$ mM) and presence
($J_{2}=2$ mM) of the inhibitory interaction of $\delta$-cells. The system at
low or high glucose levels is described by the upper left or lower right parts
of the curves on the
$(\langle\sigma_{\beta}\rangle,\langle\sigma_{\alpha}\rangle)$ plane,
respectively. Namely, when the glucose concentration is low, $\alpha$\- and
$\beta$-cells are in high and in low activity, respectively
($\langle\sigma_{\alpha}\rangle>0$ and $\langle\sigma_{\beta}\rangle<0$); this
is reversed at high glucose concentrations. It is manifested that the
inhibitory interaction of $\delta$-cells reduces simultaneous activation of
$\alpha$\- and $\beta$-cells. Compared with the result for $J_{2}=0$ mM
(dashed line), the result for $J_{2}=2$ mM (solid line) shows that the
activity of $\beta$\- or $\alpha$-cells is reduced substantially at high or
low glucose levels. In particular $\alpha$-cells remain almost silent
($\langle\sigma_{\alpha}\rangle\approx-1$) at high glucose levels. Note,
however, that those endocrine cells are not totally silent at given glucose
levels and still exhibit residual activity, which results from fluctuations in
the glucose responses of the cells. Interestingly, it was suggested that such
basal hormone secretion also plays an effective role: The minimal basal
secretion of glucagon compensates the glucose uptake in the liver while basal
secretion of insulin inhibits over-secretion of the basal glucagon
[Cherrington et al., 1976].

####  3.2.2 Enhancement of glucose dose-responses of $\beta$-cells

Another consequence of the inhibitory interaction of $\delta$-cells is the
shift of glucose dose-responses for insulin secretion to the right direction.
This is associated with the increased control of $\beta$-cells by
$\delta$-cells at high glucose levels. Figure 7 indeed shows that the shift
leads to more conspicuous glucose responses of $\beta$-cells at high glucose
levels. In general $\beta$-cells are coupled with each other through gap-
junction channels, which help the cells synchronize their behaviors [Sherman &
Rinzel, 1991]. A $\beta$-cell cluster thus tends to produce all-or-none
glucose responses [Soria et al., 2000]. In the real islet, on the other hand,
$\delta$-cells, with their inhibitory interactions depending on the glucose
level, can modify the glucose dose-response of $\beta$-cells. Accordingly,
insulin response can be more pronounced at high glucose levels ($G>0$).

It is observed that some primitive animals have only $\beta$\- and
$\delta$-cells in their islets, unlike the mammals whose islets contain
$\alpha$-cells as well as $\beta$\- and $\delta$-cells [Falkmer, 1985]. This
difference could perhaps be attributed to an evolutionary adaptation. At early
evolutionary stages, the islet might be a passive system: Without
$\alpha$-cells directly increasing the glucose level, the glucose level should
increase passively as a result of the decrease in insulin secretion. Still,
the precise glucose dose-responses at high glucose levels could be possible
with $\delta$-cells. At later stages, equipped with $\alpha$-cells, the islet
became an active system with regard to glucose regulation. It is of interest
that this evolutionary change is correlated with the fact that $\beta$\- and
$\delta$-cells are closer to each other than $\alpha$-cells in the development
of a stem cell [Kemp et al., 2003]. In addition, $\beta$\- and $\delta$-cells
have functional similarities of using ATP-dependent K+ channels in glucose
responses [Quesada et al., 2006, Quesada et al., 1999].

##  4 Discussions

The islet of Langerhans is a precise system that controls the glucose level
through the use of three main types of endocrine cells. Here it is of interest
to investigate whether the existing interactions between those cells are
beneficial for glucose homeostasis. There are some evidence for the critical
role of the interactions, which may not obviously be addressed by probing
$\alpha$\- and $\beta$-cells separately. The molecular mechanism of how
$\alpha$-cells regulate glucagon secretion at variable glucose levels is still
not clearly understood [Gromada et al., 2007]. Several works attempted to
explain this by means of the interactions between $\alpha$\- and
$\beta$-cells: At high glucose levels, glucagon secretion is inhibited by
insulin, GABA, or Zn2+ secreted from $\beta$-cells [Gromada et al., 2007,
Ishihara et al., 2003]. There is also a hypothesis that glucose has direct
effects on $\alpha$-cells through endoplasmic reticulum Ca2+ storage [Vieira
et al., 2006]. Another evidence for the role of cellular interactions in
glucose homeostasis comes from hyperglucagonomia, which occurs in diabetics at
abnormally high glucose levels. It appears paradoxical that the glucagon
levels of such patients are high even though the blood glucose levels are high
enough to make $\alpha$-cells silent [Gromada et al., 2007]. This puzzling
result can be explained on the basis of cellular interactions in an islet
[Franklin et al., 2005, Rorsman et al., 1989, Takahashi et al., 2006]. Note
that there is also another explanation of this phenomenon in terms of the
peculiar glucose dose-responses of (rat) $\alpha$-cells [Kemp et al., 2003].

In contrast, there also exist a few reports that some cellular interactions
may not exist and are not necessary for glucose homeostasis: It has been
proposed that the microcirculation from $\beta$\- to peripheral $\alpha$\- and
$\delta$-cells prohibits the paracrine action from non-$\beta$ to
$\beta$-cells [Wayland, 1997]. In addition, it has recently been reported that
islet transplantation is successful in recovering from hyperglycaemia with
only $\beta$-cell clusters [King et al., 2007].

Nevertheless, the existence of the receptors of signalling molecules such as
insulin, glucagon, somatostatin, glutamate, and GABA, which are expressed in
pancreatic endocrine cells, apparently implicates their physiological roles in
the fine control of glucose levels [Gromada et al., 2007, Strowski & Blake,
2008]. A better understanding of this tissue, therefore, will contribute to
more advanced medical treatment of diabetes than the current one relying
mostly on insulin. For example, it is conceivable to use other hormones such
as glucagon and somatostatin for more active and precise glucose control.

A variety of complicated interactions in an islet makes it difficult to
recognize their roles, and existing experiments as to those interactions have
focused mostly on static responses of the endocrine cells. However, it is
likely that the cellular interactions actually contribute to dynamical
responses to glucose. In this study, therefore, to understand the role of
intercellular interactions between $\alpha$-, $\beta$-, and $\delta$-cells, we
have proposed an islet model and investigated the effects of integrated
intercellular communications between those cells in glucose homeostasis. Our
mathematical model can systematically include all the cellular interactions
and identify their effects on static and/or dynamic responses to external
glucose changes. It also takes individual heterogeneities into consideration,
e.g., in glucose sensitivity; the basal hormone secretion at the normal
glucose level reflects that some cells can be active to secrete hormones even
though most of the cells are silent at that glucose level. The small
variations in glucose responses among homologous cells may contribute
crucially to the cellular interactions between heterologous cells, which are
actually activated in quite different glucose concentrations, because the
heterogeneous responses of homologous cells can lead to an overlap in the
activation between the heterologous cells. ¿From this model, it has been
revealed that the interactions give more stable, efficient, and accurate
control of glucose: First, asymmetric interactions between $\alpha$\- and
$\beta$-cells contribute to the dynamic stability when the glucose level,
perturbed from the normal level, recovers to the latter. Second, the
interactions of somatostatin for glucagon and insulin secretion prevent their
wasteful co-secretion at the normal glucose level. In addition, at high
glucose levels, the inhibition by $\delta$-cells modifies glucose dose-
responses of insulin secretion. For a more realistic and accurate
understanding, it would be necessary to know the physiological values of the
model parameters. In particular, the relative effects of the direct glucose
stimulus $G$ and paracrine interactions $J$ on the states of endocrine cells
should be identified.

Here it is proposed that these predictions can be verified in experiment. As
for the role of $\delta$-cells, our results may be confirmed through the use
of cell clusters of different compositions of cell-types, for which the
culture method was used in the existing study [Pipeleers et al., 1982].
Another prediction related with the asymmetric interactions between $\alpha$\-
and $\beta$-cells needs to be verified in vivo experiment on transgenic mice
because the effects should arise in the dynamics of whole-body glucose
regulation. Note that the specific cellular interaction may be blocked
selectively in knockout mice lacking specific hormone receptors in an
endocrine cell [Diao et al., 2005, Sorensen et al., 2006].

Beyond the interactions between endocrine cells analyzed in this study, there
exist reports that $\delta$-cells are also influenced by $\alpha$\- and
$\beta$-cells [Unger & Orci, 1977] and these paracrine interactions should be
considered with the microcirculation of hormones in an islet as described
above [Wayland, 1997]. It has also been reported that there exist autocrine
interactions via which a cell is affected by its own hormone secretion
[Aspinwall et al., 1999, Cabrera et al., 2008]. Furthermore, input from
exocrine cells [Bertelli et al., 2001, Bishop & Polak, 1997, Wayland, 1997]
and glucose-sensing neurons [Schuit et al., 2001] have been suggested. There
may thus be more complex communications in the pancreas for glucose
homeostasis, which are left for further study. Finally, we also point out that
the mathematical model proposed can be generalized to describe cellular
interactions in other systems, e.g., neural networks consisting of excitatory
and inhibitory couplings.

Acknowledgments
We thank D. Gardner-Hofatt and W. Heuett for useful comments on the
manuscript. M.Y.C. thanks Asia Pacific Center for Theoretical Physics, where
part of this work was performed, for hospitality. This work was supported in
part by the KOSEF/MOST grant through National Core Research Center for Systems
Bio-Dynamics and by the KOSEF-CNRS Cooperative Program.

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## Tables

Table 1: Coefficients in the transition rate. Here $k^{x}\equiv(g^{x}+j_{1}^{x}+j_{2}^{x}+g^{x}j_{1}^{x}j_{2}^{x})(1+g^{x}j_{1}^{x}+j_{1}^{x}j_{2}^{x}+j_{2}^{x}g^{x})^{-1}$ with $x$ denoting $\alpha$, $\beta$, or $\delta$. Parameters $g^{x}$, $j_{1}^{x}$, and $j_{2}^{x}$ are given in Table 2. Coefficient | Value
---|---
$w^{x}$ | $1$
$w^{x}_{\alpha}$ | $k^{x}$
$w^{x}_{\beta}$ | $k^{x}j^{x}_{1}$
$w^{x}_{\delta}$ | $k^{x}j^{x}_{2}$
$w^{x}_{\alpha\beta}$ | $j^{x}_{1}$
$w^{x}_{\beta\delta}$ | $j^{x}_{1}j^{x}_{2}$
$w^{x}_{\delta\alpha}$ | $j^{x}_{2}$
$w^{x}_{\alpha\beta\delta}$ | $k^{x}j^{x}_{1}j^{x}_{2}$

Table 2: Parameters in the coefficients of the transition rate. $x$ | $\alpha$ | $\beta$ | $\delta$
---|---|---|---
$g^{x}$ | ${\rm tanh}(G/2\Theta)$ | ${\rm tanh}(-G/2\Theta)$ | ${\rm tanh}(-mG/2\Theta)$
$j_{1}^{x}$ | ${\rm tanh}(J_{1}/4\Theta)$ | ${\rm tanh}(-J_{1}/4\Theta)$ | $0$
$j_{2}^{x}$ | ${\rm tanh}(J_{2}/4\Theta)$ | ${\rm tanh}(J_{2}/4\Theta)$ | $0$

## Figure Legends

#### Figure 1.

Schematic diagram of cellular interactions between $\alpha$-, $\beta$-, and
$\delta$-cells. The arrow represents enhancement while bars represent
inhibition. Here the intercellular interactions between two cells are present
only when both are in active states.

#### Figure 2.

Stationary glucose level $G_{\infty}$, depending on the cellular interaction
strengths $J_{1}$ and $J_{2}$. Note that $G_{\infty}$ measures the stationary
level relative to the fasting glucose level $\tilde{G}_{0}$ in the absence of
cellular interactions.

#### Figure 3.

Time evolution of the glucose level $G$, depending on the interactions between
$\alpha$\- and $\beta$-cells. Starting from the high-glucose state
[$\langle\sigma_{\alpha}\rangle=-1$ and $\langle\sigma_{\beta}\rangle=1$ at
$G=1$ mM (or $\tilde{G}=\tilde{G}_{0}$ \+ 1 mM)], the system recovers
eventually the normal glucose level $G_{\infty}$. The time constants are taken
to be $\tau=\tau_{G}=1$ for simplicity and the cellular interactions have
strengths $J_{1}=2$ mM and $J_{2}=0$ mM.

#### Figure 4.

Balance function for glucose regulation, depending on the interactions between
$\alpha$\- and $\beta$-cells. The strengths of cellular interactions are
$J_{1}=2$ mM and $J_{2}=0$ mM.

#### Figure 5.

Time evolution of the average states $\langle\sigma_{\alpha}\rangle$ and
$\langle\sigma_{\beta}\rangle$ of $\alpha$\- and $\beta$-cells, starting
initially from the high-glucose state $\langle\sigma_{\alpha}\rangle=-1$ and
$\langle\sigma_{\beta}\rangle=1$ at $G=1$ mM, during the recovery to the
normal glucose level. The asymmetric interactions between $\alpha$\- and
$\beta$-cells have the strength $J_{1}=2$ mM whereas the inhibitory
interactions of $\delta$-cells are absent in (a) $J_{2}=0$ mM but present in
(b) $J_{2}=2$ mM.

#### Figure 6.

The average state $\langle\sigma_{\alpha}\rangle$ of $\alpha$-cells versus
$\langle\sigma_{\beta}\rangle$ of $\beta$-cells for the asymmetric interaction
$J_{1}=2$ mM, in the absence ($J_{2}=0$ mM) and presence ($J_{2}=2$ mM) of the
inhibitory interaction of $\delta$-cells. The dotted line along the diagonal
represents the stationary condition
$\langle\sigma_{\alpha}\rangle=\langle\sigma_{\beta}\rangle$.

#### Figure 7.

Glucose dose-responses in the activity of $\beta$-cells for the inhibitory
interaction $J_{2}=0$ and $2$ mM. The asymmetric interactions are taken to
have the strength $J_{1}=2$ mM.

Figure 1:

Figure 2:

Figure 3:

Figure 4:

(a) (b)

Figure 5:

Figure 6:

Figure 7: