Equilibrium behaviour of feedback-coupled physiological saturation kinetics

269
J. W. Dietrich
Medizinische Klinik I,
Endokrinologie und Diabetologie,
BG-Kliniken Bergmannsheil,
Klinikum der Ruhr-UniversitŠt Bochum,
Buerkle-de-la-Camp-Platz 1,
D-44789 Bochum, F. R. Germany
johannes.w.dietrich@bergmannsheil.de
B. O. Boehm
Sektion Endokrinologie
Abteilung Innere Medizin I,
Medizinische Klinik,
UniversitŠtsklinikum Ulm,
Robert-Koch-Str. 8,
D-89081 Ulm, F. R. Germany
bernhard.boehm@medizin.uni-ulm.de
Abstract
In a simplifying manner biological feedback-
control systems use to be described as linear
circuits with subtractive comparison elements,
although in living organisms most signal
transduction pathways are known to have
saturable characteristics, as described by
Michaelis-Menten-Hill kinetics.
Therefore, while linearisation methods supply a
straightforward way for modelling biological
information processing structures they also
expose the models to charges of being arbitrary.
Furthermore, they fail to provide realistic
hypotheses on the pathogenesis of diseases.
An alternative approach that comprises
formulating parametrically isomorphic models
directly incorporates empirically obtained
properties of the involved transfer elements.
With a novel universal model for nonlinear
biological processing structures (MiMe control-
model) we show that results of both analytical
solving of involved equations and computer
simulation are in better agreement with
physiology than those of traditional models.
In conclusion it seems that parametrically
isomorphic modelling of feedback control with
Michaelis-Menten-kinetics may provide a
general and universal method for characterizing
biological information processing structures.
1. Introduction
In an often unpredictable world biological feedback-
control systems (FCS) are a sine-qua-non for maintaining
the homeostasis of critical variables that determine the
organismÔs internal environment [Bernard, 1859 and
1878; Cannon, 1926]. Irrespective of their physiological
embodiment in form of e.g. nerval impulses, hormones,
cytokines or enzyme cascades they usually act as non-
linear systems, whose elements exhibit saturation
characteristics like Michaelis-Menten-Hill kinetics.
However, for decades biological control circuits have
used to be described in an oversimplifying manner as
linear systems being comprised of proportional elements
and controllers in form of subtractive modulator elements
(Fig. 1).
+
+
x(t)
e(t) yS(t)
yR(t) y(t)
z(t)
G1
G2
Fig. 1: A simple proportional feedback loop with set point x(t)
and load z(t). See appendix for a legend of symbols.
Considering the recursive description of this linear
feedback loop with

y(t) = G1x(t) - G1G2y(t) + z(t) (1.1)
we receive the equifinal solution as
y(¥ ) = G1x(¥ ) + z(¥ )1 + G1G2 . (1.2)
In the past, models of this type enjoyed great popularity
in the description of biological information processing
structures [Cruse, 1981; Ršhler, 1973; Varjœ, 1977;
Danziger and Elmergreen, 1956; DiStefano, 1969;
DiStefano and Stear, 1968; Norwich and Reiter, 1965;
Roston 1959], when they pioneered mathematical
modelling of closed feedback loops. Their
methodological principle is based on approximating the
real system by a linearised substitution system, whose
parameters are repetitively adjusted until the simulated
output corresponds to the known behaviour of the real
system. Although this approach provided new and
valuable insights into general navigational principles of
life, it lacks of injectivity, i.e. uniqueness. Therefore,
mapping of the real system to the model remains
ambiguous, since multiple different points in the
parameter space are apt to generate similar behaviour in
rest. On the other hand, simulating perturbations and
pathological situations like diseases uses to be very
difficult in approximated models. Therefore, their
Equilibrium behaviour of feedback-coupled
physiological saturation kinetics
CYBERNETICS AND SYSTEMS
2006

270
implications didnÕt cross the border to a real explanation
of the dynamics in the intact and diseased organism.
In this article we present a way to a new general theory of
non-linear feedback loops in the living organism that
covers a large class of physiological embodiments. The
core method maps the real system to a parametrically
isomorphic model that allows for direct incorporation of
parameters that have been experimentally gathered.
2. Modelling
Transition to this injective model requires three steps.
Initially, the subtractive comparison element of the FCS
is replaced by a divisive controlling element. This choice
is motivated by the fact that intercausal networks in
living organisms are positive systems, which means that
their parameters, like neuronal spike rates or hormone
concentrations, can't get negative.
In a traditional view the input value x(t) would be
interpreted as set point, but in the divisive model it adopts
the characteristics of a target proportion of the actual
value yR(t) and the error signal e(t). With z as disturbance
variable and G 1 and G 2 as linear gain factors of feed
forward and feedback path, respectively, in this quotient
FCS the controlled variable y receives its value from the
recursive equation
y(t) = z(t) + G1x(t)G2y(t) . (2.1)
In a second step some transduction elements are replaced
by those that exhibit Michaelis-Menten kinetics (MMK).
MMK describes the output signal ya as a one-to-one
function of the input signal xe with
ya = GxeD + xe (2.2)
given the two constant and characteristic parameters G
and D. Additionally, the controller is complemented by a
type of degenerative feedback inhibition that prevents the
output signal from getting infinite if the inhibiting input
signal xe2 is 0, as it is known e.g. from enzymes that are
regulated by v-type allostery (Fig. 2).
+
1
ya
xe2 xe1
Fig. 2: Pathway for inhibition in the type of v-type allostery
(VTA).
The last step consists in accounting for level-dependent
degradation of signals. Different parameters like
enzymes, metabolites, hormones, blood cells and even
the population of whole organisms in ecosystems may be
described as being located within a leaky integrating
compartment whose extinction rate is proportional to the
instantaneous level of the respective signal. A unified
model for subsystems of this form is the ASIA element
(Fig. 3) with

dya
dt = a xe(t) – b ya(t) (2.3)
For evaluating steady state the parameters of an ASIA
element may be combined to a linear gain factor

G(¥ ) = a
b , (2.4)
as demonstrated recently in [Dietrich, 2000].
+
b º
xe
a
ya
xe
G(∞) = a/b
ya
dya
dt
b ya
Fig. 3: Processing structure of an ASIA element with leaky-
integrating characteristics (left) and a simplified substitution
system for steady-state analysis (right).
With the mentioned linear and non-linear components
inserted into the described divisive FCS (Fig. 4) solving
the corresponding equation system delivers the recursive
equation
y(t) = G1G2x(t)D2 + D2G3y(t) + G1x(t) (2.5)
or, in terms of c,
c(t) = G1x(t)
1+ G3G2c(t)D2 + c(t) . (2.6)
x(t)
+
1
D2
e(t) c(t)
yR(t) y(t)
G1
G3
G2 +
VT
A MMK
Fig. 4: MiMe-model: Processing structure with a Michaelis-
Menten type transfer element (MMK) and degenerative
feedback inhibition in form of v-type allostery (VTA).
By defining
K1 = G4G3G2D4 + G3G2 (2.7)
and
K2 = D4D2D4 + G3G2 (2.8)
multiple MMK elements in series can be merged to a
unique virtual MMK element K1c(t)/[K2+c(t)], so that the
corresponding processing structures (Fig. 5 A) may be
described with

271
c(t) = G1x(t)
1 + G5K1c(t)K2 + c(t) (2.9)
formally similar to FCS with single MMK elements.
More elaborate models cover multiple control systems
organized in parallel (Fig. 5 B), whose recursive
behaviour is given in the case of two pathways with
c(t) = G1x(t)
1+ G3G2c(t)D2 + c(t) 1+
G5G4c(t)
D4 + c(t) . (2.10)
x(t)
+
+
1
D2
D4
e(t) c(t)
yR(t) y(t)
wR(t) w(t)
G1
G3
G5
G2 +
+ G4
VTA MMK 1 MMK 2
+
x(t)
D2
+
1
D4
c(t)e(t)
v(t)yR(t)
y(t)
w(t)
G2G1
G3
G4G5
+
VT
A
MMK 1
MMK 2
A
B
Fig. 5: MiMe-System with two Michaelis-Menten type transfer
elements (MMK 1 and MMK 2) in series (A) and parallel (B)
and v-type degenerative feedback (VTA).
3. Results
The recursive description (equation 2.1) of the divisive
FCS can be solved in form of a quadratic equation to
obtain the equilibrium behaviour of the information
processing structure. Its equifinal value corresponds to
the positive one of the two solutions

y(¥ )1,2 = z(¥ )2 –
G22z(¥ )2 + 4G1G2x(¥ )
2G2 . (3.1)
Incorporating MMK and v-type inhibition delivers a
nonlinear FCS (Fig. 4) that allows to be described with
the recursive equations (2.5) and (2.6).
For t fi ° we may define the three variables
a = D2G3 , (3.2)

b = D2 + G1x(t) (3.3)
and

c = – G1G2x(t) , (3.4)
so that the interrelationship can also be expressed as
quadratic equation

ay(¥ )2 + by(¥ ) + c = 0 (3.5)
with the two well-known solutions
y(¥ )1,2 = – b – b
2
– 4ac
2a . (3.6)
As all parameters of the information processing structures
are necessarily positive we can assume that

b < b2 – 4ac > 0 , (3.7)
so that one of the solutions is positive while the second one is
negative. Certainly, only the positive one of these solutions
finds a physiological realization that corresponds to the
equilibrium behaviour of the system.
In the case of two serially organized MMK (Fig. 5) the
recursive behaviour (equation 2.9) can again be solved in
the form of the quadratic equation (3.5) with the same
solutions given by equation (3.6). The distinction lies in
the parameters a, b and c that are now defined with

a = D1 + D1G2 + x(t) , (3.8)

b = D1D2 + D2x(t) – G1x(t) (3.9)
and

c = – G1D2x(t) . (3.10)
Again, relation (3.7) proves to be true, so that the
equation system has one and only one positive solution.
The systemÕs behaviour turns out to be more complex if
control systems are organized in parallel (Fig. 5). Solving
equation (2.10) delivers the cubic equation

ec(¥ )3 + f c(¥ )2 + gc(¥ ) + h = 0 (3.11)
with
e = 1 + G5G4 + G3G2 + G5G4G3G2 , (3.12)

f = D2 + D2G5G4 + D4 + G3G2D4 – G1x(t) , (3.13)

g = D2D4 – (G1D2 + G1D4)x(t) + D2D4 (3.14)
and

h = G1D2D4x(t) . (3.15)
Parallel organization is a common feature in living
systems where we often find multiple redundant
pathways. As each additional MMK-based feedback loop
n introduces a term of the form
1+ G2n +1G2n c(t)D2n + c(t) (3.16)
in the denominator of the one-loop equation (2.6) we
obtain the universal recursive equation

272
c(t) = G1x(t)
1+ G2n +1G2n c(t)D2n + c(t)Pn = 1
k
(3.17)
for dynamical feedback-regulated systems with
Michaelis-Menten kinetics. This equation solves in the
form of polynomials with grade n = k+1, where k is the
number of parallel feedback loops:

ancn(¥ ) + an – 1cn – 1(¥ ) + … + a1c(¥ ) + a0 = 0 . (3.18)
Parameters of this general parametrically isomorphic
model can unambiguously be mapped to empirically
obtained parameters of biological information processing
structures (Table 1).
Table 1: Parameters and variables of a general model with two
Michaelis-Menten-kinetics in series (MiMeMod, see Fig. 5A)
and of two corresponding endocrine processing structures.
Values from [Byrne et al. 1994], Jones et al. [1997 and 2000],
[Kraan et al. 1998], [Lundquist and Panagiotidis 1992],
[DiBartolomeis et al. 1986], [Rizza et al. 1981], [Breuninger et
al. 1993] and [Magnusson et al. 1992]. APC: Volume of
distribution, t1/2: Halflife. G1 and G3 represent ASIA-elements.
General
Model
Glucose Control Pituitary-adrenal axis
G1
(ASIA)
Calculated from APC
and t1/2 of glucose in
insulin-deficiency:
0.14 / 3.8eÐ5 sec/L
Calculated from APC
and t1/2 of ACTH:
0.4 / 0.0002 sec/L
G2 Secretion capacity of
pancreatic beta cells:
8 pmol/sec
Secretion capacity of
adrenal cortex:
1.2 nmol/sec
D2 EC50 of glucose at beta-
cells:
5e-3 mol/L
EC50 of ACTH at adrenal
cortical cells:
1eÐ11 mol/L
G3
(ASIA)
Calculated from APC
and t1/2 of insulin:
0.07 / 2.3eÐ3 sec/L
Calculated from APC
and t1/2 of cortisol:
0.05 / 1.2eÐ4 sec/L
G4 Receptor gain
normalized to 1
Receptor gain
normalized to 1
D4 EC50 of Insulin at
peripheral tissue:
1.5eÐ9 mol/L
EC50 of cortisol at tissue
receptors:
2eÐ7 mol/L
G5 Effector gain
estimated as 100
Effector gain
normalized to 1
x(t) Glucose production rate:
10.5 µmol/sec
Hypothalamic input
(CRH signal)
e(t) Release rate of glucose Secretion rate of ACTH
c(t) Blood glucose level ACTH level
y(t) Secretion rate of insulin Secretion rate of cortisol
w(t) Insulin level Cortisol level
v(t), yR(t) 2nd and 3rd messengers Intracellular signals
Table 2: Equifinal levels of variables resulting from numerical
simulations with parameters of Table 1.
Signal S1 S2
Glucose Ð2.4 mmol/L 4.7 mmol/L
Insulin Ð220 pmol/L 120 pmol/L
ACTH Ð2.8 pmol/L 6.9 pmol/L
Cortisol Ð160 nmol/L 170 nmol/L
Analytically solving the model equations yields two
results, respectivly, one of them being negative and the
other solution being positive. The positive solution is
identical to the corresponding equifinal result of numeric
simulations. These latter results lie within well-known
physiological reference ranges (Table 2 and Fig. 6).
Table 3: Behaviour of the model from table 1 (MiMeMod)
compared with a traditional linear model (LinMod) [Bolie
1961] and the quasi-standard minimal model (MinMod)
[Bergman et al. 1979 and 1980] in extreme situations of insulin-
glucose metabolism. GIR: Glucose infusion rate, IAR: Insulin
appearance rate, [Glc]: Glucose plasma level, IS: Insulin
Secretion. *Examples are insulinoma and insulin overdosage.
Experiment LinMod MinMod MiMeMod
High finite GIR High finite IS Infinite IS Limited IS
Very high IAR* [Glc] < 0 [Glc] ³ 0 [Glc] ³ 0
4. Discussion
In this paper we introduced a parametrically isomorphic
approach for characterising biological information
processing structures with saturable subsystems, as they
are to be found in endocrine and enzymatic signalling
chains. Solving the model equations both analytically and
by means of computer simulation delivers results that
mutually agree and that are located within known
reference ranges.
Up to now, parametrically isomorphic modelling has
been applied for a few biological control systems, e.g.
pituitary-thyroid interaction [Dietrich et al., 2004;
DiStefano et al., 1975], albeit most often for open-loop
subsystems only.
Here we show a more universal way to describe closed
loop systems with parametrically isomorphic modelling.
Although biological FCS may exhibit a plethora of
different structures we could show that typical
configurations allow to be mapped to a common general
model. The mathematical aspects of this model are
represented by a universal recursive equation and its
polynomial solution.
Certainly, this methodoloy turns out to be mathematically
more complex than traditional approximation by linear
models [Bolie, 1961; Danziger and Elmergreen, 1956;
Norwich and Reiter, 1965; Roston, 1959]. Furthermore,
classical stability measures as they are applied in control
technology [Cruse, 1981; Ršhler, 1973; Varjœ, 1977;
Bateson, 1996] are no longer directly applicable.
On the other hand, non-linear technique delivers better-
founded results as most model parameters can be mapped
to empirically obtained values like dissociation constants
and maximum velocities of enzymes. Furthermore,
MiMe-models are immune to certain problems like
physiologically impossible infinite or negative results
that traditional approaches suffer from (Table 3). Finally,
for dynamic studies this approach paves a better-founded
way to approximate real systems by linear substitution
models, if the parameters of the linear equations (slope
and axis intercept) are obtained from the tangents of the
corresponding non-linear functions at the equilibrium
point found in non-linear analysis.
Therefore, parametrically isomorphic models may be
successfully applied to study dynamics of various
disorders affecting biological information processing,
where linear models failed. This can be demonstrated
with examples of common endocrine diseases like
diabetes mellitus, or thyroid disorders or homeostatic
responses of the adrenal and/or pituitary gland [Dietrich
et al, 2004].

273
A B
Adrenal
Response
Pituitary Response
0 2 4 5 8 10 12
ACTH
pmol/L
50
100
150
200
250
300
Cortisol
nmol/L
Fixpoint
Reference
Range
5 10 15 20
Blood
Glucose
mmol/L
100
200
300
400
500
Insulin
Level
pmol/L
Po
st
pr
an
di
al
R
ef
er
en
ce
R
an
ge
Fi
xp
oi
nt
Pa
th
ol
og
ica
l
G
lu
co
se
T
ol
er
an
ce
Diabetes
mellitus
Fasting
Reference
Range
Impaired
Fasting
Glucose
Beta-Cell
Response
M
et
ab
ol
ic
R
es
po
ns
e
Fig. 6: Iteration plots of the two endocrine feedback control systems from table 1 and 2. The left example (A) illustrates the effects of
an oral glucose tolerance test showing short overshooting compensation. The right pane (B) shows how a greater distance of the
fixpoint from the curveÕs saturation area causes that control of adrenocortical function is more robust than glucose control.
The solid line illustrates the response of beta-cells to blood glucose (A) or adrenal gland to ACTH (B). The grey line (mirrored at the
bisector) represents reaction of peripheral tissues to insulin or pituitary gland to cortisol levels, respectively. The intersection point of
both lines marks the fixpoint of the feedback control system with the equifinal levels of its variables.
5. References
[Bateson, R.N., 1996] Bateson, R.N., Introduction to
control system technology. 5 ed. 1996, Upper Saddle
River, NJ: Prentice Hall. 784.
[Bergman, R.N., et al., 1979] Bergman, R. N., Ider, Y. Z.,
Bowden, C. R., and Cobelli, C., Quantitative
estimation of insulin sensitivity. Am. J. Physiol.,
1979. 136(6): p. E667-677.
[Bernard, C., 1859] Bernard, C., Leons sur les propriŽtŽs
physiologiques et les altŽrations pathologiques des
liquides de lÕorganisme. 1859, Paris: Librairie J.B.
Baillire et fils.
[Bernard, C., 1878] Bernard, C., Les phŽnomnes de la
vie. 1878, Paris: Žditions Baillre.
[Bolie, V.W., 1961] Bolie, V.W., Coefficients of normal
blood glucose regulation. J. Appl. Physiol., 1961.
16(5): p. 783-788.
[Breuninger, L.M., et al., 1993] Breuninger, L.M., et al.,
Hydrocortisone regulation of inte69rleukin-6 protein
production by a purified population of human
peripheral blood monocytes. Clin Immunol
Immunopathol, 1993. 69(2): p. 205-214.
[Byrne, M.M., et al., 1994] Byrne, M.M., J. Sturis, K.
Clement, N. Vionnet, M.E. Pueyo, M. Stoffel, J.
Takeda, P. Passa, D. Cohen, G.I. Bell, and et al.,
Insulin secretory abnormalities in subjects with
hyperglycemia due to glucokinase mutations. J Clin
Invest, 1994. 93(3): p. 1120-1130.
[Cannon, W.B., 1926] Cannon, W.B., Physiological
regulation of normal states: some tentative postulates
concerning biological homeostatics. Jubilee volume
for Charles Richet, 1926: p. 91-93.
[Cascieri, M.A., et al., 1999] Cascieri, M.A., et al.,
Characterization of a Novel, Non-peptidyl Antagonist
of the Human Glucagon Receptor. H. Biol. Chem.,
1999. 274(13): p. 8694-8697.
[Cruse, H., 1981] Cruse, H., Biologische Kybernetik.
studium biologie, ed. W. Nachtigall. 1981, Weinheim,
Deerfield Beach (Fl.), Basel: Verlag Chemie. 101.
[Cypess, A.M., et al., 1999] Cypess, A.M., C.G. Unson,
C.R. Wu, and T.P. Sakmar, Two Cytoplasmic Loops
of the Glucagon Receptor Are Required to Elevate
cAMP or Intracellular Calcium. J. Biol. Chem., 1999.
274(27): p. 19455-19464.
[Danziger, L. and G.L. Elmergreen, 1956] Danziger, L.
and G.L. Elmergreen, The Thyroid-Pituitary
Homeostatic Mechanism. Bulletin of Mathematical
Biophysics, 1956. 18: p. 1-13.
[DiBartolomeis, M.J., C. Williams, and C.R. Jefcoate,
1986] DiBartolomeis, M.J., C. Williams, and C.R.
Jefcoate, Inhibition of ACTH Action on Cultured
Bovine Adrenal Cortical Cells by 2,3,7,8-
Tetrachlorodibenzo-p-dioxin through a Redistribution
of Cholesterol. J. Biol. Chem., 1986. 261(10): p.
4432-4437.
[Dietrich, J.W., 2000] Dietrich, J.W., Signal Storage in
Metabolic Pathways: The ASIA Element.
kybernetiknet, 2000. 1(3, 2000): p. 1-9.
[Dietrich, J.W. and B.O. Boehm, 2004] Dietrich, J.W.
and B.O. Boehm, Antagonistic Redundancy Ð A
Theory of Error-Correcting Information Transfer in
Organisms, in Cybernetics and Systems 2004, R.
Trappl, Editor. 2004, Austrian Society for Cybernetic
Studies: Vienna. p. 225-230.
[Dietrich, J.W., et al., 2004] Dietrich, J.W., A. Tesche,
C.R. Pickardt, and U. Mitzdorf, Thyrotropic Feedback
Control: Evidence for an Additional Ultrashort
Feedback Loop from Fractal Analysis. Cybernetics
and Systems, 2004. 35(4): p. 315-331..
[DiStefano, J.J., 1969] DiStefano, J.J., A Model of the
Normal Thyroid Hormone Glandular Secretion
Metabolism. J. Theoret. Biol., 1969. 22: p. 412-417.

274
[DiStefano, J.J. and E.B. Stear, 1968] DiStefano, J.J. and
E.B. Stear, Neuroendocrine Control of thyroid
secretion in living systems: a feedback control system
model. Bulletin of Mathematical Biophysics, 1968.
30: p. 3-26.
[DiStefano, J.J., et al., 1975] DiStefano, J.J., K.C.
Wilson, M. Jang, and P.H. Mak, Identification of the
Dynamics of Thyroid Hormone Metabolism.
Automatica, 1975. 11: p. 149-159.
[Hjorth, S.A., C. ¯rskov, and T.W. Schwartz, 1998]
Hjorth, S.A., C. ¯rskov, and T.W. Schwartz,
Constitutive Activity of Glucagon Receptor Mutants.
Molecular Endocrinology, 1998. 12(1): p. 78-86.
[Hoefig, B., et al., 1996] Hoefig, B., A. Kistner, A.
Seibold, and B. Boehm, Extended physiological
models for the simulation of the glucose metabolism.
Math Modelling Systems, 1996. 2: p. 41-54.
[Jones, C.N., et al. 2000] Jones, C.N., F. Abbasi, M.
Carantoni, K.S. Polonsky, and G.M. Reaven, Roles of
insulin resistance and obesity in regulation of plasma
insulin concentrations. Am J Physiol Endocrinol
Metab, 2000. 278(3): p. E501-508.
[Jones, C.N., et al. 1997] Jones, C.N., D. Pei, P. Staris,
K.S. Polonsky, Y.D.I. Chen, and G.M. Reaven,
Alterations in the Glucose-Stimulated Insulin
Secretory Dose-Response Curve and in Insulin
Clearance in Nondiabetic Insulin-Resistant
Individuals. Journal of Clinical Endocrinology and
Metabolism, 1997. 82(6): p. 1834-1838.
[Kraan, G.P., et al. 1998] Kraan, G.P., R.P. Dullaart, J.J.
Pratt, B.G. Wolthers, N.M. Drayer, and R. De Bruin,
The daily cortisol production reinvestigated in healthy
men. The serum and urinary cortisol production rates
are not significantly different. J Clin Endocrinol
Metab, 1998. 83(4): p. 1247-1252.
[Kruszynska, Y.T., et al., 1998] Kruszynska, Y.T., S.
Goulas, N. Wollen, and N. McIntyre, Insulin
secretory capacity and the regulation of glucagon
secretion in diabetic and non-diabetic alcoholic
cirrhotic patients. J. Heptatol., 1998. 28(2): p. 280-
291.
[Kurstjens, N.P., et al., 1990] Kurstjens, N.P., H.
Heithier, R.C. Cantrill, M. Hahn, and F. Boege,
Multiple hormone actions: the rises in cAMP and
Ca++ in MDCK-cells treated with glucagon and
prostaglandin E1 are independent processes. Biochem
Biophys Res Commun, 1990. 167(3): p. 1162-1169.
[Lundquist, I. and G. Panagiotidis, 1992] Lundquist, I.
and G. Panagiotidis, The relationship of islet
amyloglucosidase activity and glucose-induced
insulin secretion. Pancreas, 1992. 7(3): p. 532-537.
[Lynch, C.J., et al., 1996] Lynch, C.J., K.M. McCall,
Y.C. Ng, and S.A. Hazen, Glucagon stimulation of
hepatic Na+-pump activity and a-subunit
phosphorylation in rat hepatocytes. Biochem. J.,
1996. 313: p. 983-989.
[Magnusson, I., et al., 1992] Magnusson, I., D.L.
Rothman, L.D. Katz, R.G. Shulman, and G.I.
Shulman, Increased rate of gluconeogenesis in type II
diabetes mellitus. A 13C nuclear magnetic resonance
study. J Clin Invest, 1992. 90(4): p. 1323-1327.
[Marks, J.S. and L.H.P. Botelho, 1986] Marks, J.S. and
L.H.P. Botelho, Synergistic Inhibition of Glucagon-
induced Effects on Hepatic Glucose Metabolism in
the Presence of Insulin and a cAMP Antagonist*. The
Journal of Biol. Chem., 1986. 261(34): p. 15895-
15899.
[Norwich, K.H. and R. Reiter, 1965] Norwich, K.H. and
R. Reiter, Homeostatic Control of Thyroxin
Concentration Expressed by a Set of Linear
Differential Equations. Bulletin of Mathematical
Biophysics, 1965. 27: p. 133-144.
[Rizza, R.A., L.J. Mandarino, and J.E. Gerich, 1981]
Rizza, R.A., L.J. Mandarino, and J.E. Gerich, Dose-
response characteristics for effects of insulin on
production and utilization of glucose in man. Am. J.
Physiol., 1981. 240(Endocrinol. Metab. 3): p. E630-
639.
[Ršhler, R., 1973] Ršhler, R., Biologische Kybernetik.
StudienbŸcher der Biologie, ed. H. Stieve and E.
Hildebrand. 1973, Stuttgart: B. G. Teubner. 180.
[Roston, S., 1959] Roston, S., Mathematical
Represention of Some Endocrinological Systems.
Bulletin of Mathematical Biophysics, 1959. 21: p.
271-282.
[Toffolo, G., et al., 1980] Toffolo, G., Bergman, R. N.,
Finegood, D. T., Bowden, C. R., and Cobelli, C.,
Quantitative estimation of beta cell sensitivity to
glucose in the intact organism. Diabetes, 1980. 29: p.
979-990.
[Varjœ, D., 1977] Varjœ, D., Systemtheorie fŸr Biologen
und Mediziner. Heidelberger TaschenbŸcher. Vol.
182. 1977, Berlin, Heidelberg, New York: Springer.
285.
[Zech, R. and G. Domagk, 1986] Zech, R. and G.
Domagk, Enzyme. edition medizin. 1986, Weinheim:
VCH.
Appendix
Legend for IPS symbols:
x1
x2
y
Multiplication Point
x2(t)y(t) =

x1(t)
Filled sectors denote inversion
(here: division).
Empty input signals are neutral
(1 for muliplication and division).
+
x1
x2
y
Addition Point
y(t) = x2(t) – x1(t)
Filled sectors denote inversion
(here: subtraction).
Empty input signals are neutral
(0 for addition and subtraction).
G yx
Constant Gain
y(t) = Gx(t)
yx
Special Function
y(t) = ∫ x(t) dtº
Symbols used (see also [Dietrich and Boehm, 2004] for reference)