We have studied the peculiarities of the
nonlinear compartmental model for human Sr metabolism (Staub JF, Foos
E, Courtin B, Jochemsen R, and Perault-Staub AM. Am J
Physiol Regul Integr Comp Physiol 284: R819-R834, 2003),
including its physiological reliability in the context of Sr-Ca
similarity-dissimilarity. We found it to be relevant to Ca metabolism,
except for discrimination against Sr relative to Ca at urinary and
intestinal levels. The main findings are as follows: 1) the
saturable part of intestinal absorption, shared by Sr and Ca, does not
seem to be responsible for the discrimination of the transcellular
pathway; 2) although there is little discrimination in bone,
the physicochemical behaviors of Sr and Ca at the bone surface differ,
at least quantitatively; and 3) Sr behaves as a "tracer"
for Ca metabolic pathways and, under non-steady-state conditions, can
also reveal self-regulatory processes. It is suggested that they depend
on Ca2+ (cationic)-sensing receptors that are apparently
more sensitive to Sr than to Ca. Acting on gastrointestinal and
osteoblast lineage cells, these slow processes might contribute to
adaptive, rather than homeostatic, regulation of Ca metabolism.
Understanding these features could help clarify the pharmacological and
therapeutic effects of oral Sr.
strontium administration; calcium-strontium discrimination; self-regulatory process; calcium (cationic)-sensing receptor
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INTRODUCTION |
IN THE ABSENCE OF
EVIDENCE for a physiological role for Sr, the biological interest
in this mineral and its metabolism was focused on analogies and
discrepancies with Ca2+ as its closest divalent cation in
the periodic table and the essential mineral in numerous key
extracellular and intracellular processes. Sr and Ca are members of the
alkaline earth series, with comparable features of cation chemistry
(ionic radius, charge-to-size ratio, high coordination number, and
H2O exchange) (46). Thus they show
physicochemical similarities in their interactions with organic or
inorganic components. These similarities are such that the major
metabolic pathways of both cations, intestinal absorption, deposition
in and removal from bone, and excretion in urine and feces, are
believed to involve identical processes. However, in vivo and in vitro
kinetic studies, often using a tracer radionuclide of Sr2+
and Ca2+ (27, 36), have revealed, at least
quantitatively, behavioral differences at gastrointestinal (GI), renal,
and, possibly, bone levels (10, 42). They have been
interpreted in terms of discrimination between Sr and Ca
(44). Globally, retention of Ca relative to Sr is favored
at the organism level.
Physicochemical analytic studies have shown that synthetic
hydroxyapatite (HA), with partial substitution of Ca2+ by
Sr2+ into the crystal lattice, can be produced from calcium
phosphate-supersaturated aqueous solutions containing Sr2+,
the level of discrimination against Sr2+ relative to
Ca2+ being directly dependent on the rate of crystal growth
and the associated crystal perfection (24, 30). Recently,
Sr2+ in solution was described as easily adsorbed at the
crystal surface and as inhibiting the rates of HA crystal growth and
dissolution (9). In vivo data indicate that
Sr2+ can be incorporated into biological apatite (5,
12), with possible limitation in the number of substitutions.
For situations in which cations interact with organic components, the
behavioral differences between Ca and Sr may be analyzed in terms of
the physicochemical, stereochemical, and structural characteristics of
the organic species, i.e., from nearly no difference between ion
reactivities to a nearly complete specificity for Ca as a result of the
small difference between Ca and Sr chemistry.
Among the main factors involved in discrimination between Sr and Ca are
their relative abundance and availability in the natural environment
(46). Sr is a trace element, whereas Ca is a
macronutrient. Their concentrations in biological fluids also differ
greatly. There are extracellular fluids with millimolar concentrations of Ca and micromolar concentrations of Sr. The very high extracellular Ca-to-Sr molar ratio means that Sr cannot significantly compete with Ca
under physiological conditions; therefore, there has been little need
for selection, during evolution, of mechanisms specific for Ca, rather
than Sr. In vitro experiments confirm this. Sr2+ can
replace Ca2+ in activating the parathyroid
Ca2+-sensing receptor (PCaR) (25) at a
concentration that is twice that needed for Ca2+.
Similarly, the apical Ca2+ channels isolated from intestine
(31) and kidney (41) are also permeable to
Sr2+, with higher apparent permeability for
Ca2+ than for Sr2+. However, the intracellular
fluid contains 0.1 µM free Ca2+ (Cai) and
Cai is an important second messenger for numerous cellular processes (e.g., proliferation, differentiation, apoptosis, and activity). Inasmuch as Sr is not as effectively regulated as
Cai, their intracellular concentrations might be similar,
hence, the requirement for high specificity for Ca by most
intracellular proteins influencing resting Cai and
Cai signaling and regulatory mechanisms. For instance,
Sr2+ is 600-fold less potent than Ca2+ in
causing calmodulin-induced inhibition of liver inositol
trisphosphate-induced Ca2+ release, probably because
Sr2+ binds 30 times less well than Ca2+ to
calmodulin (29).
This high degree of discrimination against Sr relative to Ca for vital
intracellular functions is likely the reason for hypocalcemia and/or
hypocalcified bone and the decrease in 1,25-dihydroxyvitamin D
synthesis, which are the first manifestations of Sr administration, only if there is a drastic increase in plasma Sr concentration. Nevertheless, plasma Sr concentration can be dose dependently increased
100-fold over a large range of Sr salt administration, while a molar
Ca-to-Sr ratio higher than unity is always maintained. This has no
apparent toxic effect and causes no significant change in plasma Ca
concentration or in calcitropic hormones (17, 26).
We have reported the development of a nonlinear compartmental model for
human Sr metabolism (39). In contrast to most mathematical analyses of Sr kinetics, our model focuses on Sr metabolism per se,
postponing its comparison with Ca metabolism. Because it is based on
non-steady-state kinetic data, it includes several nonlinearities that
must be interpreted to validate our model. The primary concern of this
work is to study, within the context of the Sr-Ca
similarity-dissimilarity outlined above, the relevance of the model to
Ca metabolism and to yield and illustrate, using model refinements,
suitable explanations about the identified nonlinearities and their
role in the homeostatic and/or adaptive regulation of this metabolism.
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NONLINEAR COMPARTMENTAL MODEL |
Briefly, the compartmental model described previously
(39) was based mainly on plasma Sr concentration kinetic
data collected in postmenopausal women given twice-daily oral doses of
Sr (S-12911, Institut de Recherches Internationales Servier). These
non-steady-state data were obtained for four doses of Sr (1.95, 3.89, 7.78, and 15.57 mmol/day) and included the increase in plasma Sr
concentration during Sr administration (AdP, the first 25 days) and its
decrease after cessation of treatment [postadministration period
(PAdP), the other consecutive 27 days]. The plasma Sr kinetics and the fit to these experimental data obtained at the end of model building have been described previously (Fig. 1 in Ref. 39).
The model consists of two distinct, but interdependent, compartmental
systems (Fig. 1).
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Fig. 1.
Nonlinear compartmental model of Sr kinetics developed in
Ref. 39. Solid lines, overall Sr metabolism (system
1); dashed lines, biological variables other than Sr (z
variables of system 2 interacting with system 1).
Circles numbered 1 to 7 are kinetically distinct entities associated
with Sr metabolism in bone (compartments 2, 6, and
3), gastrointestinal (GI) tract (compartments 4 and 5), and the internal distribution pool
(compartment 1, including plasma, and compartment
7). Circles numbered (n + 1) and
(n + 2) are compartments representing
z(n+1) and
z(n+2) (system 2).
k and K denote fractional transfer coefficients
(kij) and fractional transfer functions
(Kij); F(t) represents
input pathways associated with dietary Sr and oral Sr doses. Bold
arrows denote nonlinearities that are intrinsic [Michaelis-Menten
(M-M) for K15 and Langmuir-type for
K21] or extrinsic due to reciprocal
interactions between system 1 and system 2 (curved arrows). For further explanation, see APPENDIX A.
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System 1 is directly concerned with Sr metabolism and
includes seven compartments describing whole body Sr and its relations to the surrounding milieu. These compartments are organized into three
subsystems. 1) The GI tract consists of two compartments (compartments 4 and 5): one (compartment
4) accepts the Sr from food (and the oral Sr administration), and
the other (compartment 5) is associated with the
irreversible excretion of Sr in the feces and the bidirectional
transfer of Sr to (intestinal absorption) or from (endogenous
intestinal secretion) the internal distribution pool (IDP) via
compartment 1. 2) The bone subsystem has three compartments connected to compartment 1 through
bidirectional (compartments 2 and 3) or
unidirectional (compartment 6) relations. Compartment
6 is directly concerned with irreversible Sr movements associated
with bone mineral accretion and removal. Bone formation operates from
one compartment (compartment 2) lying between
compartment 1 and compartment 6. 3)
The IDP consists of two compartments. From compartment 1, Sr
is excreted via the kidneys; compartment 1 includes plasma
and other extracellular and, probably, intracellular sectors in rapid
equilibrium with the plasma [apparent distribution volume
(Vapp) ~40 liters]. Compartment 7 is a
rapidly exchangeable pool with no precise physiological identity. This
system 1 also contains two intrinsic nonlinearities (INL)
that involve nonlinear relations to a given system 1 compartment. These INL account for a saturable Michaelis-Menten
(M-M)-type process (Eq. A1) involving intestinal absorption
and for a Langmuir-type reaction (Eq. A3) involving mineral
transfer from the extracellular fluids to the bone. This second
nonlinearity does not behave as a simple saturable process, because it
includes the inhibitory effect of one compartment that is different
from the source compartment 1, compartment 6, or
compartment 3, according to whether one refers to one or the other of both optimal models retained in the companion article (39). With reference to this inhibition variable, the
optimal model here is model L6 or model
L3.
System 2 is relative to biological variables other than Sr
[compartments (n + 1) and (n + 2); Fig. 1]. Each extrinsic variable is associated with a
one-compartment structure with an entry flow as an S-shaped growth
function known as the logistic equation, affected by the Sr
concentration in compartment 1 with a high (~3 or more)
cooperativity order (see Eqs. A5 and A6 for
formulation). It acts on system 1 through feedback
modulation of some Sr transfer fluxes. The action of system
2 on system 1 introduces additional nonlinear functions
in system 1, called extrinsic nonlinearities (ENL). The
model includes two parameter-distinct time-explicit variables:
z(n+1), acting on the intestinal
endogenous secretion rate and on the Langmuir-type process of Sr
transfer from compartment 1 to bone compartment
2, and z(n+2), acting on the
saturable intestinal Sr absorption rate.
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PHYSIOLOGICAL INTERPRETATION AND MODEL REFINEMENTS |
We have examined the meaning of the structure (Fig. 1) by testing
the physiological reliability of the optimal models (models L6 and L3) and their relevance to Ca metabolism at four
levels: 1) the initial steady state of mineral metabolism
itself (for Sr and Ca), 2) the physiological significance of
intestinal and bone INL and their applicability to Ca metabolism,
3) the kinetic and dynamic properties of the z
logistic variables and their possible extension from dependence on Sr
to dependence on Ca, and 4) the nature of the complex
interactions between INL and ENL in system 1 and their
potential meaning for the regulation of some Ca metabolic pathways and
for the effects of Sr on Ca metabolism.
The relevance of the Sr model to Ca metabolism was checked with the
assumption that compartment 1 has a constant concentration of 2.5 or 1.25 mM. The experimental data (not reported) indicate that
Sr has no effect on total and/or ionized plasma Ca concentration.
Model refinements were proposed to theoretically illustrate some
physiological interpretations. If required, parameter estimation and a
posteriori identifiability with precision of parameter values expressed
as coefficient of variation (CV, in %) were carried out as described
previously (39).
Initial Steady State: From Sr to Ca Metabolism
The mineral mass distribution and mean daily transfer rates for Sr
and Ca metabolism can be computed for system 1 with the assumption that Sr and Ca metabolism are in steady state under physiological conditions. As discussed elsewhere (39),
with use of the set of parameter, initial condition, and
Vapp estimated from model L6 (similar results
are obtained for model L3), the predicted characteristics
for the whole of Sr metabolism are in agreement with published data.
The Sr mass in the IDP is very low compared with the predicted amount
of Sr in bone (compartment 6 contains >96% of the total
body Sr mass). Estimates of the mean daily rate for ingested Sr (1.54 mg/day), urinary excretion (0.32 mg/day), and net intestinal absorption
(~20% of the ingested Sr) are consistent with known Sr physiology.
The main change required to apply the same model to Ca metabolism
concerns compartment 1 in system 1. We used the
total plasma Ca concentration (Y1 = 2,500 µM), instead of the physiological plasma Sr concentration (y1
0.5 µM). All the other model parameters
were unchanged, except k01 was divided by 2. Indeed, the urinary clearance of Ca (2.45 ml/min) was estimated to be
about one-half that of Sr (4.77 ml/min) using the Ca experimental data
collected just before, during, and after the period of oral Sr
administration. The extrinsic z variables, although
kinetically influenced by Sr (and perhaps implicitly by Ca as described
below), do not directly depend on the mineral species. Consequently,
under physiological conditions (at time 0), they act
identically on each fractional transfer function (FTF) that they
modulate (K15, K51, and
K21; Fig. 1), regardless of the mineral
metabolism concerned. Unlike a linear model, shifting system
1 from Sr to Ca metabolism does not give obvious results, because
the INL are sensitive to the absolute concentrations of the variables
they involve. However, in the initial steady state, the only effective
INL is the M-M function with its compartment 5 mineral (Sr
and/or Ca) concentration dependence. Indeed, according to the
normalized version of the other INL, Langmuir-type nonlinearity
(Eq. A3), the value of the transfer function from
compartment 1 to compartment 2 is independent of the value of the inhibition variable at time 0.
The results presented in Table 1 show
that applying model L6 to Ca metabolism gives rise to a
number of predicted theoretical values that agree with the known
characteristics of Ca metabolism. With application of the same model
parameters for Ca and Sr, except for the urinary excretion, the
relative mass distribution of Ca within the body (IDP and bone) is
similar to that of Sr. The total Ca mass is ~900 g, with >800 g in
compartment 6 and only ~30 g (3.4% of the total mass) in
the other compartments. The fluxes into and out of compartment
6 (bone mineral solid phase) give a Ca bone turnover of ~600
mg/day, 1.6 times the Ca urinary excretion (340 mg/day), with a mean
residence time in compartment 6 of ~4 yr. On the contrary,
applying the GI parameter to Ca metabolism reveals a considerable
discrepancy between Sr and Ca. The model predicts an unrealistic value
of >9 g/day for mean Ca ingestion, whereas the net intestinal balance
(340 mg/day) represents <4% of the predicted Ca daily ingestion
(~96% is excreted in feces), although its absolute value is not
inconsistent. This meaningless model prediction can be made realistic
by reducing K51, the value of the FTF linked to
the return of mineral from compartment 1 toward
compartment 5 (Fig. 2).
Dividing k51 by 8-10 gives an expected daily Ca ingestion of ~1 g, with a net intestinal absorption >30%. This causes no change in the other properties that fit the expected processes involved in Ca metabolism of these human subjects. Thus, in
addition to urinary excretion, our results suggest that some of the
mechanisms influencing the bidirectional relation of the GI compartment
to compartment 1 differ quantitatively between Ca and Sr
metabolism (see below).
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Fig. 2.
Initial steady-state prediction for daily dietary Ca
intake required when model L3 is applied to Ca metabolism as
a function of decreasing intestinal endogenous mineral secretion
(K51 divided by an increasing divisor number).
Dashed lines delimit the expected physiological range of daily dietary
Ca intake.
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GI and Bone INL
The INL include one extra assumption that may be important for the
kinetic and dynamic behavior of our model, in contrast to the ENL,
which depend on the time-explicit z variable. This assumption is the pseudo-steady-state hypothesis, which assumes that
any intermediary step involved in a substrate-product transformation (e.g., mineral transfer from one compartment to another) is in rapid
equilibrium compared with the overall transformation rates. It is
possible to test the reasonableness of such an assumption (see
APPENDIX B). This is important, because the INL (M-M and
Langmuir) in our model are modulated by the extrinsic (time-explicit) z variables. Thus we will compare experimental data with the
response of refined models, making explicit the kinetic behavior of
intermediary steps neglected under the intrinsic formulation. Only
after that, will we attempt any physiological interpretation of the
processes of intestinal mineral absorption or mineral transfer from the extracellular fluids to bone.
Intestinal M-M-type process: Sr-Ca similarity-dissimilarity.
The transfer function K15 from GI
compartment 5 to compartment 1 in our model (Fig.
1) is governed by an M-M equation in addition to a simple linear
process (Eq. A1). This kind of representation has been used
to account for in vitro data on Ca intestinal absorption (45), demonstrating that the transfer of mineral from the
GI compartment to extracellular fluids is the sum of two distinct processes: one is saturable and mediated by species required for the
intracellular mineral transfer but present in limited amounts; the
other is unsaturable and remains proportional to mineral concentration in the lumen.
Attempts to explicitly introduce the intermediary species presumably
involved in the saturable carrier-mediated process (see APPENDIX
B) have failed to give results that improve the fit to the
experimental data or give more precise supplementary parameter values.
These parameters are also high, in agreement with the rapid equilibrium
(pseudo-steady-state) assumption. Thus the M-M equation seems adequate.
Figure 3 illustrates the type of behavior
predicted from the FTF K15 parameter values
identified for model L6 at time 0, i.e., under
physiological conditions, and shows the linear and nonlinear dependence
of the intestinal absorption rate on the apparent mineral concentration
in compartment 5. The maximum rate of the saturable process,
k
z(n+2)(0), can be estimated to be 21.6 mmol/day, using Vapp, or >1.8
g/day of Sr. This value is >103 times the predicted Sr
daily intake and, thus, has no reliable meaning in terms of the
regulation of Sr intestinal absorption in physiological conditions.
Similarly, k
, which defines the
concentration in compartment 5, from which the rate of
transfer to compartment 1 is half-maximal, is ~0.8 mM if
the intestinal volume is assumed to be on the order of 1 liter. This
millimolar estimate for k is much higher
than the expected physiological Sr concentration in intestinal juice.
On the contrary, the maximum rate and k values are quite consistent with a physiological process directly concerned with intestinal Ca absorption. The half-maximal concentration of the saturable process agrees with the mechanism of facilitated entry
that dominates at low luminal Ca concentrations (38) and saturates at 0.4-1 mM (14, 31). Furthermore, the
maximal daily rate for Ca, ~860 mg/day, is consistent with, but
slightly lower than, the normal daily Ca intake.
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Fig. 3.
Predicted dependence of linear (solid line) and nonlinear
(dashed line) Sr intestinal absorption rates on compartment
5 concentration. Values are expressed as apparent concentration.
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Nevertheless, the discrepancy between Sr and Ca (see Initial
Steady State: From Sr to Ca Metabolism), which is apparently linked to the amount of mineral secreted from compartment 1 to the GI compartment, prompted us to look for a plausible biological mechanism for the bidirectional relation of the GI compartment to
compartment 1 indicated by our model. Although it is
generally accepted that mineral may be secreted from extracellular
fluids into the gut, there is still debate as to whether this mineral is reabsorbed by the intestine and whether Ca and Sr are secreted into
the intestinal lumen via paracellular and/or transcellular routes
(23). The models developed here indicate negligible fecal loss of Sr directly from compartment 1 but significant
intestinal reabsorption of endogenous Sr. This, together with the fact
that k and k51,
which describe the maximum rate of saturable absorption and the
endogenous intestinal secretion, are influenced by the same kind of
ENL, led us to examine a model that dissociates the transcellular and
paracellular pathways and, consequently, the saturable and nonsaturable
parts of intestinal absorption. We included an intermediary compartment
(compartment 1') between compartments 1 and
5, possibly representing the mineral within intestinal
epithelial cells. Figure 4 shows the
three assumptions: 1) compartment 1' is in linear
exchange with compartment 1, 2) the linear part
of intestinal absorption (the paracellular route) operates directly
from compartment 5 to compartment 1, and, thus, the fractional transfer function from compartment 5 to the
intermediary compartment, K1'5, obeys M-M
kinetics, and 3) the transfer of mineral from the
intermediary compartment to compartment 5 and the maximum
rate of the saturable process are modulated by the extrinsic variables
z(n+1) and
z(n+2). These assumptions allow the
model response [quite identical to that obtained from the initial
model L6 (see Fig. 1 in Ref. 39)] to be
correctly fit to experimental data without any significant change in
parameter values other than those directly related to the intermediary
compartment. However, the great inaccuracy of some parameter values
precludes any quantitative interpretation of these results.
Consequently, only the high turnover rate between the intermediary
compartment (compartment 1') and compartment 1 and its initial value, which is much lower than that of
compartment 1, must be pointed out. The parameters defining
the saturable process did not change significantly.
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Fig. 4.
Refined nonlinear compartmental substructure for GI Sr
metabolism. Compartments are as shown in Fig. 1, except for
intermediary compartment 1', which is associated with
intestinal intracellular mineral. Rate-limiting (M-M-type) process of
transcellular mineral transport from GI compartment 5 to
compartment 1 is the apical entry into epithelial cells
indicated by K1'5 that, as for
K51', is modulated by a z extrinsic
variable. k15 denotes paracellular nonsaturable
(linear) intestinal absorption. See Fig. 1 legend for explanation of
symbols.
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Even if there is no justification from a strict modeling point of view,
this model can lead to interesting physiological interpretation. In
agreement with the GI model shown in Fig. 4, the first step in the
transcellular process of intestinal absorption, the Ca uptake by
enterocytes (K1'5), is given as a first
saturable pathway and a decisive rate-limiting step in the overall
process (41). The two subsequent steps in transcellular
intestinal absorption, cytosolic facilitated diffusion
[calbindin(s)-dependent] and active extrusion across the basolateral
membrane (via Ca2+-ATPase and/or
Na+/Ca2+ exchanger) (45), are
parts of the rapid transfer of mineral from the intermediary
compartment to compartment 1, k11'. Our model indicates that there are pathways opposing this process, with
the transcellular transfer from compartment 1 to
compartment 5 passing through the intermediary compartment,
with the possible reabsorption of secreted mineral. This expands the
regulatory potential of the GI mineral metabolism. This feature is
clear, because the amount of Sr transferred by the unsaturable
paracellular process (0.820 mg/day) exceeds the net balance at the
intestine (0.371 mg/day) as estimated at the initial steady state from
the identified parameters of the satisfactory model L6. In
other words, the intestinal secretion of Sr (3.10 mg/day) is greater
than the amount of Sr absorbed through the saturable process (2.60 mg/day), so that the net balance of the bidirectional transcellular
pathway is negative (
0.503 mg/day). This explains the difficulty in
applying the GI parameters to Ca metabolism. This problem can be
overcome by assuming that it is the ion transfer from the intermediary compartment to compartment 1, by cytosolic diffusion and/or
basolateral extrusion, that discriminates against Sr, contrary to our
initial suggestion that more Sr than Ca is secreted toward the lumen
(see Initial Steady State: From Sr to Ca Metabolism). A
higher value of this already high transfer coefficient,
k11', for Ca than for Sr may reduce the
steady-state value of the intermediary compartment (Cai
concentration becoming lower than intracellular Sr concentration) and,
thus, the Ca efflux from this compartment toward the lumen. For
instance, daily Ca intake will be normal and there will be some
endogenous Ca intestinal secretion if k11' is 8- to 10-fold larger. The mechanisms underlying reversibility, at least
partial, of the processes involved in the mineral relations between the
lumen and the intracellular milieu remain to be studied (see An
integrative mechanism for intestinal secretion of endogenous mineral).
Bone Langmuir-type function, an Sr-specific process.
The second INL is the so-called Langmuir-type function. It operates on
K21, the FTF related to the transfer of mineral
from compartment 1 to the major bone compartment
6 via compartment 2 (Fig. 1). The interpretation of the
present nonlinearity differs from that of intestinal absorption,
because it concerns the bone, where not only the cellular and organic
components may interact with mineral, but also the various mineral
physicochemical reactions may be directly responsible for the
nonlinearity. For instance, the adsorption of mineral species at the
surface of the bone solid phase or other deeper sites, such as ion
integration or substitution inside the crystal lattice, may be
saturable processes, because they depend on free sites, the number
(concentration) of which may be restricted. Similarly, the reactivity
of these sites may change with the composition of the liquid or the
solid phase. Impurities (foreign ions) may act as solutes or as
constituents of the crystal lattice and so inhibit some of the numerous
steps involved in bone mineralization (1-3, 9).
As reported in APPENDIX A, the intrinsic form for this
Langmuir-type nonlinearity (Eq. A3) is derived from a more
general equation (Eq. A4) that anticipated complex nonlinear
behavior due to a saturable process dependent on the mineral
concentration of the source compartment and the inhibition by a
compartment other than the source compartment. The model refinements
undertaken here to clarify the physicochemical processes involved in
the bidirectional transfer of mineral across the solute ions-bone solid
phase interface were based on a scheme including a priori the two
nonlinear components of the general equation. For this, we used a
procedure similar to that reported in APPENDIX B, in which
the rapid equilibrium (pseudo-steady-state assumption) of possible
intermediary species, so far neglected in any intrinsic form of the
Langmuir-type nonlinearity, is questioned. Obviously, the complexity of
the refined structures increases: at the most, three compartments (2 for system 1 + 1 for system 2) and three parameters could be added. Model refinement was also attempted to check
Ca and Sr interactions in their binding to the same absorption sites as
illustrated in APPENDIX C.
Conditions for numerical parameters similar to those reported
previously (39) (the 4 Sr doses were considered
simultaneously) were used in several models to study their ability to
fit the experimental data. Indeed, the Langmuir reaction can be
inhibited in various ways (APPENDIX B) and so have various
compartmental representations. Nevertheless, they have several features
in common. The first feature is explicit formulation of a Langmuir-type
process as the first step in the transfer of mineral from
compartment 1 to the bone bulk solid phase
(compartment 6) via compartment 2. An additional
compartment (compartment 1', called the interfacial compartment), inserted between compartment 1 and
compartment 2, accounts for the reversible attachment of
mineral ions at particular binding sites on the bone surface (Fig.
5). This reaction may be simple
adsorption and is assumed to be of first order in its dependence on the
concentration of mineral ion (y1) and free sites [z(n+3)] for uptake. The release of
mineral ions from the bone surface and the resulting recovery of free
sites are proportional to the concentration of mineral bound to sites (y1'). The second feature is inclusion of an
irreversible flux from the interfacial compartment 1' to
compartment 2 as a second step of the mineral transfer
toward bone. This transfer is believed to be related to the formation
of an initial mineral ion association that is relatively unstable,
because it can dissociate into solute ions (direct transfer of mineral
from compartment 2 to compartment 1; Fig. 5).
This results in the recovery of one free site for each ion
incorporated. According to our model, the extrinsic logistic
z(n+1) function (see below for its
physiological interpretation) modulates the rates of mineral incorporation and the recovery of the associated free sites.
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Fig. 5.
Refined nonlinear compartmental substructure for bone Sr
metabolism. Explicit representation of mineral adsorption at the bone
surface is shown as the first reversible step of Sr (and Ca)
incorporation into the bone solid phase. System 2 compartment (n + 3) characterizes free adsorption
sites. Compartment 1 includes plasma. Intermediary
compartments 1' and 3 are associated with mineral
bound to adsorption sites and compartments 2 and
6 with mineral incorporated into the first and deep layers
of bone solid phase. See Fig. 1 legend for explanation of symbols.  ,
Fractional transfer associated with recovery of free sites.
Compartment 3 acts as a noncompetitive inhibitor. For
further explanation, see APPENDIX B.
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Introduction of the Langmuir reaction plus an inhibition into the model
results in a satisfactory fit, whatever the type of inhibition chosen.
However, a posteriori identifiability study reveals that some of the
identified parameter values are very inaccurate. Only when the
inhibition is noncompetitive do the structures give reasonably precise
parameter values (CV < 100%). Interestingly, under these
conditions, few additional compartments and parameters are required,
because compartment 3, in linear relation to
compartment 1 in the initial structure (Fig. 1), is no
longer required. It seems that the turnover rate between
compartment 1 and the interfacial compartment 1'
is high enough that the kinetic effect of compartment 3 is
maintained, despite its structural translocation. For example, the bone
substructure shown in Fig. 5 has compartments 1' and
3 in system 1 and compartment (n + 3) (free site concentration) in system 2 directly
belonging to an explicit form of the Langmuir process plus a
noncompetitive inhibition. We can account for the noncompetitive
inhibition by assuming that compartment 3 is produced from
compartment 1' without recovery of free sites, with the
sites recovered later when compartment 3 supplies mineral to
compartment 2. This last process is linear, in contrast to
that providing material directly from compartment 1': the
mineral transfer from the interfacial compartment 1' to compartment 2 is modulated by the
z(n+1) logistic function. Under such
conditions, the identified values for parameters of system 1 (GI, IDP, and bone) and system 2 [z(n+1) and
z(n+2)], other than those directly
involved in the new ENL [z(n+3)],
are broadly similar to those estimated for the initial simple
models L3 and L6. Consequently, the predicted Sr
mass distribution within the model and the transfer rate associated
with the main Sr metabolic pathways are essentially identical to those
at time 0 in models L3 and L6. Now,
when the predicted model characteristics directly related to the
inhibited Langmuir representation are taken into account, it emerges
that, at the initial steady state, i.e., under physiological
conditions, the turnover between compartment 1 and the
interfacial compartment 1' is more rapid than that of
compartment 2 (k11'/K21' = 30.5)
and very similar to that of compartment 7, the sole IDP compartment in linear relation to compartment 1. This
turnover is also ~40 times higher than bone metabolism, involving
compartment 6 with its mineral transfer rates similar to
bone mineral accretion and removal. Also, compartment 3,
which describes mineral species (Sr2+ or small clusters
such as ion pairs containing Sr) adsorbed at the bone surface, as does
compartment 1', resembles a relatively slowly exchanging
pool of mineral bound to sites with a mean residence time
(1/k23) of ~28 days. It is the largest portion
of the Sr at the liquid-solid bone interface (85% of
compartments 1' and 3). Finally, the estimated
quantity of free sites, compartment (n + 3),
represents >99% of the total number of sites, with only ~1%
occupied by Sr. Briefly, using the time-explicit formulation of the
Langmuir-type nonlinearity, the above properties of our model seem to
indicate that interfacial mineral dynamics are important in the overall
process of Sr incorporation into the bone mineral solid phase.
Nevertheless, because most of these processes are also relevant to bone
Ca metabolism, another model refinement was examined that accounts
mainly for Ca2+ and Sr2+ interaction in their
binding to the same adsorption sites at the bone surface. This study
seems to be all the more appropriate, inasmuch as the identified
apparent concentration of free sites is very low compared with the
plasma Ca concentration. The procedure illustrated in APPENDIX
C was applied to the model structure given in Fig. 5:
compartments 1 (free mineral ions), 1', and
3 (mineral bound to sites, with compartment 3 related to inhibition) were considered explicitly for the Sr and Ca
concentrations (systems 1 and I,
respectively, see APPENDIX C); compartment (n + 3) is common to Sr and Ca metabolism, because it represents the free adsorption sites that bind Ca and Sr.
These conditions and the compartment 1 Ca concentration
(Y1) constant (2,500 µM) gave a satisfactory
fit using the bone compartmental substructure given in Fig. 5 with the
same set of parameter values for Sr and Ca (kij = 
ij), except for the parameter linked to the
recovery of free sites from compartment 3 (incorporation of
mineral into the first mineral solid phase, compartment 2). Contrary to Sr, for which k23 is relatively low
(2.2 × 10
3 h1), 23,
which defines the same transfer process, except for Ca, must be 10
times larger than k23 for the model response to
fit experimental data (fit not significantly different from that
obtained when Sr alone is considered or with the original satisfactory model L3 or L6). This result can be easily
interpreted if the processes involved in Ca dynamics at the
liquid-solid bone interface are operating without inhibition of the
overall process of mineral incorporation into the bone solid phase.
Thus this inhibition is specific for Sr, perhaps shared with other
foreign ions, but not with Ca. It seems to involve processes of Sr
adsorption onto forming or growing apatite nuclei and/or onto the
surface of existing bone mineral solid phase.
Unfortunately, this last Ca- and Sr-refined model is not accurate
enough for some of the identified parameter values; thus a detailed
examination of its properties is ruled out. Nevertheless, there was no
significant variation in the set of parameter values relative to
metabolic pathways other than those directly concerned in the Langmuir
nonlinear expression, similar to the previous version that considered
the interfacial dynamics of Sr alone. Moreover, the very low apparent
concentration of free sites (~20 µM, not significantly different
from zero) is probably the origin of the large inaccuracy observed on
the Langmuir FTF. The reason for this seems to be that Ca occupies most
of the adsorption sites in the initial steady state (~96% of the
total number of sites vs. 3.5% of free sites and <0.5% of sites
occupied by Sr). Besides its noncompetitive inhibitory effect, Sr acts
mainly by diminishing the number of sites associated with Ca. Now, if
only the competition between Ca and Sr at the binding level is
considered and identical parameter values for Ca and Sr relations are
assumed between compartment 1 and the interfacial
compartment 1', the model predicts that the total amount of
mineral (Ca + Sr) incorporated into bone solid phase should be
maintained in a range not significantly different, regardless of the
increase in the compartment 1 Sr concentration. Thus only
the noncompetitive inhibition seems to be responsible for
physicochemical discrimination against Sr. This is the case if the
predicted apparent Ca and Sr concentrations in compartment 3 are examined. Because of the difference between the Ca and Sr fractional transfers from compartment 3 to compartment
2 (23 vs. k23), the relative
concentration of Sr in bone surface compartment 3 is higher
than that of Ca (high Sr-to-Ca molar ratio for compartment 3 in contrast to that of other compartments), a discrimination that is
also predicted during Sr administration.
Other structural arrangements of the bone interfacial substructure
(such as competitive, rather than noncompetitive, inhibition; see Fig.
12) cannot be ruled out, because they can produce model responses
correctly fitting the experimental data. Interestingly, it was possible
to refute the hypothesis that inhibition of the Langmuir process
results from a simple competition between Ca and Sr for the same
adsorption sites. We used the above model with compartment 3 directly linked to compartment 1 (as in the initial
model L3; Fig. 1), eliminating the noncompetitive
inhibition, but we could not identify parameter values that correctly
fit the experimental data with reliable model behavior for Ca bone metabolism.
Finally, there is no reason to reject the simpler initial models
L3 and L6, even if the refined model, including an
explicit representation of the interfacial dynamics of Sr alone, has an interesting heuristic potential and can be justified from a modeling point of view. Simple and refined models have quite similar overall properties. The mineral mass distributions and the predicted main transfer rates are essentially identical whatever the (time-implicit or
time-explicit) formulation used for the Langmuir-type function. However, model L3 appears to be better than model
L6. It is indeed physiologically difficult to reconcile the high
mineral mass of compartment 6 to a dynamic behavior
associated with the inhibition operating at the bone surface.
ENL Acting on G1 and/or Bone
It is necessary to analyze the characteristics of the
time-explicit (differential) form of the logistic
z(n+1) and
z(n+2) variables that modulate a
number of transfer rates inside our model (Fig. 1) and their nonlinear
dependence on compartment 1 Sr concentration (Eqs.
A4 and A5) for an understanding of their physiological meaning.
Kinetic and dynamic behavior of the logistic z variables.
As reported for z(n+1) in model
L3 (Fig. 6A), the kinetic
behavior of this variable reveals important differences between the
various Sr doses. There is almost no change over time with the smallest
dose (D1), whereas z(n+1)
increases acutely during oral Sr administration (AdP) with the highest
one (D4), to reach its maximum after ~40 h, with only
slight changes. For the other two doses (D2 and
D3), z(n+1) produced a
more moderate change, intermediate between D1 and
D4. During PAdP, z(n+1)
tended to decrease at a rate that depended on the value reached at the
end of AdP. The kinetics of z(n+2) (not shown) are rather similar to the kinetics of
z(n+1), with some differences linked
to distinctive parameter values (Table 2): the slopes of the sharp rise during
AdP and the fall during PAdP are more pronounced for
z(n+2). From a dynamic point of view,
the expected sigmoidal curves (Fig. 6B) obtained when the
asymptotic z values (computed from the identified parameter values given in Table 2) are plotted against y1
increasing values also differ according to whether
z(n+1) or
z(n+2) is considered, with
concentrations for the half-maximal activating effect (HMC) slightly
above 30 or 50 µM. These curves clearly indicate that the modulation
of Sr metabolism (system 1) by the logistic z
functions through their effects on the target FTF
(K15, K51, and
K21) is not due to kinetic features alone but is
also dose dependent. The asymptotic values of y1
for the different Sr doses, i.e., the compartment 1 values
obtained after prolonged simulation of model L3 with
continued exogenous Sr, only gave the maximal increase (the
z unit value) with the two highest doses for
z(n+1) and
z(n+2) (Fig. 6B).
D1 shows a smaller increment of
z(n+2) than
z(n+1).
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Fig. 6.
Kinetic and dynamic behavior of system 2 logistic variables. A: predictions, from model
L3, of z(n + 1) variations
over experimental duration [during administration period (AdP) and
after cessation of treatment (PAdP)] for the 4 oral Sr doses:
D1 (solid line), D2 (dashed line),
D3 (dashed-dotted line), and D4
(dashed-dotted-dotted line). B: computed dependence of the
asymptotic value of z(n+1) (solid
line) and z(n+2) (dashed line ) on
compartment 1 Sr concentration. Vertical arrows, asymptotic
compartment 1 Sr concentration reached for each Sr dose.
Each z variable is expressed in normalized concentration
(defined in APPENDIX A).
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Ca as an inducer of the z function: evidence for mineral
self-regulation.
Evidently, the high level of cooperativity revealed by the identified
value of p(n+1) and
p(n+2) (~3 and 4, respectively;
Table 2), defining the order at which y1
activates the z logistic functions, accounts for the main
peculiarities reported above. Another interesting characteristic is
that, despite showing large inaccuracy (Table 2), the identified values
for g
and
g
, i.e., the constant
parameters independent of y1 in Eq. A6, have CV < 100%. Thus some organic and/or mineral
species other than Sr and assumed to be constant throughout the
experiment could influence the logistic functions. Inasmuch as Ca
influences many cellular processes, we investigated whether the Sr
dependence of extrinsic z functions could result from a
direct interaction between Sr and Ca. We have merely conjectured that
Sr mimics the inducing effects of Ca on some Ca2+-dependent
physiological processes. We changed the nonlinear Eq. A6, as
shown in APPENDIX C, by Eq. C1 to clarify the
dependence of Ca and Sr on the z functions.
Equation C1 was applied to each of the z
functions to determine whether this form was consistent with a model
response fitting the experimental data and then to predict the link
between the z functions and Ca concentrations, which would
indicate a plausible mineral self-regulatory mechanism. With
each system 1 parameter maintained at a fixed value (that
previously identified when the initial form of Eq. A6 was
used), the system 2 parameters alone were estimated from
model L3, with Y1 = 1,250 µM,
a constant value corresponding to the extracellular free
Ca2+ concentration. Under such conditions, the model
response correctly fit the experimental data. However, the a posteriori
identifiability study showed an indetermination between the parameter
values defining the Sr-to-Ca molar ratio efficiency
[g
and
g
] and the cooperativity orders [p(n+1) and
p(n+2)]. Complete optimization was
therefore performed with fixed p(n+1) and p(n+2) (5 and 6, respectively),
chosen to be physiologically representative and slightly higher than
the previously identified cooperativity orders (Table 2). A correct fit
to data was obtained with CV < 100% for system 2 parameter values. There were only minor variations of system
1 parameter values that did not significantly differ from those
previously identified. The dynamic behavior of the z
logistic functions could be predicted through their theoretical
dependence on not only Sr, but also Ca, concentration.
Hence, Sr and Ca activate the z logistic function; the
S-shaped curves obtained with increasing Sr concentration have
half-maximal concentrations of ~75 and 87 µM Sr for
z(n+1) and z(n+2), respectively, in the absence
of Ca (Fig. 7A). In the
presence of physiological plasma Ca2+ concentration (1,250 µM), these curves were shifted toward lower Sr concentrations, with
HMC close to 32 and 52 µM Sr, which are quite similar to the values
obtained for model L3 with the initial formulation of the
ENL. Moreover, when the Ca dependence is investigated, the sigmoidal
curves cover concentrations >25 times Sr concentration [Ca HMC of 2.2 and 3.1 mM for z(n+1) and
z(n+2), respectively; Fig.
7B]. This finding agrees with the identified values of the
Sr-to-Ca molar ratio [g and g], which are ~30 for z(n+1) and 35 for
z(n+2). This indicates that Sr is a
better activator of the z logistic functions than Ca.
However, this does not mean that Sr acts physiologically to induce such
nonlinear functions. Although a "normal" Ca concentration influences Sr dependence (Fig. 7A), physiological Sr
concentration (0.5 µM) does not change the predicted Ca dependence
curves in Fig. 7B. Moreover, the Ca-response curves suggest
that neither z(n+1) nor
z(n+2) is sensitive to values close
to, or lower than, the physiological extracellular Ca concentration.
The maximal sensitivity to Ca is obtained at 1.5-3.0 mM for
z(n+1) and 2.2-4.0 mM for
z(n+2) (Fig. 7B). This
suggests that these functions operate only for Ca concentrations above
the normal range.
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Fig. 7.
Theoretical dependence of z(n+1)
(solid line) and z(n+2) (dashed line)
asymptotic value on Sr (A) and Ca (B)
compartment 1 concentration when Sr and Ca activate the
z logistic functions. Arrows, half-maximal plasma mineral
concentration for each computed curve. In A, thin and thick
curves are obtained in the absence and presence, respectively, of a
physiological plasma Ca concentration. In B, stippled
vertical bar represents range of normal plasma free Ca concentration.
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Interaction Between INL and ENL: How Sr Affects Sr and Ca
Metabolism
One of the peculiar features of the retained structure
(model L3) is that it includes three nonlinear transfer
functions, two of them combining INL and ENL (Fig. 1; see
APPENDIX A). Indeed, in contrast to intestinal secretion,
for which K51 is purely ENL [only
z(n+1) in Eq. A2],
K15 for the intestinal absorption and
K21 for the influx of mineral from
compartment 1 to bone compartment 2 result from
mixing an M-M or a Langmuir-type equation with a z logistic
function (Eqs. A1 and A3). This intricacy gives
rise to complex kinetic variations of these FTF, as predicted from the
simulation of model L3 during the experiment, depending on
the Sr dose.
Time-varying FTF.
Although variations in K51 only reflect
variations of z(n+1) (Fig.
6A) with, for D3 and D4, a maximal
increase during AdP of about six times its initial value,
K21, although modulated by the same logistic
function [z(n+1)], changes
differently with time, with a maximum for D4 of less than three times the initial value (Fig.
8A). This difference is due to
the greater influence of the Langmuir-type than the logistic function:
the Langmuir-type function tends to progressively decrease K21 when compartment 3 (the
inhibitory variable in Eq. A3) rises. Conversely,
z(n+1) increases this FTF, but with its own y1-dependent kinetics.
K21 continuously decreases during AdP at the
lowest Sr dose (Fig. 8A) because of the very small z(n+1) increment (Fig.
6A). In contrast, K21 increases sharply at the two highest doses, further counterbalanced by the opposite effect of the Langmuir-type nonlinearity. The relative weights
of the nonlinearities are such that K21 for
D4 has a value that is lower than that for the intermediary
D2 and D3 at the end of AdP. This could be
interpreted as D2 and D3 being more efficient
than D4 for the transfer of mineral toward bone. This result could be important for the effect of Sr on Ca transfer to bone
via bone formation/mineralization, because this metabolic process,
characterized by K21, is common to Ca and Sr,
and the extracellular Ca concentration (Y1) does
not change significantly under our experimental conditions.
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Fig. 8.
Time variations of nonlinear fractional transfer
functions predicted from model L3 over experimental time
(AdP and PAdP) for D1-D4 (see Fig. 1
legend for explanation of lines). A: complex kinetic
behavior of mineral transfer from compartment 1 to bone
mineral solid phase, resulting, at K21 level,
from interaction of a Langmuir-type intrinsic nonlinearity (INL) with a
logistic extrinsic nonlinearity (ENL; see Eq. A3).
B and C: because of the interaction of an
M-M-type INL with a logistic ENL (see Eq. A1),
K15 also has peculiar kinetics with complex
dose-dependence relationship. B: long-term variations in
K15. C: short-term (24-h) kinetics
for K15 compared with quasi-constant value of
K51 (see Eq. A2) observed late in AdP
for D3 and D4.
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There are problems with the time variations of
K15, because the kinetic expressions of the INL
(M-M) and ENL [z(n+2)] do not
match. Although z(n+2) is relatively
slow, the M-M process depends on compartment 5, which has a
high turnover rate and is directly influenced by the exogenous Sr.
Hence, K15 varies over the short term and, thus,
directly influences the comparison of the model response with the
detailed plasma kinetic data collected during AdP (see Fig. 1 in Ref.
39). We use a value for K15 during
AdP obtained just before the morning Sr dose, its maximal 24-h value,
which fit the long-term kinetics of this FTF (Fig. 8B). The
variations in K15 largely reflect those of the
z(n+2) logistic function, with a
highly nonlinear dose-dependent increase during AdP, followed by small
changes during PAdP. However, the M-M INL also has an effect. First,
K15 increases during the first part of PAdP,
mainly for the two highest doses, as expected from the fast fall in
compartment 5 concentration resulting from the cessation of
exogenous Sr. Second, K15 reaches lower values
for D4 than for D3 during the last part of AdP,
in agreement with the unexpected dose dependence of
K21 (Fig. 8A). The mineral transfer
from the GI compartment to compartment 1 tends to become
saturated (K15 diminishes), because the
compartment 5 concentration exceeds the
k value (Eq. A1). This also
occurs when short-term variations in K15 are
considered (Fig. 8C). Despite large variations linked to the
effect of exogenous Sr on compartment 5, the
K15 FTF associated with D4 is always smaller than that for D3 during the last part of AdP. Thus
K15 varies daily between 1.7 and 3.2 times its
initial value for D4 and between 2.6 and 3.9 times its
initial value for D3. In addition, K51 remains nearly constant at about six times
its initial value for D4 and D3. Globally, the
dissimilarity between the dependence of
z(n+1) and
z(n+2) on y1
(Fig. 6B) could be important for variations in the net
mineral intestinal absorption capacity induced by Sr, in addition to
the rather instantaneous effects of the M-M INL on
K15.
Dose-dependent effects of Sr on Ca metabolism.
The above analysis has revealed an attractive property of the model
related to the complex dose dependence of GI and bone FTF. If Sr and Ca
share the same metabolic pathways, then the variations in FTF in
response to exogenous Sr must also affect Ca metabolism. We have
therefore departed from a strict physiological framework to examine the
effects of exogenous Sr on the metabolism of Ca. The asymptotic
behavior of the model variables was computed for Ca and Sr (see
APPENDIX C) using different values for compartment
1 Sr concentration (y1) and a constant
compartment 1 Ca concentration
(Y1 = 2,500 µM) plus the set of parameter
values previously identified from model L3.
Thus, for bone mineral metabolism, the inducing effect of Sr on the
z(n+1) function and the Sr-specific
inhibitory effect on the transfer of mineral from compartment
1 to bone (K21) through the Langmuir-type
nonlinearity are taken into account. A range of Sr concentrations,
y1 = 30-110 µM (Fig.
9), were found, for which the asymptotic
Ca mass in compartment 6 (mineral solid phase associated
with mature bone) is increased, with the assumption of no other
interaction between Sr and Ca. The maximal effect is obtained for
y1 = 55 µM, with a 35% increase in bone
Ca mass compared with that obtained at physiological Sr concentration. Compartment 6 has an asymptotic value below the
physiological value at other values of y1. The
Langmuir-type nonlinearity is indeed more efficient than the
z(n+1) logistic function in two
situations: when y1 < 30 µM and there is
only a small increase in z(n+1), and
when y1 > 120 µM and the Langmuir-type function continues to decrease the mineral transfer to bone while z(n+1) approaches its maximal unit
value (Fig. 9).
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Fig. 9.
Complex dose-dependent effect of Sr on GI and bone Ca metabolism
according to model L3. A: asymptotic Ca mass in
compartment 6 (bulk bone solid phase) as a function of
compartment 1 Sr concentration. B: asymptotic
oral Ca intake required to maintain compartment 1 at
physiological plasma Ca concentration (2.5 mM) as a function of
compartment 1 Sr concentration.
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Using a fixed value for each parameter with, for Ca,
k01 (urinary excretion) divided by 2 and
K51 divided by 16, we then applied the same
theoretical procedure to GI metabolism (Fig. 2). The asymptotic steady
state was computed by monitoring the mean daily Ca ingestion required
to maintain the compartment 1 Ca concentration constant
(Y1 = 2,500 µM), whatever the value of
the Sr concentration (y1). The changes induced
by the K51 and K15
y1 dependence through z(n+1) and
z(n+2), together with the K15 M-M nonlinearity influence, caused
intestinal mineral absorption capacity to vary, and this must be
counterbalanced by the dietary Ca intake. The predicted values of the
mean daily Ca ingestion as a function of y1 are
reported in Fig. 9B. The maximum value was ~1,200 mg/day
for y1 = 35 µM, with values lower than
the initial value for y1 > 65 µM. These
values can be within a normal physiological range, despite the large
dose-dependent increases in K51 and
K15 related to the modulation by
z(n+1) and
z(n+2). Thus the Sr concentration
that has the maximal effect on bone formation/mineralization
(y1 = 55 µM) requires only a small
increment in the daily Ca intake. The required Ca intake, 989 mg/day,
remains in the normal range and corresponds to only a 14% increase
over the value (868 mg/day) computed using a physiological Sr concentration.
An integrative mechanism for intestinal secretion of endogenous
mineral.
Finally, we examined the bidirectionality of the relation of GI
compartment 5 to compartment 1. The satisfactory
initial models (Fig. 1), as well as the refined GI compartmental
structure with the intermediary intestinal cellular compartment
explicitly considered (compartment 1' in Fig. 4), require
reversibility between lumen and plasma. This kinetic reversibility
could be due to peculiar features of some process involved in the
facilitated membrane transport of mineral, as suggested for intestinal
Ca absorption. Two mechanisms of Ca entry from the gut lumen into the
enterocyte via the apical membrane have been proposed: the first
mechanism may be essentially irreversible, with a Ca2+
channel similar to that recently detected in the proximal small intestine [mainly in the duodenum (19)]; the second
mechanism, associated with a lipid-soluble mobile carrier, could be
reversible to some extent (47). Each of these processes,
channels (19), channel-like transporters
(31), or mobile carriers (47), is also
saturable, with an apparent M-M constant in the same range as that
estimated by our model. Finally, multiple signaling pathways may
regulate them, a property consistent with their modulation, in our
model, by extrinsic z functions.
A satisfactory fit to experimental data is obtained only when the
saturable part of the intestinal absorption and the endogenous secretion are modulated by one distinct z function, as
reported elsewhere (39). Hence, our model agrees with
these two processes, which operate in opposite directions, involving
different transport systems, which can be neither refuted nor supported
because of uncertainty about the molecular mechanism(s) of intestinal
endogenous secretion. Is this last process active, passive, hormonally
dependent, or related to exsorption? However, another proposition is
that part of the entering mineral may be reversible (reversible mobile carrier), so that Cai or intracellular Sr may be
countertransported out of the cell to the intestinal lumen, in addition
to an irreversible process (channel-dependent process). This was
checked using a refined model. We assumed that two M-M equations
operated simultaneously on the transfer from compartment 5 to compartment 1, in addition to the nonsaturable part of
intestinal absorption (paracellular transfer). Each M-M equation
depended on one distinct extrinsic logistic function
[z(n+2) or
z(n+3)], and we assumed
z(n+3) modulating one M-M equation
and also the mineral transfer from compartment 1 to
compartment 5 (intestinal endogenous secretion). Obviously,
there were many more unknown parameters (addition of 6 parameters and 1 system 2 variable). Also, optimization was undertaken with
numerous parameter values set at those identified for model
L3; only the parameter values directly related to bone and GI
nonlinear behaviors were reevaluated, with
p(n+1) and
p(n+2), respectively, the
cooperativity order of the z functions acting, on the one hand, on bone and, on the other hand, on the GI irreversible M-M equation. These conditions gave a correct fit to experimental data (not
significantly different from that obtained from model L3),
even with the use of a reversible process accounting for a large part
of the total saturable transfer. For instance, if the same maximal rate
for reversible and irreversible parts of the saturable transfer under
physiological conditions (at time 0) is assumed, the set of
identified parameter values shows similar M-M constants that were also
close to the corresponding value for the initial model L3
(k in Eq. A3). The main
changes were in the S-shaped y1 dependence
curves computed for each z function (Fig.
10). Thus, contrary to the results in Fig. 6B, z(n+1) and
z(n+2) differ more in the slope of
the curve, which is smoother for
z(n+1) than for
z(n+2), than in their HMC values
(~42 and 50 µM). Despite a slightly lower cooperativity order
(2.62) for z(n+3) than for
z(n+1) (2.96), the z(n+3) function modulating the
reversible part of the intestinal absorption appears to be more
sensitive to the compartment 1 Sr concentration, with
an HMC of ~25 µM.
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Fig. 10.
Computed S-shaped curves of y1
dependence for 3 logistic variables:
z(n+1) (solid line), which modulates
bone fractional transfer function, K21;
z(n+2) (dashed line), which modulates
a first saturable irreversible process of intestinal Sr absorption; and
z(n+3) (dashed-dotted line), which
acts on another intestinal absorption process that is reversible
because of modulation of intestinal endogenous mineral secretion by
this same extrinsic z(n+3)
variable.
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The complexity of this model refinement precludes any other development
in the absence of additional data for the GI compartment. Nevertheless,
this apparent complexity might originate from the heterogeneity of the
mineral absorption capacity along the intestine, which is not taken
into account in our model (the single compartment 5 includes
all segments of the intestine that can transport mineral between lumen
and blood; Fig. 1). For instance, the proximal part of the intestine
(duodenum) could make a major contribution to the irreversible process,
whereas the reversible component could be representative of the more
distal segments (e.g., ileum) and, thus, be mostly responsible for the
intestinal secretion of endogenous Sr and Ca (43).
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DISCUSSION |
The present study was prompted by the need to analyze the
relevance of a nonlinear compartmental model, developed in the
companion paper (39) to describe human Sr metabolism, for
Ca metabolism. Given the model structure (Fig. 1) and the new
quantitative and mainly qualitative information it offers, our purpose
was to test the feasibility of physiological mechanisms that could
underlie
TOP
ABSTRACT
INTRODUCTION
