The Mathematical Model
From the viewpoint of systems analysis, iron metabolism of the body is an open black-box with input, internal processing and output. Absorption fluxes and losses are relatively slow compared to internal iron circulation [34, 35]. The most relevant internal dynamic events occur within the first few days after tracer injection. They involve the plasma iron turnover and the turnover of the erythron. On this time scale the system is nearly closed, with input (absorption) and output (excretion, desquamation) being slow compared to the dominant rates of inner metabolism. Iron absorption from guts is in the range of 0.5 μg per day (calculated to whole-body scale from the data of Bahram et al. [36] and Lebeau et al. [21], whereas the rate of total plasma iron exchange with the body periphery is in the range 20-30 μg per day (Additional file 1: Table S8). Such a system will approach an inner steady-state with slowly drifting concentrations and fluxes.
To study such a steady-state one can measure stationary content variables and inject a tiny dose of 59Fe, preferentially bound to its transferrin carrier [37], into the central compartment of radioactive iron. In the initial hours after mixing of the injected iron, when the periphery does not appreciably return tracer, the flow of blood into the organs is proportional to the flow of tracer. This initiates distribution dynamics of the tracer in conformity with the pools and stationary rates of the unlabelled bulk of iron in the steady-state, which is not disturbed by the addition of trace amounts. The time course of the ensuing run-off of tracer distribution obeys a system of ordinary differential equations (methods and Additional file 3).
Iron status of the adult mice on different dietary regimes
The experimental data which form the basis for the model calculations presented here are derived in adult mice that were on different dietary regimes during growth. The adult animal develops a steady-state which is maintained during its further adult life, i.e. for approximately 1-2 years. This assumption is prerequisite for the ferrokinetic model. The empirical background for the model consists of the static iron status and of tracer dynamic data.
Modelling iron metabolism by the 59Fe tracer method
The literature contains numerous papers [2, 5, 6, 8, 38] which measured the dynamics of iron metabolism on the intact organism with the help of the tracer 59Fe. The basic rationale is that the tracer, while being always measurable as radioactivity, due to its tiny relative amount does not perturb appreciably the iron status of the body. Most of the mathematical models derived from such measurements were obtained for humans, dogs and rats. Vácha et al.[4] derived a whole-body model for the normal mouse (related to our strain), collected from measurements in blood, liver and spleen, without systematic control of dietary regime. The model was based on a number of ad-hoc assumptions of fluxes, which partly could not be substantiated by cellular mechanisms, and the parameters were in part estimated without a whole-model statistical fitting procedure (computer capacity-limited). We can confirm, with the experimental evidence now available, that their ingenious model, in spite of some speculative elements due to limited molecular knowledge, gives a remarkably adequate description of the global dynamic structure of murine iron metabolism.
Static iron status
This is defined as the iron content of the iron fractions in the different organs and tissues. It reflects the expression of protein carriers to which iron is bound (transferrin, ferritin, heme proteins etc.). We can assume that this status is stable during the observation period.
Dynamic fluxes
The second data type is the time course of iron flow through body organs after administration of a radioactive tracer. The data base stems from partly published measurements of our own labs [9] and from studies done elsewhere under a comparable experimental design. The aim is to integrate the entire data set into an integrative model, thereby displaying the systemic structure which is not obvious from inspection of raw data.
Kinematic model of iron steady-state
The crucial supposition for a representative model is that iron fluxes in the body are (approximately) balanced and cellular pools do not fluctuate violently during the experimental period. Tracer injection permits to collect data on internal fluxes without upsetting the steady-state. Analysis of the fate of tracer-59Fe in the tissues in terms of a linear system of differential equations describing influx, outflux and internal metabolism of the system gives a picture of the prevailing "kinematics" of the system, i.e. it describes what happens, not the causes and controlling mechanisms. Such a phenomenology is the prerequisite for any in-depth systemic description.
Inhomogeneity of compartments
For some important tissues under consideration the assumption of homogeneity is not valid. This applies to liver, which consists of parenchymal cells (hepatocytes) and cells of the RES (Kupffer-cells). Both types have different iron regulation. Similarly, the murine spleen has subcompartments, of erythropoietic and of macrophage cells. Muscle tissue contains a large fraction of iron in myoglobin, the turnover of which is different from that of the macrophages in muscle. On the whole, the macrophages of RES are spread over a multiplicity of organs and change their distribution in inflammation. For the sake of model calculation, nevertheless, we treated these organs as compartments.
Numerical parameter estimation
Tracer motion in a steady-state system of homogenous pools (like in figure 1) can be modelled by ordinary linear differential equations. In theory, the concentration of tracer in these pools follows a time course described by superimposed exponentials. In the initially labelled central compartment (plasma) the tracer content falls monotonously. In all the other compartments, initially void of tracer, the concentration rises to a maximum and then turns into a monotonously decreasing phase of recycling into plasma together with outflux out of the body. The parameter values of the interconnected system can be obtained as "best fit" according to a suitable distance criterion. In practice this estimation process may run into two types of difficulty: statistical scatter and redundancy of the parameter space.
The scatter of measured data in most biological systems is considerable and cannot be avoided. The reason is partly individual variation between subjects and partly the technical difficulty to exactly repeat the same experiment. A statistical model of this situation can, at best, be a close approximation to the measured data in the form of an idealized curve. We repeated the estimation procedure on sets of artificially generated data which keep the error structure of the observed data. The range of parameter fluctuation was revealed in this way.
Interdependence (correlation) of parameter estimates
This is an unavoidable problem of complex biological models. It became clear from the ACE analysis (see methods) that our data contained two causes of parameter interdependence: insufficient resolution at very early time after tracer injection, and cases of double output of tracer back into plasma and out of the body. We overcame these problems by prescribing an approximate value of the total plasma clearance calculated from the data of Trinder et al. [10]. Furthermore, we replaced double outfluxes by a single lumped one, thus not specifying the precise fractional contribution of each pathway (see dotted outflux arrows in figure 1). In this way we obtained parameter estimates with a reasonable range of scatter avoiding strong intercorrelation.
Further parameters of the model
The set of clearance-and rate-parameters resulting from the parameterization is given in table 1, and visualized in Additional file 2: Figure S1. The quality of the fit is satisfactory, as demonstrated in figures 2, 3, 4. Table 1 contains the most compact representation of the information content of the empirical data. It can be used to calculate flux rates (Additional file 1: Table S8), pool sizes (Additional file 1: Table S10) and as well as a characteristic temporal structure of the system (Additional file 1: Table S11). Additional file 2: Figure S2 through S4 visualize these quantitative estimates. These indirectly derived data indicate scatter intervals of system-relevant parameters. Their totality is amenable to physiological interpretation of the static and dynamic state of the iron system in the mouse in the different "lifestyle" regimes studied.
Physiological interpretation of the global system
Iron metabolism can be described as a closed compartment system
The quasi-closed state of the iron system together with the ensuing internal steady-state makes into possible to simplify the non-linear structure to a system of ordinary linear differential equations. The dynamics of tracer motion depicts the statics of the underlying stationary flux-and-pool network. We could build on a number of previous attempts to model iron metabolism in this way [5–8, 38], reviews in [2, 38]. The novel aspect here is the detailed reversible balance in a network of peripheral tissues that were previously combined ad hoc to black boxes.
Iron metabolism is organized as temporal hierarchy on five time scales
Analysis of the clearance parameters of our experiments (table 1, transformed into residence times-Additional file 1: Table S11) and of literature data [10, 18, 19] lead to the following grouping of characteristic time periods:
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Rapid mixing time of the transferrin-bound plasma/ECF pool: below 1 h
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Total clearance time of plasma iron due mainly to flux into bone marrow, liver and muscle: ~ 1 hour.
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Clearance time of compartments that return tracer into plasma (descending branches in figure 3, and Additional file 2: Figure S1): between ~2 days in the intestinal tract and ~8 days in parenchymal organs.
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Incorporation time into iron-carrying proteins in red blood cells, muscles, integument or fat: ~ 1 month.
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Characteristic life-time of iron molecules in the whole body: approaches 200 days in the adult mouse. If the mouse has acquired a store in the adolescent stage (adequate diet), this would be sufficient for its whole life. It is therefore difficult to induce iron-depletion anemia in the mouse (contrast to humans).
The time hierarchy does not change appreciably between different iron statuses in normal mice (confirming the conclusion in [10]).
Iron turnover in the plasma compartment depends on the iron status
The concentration of transferrin-bound plasma iron in plasma is in the range of 100-200 μg/dL in the mouse (Additional file 1: Table S9). This is similar to other mammalian species (e.g. [12, 39–42]). The iron concentration tends to lower values in iron-deficient and to higher values in iron-loaded mice. The iron clearance from plasma defines a half-time of renewal in the range of 1-2 hours, again similar for species otherwise as different as Mus musculus and Homo sapiens. Rats [41] and dogs [5, 39] are also in the same range. In rats, however, iron deficiency does not lower the plasma concentration [41].
Iron distribution into body periphery is a three-level hierarchy of flux rates
The initial tracer concentration in plasma becomes rapidly cleared within a few hours after administration and stays at a low, but steady value afterwards. This coincides with the ascending tracer curves in the peripheral compartments (figures 2 and 3. The initial distribution is complete at the first time of measurement (12 h). The position of the maximum fixes the time point when plasma tracer is nearly washed out and the periphery begins to return some of the previously accumulated tracer iron into the plasma (figures 2 and 3. The continuous decrease of organ tracer content begins after 12-24 hours. It is an expression of the fact that "fresh" cellular iron is not only stored or channelled into biosynthesis, but also shows an appreciable back-flow into the plasma.
The descending branch of the peripheral tracer curves show that all tissues return the radioactivity into the plasma, unless they lose it by desquamation, which is the case for intestine and integument. This characteristic pattern proves that iron flux into the periphery and reflux into plasma take place simultaneously.
The quantitative level of all the superimposed fluxes can result only from a deeper analysis of the corresponding mathematical model. This analysis yields a set of fractional clearance parameters (table 1). From these values and by application of the steady-state assumption all iron fluxes can be estimated when the iron content of the central compartment is available. Data by Trinder et al. [10] contribute an estimate of the total plasma turnover clearance rate (Additional file 1: Table S9).
Three clusters of flux rates may be distinguished
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flux through the erythron (range of 10 to 20 μg/day per mouse)
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flux through peripheral compartments with storage function (liver, muscle, integument, intestinal tract, kidneys, lungs, heart: 0.5 to 4 μg/day per mouse)
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flux through organs with slow iron turn-over (testicles, fat, brain) -0.05-0.08 μg per day per mouse.
Share of flux into tissues mirrors transferrin receptor expression
The clearance time of plasma iron is in the range of 1 h, largely independent of the plasma iron content and hence the state of the animal's iron supply. This linear kinetics suggests that the total population of TFR1 receptor molecules (responsible for most of the iron uptake) works far below its maximal capacity in all cells. The share of radioiron going into the body organs reflects this tissue-specific transferrin receptor expression. In contrast to the rather stable total clearance time the share of radioiron is dependent on the physiological state (Additional file 2: Figure S2). In the states of iron depletion and of normal iron supply more than two thirds of the plasma iron turnover is directed to the erythropoietic bone marrow and is rapidly incorporated into hemoglobin. This is, again, similar to other species [2, 13, 14, 31]. The corresponding fraction of tracer iron passes through the immature cells of the erythropoietic lineage until it reaches the erythrocyte compartment.
Tracer distribution iron-rich condition reflects the switch-over to the storage mode
The flux through the storage pathway into parenchymal organs increases from 25% to 49% of plasma turnover (table 1; Additional file 1: Table S8; visualized in Additional file 2: Figure S2 and S3). Stores are filled up in liver, kidney, spleen/RES (Additional file 1: Table S12), to a lesser extent also heart and skeletal muscle, but not integument and brain.
Tissue cells equilibrate influx and reflux of iron to maintain the iron pool
An adult mouse does not grow much during its life-time of ~2 years (if not killed before). Iron is taken up by cells with a time characteristic of a few days and must, therefore, be balanced by corresponding iron-release. Muscle, fat, heart, lungs, brain and testicles excrete iron into plasma or extravascular fluid. The influx of tracer is mediated by transferrin receptor. It is not clear from tracer data whether the export is mediated only by the ferroportin channel [11], or also via catabolism of heme-bound iron. Ferroportin is dominantly expressed in liver, duodenum, and macrophages, and to a lesser extent also in other tissues [43]. Ferroportin is not involved in the case of loss of whole cells (erythrocytes, intestine, and integument). The tracer data as used here cannot distinguish between export of iron and loss of whole cells. They yield only an estimate of the total flux out of the compartment.
Intracellular residence time of iron is longer than the life time of its protein "carriers"
The life-time of the iron-storage proteins (such as apoferritin/holoferritin) is in the range of one day in the liver [33]. Up to 4500 iron ions can be stored in one ferritin molecule [44], and become released on proteolytic ferritin degradation. The residence time of iron in the liver cell, however, is much longer-in the range of 1-2 weeks (Additional file 1: Table S11). This shows that iron released into the very small labile iron pool does not leave the cell, but is re-utilized. This slow export conforms to well-known data showing how slowly iron is mobilized from ferritin stores to replace iron losses, e.g. after phlebotomy (in men: [32, 45]). Intracellular iron stores are no inert long-term reserves, but are continuously turned over within the cell and may therefore be directed, in accordance with changing requirements, into the three competing pathways (biosynthesis, storage, export).
Readily accessible tissue iron pools are a fraction of the non-heme iron
These iron pools are stored in different subcompartments, mainly in non-heme form. The iron-loaded liver stores ferritin in the hepatocytes and a less mobilizable (hemosiderin?) form in the Kupffer cell [46–48]. The labelled and unlabelled iron data from whole organs do not permit to differentiate quantitatively between parenchymal and macrophage iron in such mixed cases. Tracer dynamics identifies iron pools that become quickly labelled. Their pool sizes have been estimated from the fractional plasma iron turnover and the tissue clearance rates (Additional file 1: Table S9 and table 1). Three groups may be distinguished. Red blood cells contain as hemoglobin the largest readily labelled iron pool (~ 300 μg Fe per mouse, about 50% of total haemoglobin-iron, see calculation in Additional file 1: Table S9). There is a second cluster of pools (integument, liver, bone marrow, skeletal muscles, skin), each containing about 20-40 μg Fe. In particular the hepatic iron pool is expandable in iron overload to reach a kinetic pool level of ~ 100 μg Fe, presumably in ferritin form. A still larger store can possibly accumulate on a longer time scale, which is not covered here. There are additional pools with an iron content (lungs, kidneys, intestine, heart, and spleen) of about 3 μg Fe each, which can moderately expand up to 4-14 μg (Additional file 2: Figure S4, lower left). Other organs, such as fat, testicles or brain, are not able to store more iron in overload. Additional file 1: Table S13 shows that in some tissues the readily accessible pools are only a fraction (6 to 40%) of cellular non-heme iron.
There are two kinetically distinct major iron pools in the mouse body
The total tracer-accessible iron amounts to ~400 μg (Additional file 1: Table S10). The residence time in the main compartments excluding intestine (Additional file 1: Table S11) is between 5 and 25 days. This comprises about 20% of the total iron (i.e. of ~2 mg per 25 g body [18]). The reminder is not readily accessible. The residence time of molecules here is ~200 days [18, 19].
Iron turnover occurs at similar rate in intestine and skin, but assignment to iron loss vs. iron reflux is only indirectly estimable
Physiologically iron enters the body via duodenal and (less) small-intestinal absorption in a tightly controlled way. It leaves the body by desquamation, exfoliation of epithelial cells, by blood losses, and to a lesser extent via bile and urine [49]. The net amount leaving the murine body is, according to literature references, 2 to 5 times larger than in other animals and man [18, 20, 40, 49–51]. A consequence of this higher excretion is that heavy iron-load is sometimes difficult to attain in mouse models.
Net iron losses cannot be measured by the tracer method as applied here. However, the fractional clearance rates (table 1) yield indirect information on iron fluxes through intestine and integument (Additional file 1: Table S8). Iron clearance of the epidermis integument is about 5% per day and that of the stomach-intestinal epithelium ca. 36% per day (calculated from table 1). From the fractional uptake from plasma one can calculate influx rates of ~1.7 μg per day into epidermis, and a sum of ~1.5 μg per day into intestine plus stomach (all for iron-adequate mouse, see Additional file 1: Table S8). These values are about 39% lower and higher in iron deficient and iron-load regimes, respectively. The data do not support a calculation of the rate of net iron loss through these compartments, because there may be a fraction that is recycled into plasma. The iron residence times for intestine are similar to the known exfoliation times of epithelium (3-5 days), which suggests that the main fraction goes into loss. For skin integument (iron residence about 40 days) such external information was not available.
Murine erythrocyte iron turnover has a random elimination component together with a lifespan-determined removal component
During one month after administration 60% of the tracer (40% in iron-loaded state) accumulates in the red blood cell compartment (figures 2, 3. The first quick uptake of 59Fe reflects passage through bone marrow and incorporation into hemoglobin at a steady rate. The uptake reaches a saturation phase which is clearly visible in the RBC curve of figures 2 to 3. This behaviour proves the existence of a reflux caused by a random component of erythrocyte catabolism independent of the cell age. Without reflux iron would be further incorporated even at a very low plasma radioactivity. The erythron cycle transports (Additional file 1: Table S8) 15;19;14 μg Fe/d into bone marrow in iron-deficient, -adequate, and -loaded animals, respectively, of which 12;17;12 μg Fe/d pass through the RBC compartment back via into RES into plasma. This turnover rate is quantitatively analogous to ~ 25 mg Fe/d per 70 kg in iron-adequate humans.
The life span of mouse erythrocytes has been studied in mathematical detail by Horký et al. [52]. They also formulated an age-independent linear elimination component acting simultaneously with a lifespan-determined senescence process. Our elimination rate (between 0.03 and 0.06 d-1) is somewhat higher than obtained in [52] (0.012d-1). However, these estimates are not very reliable, as they stem from an indirect deduction. This applies also to the size of the "readily accessible" iron pool in red blood cells (300 μg instead of the 568 μg calculated from the hemoglobin pool of the mouse, see Additional file 1: Table S9).
The spleen is a mixed indicator of erythropoiesis and RES activity
The murine spleen is an erythropoietic organ [53]. Therefore, one subcompartment of iron in the spleen is expected to behave similar to iron in the bone marrow. Figures 2, 3, and Additional file 1: Table S5 to S7 show a similarly quick uptake phase in both bone marrow and spleen. The ratio of tracer iron content between both organs after 12 h is about 50 to 60 in adequate and iron-rich mice, and 20 in iron-deficient animals. Thus, the quantitative contribution of the spleen to total murine erythropoiesis is not high. Furthermore, the iron-deficient spleen loses iron as quickly as the bone marrow, reflecting the rapid flow into "iron-deficient" erythropoiesis. In contrast to the bone marrow, the iron-adequate, and even more so the iron-loaded spleen retains 59Fe for long periods. This reflects a storage behaviour which is similar to that of RES cells in the liver and elsewhere. The spleen contains 5% and the liver 16% of the whole population of macrophages [54]. The RES system serves as scavenger to remove senescent erythrocytes together with their hemoglobin and colloidal iron from the circulation [8, 55]. Part of this RES iron is rapidly recirculated into plasma, thereby completing the iron-recirculation back to the erythrocyte pool. Except in iron deficiency, another part of the RES iron is stored as ferritin or hemosiderin [44].
The quantitative contribution of both spleen compartments to whole body iron turnover is low. The spleen is therefore an indicator, but not the main quantitative locus of the total erythropoietic and macrophage activity. In iron-deficiency splenic iron clearance is very rapid (15% d-1, see table 1 and Additional file 2: Figure S1 and S2), while it is distinctly slower (down to 1.9% d-1) in iron loaded mice. This may reflect distinct differences in the role of the spleen depending on the state of iron-repletion. A precise quantitative partition of splenic iron fluxes into a RES-and an erythron-fraction would require separation of the cells.
Experimental design for characterizing the iron status and the dynamic turnover of the C57BL6 mouse strain
The C57BL6 mouse is a widely used strain for genetic modifications to address the regulatory networks of iron metabolism. Any such transgenic strain needs a characterization of its iron kinetics (examples in [56, 57]). This includes a survey of static and dynamic characteristics of iron metabolism under the limitations set by thrifty experimental expense. The turnover model developed here permits to derive diagnostic requirements for healthy or diseased mice, after a steady-state has been established and maintained for the time of at least one red blood cell turnover. The following data should be scaled up to the total body level:
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plasma iron steady state, measured by transferrin level and its iron saturation
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liver and spleen total iron and non-heme iron (may be replaced by plasma ferritin as indicator)
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hemoglobin iron content, if possible red blood cell turnover (as indirect control of iron turnover
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hepcidin and erythropoietin levels
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plasma iron clearance rate constant (only possible with tracer injection and several measurements during the first 12 h)
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percentage uptake of tracer from duodenum (after a bolus dose into the stomach)
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organ content of tracer iron, blood-corrected, by several measurements between 12 and 72 h at least in liver, spleen, red blood cell compartment and bone marrow
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long-term rate of iron loss.
The biochemical parameters yield a survey of the static of iron metabolism and its steady-state level. Ferrokinetics yields the fluxes. This full programme can be reduced, if in a particular situation preliminary analysis of data and their comparison with the mathematical model indicate that certain features of the iron status are not changed or are negligible.