2 - Models and Experimental Methods

2.1 General Structure of Iron Metabolism

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Iron metabolism has been reviewed in several monographs from which the general structure can be extracted as basis for a mathematical model [15 ].

Iron is contained in every cell of the mammalian body and also in the extracellular space. Only tiny amounts appear as free ionic iron. All the other iron is in bound form, as hemoproteins, as oxygen carriers, as iron-sulfur proteins, in non-haem enzymes (such as transferrins), or as iron transporters in cellular membranes.

2.2 General Flux Network of Iron in the Organism

Mammalian organisms absorb iron from the intestinal tract, mainly via the duodenal epithelium, and loose it predominantly by exfoliation of intestinal epithelium, by desquamation of the skin, by occasional or repeated loss of blood, and to a lesser extent via excretion of a non-reabsorbed fraction of bile and urine. Duodenal absorption transfers iron into plasma where it is bound to transferrin. Transferrin-bound iron is distributed to peripheral tissues in accordance with their expression level of the transferrin receptor (TFR1). This stream into the periphery can be measured after injection of radioactive 59Fe into plasma as rate of appearance of the tracer in the periphery. This intake of iron into cells is balanced by an out- stream back into plasma of similar strength, mediated by iron export protein ferroportin [18 ].

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This global scheme of iron uptake and distribution into the periphery and its reflux into plasma is adequately represented by a “mammillary” compartment system with reversible flux. The layout of the general flux model is depicted in fig. 2.1.

Plasma and extracellular fluid contain transferrin-bound iron that equilibrates quickly (< 1 day) between both compartments [6 ], which are therefore treated as one central compartment. Other tissues form peripheral compartments. The erythropoetic compartment of bone marrow has a high expression level of the transporter TFR1 and rapidly integrates iron into hemoglobin [20 ]. Both fluxes as well as the filtering-out of senescent blood cells are irreversible. In contrast, the expression level of ferroportin in bone marrow is low. This allows us to model the iron pathway from plasma over bone marrow, erythrocyte pool, and RES (emphasized by thick arrows in fig. 2.1) as a circular irreversible flux without reflux. A smaller flux from bone marrow into the Reticulo-Endothelial System (RES) has to be included. It represents partly spleen erythropoiesis in mice and partly “ineffective erythropoiesis” [22 ].

Figure 2.1: Global flux model layout.

In green are shown the compartments representing each organ, the arrows are iron flows between compartments and plasma or directly between organs (bone marrow, RBC and Spleen). Dashed arrows represent fluxes that are known to exist but were omitted in our model because they were combined with the dominant outflux from the same compartment (to avoid parameter indeterminacy).

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Some of the peripheral compartments have two outlets, one leading back into the central compartment and the other one leaving the system by loss of cells or excretion. The clearance of radioactive tracer can be estimated in these cases, but not the partition between these two outlets. Loss of tracer from the body is difficult to measure, and the plasma curve is not sufficiently sensitive to small variants of back-flux from those tissues. For parameter estimation an a priori decision had to be made about which of the double outflows is quantitatively the more important. In the intestine and integument, with the exception of the duodenum, iron losses from the body are assumed to be the main route. This reflects the rapid exfoliation of intestinal epithelium (around 4 days), as well as the slower, but of larger volume, desquamation of skin and integumental adnexes. In the duodenum the 59Fe influx from plasma counters the physiological uptake of unlabelled iron from the lumen. In erythrocytes, liver, and kidney it was decided that the main pathway balancing iron uptake is reflux into plasma rather than loss out of the body.

The compartments in fig. 2.1 represent organs and tissues. They are not kinetically homogenous, as every organ consists of cell types that may differ in their iron metabolism. Iron content in such compartments and intercompartment fluxes represent therefore a weighted mean over different cell types. Mixture compartments of this type are liver, spleen and muscle.

2.3 Iron balance: absorption from duodenum and loss from the body

Iron is being distributed in various amounts over all body cells. The exchange traffic is mainly mediated via the transferrin pool in plasma (about 1.5 µg iron per mouse body). The absorption of iron takes place in the duodenal and upper jejunal lumen. The rate of absorption is tightly controlled and amounts to about 3 µg per mouse body per day. The adult healthy mouse is in a steady state, because the rate of iron loss is the same, amounting to 0.5% of the total body iron per day [23 ]. This happens mainly through desquamation of skin cells, loss of hair, and shedding of stomach and intestinal mucosa. In our model it will be formulated as aggregated iron ‘leakage’ where the whole cell is lost from the body, loosing its iron content. The rate of iron leakage is small and varies with different cell types.

2.4 Numerical Scales of Pools and Turnover Rates.

2.4.1 Scaling of iron content to the whole mouse organism

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The generic model of fig. 2.1 has to be quantitatively specified. The assumed reference organism of all data in this paper is an adult mouse of 25 g body mass; all other references are transformed to this scale. Comparing relevant data from other species involves a scale factor of approximately 10 from mouse to rat and 2500 to 3000 from mouse to humans. Compartments are envisaged as iron aggregates (“pools”) the iron content of which is a systemic variable, expressed in units of µg iron per animal. Fluxes into and out of a compartment are expressed in units of µg iron per animal per day.

The systemic structure of this model is specified by “content” data (“concentration” of iron in its various biochemical forms in the different compartments), and by “turnover” data, usually obtained from tracer studies. Both data sets must be scaled up to the whole organism. The model must have a mathematical structure that reflects statics and dynamics. Its parameters are to be estimated from the empirical data.

An important indicator of the iron status of cells or of the whole body is the iron content in the various biochemical fractions of iron.

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The quantitatively dominating biochemical form of iron in the mammalian body are:

The biochemical status of free ionic iron in the cell is not quite clear. One fraction is called Labile Iron Pool (LIP), ([25 ]) can be extracted by chelating agents, and is said to the crossroads of biosynthetic and biodegrading pathways in the cell. Its concentration is very small in all cells, well below 1 µM, e.g. in liver [26

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LIP can serve as theoretical indicator of the iron state in mathematical models. A better measurable indicator of cellular iron is the amount bound to ferritin, which can be extracted and measured. Here we record both variables, LIP and holoferritin.

2.4.2  Contribution of organs and tissues to whole body mass

An important scale factor is the fractional contribution of each organ or tissue to the mass of the whole body. Table 2.1 provides data for organ weights of C57BLC mice litter mates [27 ]. We include the estimates of plasma and blood volume of the whole body corrected for the bias of peripheral venous measurement [28 ].

Table 2.1: The values were taken from [27 ].

Organ weight in g per animal  

(ca 25 g)  

Standard deviation

% of body weight

Liver*

1.22

0.10

4.75

Spleen

0.07

0.01

0.27

Bones

1.80

0.25

7.01

Heart

0.14

0.02

0.55

Kidneys

0.38

0.05

1.48

Lungs

0.13

0.05

0.51

Stomach

0.18

0.04

0.70

Intestine

1.22

0.19

4.75

Duodenum

0.04

0.01

0.16

Integument (fur)

3.97

0.73

15.50

Fat

0.31

0.07

1.21

Muscle

13.42

1.21

52.30

Brain

0.47

0.01

1.83

Testicles

0.24

0.07

0.94

Plasma

1.36

0.03

5.30

Blood cells

0.71

0.02

2.77

Whole blood**

2.07

0.04

8.07

Bone marrow***

0.21

0.03


0.84

Blood values were scaled to the whole body hematocrit according to [28 ]. It is being assumed that the organ weights do not considerably change between different dietary regimes in a healthy mouse. * Liver weight per mouse. Data from [27 ] ** Barbee et al. [29 ] give 2.3 ml / 25 g mouse *** Mass of bone marrow in mouse (body weight 23 g) calculated from Lee et al. [30 ] p.484) is 192 mg, here scaled up to a mouse body of 25 g.

2.4.3 Upscaling of iron content to the whole organism 

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To scale up the specific iron concentration (f, expressed as μg iron per g wet organ) to the total contribution of an organ or tissue compartment we multiplied such data by the total weight w (in g) of the organ(s) per body, i.e. f * w. The standard deviation was calculated from the standard deviations h(f) and s(w) of the factors according to the formula 2.23 on p.7 of [31 ]:

which assumes that the measurements of organ mass and iron content are independent random variables.

2.5 Ferrokinetic study of tracer distribution

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As pointed out in the introduction it is intended to specify the model to the iron status of the mouse strain C57BL6. There are two basic sources of quantitative information available: biochemical assays of the “iron status” of the animal, and tracer-kinetic measurement of the fluxes between the iron compartments of cells and tissues. The former type of information has been assembled from a detailed study of the published literature on this mouse strain. The latter type of information was derived from experiments done in the laboratory of Professor K. Schümann, TU Munich, complemented again by literature data.

2.5.1 Experimental Setting

The data used for the model have been derived from ferrokinetic data [27 ]. In brief, male young adult mice (C57BL6 strain; 18-20 g) underwent a 5-week period (growth up to 25 g) with a diet controlled for iron content (iron content of diet induction of deficiency: 6 mg/kg; for adequate supply: 180 mg/kg; for iron overload 25000 mg/kg). The experiment was started by intravenous administration of ionic radioactive tracer (Fe59 nitrate in complex with nitrolotriacetic acid). It is known that this equilibrates quickly with the transferrin-bound iron pool [32 ]. The single tracer dose contained about 0.285 µg per mouse, which is less than 2% of plasma iron and in the range of 0.01% of body iron. At certain intervals between 12 hours and 28 days animals (n = between 3 and 7) were sacrificed, blood was collected and their main organs were dissected, weighed, and their iron status (non-heme iron) was measured. Hematocrit and haemoglobin content of blood were measured, and aliquots were separated into plasma and red blood cell compartment. The weight of organs and tissue samples was measured and normalized to the whole body (25 g). The ambient Fe59 content of organs and blood compartments was measured by scintillation counting and also converted to a whole body value. The tissue contents of tracer were corrected by subtraction of the tracer in the residual blood, by a calculation scheme that was derived from parallel model experiments with Fe59-labeled erythrocytes as indicator [27 ].

All tracer data were corrected for the decay of radioactivity during the experiment by normalizing them with the help of the radioactivity of the injection solution measured at the same time.

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Data concerning the total iron clearance rate were obtained from Trinder et al. [33 ], as they were obtained on control mice of the same strain under very similar dietary regimes.

2.5.2 Raw data corrected for blood content

The raw data obtained from the experiments are given in (see Appendix A tables 1, 2 and 3). In iron-deficient mice the tracer content of the spleen dropped to zero after a short time and even reached apparent negative values. This is clearly an overcorrection for blood-related Fe59, as the spleen contains a large extravascular blood pool that cannot be removed completely by perfusion. The true tracer content of the splenic tissue was obviously close to zero at those time points (see Appendix A, table 1). Therefore, we set these small negative values equal to zero. This manipulation did not appreciably change the fractional clearance parameter of the iron-deficient spleen.

2.5.3 Averaged tracer content in the intestine

As the precise organ weights of intestinal subsections, such as ileum/coecum/colon were not established, we replaced the tracer concentrations in Appendix tables 1, 2 and 3 by a mean ascribed to the intestine as a whole. The standard deviation of this derived quantity was calculated according to the SQR-forluma sketched above.

2.5.4 Normalization of the data set

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It is technically difficult to inject exactly the same small fluid dose of radioactive iron-solution. Therefore, the Fe59 content of an organ at a given time point was expressed as the Fe59 concentration normalized according to the formula:

Sum of radioactivity in the body at time t = 100% * exp h ,

were t is expressed in days.

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This describes the long-term rate of iron loss from the murine body [23 , 34 ] give a somewhat lower rate of ~ 0.004 per day. The process of normalization smoothed the ups and downs of determined tracer contents. It indicates a loss of about 13% of the injected tracer over an experimental period of one month, which is in accordance with previous estimates (10-15%) [35 ].

2.5.5 Mathematical structure of the compartment model of tracer distribution

The generic structure of the model is a set of balance equations describing time course and steady-state of the iron content of kinetically relevant pools:

(1)

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where

Ci ( ≥ 0) - “pool size”: iron content of the i-th compartment, i = 1…n (number of pools) - chosen scale is μg per body

vij ( ≥ 0) - rate of iron (in-) flux from compartment j to i (j = 1…n; j ≠ i ) (μg per body per day)

↓17

vji  ( ≥ 0) - rate of iron (out-)flux from compartment i to j (μg per body per day)

vio ( ≥ 0) - rate of iron flux from outside the system into compartment i (μg per body per day); only influx into duodenum of “cold” iron has a value > 0

voi ( ≥ 0) - rate of iron flux from compartment i out of the system (μg per body per day; values > 0 only for intestine, stomach, integument).

2.5.6 Clearance mode of model description and derivation of motion equations.

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Each of the rates is a function of the status of some of the pool sizes and of kinetic parameters. One may introduce fractional clearance coefficients (letter k) by the following definitions:

 

 

(2)

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These coefficients describe the ambient tendency of a given flux to clear the pertinent source pool. They apply also to the dynamics of the tracer content. If fluxes and pool sizes do not change substantially during the experiment (steady-state of the bulk of “cold” iron fluxes and pools during the experiment by the tiny amount of tracer), the k´s become constant and a system of ordinary linear differential equations with constant coefficients describes the motion of the tracer content (x – being measured as specific radioactivity, i.e. counts per mass of compartment iron, normalized to initial tracer dose):

 

 (i = 1…n) 

(3)

↓20

where a term kio * xo has been dropped because re-absorption of excreted radioactivity can be neglected, if the tracer has been applied to the plasma compartment. The k-values for non-existing fluxes (out of the system or non-reversible) are set to zero. The system of differential equations describing the tracer motion is found in the next sections.

2.5.7 Residence time

Time scale is an important aspect of any metabolic model [102]. Interpretation of systems dynamics is simplified by introduction of expected residence times of molecules of defined biochemical state within a compartment:

(j=1…n; j ≠ i)

(4)

2.6 System of Ordinary Differential Equations for Tracer Motion 

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We established a mathematical description of the tracer content and the inter-compartmental tracer flow using linear ordinary differential equations. This choice is justified as it can be assumed that pools and fluxes are approximately constant during the experiment. Under these conditions are the linear coefficients of tracer clearance constant quantities. The system of equations is shown below. For designations, refer to Figure 2.1. Parameter names in the computer program were assigned according to the convention ka_b, meaning clearance parameter of flux from a into b:

At time = 0 the injection of a tracer dose (scaled to 100%) into plasma sets the boundary condition of this system (plasma (0) = 100% all other compartments = 0%). This sets a relaxation into motion that follows the fluxes of the bulk iron through the body.

2.7 Parameter optimization pipeline

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With a system of linear ordinary differential equations representing the major ferrokinetic processes, the next step was to find reasonable parameters values that correctly described the physiological exchange of iron in the body. For this purpose we created an optimization pipeline (Figure 2.2).

Figure 2.2: Optimization pipeline developed in this study.

Starting with differential equations with arbitrarily chosen parameters the system is run and its results continuously compared to measurements. The difference between them is used by the optimization algorithm to improve the selection of parameters values for further runs.

The first step was to create a system of linear ordinary differential equations and define reasonable initial parameter values, which were mainly derived from literature extracts.

↓23

With this system of equations we produced simulated curves for each organ of our model (step number 1 in figure 2.2). We used an ODE solver from Matlab for stiff systems, since our system comprises different time scales, varying from minutes to days.

We then compared the generated curves with real ferrokinetic measurements used as input for our model (step 2 in figure 2.2). Three different distance measures were tried: simple squared distance, squared distance divided by the standard deviation and squared distance divided by variance. Our experiments demonstrated that in our model the choice of one distance measure did not produce strong differences in parameter estimates, so we chose the second, defined by the equation :

(5)

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where is the estimated value produced by our model, and is the measured value obtained from radioiron injection and std stands for the standard deviation of the measured point . The index i runs from 1 to7 and corresponds to the 7 points of measurements in our experiments. The index j runs from 0-16 and corresponds to each organ of our system and experimental settings.

We then estimated the parameters of the equation system in order to minimize the distance between the model curves and the measurements (step 3 in figure 2.2).

The solution was to use a constraint optimization algorithm named Active-Set. This is a hill-climbing algorithm which receives as input: the initial parameter values and the system constraints. In our case only one constraint was defined, the sum of the parameters leaving the central compartment (Plasma) to the peripheral organs should be 20. This is based on the principle that the iron clearance time is below 1 hour in mice [33 ].

2.7.1 Parameter Estimation by Convergence from different Starting Points

↓25

The parameter optimization step was performed using as input the datasets with measurements of organ radioactivity under different diets. It was checked that the algorithm always converged to a parameter set giving an equal value of the distance minimum. In order to test the statistical uncertainty of these parameter estimates we randomly perturbed the measured compartment radioactivities within the measured standard deviations around their mean values. This change was performed for every measurement in the three datasets and the optimization pipeline of figure 2.2 executed. This complete procedure was repeated 100 times for each dataset.

2.7.2 Quality of final fit

As an intuitive measure of the quality of the final fit we chose the root of mean of squared weighted deviation between prediction and the mean of measurements (table 3.2; “sq root of mean weighted squared dev”). We document the parameter set of the best fit together with an upper and lower bound obtained from a sextile-truncated sample of the empirically found parameter variation. In the case of a Gaussian distribution the interval thus defined would be twice the standard deviation.

2.7.3 The problem of interdependence of parameter estimates

Some parameters or sets of parameters are not identifiable by measurement in a model of given design. For instance, if ferrokinetic measurements are available only for the time course of changes in plasma concentration of tracer, the total clearance rate can be adequately assessed, but not the distribution into the network of body compartments. In contrast, if the first measurement was obtained after most of the tracer has left the plasma compartment, the relative distribution to the various organs can be assessed, but the flux dynamics of this distribution cannot. Even if parameters seem to be identifiable in the Laplace domain [36 ], the error structure of the data will make the ensuing algebraic treatment ill-conditioned and will expand the range of parameter estimates. They become meaningless. We tackled this problem by studying the parameter space using the alternating conditional expectation algorithm (ACE) [37 ].

↓26

The essence of this strategy is to explore the total domain with all parameter values that support an acceptable fit, i.e. the criterion value of this fit is sufficiently close to the optimum. This is done by running a large number of parameter estimations from a large range of starting points. The result is a set of sometimes widely differing parameter vectors that yield very similar optimal values of the fit criterion. Interdependent and hence non-identifiable parameter combinations can be addressed by the information furnished by the ACE method. On the basis of this analysis one or more suitable parameters are chosen to be fixed or held in definite proportion to each other so that the other parameters became identifiable. The choice of suitable parameters for this reduction of parameter redundancy and of their fixed values is not always easy and requires a theoretical understanding of the dynamic structure of the model. Maiwald et al. [39 ] have developed software (“Potter´s wheel”) the tools of which support this computer-time-intensive study very effectively. Specific assumptions made to remove strong parameter intercorrelation will be mentioned explicitly below. 

2.8 Flux rates and pools sizes derived from clearance parameters

2.8.1 Calculation of absolute flux rates from fractional clearances

The tracer data alone can be used to estimate fractional clearance parameters (per day) of flux out of plasma (ki_plasma). To calculate flux rates (µg per day per mouse) one needs the iron content of plasma plus extracellular fluid (Cplasma/ECF):

(6)

↓27

The iron pool of the plasma plus ECF is calculable from the iron concentration and the plasma volume plus the accessible volume of ECF.

2.8.2 Estimation of peripheral pool size from countercurrent clearance parameters and plasma pool

The iron content of body organs can be measured by chemical methods, mainly as hemoglobin, myoglobin, and non-heme iron. Cellular heme iron content as oxidoreductases and other proteins is less important quantitatively. Such direct measurements may be contrasted with tracer-accessible iron pool sizes. When in the steady-state influx and outflux are equal, the following equation relates the pool size Ci of a peripheral compartment to that of plasma

(7)

↓28

If the data support estimates of the two rate constants (ki_plasma defining the early phase and kplasma_i  characterizing the late phase of tracer distribution) and of the plasma/ECF iron content, then the size of the “accessible” pool Ci  may be inferred.

2.8.3 Scaling of the system variables and parameters 

All parameters and variables were scaled to be dimensionless. On rescaling to the in-vivo state variables and fluxes were transformed to units of µg per mouse (24 g) and µg per day per mouse, respectively. Turnover times and the inverse of related kinetic constants are expressed in days.

2.9 The Cellular Model of Iron Metabolism

So far we obtained a description of the physiological distribution of iron in mice. It aims at estimation of the cellular iron pools and iron fluxes between organs in the steady-state of the whole organism. In this section the layout a kinetic description of the iron turnover within and between cell types of the organism will be developed with the goal of predicting the steady-state of the whole organism of the mouse as a result of kinetic interaction of iron species in different cell types. The model will also comprise the most important regulatory mechanisms of iron metabolism. It will be based on quantitative literature data concerning the fine-tuning of iron metabolism. For obvious reasons it is not feasible to formulate such a model on the most elementary molecular and cell-biological level. This is because most types of kinetic measurements cannot be done on the intact tissue or cell in its natural environment. The reductive approach of experimental cell biology therefore provides in many cases qualitative data (molecular “mechanisms”), but no quantitative description. It will however be feasible, on the basis of existing physiological data and the established network structure of iron metabolism to derive a description of the core regulatory structure of the system.

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The mechanistic skeleton of iron metabolism is very much the same in each cell type, because the genes and their products are the same. We therefore derive a generic model of iron metabolism within the cell. The specificity of tissue and cell types will be introduced in addition by modifying parameters of the regulatory structure of this generic model. Such core parameters are the iron status and the expression level of iron-related mRNA and/or protein of a specialized cell type.

2.9.1 Transfer across the cell membrane

2.9.2 Intracellular processes

2.10  Iron flux network

2.10.1 Intracellular and transmembrane iron flux

↓30

The fig. 2.3 depicts how the different elements of intracellular iron metabolism interact with each other. The figure refers to a generic cell, i.e. to a cell that contains the main elements present in every body cell. Some special features of cell types that have a special role in iron metabolism are also sketched.

The upper part of the picture shows the transferrin cycle (fluxes v1, v2, v3 and v4) as present in almost every cell. Depicted on the the left side is the iron uptake by DMT1 (v5) which is expressed in duodenal cells, as well as non-transferrin-mediated iron uptake (v7) which happens mainly in hepatocytes.

Iron taken up by the mechanisms modelled as v1, v5 and v7 is further transferred into the Labile Iron Pool (LIP) (fluxes v3, v6 and v7).

↓31

LIP iron molecules undergo four possible fates: stored in ferritin (w1), deployed in heme synthesis (w9), biosynthesis of Fe-S cluster (w3), and export by ferroportin transferred onto holotransferrin (w5 and w6). Stored iron may be mobilized (flux w2). Also iron bound in Fe-S clusters are re-utilized during the general protein turnover occurring with different time course in all cells that do not undergo cell loss. Heme iron may only be reutilized (w12) if the cell expresses the corresponding catabolic pathways (e.g. RES macrophages). Iron taken up into the erythropoetic pathway in bone marrow is converted to hemoglobin and added to the red blood cell compartment (w1 0). Red blood cells become senescent after a certain functional lifetime and the heme is transferred to macrophages of RES. There exists also a direct shuttle from bone marrow to RES [22 ] which has been explained as “ineffective erythropoesis”.

At last, on the right side of the picture, is a leakage term which is used to represent the iron loss of the system due cell death and removal, mainly of intestinal epithelium and skin and other integument cells. Since labile iron pool is present only in tracer amounts, this leakage will visibly affect only the holoferritin level, which is in equilibrium with LIP.

Figure 2.3: The figure shows the general cellular model shared by all organs in the model.

What distinguished the cell-types are the parameter values defined for the equations. There are three pathways of iron input in the cell: TfR1 cycle, dietary uptake through DMT1 and non-TF bound iron. Inside the cell iron has some possible fates: be stored within ferritin, be exported to blood plasma by FPN1, participate in heme synthesis, be part of Fe-S clusters or in the case of bone marrow, be transferred directly to RES, due ineffective erythropoiesis.

2.11 Regulated turnover of iron-processing macromolecules

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The macromolecules taking part in the regulatory scenario of this cellular model undergo continuous biosynthesis and degradation. This involves gene transcription into mRNA, translation of mRNA into protein and, after a certain characteristic life time within the cell, the degradation of mRNA catalyzed by RNAse and of proteins by the protein-degrading machinery of the cell. We simplify the enormous complexity of these processes by formulating for each protein an overall rate of biosynthesis and a rate of degradation. The latter is assumed to be proportional to the ambient cellular expression level. Both processes are influenced by kinetic signals of specific regulatory factors. The different levels of regulation (e.g. transcription regulation by a signal cascade involving plasma iron and hepcidin, or translation of mRNA into protein regulated by the IRP machinery) are integrated into these two overall rates. Figs. 2.4a and 2.4b illustrate how the aggregation of the mRNA and protein level together with a qualitatively equivalent transformation of the regulatory levels was modelled.

Fig. 2.4:

a) EPO increases transcription of TFR1-mRNA in bone marrow; IRP stabilizes mRNA and thereby indirectly increases TFR1 expression level. b) Hepcidin increases degradation of ferroportin protein; IRP inhibits translation of ferroportin mRNA, so indirectly decreasing ferroportin expression level

Figure 2.5 shows the cellular processes of steady macromolecular synthesis and steady decay that were implemented in this model. The “u” arrow stands for the integrated rate of translation and translation of a certain protein, whereas “z” represents the rate of degradation of a given protein.

↓33

Included in this schematic picture is the IRP regulation. In the case of TfR1 and DMT1 the IRP proteins inhibit m-RNA degradation and hence enhance protein synthesis indirectly by the increased level of the mRNA.

In the case of ferroportin, apoferritin and other cellular iron proteins the IRP system inhibits the translation of mRNA, thereby reducing the rate of synthesis of the respective protein in the cell. The Labile Iron Pool (LIP) plays a significant regulatory role as intracellular signal of the iron status. The exact nature of the iron signal is not known, but there is evidence that the LIP is correlated with it. The key regulator IRP1 protein has two states: active and inactive. The LIP has a stimulatory effect in the transition of the IRP1 protein from active to inactive state. When there is plenty of iron in the cell the synthesized Fe-S clusters bind to the IRP1 protein and convert it to a cytosolic aconitase which is not able to exert its role as an iron regulatory protein (reviewed in [40 ]).

In the case of the second regulator IRP2 it is not a conversion between states that is regulated but rather the degradation rate of this protein in the cell. Therefore in our model LIP has a positive effect on the degradation of IRP2.

↓34

One important aspect of our model concerns the descriptive combination of the IRP1 and IRP2 proteins in order to formulate an indicator variable “IRP effective”. The “division of labor” between these two closely related versions of intracellular iron regulation is not completely clear. We represent the two proteins here as a “pool” of IRP and posit a combined effect of the IRP´s, represented by a simple formula:

IRP = IRP1act/ IRP1reference + IRP2 / IRP2reference

where IRP1act stands for the active part of the expression level of IRP1, and IRP2 stands for the total expression level of the protein. The reference parameters allow for different activities of the two versions, the empirical basis of which is the fact that knockouts of IRP1 in mice are lethal and those of IRP2 are not lethal, but lead instead to an iron storage disorder [42 ]. The transition for active IRP1 into its inactive aconitase form is taken into consideration as rapid equilibrium, whose position is influenced by the level of the labile cellular iron pool (as well as by other physiological parameters like oxygen level, H2O2 and NO levels). [43 ].

↓35

Fig. 2.5: This is a simplified scheme of the turnover of the iron-processing proteins in the generic cell.

The transcription/translation process is represented in aggregated form by arrows and fluxes dubbed u (with index) and the decay process by arrows and rate variables z. The latter refer either to proteases or RNAses that remove the respective iron-processing proteins or their mRNA precursors. The lower-case letters at the macromolecules refer to the variable names of the differential equation system.

Some processes are subject to global regulation by hepcidin. Hepcidin expression is regulated by the iron saturation level of plasma transferrin (symbolized as a direct stimulation of hepcidin synthesis). Hepcidin enhances the degradation of ferroportin predominantly in RES and duodenum, also in liver. It leads to a rapid internalization of membrane-resident ferroportin (f0 into f2) with slower proteolytic degradation afterwards.

EPO has a positive stimulatory effect on the synthesis of TfR1 in bone marrow. The synthesis of EPO is negatively regulated by the levels of RBC: when the RBC level is high, there is no need for EPO, when RBC level diminishes, more EPO is synthesized to stimulate erythropoiesis.

↓36

All these processes tend towards a steady state ( u = z) which is reached after a characteristic time set by the ratio of ambient protein level / cellular degradation rate.

2.12 Nomenclature: variables and rates

Table 2.1 summarizes the molecular elements and table 2.2 describes the fluxes in the model. Every organ has the same basic set of fluxes and the same set of differential equations; what distinguish them are the kinetic parameter values relating to the expression level and its regulation. There is only one transferrin synthesis and degradation term for the whole system (table 2.3), because the carrier protein transferrin is mainly synthesized in the liver and not in other cells.

Table 2.1: Species shared by all cell-types in the model.

Symbol

Description

α

TFR1 on cell membrane surface

β

TFR1 / holoTF complex on cell membrane surface

γ

TFR1 / holoTF complex internalized

δ

TFR1 / apoTF complex in membrane

τ0

apo TF in plasma

τ1

holo TF in plasma

λ

LIP (labile intracellular iron pool)

φ0

apoferritin

φ1

holoferritin (with iron)

s

FeS cluster

f0

free ferroportin exporter in cell membrane

f1

iron-loaded ferroportin exporter in cell membrane

f2

ferroportin in inactive form

h

heme level in cell

m0

DMT1 as duodenal entry of iron, and as activator of holoTF/TFR1-processing in the lysosomes

m1

iron-binding state of DMT1

y0, y1 

IRP1 level, in active and inactive form

y2

IRP2 level

EPO

Erythropoietin

Hep

Hepcidin level

RBC

Circulating red blood cells, iron content

The exceptions are Hepcidin, EPO, RBC and transferrin (τ0, τ1), which are defined once for the whole system.

↓37

Table 2.2: Model fluxes.

Flux

Description

yeff

effective activity of IRP system

w1

uploading LIP iron on apoferritin

w2

mobilization of ferritin iron into LIP

w3

synthesis of FeS cluster

w4

decay of FeS cluster protein, liberation of iron into LIP

w5

uploading LIP iron onto ferroportin

w6

export of iron onto apotransferrin (via FP)

w9

heme synthesis

w10

heme degradation

w11

flow of iron between the bone marrow and RES (ineffective erythropoiesis

w12

Return of iron from heme to LIP (catabolism)

w13

iron loss due epithelial cell desquamation (skin / intestine)

v1

binding of holoTF (plasma) onto TFR1 (membrane)

v2

internalization of TFR1 / holoTF complex

v3

release of iron into LIP from TFR1

v4

return of apoTF into plasma & TFR1 onto membrane surface

v5

TFR1-independent inflow of iron into certain cells

v7

Non-transferrin or DMT1-mediated iron uptake

This are the essential operations which involve transfer of ionic iron between different model parts, e.g. from outside the cell to inside or from LIP to apotransferrin. They are depicted in figure 2.3.

Table 2.3 summarizes the turnover rates of macromolecules (synthesis and degradation).

Symbol

Description

ui1

translation of IRP1-mRNA

ui2

translation of IRP2 –mRNA

ui3

translation of TFR1 –mRNA

ui4

translation of DMT1-lyso – mRNA

ui5

translation of ferroportin mRNA

ui6

translation of apoferritin-mRNA (in liver)

ui7

synthesis of apo-transferrin

zi1

Decay of IRP1-mRNA

zi2

Decay of IRP2 –mRNA

zi3

Decay of TFR1 –mRNA

zi4

Decay of DMT1– mRNA

zi5

Decay of ferroportin mRNA

zi6

Decay of apoferritin-mRNA

zi7

Decay of plasma transferrin

It should be noted that the elements in this table also contain the index “i” which refers to the different cell types (tissue types).

2.13 Balance equations

We derive the kinetics of iron interaction within and between cell types of the body according to the following general concept:

↓38

These is a kinetic “hybrid” model (in the spirit and wording of [45 ]). For practicability purposes we combine simplified rate laws as basis of the model and insert special terms for signalling effects.

2.13.1 Balance equations in the plasma compartment

In the central compartment the transferrin-bound iron and the hormones hepcidin and erythropoietin are dispatched to their effector locus.

↓39

0/dt = u7  - z7 + Σi (v i4 – wi6)

apo-transferrin

1/dt = v0i (wi6 – v i1)

holo-transferrin

The indexed fluxes have the following meaning:

v i1 - entry of holo-transferrin into the endocytotic cycle

↓40

v i4 – release of apo-transferrin after endocytotic deloading of iron

wi6  - transfer of cellular iron onto apo-transferrin (ferroportin-mediated) 

The index i applies to the cell/tissue types:

↓41

 i=1: bone marrow,

i=2: liver

i=3: RES

↓42

i=4: muscle

i=5: integument

i=6: intestine

↓43

i=7: duodenum

 and the sum Σ is defined for the w6 and v1 of the respective cell type or tissue.

dHEP / dt = u8 – z8

hepcidin

dEPO / dt = u9 – z9

erythropoetin

↓44

Balance equations in the blood cell compartment

dRBC / dt = w10 - wRBC

2.13.2 Balance equations in the cell, with cell type parameter specification

Fig. 2.2 symbolizes the events within the generic cell and its borders. We now write down the balance equations pertinent to this scheme. The individual rate equations for special cell types will be derived below, but we indicate in bracket if a flux applies only to certain cell types (i.e. has a non-zero value only for the cell types indicated in bracket). The designations are defined in tables 2.1-2.3 (apo- and holo- applies to free and iron-loaded entities, respectively).

↓45

0 / dt

=

w2 – w1 + u6 – z6

apo-ferritin

1 / dt

=

w1 – w2

holo-ferritin

df0 / dt

=

u5 – za5 – w5 + w6

apo-ferroportin

df1 / dt

=

w5 – w6

holo-ferroportin

df2 / dt

=

za5 – z5

ferroportin for degradation (phosphorylated)

dm0 / dt

=

u4 – z4 + v6 – v5

apo-DMT1

dm1 / dt

=

v5 – v6

holo-DMT1

dα / dt

=

u3 – z3  + v4 – v1

TFR1 in membrane

d β / dt

=

v1 – v2

holo-TF-TFR1 in membrane

d γ / dt

=

v2 – v3

holo-TF/TFR1 internalized

d δ / dt

=

v3 – v4

apo-TF/TFR1 in membrane

dy1 / dt

=

u1 – z1

IRP1

dy2  / dt

=

u2 – z2

IRP2

ds / dt

=

w3  - w4

dh / dt

=

w9 - w10 (bone marrow) - w12 – w11

The expression for cellular free iron (LIP) reads

2.14 Rate equations of iron transfer between iron-processing proteins

↓46

Kinetic theory of biochemical catalysis in aqueous dilution is not applicable to iron metabolism. Iron occurs in coordinated states bound to carrier protein or in complex as component of prosthetic groups (e.g. heme). The kinetic description was therefore chosen in terms of on- and off- rate constants times the cellular content (“concentration”) of respective reactants of a transfer or binding reaction. For the transferrin-receptor-mediated endocytosis a kinetic theory was obtained from data on isolated cells [41 ]. We assume that the time characteristics of intra-cellular iron transfer is of the same range for all metabolically active cells. Variables are scaled to dimensionless quantities, setting them equal to unity in what we call the reference state of the organism (adult male mouse, 25 g, on adequate iron diet). Tissue contents of ferritin (“non-heme iron”) were available from the literature and from measurements of Schümann et al. [27], as were flux rates between tissue compartments after analysis of ferrokinetic data. These quantitative data can be converted to tissue-specific kinetic rate parameters. We applied power-rate laws in some cases, when the subsystems of iron transfer or endocrine signalling to cells have a high logarithmic gain in vivo (i.e. small relative concentration changes lead to high relative effects).

2.15 Kinetic Description of Iron-Transfer and Regulatory Signals

The cell manages ferrous iron in bound form, attached to specialized carrier proteins, such as ferritin. Iron-containing prosthetic groups (such as heme group) are bound to protein carriers. Uptake and secretion of iron is also catalyzed by proteins which organize the transfer of iron into and out of the cells. Plasma iron is bound to the transport protein transferrin. All these processes take place in a cellular or membrane medium where the methodology of kinetic analysis in biochemistry with its concepts of “concentration” and “steady-state enzyme kinetics” are applicable at best by loose analogy. Well-known methods to achieve this are the methods of approximate kinetic formalisms [46 ]. In effect, these concepts allow for a quantitative description of metabolic turnover that integrates the mechanistic molecular detail as black-box which is assumed to be in a dynamic steady state.  [46-52 ]

Many of the iron-transfer reactions take place within a narrowly restricted range of “concentrations” of the partaking components. We introduce expansions (linear for protein turover, bilinear for iron processing, power law for reguatory signals) of the complex rate laws which are approximately valid in the neighborhood of a generic reference state, which is conceived as an idealized model of the “normal” healthy organism. All iron-processing proteins are assumed to exist in an iron-free and an iron-bound state (e.g. apoferritin and holoferritin), the sum of both being fixed at a given moment. This ensures that there are upper and lower constraints to the rates in the simplified kinetic description.

↓47

For convenience of controlling different simulation runs with systematic parameter variation we adopted a formulation of the bi-linear kinetics of reversible binding reactions as reversible rates with kon and koff as rate parameters:

Binding rate = kon * substrate level * free protein carrier level

and

↓48

Release rate = koff * occupied protein carrier level,

with

free protein carrier + occupied protein carrier = ambient expression level of protein carrier.

↓49

The signal cascades operating in iron metabolism involve a number of components to which the details of the accepted biochemical formalism cannot be specified. In contrast to the kinetic formalism of metabolic transfer the black-box formulation of the signaling cascades has to include an amplifier mechanism, because otherwise the operational gain of a signal (log final effect /log effector level) becomes very small at the end of a long and branched cascade. Regulatory cascades in vivo have inbuilt strong amplifier mechanisms (sometimes even to the extent of an all-or-none effect), but we will not produce this by meticulous formulation of details, but rather choose a general black-box formula in the form of a simple power rate law. For an activator signal we adopt the representation

activator signal = (activator level / Kapp) n  

and for an inhibitor signal:

↓50

inhibitory signal = (Kapp / effector level)) n 

The effector levels and the signal strength are expressed in dimensionless form and included into the rate equations for the respective reaction. The term Kapp (= apparent) refers the ambient effector level to that of the idealized (“normal”) reference state. The exponent n realizes the required amplifier gain. This particular form of signal kinetics was chosen for the sake of ensuring that the logarithmic gain of activator and inhibitor signals of different regulatory loops is symmetric and may be qualitatively related to each other by choice of the apparent kinetic parameter Kapp and the logarithmic gain factor n. For illustration, consider the fate of a macromolecule x which is synthesized and degraded by a steady-state dynamics:

x

↓51

If there is an activator effect operative enhancing the expression level of x, this can be achieved either by an activator term (act) on the input reaction

dx/dt = 0 = U * act – Z * x with stationary x = U * act / Z

or, equivalently (when there is no information on the precise molecular mechanism) by an inhibitor term (inh) on the output reaction

↓52

dx/dt = 0 = U – Z * x * inh with stationary x = U / (Z * inh)

Equivalence of both effects is only given if

act = 1 / inh

↓53

With a logarithmic gain of n this requires that

act = (effector / Kapp)n  and inh = (Kapp / effector) n

Any of the usual Hill or allosteric equations for activation and inhibition cannot guarantee this equivalence. For the sake of comparison of interacting global and cellular signals we chose therefore this simple form. It guarantees that the reference state of the system (arranged to effector/Kapp=1) does not change when different n-values are tested for efficiency. And it guarantees that activation of one partial reaction is symmetric to inhibition of the other partial reaction involving synthesis and decay of an effector, so that quantitative comparison of effector strength becomes possible, with Kapp and n as modifying parameters. In numerical simulations one must take care of not falling into regions far from the reference state where the simple formula is no longer even qualitatively valid (e.g. approach of positive or negative infinity of the ratio). 

2.16 Modelling the hepcidin effect on ferroportin expression

↓54

Hepcidin is being synthesized in the liver and reaches the locus of its signalling effect via plasma. Effector cells are mainly duodenum, macrophages of RES and hepatocytes. Hepcidin interacts with the membrane-resident ferroportin and causes its internalization, which is then followed by intracellular proteolytic degradation. From hepcidin injection experiments it is known that the internalization occurs rapidly (within a few hours [53 ], whereas the ensuing proteolytic degradation has the general time characteristics of several days. This regulation was therefore modelled as depicted in fig. 6.

Fig. 6: Hepcidin increases degradation of ferroportin protein by binding and internalizing.

The balance equations of this subsystem read

↓55

df0/dt = u5 – z5a – w5 + w6

df1/dt = w5 – w6

df2/dt = z5a – z5 .

↓56

Assuming linear kinetics one gets

z5a = Z5a * f0 * hepact  

z5 = Z5 * f2 

↓57

(hepact is the kinetic activation term of hepcidin level) 

and for the steady state

f0 = u5 / (Za5 * hepact)

↓58

f2 = u5 / Z5

f1 is proportional to f0 , depending on cellular iron and free plasma transferrin 

(λ and {τ0 / (τ01)}

2.17 Rate equations of iron uptake and iron release by the cell

↓59

v1 = K1 * α * τ1

(K1 = 0.4; calculated from [41 ])

v2 = K2 * β

(K2 = 0.1; calculated from [41 ])

v3 = K3 * γ * m1

(K3 = 0.05; calculated from [[41 ])

v4 = K4 * δ

(K4 = 1.25; calculated from [41 ])

v5 = K5 * m0

(K5  = 2.5; iron in intestinal tract)

v6 = K6 * m1

(K6  = 2.5; iron uptake via DMT1)

v7 = K7

(K7 = 0.5; parameter for non-TFR1 iron uptake by liver)

2.18 Rate equations of internal transfer

w1 = H1 * φ0 * λ

(H1 = 10)

w2 = H2 * φ1

(H2 = 1)

w3 = H3 * λ * / Yeff ^ q3

(H3 = 5; q3 = 1; signal Yeff  see below)

w4 = H4 * s

(H4 = 5)

w5 = H5 * λ * f0 ^ 3

(H5 = 1.8)

w6 = H6 * f1 * τ0

(H6 = 2.4)

w9 = H9 * λ / Yeff ^ n9

(H9 = 20.1 in bone marrow; 2.22 in muscle else=0; n9   =1)

w10 = H10 * h

(H10 = 18.1 bone marrow; else = 0)

w11 = H11 * h

(H11 = 2.0 bone marrow; else=0; w11 identical to wshunt)

w12 = H12 * h

(H12 = 2.2 in muscle heme turnover )

w13 = H13 * λ

(H13 = 1.65 in integument ; 1.35 in intestine)

wRBC = kRBC * RBC/ RBCref

(kRBC = 18.1; RBCref = 1)

The parameters of wRBC were derived by normalization. The actual reference pool size of RBC is 301.7 µg per mouse, and the rate corresponds to a erythrocyte renewal rate of 0.06 per day.

2.19 Rate equations of combined transcription/translation (protein biosynthesis)

↓60

u1 = U1

(U1 = 1) expression of IRP1

u2 = U2

(U2 = 1) expression of IRP2

u3 = U3 * epo ^ nepo * yeff ^ n3

(n3 = 1, nepo=5) expression of TFR1

Tissue

U 3 Parameter value in different cell types

bone marrow

420

liver

70

RES

0

Muscle

44

Integument

34.4

intestine

28

duodenum

1

The EPO activation term is only valid for the bone marrow, for the other organs it is to be cancelled out. The U3 values were calculated from ferrokinetic flux estimates (see below, results). RES does not receive iron over the TFR1-pathway, but rather by phagocytosis of erythrocyte hemoglobin. The U3 values were calculated from ferrokinetic flux estimates (see below, results).

u4 = U4 * yeff ^ n4

(U4 = 1; n4 =1) expression of DMT1

u5 = U5 / yeff ^ n5

(U5 = 0.5; n5 =1) expression of ferroportin

u6 = U6 / yeff ^ n6

(n6 =1) expression of apoferritin

Tissue

U 6 Parameter value in different cell types

bone marrow

1.63

liver

2.84

RES

6.5

muscle

1.45

integument

3.92

intestine

0.38

duodenum

0.02

↓61

u7 = U7

(U7 = 1) expression of transferrin

u8 = U8 * (τ1/( τ1+ τ0) / 0.15 - 1.0)

(U8 = 0.3) expression of hepcidin, descriptive formula calculated from figure 1 of [55 ])

u9 = U9 * (RBCref / RBC) ^ n9

(U9=1; RBCref =1; n9 =4) (expression of erythropoietin is activated when RBC decreases)

2.20 Rate equations of protein degradation

z1 = Z1 * y1

(Z1 = 1) degradation of IRP1

z2 = Z2  * y2 * λ ^ n2

(Z2 = 1; n2 = 3) degradation of IRP2

z3 = Z3 * α

(Z3 = 2) degradation of IRP1

z4 = Z4 * m0

(Z4 = 1) degradation of DMT1

z5 = Z5 * f2

(Z5 = 0.5) degradation of ferroportin

za5 = Za5 * f0 * hepact

internalization of ferroportin

tissue

Za 5 Parameter value   in different cell types

bone marrow

1

liver

0.225

RES

0.044

muscle

0.409

integument

12.857

intestine

18

duodenum

0.3529

z6 = Z6 * φ0

(Z6 = 1) degradation of ferritin

z7 = Z7 * τ0

(Z7 = 1.33) degradation of transferrin

z8 = Z8 * hep

(Z8 = 0.2) excretion of hepcidin

z9 = Z9 * epo

(Z9 = 1) degradation of erythropoietin

2.21 Kinetics expressions for autocrine and endocrine signalling

Active IRP1: Y1_act  = Y1 / λ ^ 3 (LIP inactivates IRP1)

↓62

Combined effect of IRP-system, with IRP2 dominating and the IRP1 equilibrium poised towards aconitase in the reference state; [56 ])

Yeff = Y_act/ Y1reference + Y2 / Y2reference (Y1_reference = 20;Y2_reference = 1.0526)

Erythropoetin signalling to bone marrow:

↓63

epoact = epo ^ nepo

(nepo = 5)

 

Hepcidin signalling to RES, liver, duodenum, activating ferroportin degradation:

↓64

hepact = hep ^ nhep

(nhep = 3)

2.22 Parameter portrait to simulate physiological or pathological deviation

Parameter portraits plot the steady-state value of system variables against systematic changes of one or several of the parameters. This simulates physiological challenge or genetic perturbation of the basic reference state. The dynamic stability of the steady-state was checked as return to it after a series of random-perturbations of variables and, in critical cases, by evaluating the eigen-system around the steady-state.

2.23 Numerical solution of dynamic systems (ordinary differential equations)

With parameter values and the initial state of variables specified the dynamics of approach to steady-state was calculated using MatLab integration routines, mainly a Runge-Kutta algorithm.

↓65

The attainment of steady-state was checked by calculating the absolute values of right-hand sides of the differential equations divided by the level of the corresponding variable. A steady-state criterion was a level below 10-8 of this indicator.


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