New insights into the human body iron metabolism analyzed by a Petri net based approach

BioSystems 96 (2009) 104–113

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New insights into the human body iron metabolism analyzed by a Petri net based approach Andrea Sackmann a , Dorota Formanowicz b , Piotr Formanowicz a,c,∗ , Jacek Blazewicz a,c a b c

Institute of Computing Science, Pozna´ n University of Technology, Piotrowo 2, 60-965 Pozna´ n, Poland Department of Clinical Biochemistry, Pozna´ n University of Medical Sciences, Grunwaldzka 6, 60-780 Pozna´ n, Poland Institute of Bioorganic Chemistry, Polish Academy of Sciences, Noskowskiego 12/14, 61-704 Pozna´ n, Poland

a r t i c l e

i n f o

Article history: Received 28 February 2007 Received in revised form 1 December 2008 Accepted 18 December 2008 Keywords: Iron metabolism Hepcidin Modeling Petri nets

a b s t r a c t Iron homeostasis is one of the most important biochemical processes in the human body. Despite this fact, the process is not fully understood and until recently only rough descriptions of parts of the process could be found in the literature. Here, an extension of the recently published formal model of the main part of the process is presented. This extension consists in including all known mechanisms of hepcidin regulation. Hepcidin is a hormone synthesized in the liver which is mainly responsible for an inhibition of iron absorption in the small intestine during an inflammatory process. The model is expressed in the language of Petri net theory which allows for its relatively easy analysis and simulation. © 2009 Elsevier Ireland Ltd. All rights reserved.

1. Introduction The body iron homeostasis is one of the most important processes in the human organism. Since iron is responsible for many crucial cellular functions and proper growth of tissues, maintaining its proper level should be guaranteed by some molecular mechanisms. Indeed, a very complex biochemical process regulates the iron level. This level can be deviated from the physiologically normal one by a number of factors as inflammation, kidney diseases, hematological disturbances, etc. Until recently, only rough descriptions of the homeostatic process could be found in the literature. New insights into the iron metabolism have challenged our knowledge of the previous models of the body iron homeostasis. Rather than a simplified model based on a single system of cellular iron uptake through transferrin receptor (TfR) and storage (Ferritin), it is now understood that complex mechanisms in specialized cells participate in regulating iron balance through the expression of various genes. The maintenance of stable extracellular iron concentrations requires the coordinated regulation of iron transport into plasma from dietary sources in the small intestine, from recycled senescent erythrocytes in the one-nuclear cells and from storage in hepatocytes. Highly specialized mechanisms within the body maintain strict con-

∗ Corresponding author at: Institute of Computing Science, Poznan´ University of ´ Poland. Tel.: +48 61 8528503x276; Technology, Piotrowo 2, 60-965 Poznan, fax: +48 61 8771525 E-mail address: [email protected] (P. Formanowicz). 0303-2647/$ – see front matter © 2009 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.biosystems.2008.12.003

trol of cellular iron uptake, storage, release and intracellular iron management. In the series of our very recent papers (Formanowicz et al., 2007; Sackmann et al., 2007; Formanowicz et al., submitted for publication) a formal model of the main part of the body iron homeostasis process has been proposed. The model has been expressed in the language of Petri net theory which allows for clear descriptions of all components of the process and interactions among them. The concept of Petri nets was introduced by Petri (1962) in the context of technical systems. The first biological applications of Petri net theory were presented by Reddy et al. (1993) and Hofestädt (1994). Since then, various extensions to the original concept were made in order to model different kinds of biological networks; an overview is given e.g. by Chaouiya (2007) and Hardy and Robillard (2004). The process modeled by a Petri net can be relatively easily analyzed and simulated what is important for possible drawing some biological conclusions on the basis of the model. Here, our previous model is extended by including the known mechanisms of hepcidin synthesis and its regulation. The recent discovery of this hormone (Park et al., 2001) has led to a revision of the knowledge concerning the body iron homeostasis process. Including the current knowledge of the hepcidin regulation mechanisms makes the iron homeostasis model presented in this paper, to the best of our knowledge, the most complete and accurate one. The organization of the paper is as follows. In Section 2 the body iron homeostasis process is described in an informal way in order to provide an overview of the entire process. A very brief introduction to Petri nets is given in Section 3, while in Section 4 the Petri net based model of the body iron homeostasis process is presented.

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The model is then analyzed in detail in Section 5. The results of this analysis are discussed in Section 6. Some additional results of the formal analysis are presented in Appendix A.

2. Informal Description of the Body Iron Homeostasis After absorption in the small intestine, iron (Fe2+ ) is mainly transported to the serum (Fe3+ ), where it binds to transferrin (Tf). Next, iron is transported to the various cells, i.e. cells that use iron for a heme production (the preerythrocytes) and cells that acquire and store iron (the one-nuclear cells). On the cell surface iron-bound Tf binds to transferrin receptor 1 (TfR1) and is internalized as the Tf(Fe3+ )–TfR1 complex by a receptor mediated endocytosis (RME). Then, inside the cells, iron is distributed into three compartments, which are: the “transit” compartment of intracellular free iron, called labile iron pool (LIP), the functional compartment, where the mitochondrium plays an important role in preparing iron co-factors for proteins that use iron for their metabolic activity and the storage compartment (Ferritin). In order to achieve an equilibrium between these three cellular iron compartments, iron uptake, storage, utilization and export must be a coordinated process. This process involves interactions between the iron responsive elements (IREs) and the iron regulatory proteins (IRPs). The expression of TfR1 on the cells’ surface is regulated at the transcriptional level by the status of cellular proliferation and oxygen saturation. Another well-characterized mechanism of TfR1 expression regulation is performed at the post-transcriptional level. When the iron concentration is insufficient, the iron regulatory proteins (IRPs) bind to iron responsive elements (IREs) of TfR1 mRNA, resulting in a stabilization of this transcript. During an excess of the iron concentration, IRPs are released from IREs and the transcript is degraded. The expression of another transferrin receptor, transferrin receptor 2 (TfR2), which acts as a TfR1 competitor, is not regulated by the cellular iron content via the IRE/IRP regulatory system (Trinder and Baker, 2003; Robb and Wessling-Resnick, 2004). In 2001 the discovery of the hepcidin peptide and characterization of its gene, HAMP (hepcidin antimicrobial peptide) has led to the revision of the previous models of the iron metabolism and the realization that the liver plays a key role in determining the iron absorption from the small intestine and the iron release from the recycling and storage sites (Park et al., 2001). Hepcidin is the principal regulator of the iron absorption in humans. It is a 25-amino-acid hormone exclusively synthesized by the liver, initially identified as an antimicrobial peptide (Park et al., 2001). In order to describe the postulated major role of hepcidin, it is necessary to understand the function of ferroportin (Fpn), the major iron export protein located on the surface of the small intestine cells and the one-nuclear cells. The major iron recycling pathway is centered on the degradation of senescent erythrocytes by the one-nuclear cells. The exit of iron from these cells is controlled by an iron efflux mechanism, involving Fpn. Hepcidin binds to Fpn and induces its internalization and degradation (Nemeth et al., 2004; Ganz, 2005). The expression of hepcidin is regulated by a pathway that involves three proteins: the hemochromatosis protein (HFE), transferrin receptors (TfR1 and TfR2) and hemojuvelin (HJV), which positively regulates the hepcidin mRNA expression and is inhibited by an increase of the serum iron concentration (Deicher and Horl, 2006). In a recently suggested model (Frazer and Anderson, 2003; Formanowicz et al., 2007) hepatocyte surface HFE competes with Tf(Fe3+ ) for the binding on the TfR1 surface. HFE needs beta-2 microglobulin (Beta2M) for its normal expression. The dimer HFEBeta2M forms a complex (HFE-Beta2M-TfR1) together with TfR1. Unbound surface of HFE and a higher amount of the Tf(Fe3+ )–TfR2 complex, in comparison to Tf(Fe3+ )–TfR1, were proposed to increase

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hepatic hepcidin expression and its release. TfR2 may be a signaling molecule that senses transferrin-bound iron, thereby influencing the expression of hepcidin (Kawabata et al., 2000; Formanowicz et al., submitted for publication). According to this model, iron deficiency would lead to a decrease of the circulating Tf(Fe3+ ), and the number of free surface TfR1s would increase, resulting in a decreasing fraction of the free surface of HFE, and a lower concentration of the Tf(Fe3+ )–TfR2 complex. Other factors which are involved in the regulation of the hepcidin synthesis are anemia and hypoxia, that both trigger a decrease of the hepcidin level. This was postulated to permit the rapid mobilization of iron from the one-nuclear cells (as a short term response) and the small intestine cells (as a long term response) necessary to allow for the increased erythropoietic activity triggered by the erythropoietin (EPO) release. It was also revealed, in the case of mice, that a down-regulation of the hepcidin synthesis can be triggered by hypoxia alone (Nicolas et al., 2002). During anemia and/or hypoxia, the organism takes action against this situation and increases the synthesis of the erythrocytes in the bone marrow to increase the number of the erythrocytes in the serum. After that, depending on the condition of the iron stores in the one-nuclear cells (if they are full or empty), the low intestinal expression of hepcidin and the high intestinal absorption of iron is maintained or not. Anemia and/or hypoxia suppress the expression of hepatic hepcidin, but inflammatory stimuli that are strong enough to induce acute-phase responses induce its release even in the setting of anemia. The synthesis and release of hepcidin is rapidly mediated by a bacterial lipopolysaccaride and cytokine release, especially interleukin-6 (IL-6) (Nemeth et al., 2003). Thus, the hepcidin gene is an acute-phase responsive gene which is overexpressed in response to an inflammation. A cytokine mediated induction of hepcidin caused by an inflammation or infection is now thought to be responsible for the anemia of chronic disease (Formanowicz et al., submitted for publication), where iron is retained by the key cells that normally provide it, namely the small intestine cells and the one-nuclear cells. Retention of iron leads to the hallmark features of the anemia of chronic disease, a low transferrin saturation, an iron-restricted erythropoiesis and a mild to moderate anemia. The nature of the hepcidin receptor is presently unknown, however an exciting future prospect may be the development of agents to block the receptor with the aim of treating the anemia of chronic disease, a common often intractable clinical problem.

3. Petri Nets In this section, a short introduction to Petri nets is given. An interested reader will find more formal definitions and an overview of the broad range of Petri net types and their applications, e.g. in (Murata, 1989; David and Alla, 2005). Petri nets are directed bipartite graphs. Therefore, they consist of two kinds of nodes and directed arcs connecting nodes of different types. One kind of nodes are the places modeling passive system elements, as biological species or conditions. In graphical representations they are depicted as circles. The second kind of nodes are the active system elements, called the transitions. They represent events, e.g. biological reactions, and they are depicted as rectangles. The dynamic elements of a Petri net are given by tokens. They mark places and indicate if the corresponding biological species is present, and if this condition is fulfilled, respectively. Tokens can be removed or generated by firing of a transition. Since we refer in the following to ordinary Petri nets, all arcs are weighted with one, i.e. firing of a transition moves only one token via one arc. A transition may fire, i.e. it is enabled, if all its pre-places are marked with at least one token. With its firing one token from each pre-place is removed

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and (at the same time) one token is put to each post-place. If the marked state of a place is a pre-condition for a transition to fire but its firing should not remove this token, these nodes are connected via two converse arcs, represented by bidirectional arrows, which in the following are called read arcs. A Petri net with the places P = {p1 , . . . , pn } and the transitions T = {t1 , . . . , tm } has an (n × m) incidence matrix C. Each entry cij indicates the token change on place pi by firing of the transition tj . Obviously, node connections via read arcs are not realized in the incidence matrix. A p-invariant is defined by a vector y ∈ Nn satisfying the equation y · C = 0. Analogously, a t-invariant is defined by a vector holds C · x = 0.

(1) x ∈ Nm

which (2)

The entries of a p-invariant (t-invariant) which are unequal to zero are called its support and name the places (transitions) contained in this invariant with or without a factor (depending on the value of the corresponding entry). Over the places of a p-invariant the weighted sum of tokens is constant. The (a required number of times) firing of the transitions of a t-invariant reproduces a given marking. A marking is the distribution of tokens over all places, and it therefore represents the system’s state. The initial marking is that one before any transition has fired and thus it also represents the initial state of the system. An invariant is called minimal if it does not contain another invariant by its support and the greatest common divisor of its support is one. According to Sackmann et al. (2006) a t-invariant is feasible if all its transitions are enabled either in the initial marking or during a firing sequence of this t-invariant’s transitions. The feasibility can be endangered by the application of read arcs. Thus, there has to be a processing of those minimal tinvariants containing transitions connected with an empty place via read arcs if this place has no pre-transition contained in the considered t-invariant. A net is covered by t-invariants (p-invariants) if all its transitions (places) are contained in a t-invariant (p-invariant). In the graphical representation of a Petri net the concept of logical nodes can be used to avoid confusing arc crossings. A logical node may exist in multiple copies in the net which are defined by their (identical) ID and which are identified as one node. Fig. 1 depicts a small Petri net as an illustrating example. In the initial marking place p0 is marked with one token. This enables transition t0. With its firing it removes this token and generates one token at each of the places p1 and p2. After the subsequently firing of t1 and t2 the initial marking is reproduced. Therefore, the vector (1, 1, 1)T is a t-invariant of the net. It is minimal and feasible.

Fig. 1. A Petri net example with four places and three transitions. Matrix C is the nets incidence matrix. In the initial marking the net is marked with one token. It has one minimal t-invariant by which the net is covered and two minimal p-invariants, each containing the places p0 and p3 and additionally either p1 or p2. Thus, the net is also covered by p-invariants.

The net has the two minimal p-invariants (1, 1, 0, 1) and (1, 0, 1, 1), respectively. Both of them contain one token circulating between the places of the corresponding p-invariant. The net is covered by t-invariants and by p-invariants. 4. The Petri Net Model The core of the proposed model is based on a former model biologically described in (Formanowicz et al., 2007) and mathematically analyzed in (Sackmann et al., 2007). The presented model is extended by the anemia process and the protein HJV as well as by a refinement of the regulation of hepcidin and that one of the receptors TfR1 and TfR2. Fig. 2 is a graphical representation of the model. Each node is labeled with its ID. Logical places are colored in gray. Tables 1 and 2 assign the corresponding names of places, and transitions, respectively, to their IDs. As indicated by the subscript numbers in Table 1, some places are marked with tokens already in the initial marking. This marking therefore defines the initial system state. In the context of the here discussed iron homeostasis, the initial state is this one assumed to be the physiologically balanced one, i.e. the state whose iron distribution is balanced. There are some places whose names include the word signal. This should indicate that there exists a strong influence between the processes modeled by the adjacent transitions, as e.g. a kind of an essential consequence. For example firing of transition t25, anemia process, stands for an anemic activity in the system. Therefore, it removes a token from its pre-place p28, long lasting anemia, and puts a token on each of its post-places. Beside place p29, beginning of anemia, these are places p30, anemia signal1 and p31, anemia signal2. The first mentioned signal leads via transition t38 to a decrease of the hepcidin level. The last mentioned one causes via transition t13 a decrease of the erythrocyte amount or via transition t14 an increase of the EPO level (compare Fig. 2). 5. The Analysis This section is dedicated to the validation of the model by verifying its structure and the model behavior. The model is unbounded, i.e. there exists no upper bound of the number of tokens in the net Table 1 The names and IDs of the places of the model. ID

Place name

ID

Place name

01 1 2 31 4 5 6 7 8 91 10 111 12 13 14 15 16 171 18 19 20 211 22

Fe2+ available in small intestine Fe3+ in serum Fe serum high Fe serum medium Fe serum low High Fe signal1 Low Fe signal1 Low Fe signal2 Tf(Fe3+ ) Free TfR1 Bound TfR1 Free TfR2 Bound TfR2 TfR2 signal Hem(Fe2+ ) Iron in one-nuclear cell Much EPO Less erythrocytes Erythrocyte LIP (Fe3+ )Ferritin Empty store Free HFE

23 24 25 261 27 281 29 30 31 32 331 34 35 36 37 381 39 40 411 42 43 44 45

HFE-Beta2M-TfR1 TfR hepcidin signal Much HJV Less HJV HJV signal Long lasting anemia Beginning of anemia Anemia signal1 Anemia signal2 Anemia signal5 No inflammation Inflammation Inflammation signal IL-6 cytokine Much HAMP protein Less hepcidin Much hepcidin Negative Fpn signal Much Fpn Less Fpn Positive Fpn signal Anemia signal3 Anemia signal4

The subscript number 1 rearward an ID indicates if a place marked by a token already in the initial marking.

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Fig. 2. The Petri net modeling the body iron homeostasis. Logical places are represented by their IDs and exist in multiple copies in the net which are logically identical; they are shown in gray. Via its logical places, the model is a single connected Petri net. The names of the places and transitions are listed in the Tables 1 and 2, respectively.

Table 2 The names and IDs of the transitions of the model. ID

Transition name

ID

Transition name

Transport out of the small intestine

21

1 2 3 4 5 6 7 8 9 10 11

Increasing medium Fe level Increasing low Fe level Binding Tf Binding Tf Endocytosis in the preerythrocyte (RME) Endocytosis in one-nuclear cell (RME) Endocytosis in preerythrocyte Endocytosis in one-nuclear cell Erythrocyte synthesis Erythrocyte synthesis Signal EPO synthesis (in the kidney)

22 23 24 25 26 27 28 29 30 31 32

12 13 14 15

Phagocytosis in one-nuclear cell Erythrocyte decrease EPO increase Store (apoferritin)(IRP)

33 34 35 36

16 17 18 19 20

Fe2+ release Synthesis in the liver (hepatocytes) Iron becomes available Transport out of one-nuclear cell Binding TfR1 HFE-Beta2M

37 38 39 40 41

TfR1 synthesis (RNA stabilization via IRP) Hepcidin regulation HJV decreasing HJV increasing Anemia process Anemia balance Inflammatory process Regeneration HFE expression Increasing IL-6 level HAMP expression Hepcidin increase (expressed in the liver) Hepcidin inhibition Inhibition of Fpn decrease Fpn increase Hepcidin increase (expressed in the liver) Hepcidin increase Hepcidin decrease Iron mobilization Further iron mobilization Signal empty store

0

Some of these transitions are equally named, e.g. t19 and t20, but they are identified as different nodes because they were not defined as logical ones.

and therefore, the number of reachable states is infinite. Since each reachable marking defines one node in the net’s reachability graph, this graph is also infinite. Thus, the dynamic properties whose calculation is based on the reachability graph are not decided for the model. But the analysis of the invariants also provides information about the dynamic behavior of the model (Heiner et al., 2004). The definitions of the net invariants as named above in Eqs. (1) and (2) were introduced by Lautenbach (1973). Schuster et al. (1996) introduced the established concept of elementary modes to analyze metabolic networks. In Petri net theory, these elementary modes correspond to minimal t-invariants (Zevedei-Oancea and Schuster, 2003). The minimal t-invariants are considered to represent the basic behavior of the model (Heiner and Koch, 2004). First, let us mention some basic structural properties of the net, before turning to the main part of the analysis discussing the invariants. Detailed definitions and explanations of Petri net properties are given e.g. by Starke (1990). Since all arcs are weighted with one, the Petri net is ordinary and homogeneous. The used read arcs structurally are loops, i.e. the net is not pure. The sum of the ingoing arcs is not for all transitions equal to the sum of the outgoing arcs, and therefore the number of tokens in the net is not constant, i.e. the net is not conservative. Because of transitions which are in static conflict about the tokens on their common pre place, the net is not statically conflict-free. As mentioned above, the net is not bounded. And therefore it cannot be structurally bounded (what means being bounded in every initial marking). The model is not only connected but strongly connected since

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A. Sackmann et al. / BioSystems 96 (2009) 104–113 Table 5 The non-trivial MCT-sets of the model and their contained transitions.

Table 3 The minimal p-invariants of the model. p-Invariant

Places

Biological meaning

MCT-set

Transitions

Biological meaning

1 2 3 4 5 6 7 8 9 10 11

0 2, 3, 4 9, 10 17, 18 20, 21 41, 42 38, 39 33, 34 11, 12 25, 26 28, 29

1

2, 4, 11, 19, 21

2

1, 3, 15, 16

3

30, 31, 32

4

34, 35

5

27, 28, 29

6

17, 20, 22, 36

12

28, 30, 32, 44, 45

Iron in the small intestine Iron level in serum TfR1 level Erythrocyte level Iron level in store Fpn level Hepcidin level Inflammation vs. non-inflammation TfR2 level HJV level Beginning of anemia vs. long lasting anemia Anemia signals (for hepcidin regulation, iron mobilization, anemia balance)

7

23, 24, 37

8

25, 26, 38, 39

Low iron level, iron release from one-nuclear cell, increased TfR1 and EPO level High iron level, iron stored and released from storage Hepcidin increase caused by inflammation, increased IL-6 level Fpn regulation depending on hepcidin regulation Inflammation, regeneration, increased HFE level HFE binds TfR1, more TfR2 used as surface receptor, corresponding hepcidin increase HJV regulation, corresponding hepcidin increase Anemia, corresponding hepcidin decrease

Each of p-invariants contains a place marked by a token in the initial marking of the net (compare Fig. 2 or Table 1). And since all of these p-invariants are binary vectors there circulates only one token between the corresponding places named in middle column of the table.

between each pair of nodes there exists a directed path connecting them. As discussed below in detail, the model is covered by t-invariants but not by p-invariants. The net has twelve minimal p-invariants listed in Table 3 whose entries are each either one or zero. That means that exactly one token circulates between the places of each minimal p-invariant. Thus, these subnets establish each a kind of switcher whose state is indicated by the current location of the token. Most of the p-invariants serve as an indicator of the concentration level of a special species, see right column of Table 3. As the subscript numbers in Table 1 show, the initial marking is marked with eleven tokens. The minimal p-invariants 11 and 12 both contain the place p28, long lasting anemia, marked by one token in the initial marking. Thus, in the initial marking all minimal p-invariants contain a place marked with a token. Therefore, it is ensured that all minimal p-invariants may contribute to the nets behavior. The eleven tokens in the initial marking are put in the net in a way that this marking represents a medium iron leveled state since this marking gives the initial system state. The net has 74 minimal t-invariants by which it is covered. Because of the usage of read arcs, the minimal t-invariants have to be processed to get feasible ones (Sackmann et al., 2006). There are approaches transforming non-pure nets into pure ones (i.e. without loops) but these approaches consider loops with different arc weights, i.e. loops not being read arcs, see e.g. (Lautenbach, 1986). Table 4 lists in left column the transitions, each of them connected with an empty place via a read arc. The right column names transitions providing tokens at these places. Thus, a t-invariant is non-feasible if it contains a transition from left column of the table without containing the required one from right column. Each such t-invariant is merged to one which contains the crucial transition from the right column. Doing that iteratively, 298 feasible t-invariants are built. These invariants may contain redundant Table 4 Transitions which determine the processing to get feasible t-invariants. Transitions

Depending on firing of transitions

23 16, 21, 24

1 4

The transitions listed on the left are connected via read arcs with places not marked by tokens in the initial marking. Thus, the t-invariants including these transitions and excluding transitions providing tokens at the corresponding places are not feasible. In order to make them feasible they are joint to t-invariants which contain these latter named transitions listed on the right. This processing leads to feasible t-invariants.

The right column names the biological meaning of each set. The 14 MCT-sets containing only one transition, i.e. the trivial ones, are not named here.

information because of the combinatorial way they are constructed. In order to reduce the number of feasible t-invariants, the following biological constraints are set. All of these invariants should contain: • in the case of a beginning anemia, either a further iron mobilization from the iron store, i.e. transition t39, or the signal that the store is already empty, i.e. transition t41, • only one support of the erythrocyte synthesis, i.e. either transition t9, erythrocyte synthesis during a medium iron level in the serum, or transition t10, erythrocyte synthesis during a low iron level depending on an increased level of EPO, • only one consequence of an increased EPO level, i.e. either transition t10, erythrocyte synthesis, or transition t33, hepcidin inhibition, both caused by a high level of EPO, • apart from the above mentioned essential consequences, only one of the possible consequences of a beginning anemia, i.e. either transition t13, erythrocyte decrease, or transition t14, EPO increase. After performing the corresponding reduction steps, 102 feasible t-invariants remain. To discuss their biological plausibility, at first the transitions are structured into maximal common transition sets (MCT-sets) (Sackmann et al., 2006). Each of these sets contains transitions which occur only together, i.e. depending on each other, in the feasible t-invariants. Thus, MCT-sets give functional units of the net, which all are is biological meaningful. Table 5 shows the eight non-trivial MCT-sets of the model, i.e. these which contain more than one transition, with their biological interpretation. The construction of MCT-sets is not affected by the above explained reduction step. Thus, the sets of feasible t-invariants before and after this step provide the same MCT-sets. After building the MCT-sets, the feasible t-invariants are clustered by an UPGMA approach, according to their contained transitions (Grafahrend-Belau et al., 2008). For that purpose, a distance matrix is built using the Tanimoto coefficient as a similarity measure (Backhaus et al., 2000). Based on that distance matrix, the UPGMA approach as an agglomerative hierarchical clustering algorithm is used to merge in each iteration the most similar clusters. Table 6 contains the resulting 15 so-called t-clusters (built with an accordance of 80% within one cluster) showing which processes are included in each clusters. Fig. 3 depicts the t-clusters as a tree. All t-clusters contain a low iron level in serum and its influence on the EPO and the TfR1 concentration as well as the iron release out of the one-nuclear cell (MCT-set1). Apart from t-clusters 1, 2 and 15, all tclusters contain a high level of iron and the iron storage by Ferritin

A. Sackmann et al. / BioSystems 96 (2009) 104–113 Table 6 The 102 feasible t-invariants of the model clustered by UPGMA (80%). t-Cluster No. of invar. Contained processes MCT-sets 1 2 3 4 5

1 1 6 4 10

6

8

7 8 9

2 2 24

10 11

2 25

12

8

13 14 15

2 6 1

Single transitions

1, 3, 4, 5 1, 3, 4, 5 1, 2, 3, 4, 5, 8 1, 2, 3, 4, 5, 72 , 8 1, 2, 3, 4, 5, 7, 8

6, 18, 33 5, 9, 12, 33 5, 63 , 94 , 102 , 12, 14, 183 , 334 , 403 , 413 6, 14, 18, 33, 402 , 412 06 , 5, 66 , 9, 128 , 136 , 144 , 186 , 33, 405 , 415 1, 2, 3, 4, 5, 8 0, 5, 66 , 96 , 102 , 124 , 13, 186 , 336 , 404 , 414 1, 2, 4, 7, 8 5, 61 , 10, 12, 14, 181 , 401 , 411 1, 2, 4, 7, 8 0, 5, 6, 10, 13, 18, 401 , 411 1, 2, 3, 4, 5, 6, 8 0, 517 , 620 , 712 , 814 , 922 , 1218 , 1312 , 1412 , 1815 , 33, 4011 , 4113 1, 2, 3, 4, 5, 6, 7, 8 0, 51 , 6, 7, 9, 12, 14, 181 , 33, 41 1, 2, 4, 5, 6, 7, 8 0, 517 , 621 , 713 , 814 , 923 , 1219 , 1312 , 1413 , 1815 , 33, 4011 , 4114 1, 2, 4, 5, 6, 8 0, 56 , 67 , 74 , 84 , 10, 124 , 134 , 144 , 186 , 404 , 414 1, 2, 4, 5, 6, 71 0, 6, 8, 18, 33 0, 54 , 64 , 73 , 83 , 9, 12, 182 , 33 1, 2, 4, 5, 6, 74 1 5, 10, 12

The two columns on the right list the processes contained in the t-clusters, split into occurring MCT-sets and single transitions. A superscript number n indicates in how many invariants of that t-cluster the corresponding process is included. No specification means an occurrence in all invariants. The column on the left of this gives the total number of invariants in the t-cluster. Fig. 3 shows a dendrogram of the t-clusters.

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t-clusters including the corresponding endocytosis by TfR2 (transitions t7) contain an erythrocyte synthesis (because of the invariant reduction step this is either transitions t9 or transition t10). And all of these t-clusters contain an erythrocyte decrease caused by anemia (transition t13) or a phagocytosis of the erythrocyte into the one-nuclear cell (transition t12). In all t-clusters which include an iron uptake directly from the serum into the one-nuclear cell (by transitions t6 or t8), the iron becomes available (transition t18) if it is not used only for TfR2 synthesis (MCT-set6). Some of the above mentioned t-clusters containing an inflammation also include the corresponding increase of the hepcidin level (MCT-set3), which are t-clusters 1–6, 9 and 10. One possible way to decrease the hepcidin level is via the increased level of EPO (transition t33). The other way is via anemia. The anemia process with its hepcidin regulation and iron mobilization (MCT-set8) is contained in the t-clusters 3–12. All of these t-clusters include either an erythrocyte decrease (transition t13) or an EPO increase (transition t14). These processes are mutually exclusive because of the invariant reduction step. The t-clusters containing anemia also include either the iron mobilization out of the store (transition t40) or the signal that the store is empty (transition t41). An iron transport out of the small intestine (transition t0) is contained in t-clusters 6, 8–14 and in some invariants of t-cluster 5. The HJV regulation which results in a hepcidin regulation (MCT-set7) is contained in t-clusters 5, 7, 8, 10 and 11, and in some invariants of t-clusters 4, 13 and 14. 6. Results and Discussion

(MCT-set2). Except t-cluster 15, all t-clusters include the Fpn regulation (MCT-set4). The inflammation process with a regeneration and an increased HFE expression (MCT-set5) is involved in t-clusters 1–6 and 9–14. The six last mentioned t-clusters contain an increased TfR2 level and the corresponding hepcidin regulation (MCT-set6). Accordingly, those t-clusters comprise the endocytosis of iron from the serum by TfR2 (transitions t7 and t8, respectively). The iron endocytosis by TfR1 into the preerythrocyte or the one-nuclear cell is included in all t-clusters (transitions t5 and t6, respectively). All those concerning the preerythrocyte (transition t5) and the

The main aim of our research, in the light of recent discoveries concerning hepcidin, a link between inflammation and anemia, was the analysis of the behavior of the model of the human body iron metabolism in two different states, i.e. under the influence of an inflammatory process and during anemia. First, we have focused on the regulation of the hepcidin synthesis. We have looked for factors contributing to the changes of the concentration of this hormone. On the basis of the analysis of the model, we have found three reasons that may lead to an increase of the serum hepcidin level: • hepcidin increase (expressed in the liver) [HAMP/inflammation], transition t32 in MCT-set3, in t-clusters 1–6, 9 and 10, • hepcidin increase (expressed in the liver) [TfR2], transition t36 in MCT-set6, in t-clusters 9–14, • hepcidin increase [HJV], transition t37 in MCT-set7, in t-clusters 5, 7, 8, 10 and 11 (partially 4, 13 and 14). The first one (transition t32) is the presence of an inflammatory process in the organism connected with an increase of free HFE, IL-6 and HAMP protein. The second cause (transition t36) are the changes in the TfR (TfR1 as well as TfR2) regulation, which will be discussed below. An inflammatory process contributes to an increase of the concentration of free HFE which binds to Beta2M and TfR1 (the HFE-Beta2M-TfR1 complex). Thus, TfR2, a TfR1 competitor, wins the competitions with TfR1 at the cells’ surface and finally it endocytoses iron in the preerythrocyte or in the one-nuclear cell. This phenomenon gives a signal (place p13, TfR2 signal) to increase the hepcidin synthesis in the liver. The third factor (transition t37) are the changes in regulation of HJV, i.e. because of a low iron level the HJV concentration increases, what may also lead to an increase of the hepcidin concentration. Moreover, we have found two different factors which may contribute to a decrease of the hepcidin level in the serum:

Fig. 3. The UPGMA based 15 t-clusters of the 102 feasible t-invariants built with 80% accordance within a t-cluster. For more details see Table 6.

• hepcidin inhibition [EPO], transition t33 in t-clusters 1, 2, 4, 5, 9–11, 13 and 14 (partially 3 and 6),

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• hepcidin decrease [anemia], transition t38 in MCT-set8, in tclusters 3–12. The first factor (transition t33) is a high level of serum EPO (place p16, much EPO); either caused by anemia (transition t14, EPO increase) or by a low iron level in serum (transition t11, signal EPO synthesis (in the kidney)). The second fact which has to be taken into account concerning a hepcidin decrease (transition t38) is the anemia process (transition t25), which causes directly via a signal (place p30, anemia signal1) a decrease of the hepcidin synthesis. Furthermore, on the basis of our model we have explored differences in the model behavior during inflammation and anemia. First, on the basis of the model behavior, these two processes will be discussed: • inflammatory process, transition t27 in MCT-set5, in t-clusters 1–6 and 9–14, • inflammation without anemia in t-clusters 1, 2, 13, 14. In the presented model the inflammation includes an inflammatory and a regenerative processes as well as an increased HFE expression. This phenomenon is included in 12 of the 15 t-clusters. The following processes take place dependent on inflammation: inflammation caused hepcidin regulation (MCT-set3), TfR regulation via free HFE and a resulting hepcidin regulation (MCT-set6), endocytosis by TfR2 in the one-nuclear cell (transition t8), or in the preerythrocyte (transition t7), respectively. These above mentioned processes take place only when there is/was inflammation. Concerning anemia the model provides the following occurrence: • anemia process, transition t25 in MCT-set8, in t-clusters 3–12, • anemia without inflammation in t-clusters 7 and 8. Anemia (a low serum concentration of red blood cells) occurs in 10 of the 15 t-clusters. It is represented by MCT-set8 which includes, besides the anemia process and the anemia balance, some regulations triggered by anemia signals, namely a hepcidin decrease (transition t38) and an iron mobilization from the LIP into the serum (transition t39). Other regulations which are also triggered by anemia and therefore depend on it are: either a decrease of the erythrocyte number (transition t13) or an increase of the EPO level (transition t14) and either a further iron mobilization from the store into the serum (transition t40) or the signal that the store is already empty (transition t41). Additionally, in all situations concerning an inflammation and/or anemia we have found that there is a regulation of Fpn (MCT-set4). All invariants which include an inflammation without including anemia contain a hepcidin inhibition by much EPO (transition t33) and no erythrocyte synthesis support by much EPO (transition t10). Vice versa, those invariants including anemia and no inflammation all contain an erythrocyte synthesis supported by much EPO (transition t10) (and they all do not contain this synthesis taking place during a low iron level, i.e. transition t9) and a hepcidin increase via the HJV regulation (MCT-set7). Additionally, all of these invariants include the iron endocytosis in the preerythrocyte (RME) by TfR1 (transition t5) and no endocytosis by TfR2 (neither in erythrocytes nor in one-nuclear cells, transitions t8 and t7). These invariants (anemia without inflammation) neither include a hepcidin regulation caused by much HFE directly via the HAMP protein (transition t32 in MCT-set3) nor indirectly via a TfR regulation (transition t36 in MCTset6). They all do not contain a hepcidin regulation via much EPO (transition t33). In the feasible t-invariants we have not found any significant dependencies between the occurrence of anemia and/or an inflammation and the cell type in which the iron from the serum is endocytosed (i.e. preerythrocyte or one-nuclear cell). Neither there are significant dependencies between those processes and

the receptor type used for the endocytosis (i.e. TfR1 or TfR2). Thus, under the assumption of anemia the preference for the receptor type and the iron endocytosing cell type is statistically the same as under the assumption of an inflammation. But under both assumptions the occurrence of TfR1 is approximately twice as big as the occurrence of TfR2. And there are no invariants in which only TfR2 is used for the endocytosis (see below). Another purpose of our research was to explore the factors determining the TfR concentration. This issue is very important, because TfR is proposed to be a parameter sensitive and specific enough to completely describe the distribution of iron in the human body, especially during an inflammatory state. The routinely used laboratory tests, such as serum iron, total iron-binding capacity, transferrin saturation and serum Ferritin are good indicators of iron available for erythropoiesis and iron stores but they are considerably influenced by a number of non-related conditions, e.g. acute-phase reactions, which may complicate the clinical interpretation of their results and create an inaccurate picture of the body iron metabolism (Formanowicz et al., submitted for publication; Formanowicz and Pietrzak, 2006). We have found that an increase in the TfR1 concentration occurs in all t-clusters, i.e. a synthesis enhanced via IRP (transition t21), which is triggered by a low iron level in the serum: • TfR1 synthesis (RNA-stabilization via IRP), transition t21 in MCTset1, in all t-clusters. On the other hand, we have disclosed that a decrease of the TfR1 concentration is split by the way via which this receptor is bound, i.e. there are three possibilities: • binding TfR1+Beta2M-HFE, transition t20 in MCT-set6, in the tclusters 9–14, • endocytosis in preerythrocyte (RME), transition t5 in t-clusters 2, 3, 5–8 and 15 (partly 9–12 and 14), • endocytosis in one-nuclear cell (RME), transition t6 in the tclusters 1, 4, 8, 10 and 13 (partly 3, 5–7, 9, 11, 12 and 14). On this basis, we can suggest that there is a connection between an inflammatory process and the TfR1 serum concentration, but this process does not influence TfR1 directly, but only via the changes in HFE (free HFE increases because of an inflammation). We have found that an increase in TfR2 requires iron in the one-nuclear cell (transition t17): • synthesis in the liver, transition t17 in MCT-set6, in t-clusters 9–14. Furthermore, we have found that a TfR2 decrease (which occurs only in these invariants) is split by the location of its activity: • endocytosis in preerythrocyte, transition t7 in t-cluster 10 (partly 9, 11, 12 and 14), • endocytosis in one-nuclear cell, transition t8 in t-cluster 13 (partly 9, 11, 12 and 14). Summarizing, we suggest that TfR2 is synthesized when TfR1 is bound to a Beta2M-HFE complex. The last mentioned complex is built only in the situation when there is much free HFE. We have also explored the factors determining the LIP levels. Operationally, LIP can be regarded as that component of the cell iron that is accessible to a particular chelator or a class of chelators. Topologically, it is identified primarily with the cytosol and as such, it is regarded as the crossroad of the cellular iron traffic. Biochemically, LIP has been defined as the regulatory or regulated pool of

A. Sackmann et al. / BioSystems 96 (2009) 104–113

cell iron, since depending on the cells system, its level is apparently maintained within a constant range (Formanowicz et al., 2007): • phagocytosis in one-nuclear cell, transition t12 in t-clusters 2, 3, 7, 10, 14 and 15 (partly 5, 6, 9, 11 and 12), • store (apoferritin)(IRP), transition t15 in MCT-set2, in t-clusters 3–14, • (Fe2+ )release [from store], transition t16 in MCT-set2, in t-clusters 3–14, • iron-mobilization [anemia], transition t39 in MCT-set8, in tclusters 3–12, • iron becomes available [in one-nuclear cell], transition t18 in tclusters 1, 4, 8 and 13 (partly 3, 5–7, 9–12 and 14), • transport out of one-nuclear cell, transition t19 in MCT-set1, in all t-clusters. In our model we consider only the LIP (place p19) which is located in the one-nuclear cell (it appears in the form of three copies of one logical node in the Petri net). One way for the iron to reach this LIP is directly via an endocytosis into the one-nuclear cell (transitions t6 or t8, see above), where the iron becomes available (transition t18) after triggering a TfR2 synthesis (transition t17), if required. The second possible way is via a phagocytosed erythrocyte (transition t12). Competitively, the number of erythrocytes may be reduced by an anemia signal (transition t13). In the case of a low iron level (place p7, low Fe signal2) the iron from LIP may be transported out of the one-nuclear cell (transition t19) if the necessary Fpn is present. Or in the case of a high iron level (place p5, high Fe signal1) the iron is stored (transition t15). If there is no inflammation (place p33) and the iron concentration switched to a low level, the iron is released from store again to LIP (transition t16). In the case of anemia (MCT-set8) the iron is mobilized from LIP and directly released into the serum (transition t39). We have also looked for the factors influencing Ferritin. We have disclosed that the iron from the LIP is partly stored by apoferritin if the iron level in serum is high (transition t15) and from there it is released into a low iron level if there is no inflammation (transition t16). • store (apoferritin)(IRP), transition t15 in MCT-set2, in t-clusters 3–14, • (Fe2+ )release, transition t16 in MCT-set2, in t-clusters 3–14.

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If an anemia caused iron mobilization has taken place, there are two possibilities: either the iron stores are already empty, measured via a signal (transition t41) or there is still iron in the store which then is mobilized (transition t40) by its release directly into the serum: • further iron-mobilization (anemia), transition t40 not contained in all invariants of any t-cluster (partly in t-clusters 3–9, 11 and 12), • signal empty store, transition t41 in t-cluster 10 (partly 3–9, 11 and 12). The analysis of the formal model properties may lead to discoveries of some unknown features of the studied phenomenon. In this sense the results of such an analysis are important complement to the results of clinical and laboratory research. On the basis of the behavior of our model we have found factors influencing hepcidin concentration as well as transferrin receptors. The changes of the concentration of these markers are very important in anemia diagnosis, especially among patients suffering from anemia of chronic disorders, e.g. in chronic kidney diseases. Acknowledgments This research has been partly supported by the EU grant Bioptrain, and by the Polish Ministry of Science and Higher Education. Appendix A Based on the set of t-invariants a lot of statistical examinations are possible. As mentioned above, one interesting question is, if there exists a significant dependence between the state of the system considering the possible processes of inflammation and/or anemia on the one hand and on the other hand the endocytosis of iron from the serum in preerythrocytes and/or one-nuclear cells and via which receptor (TfR1 and/or TfR2) this endocytosis takes place. Thus, the Pearson’s chi-square test, a statistical test of independence, is executed, see e.g. (Dytham, 2003). The corresponding frequencies are put into contingency tables where the marginal totals and the grand total is calculated. Based on the hypothesis that the variables are unrelated, i.e. only randomly related, the values of 2 are calculated. Taking into account that there is one degree of freedom in the contingency tables (Tables A.3–A.6), the 2 val-

Table A.1 The distribution of several processes over the 15 t-clusters. t-Cluster

Included processes Preeryth., TfR1 t5

Preeryth., TfR2 t7

One-nucl. cell, TfR1 t6

One-nucl. cell, TfR2 t8

Inflam. t28

Anemia t25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0 1 6 0 10 8 2 2 17 1 17 6 0 4 1

0 0 0 0 0 0 0 0 12 2 13 4 0 3 0

1 0 3 4 6 6 1 2 20 2 21 7 2 4 0

0 0 0 0 0 0 0 0 14 0 14 4 2 3 0

1 1 6 4 10 8 0 0 24 2 25 8 2 6 0

0 0 6 4 10 8 2 2 24 2 25 8 0 0 0

Total sum

75

34

79

37

97

91

The columns name the cell type in which the iron from the serum is endocytosed by which receptor, and the inflammation and anemia process (each with the corresponding transition ID). The rows name how many invariants of each of the 15 t-clusters contain the considered process.

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Table A.2 The distribution of the processes in the 102 discussed feasible t-invariants. Preerythrocyte

One-nuclear cell

TfR1

TfR2

TfR1

TfR2

− − − − + + + + − − − − + + + +

− − − − − − − − + + + + + + + +

− + − + − + − + − + − + − + − +

− − + + − − + + − − + + − − + +

Inflam., anemia

Total sum

Inflam., no anemia

No inflam., anemia

No inflam., no anemia

Total sum

0 4 0 4 9 15 9 15 0 12 0 2 0 15 0 2

0 1 0 2 1 0 2 1 0 2 0 0 0 1 0 0

0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0

0 5 0 6 12 18 11 16 0 14 0 2 0 16 0 2

87

10

4

1

102

The columns on the right refer to inflammation and/or anemia while the columns on the left specify in which cells (preerythrocyte and/or one-nuclear cell) the iron from the serum is endocytosed by which receptor (TfR1 and/or TfR2).

ues provide the probabilities of the observed frequencies under the assumption of independency, i.e. their p-values. With a significance level of ˛ = 5% it can be decided if the hypothesis of independency has to be rejected. Firstly, Table A.1 gives an overview of the numbers of feasible t-invariants including the endocytosis in the different cell types by the different receptors as well as the inflammation and the anemia process. To make an exact examination possible, Table A.2 splits the number of feasible t-invariants more detailed. Here, it is distinguished which endocytosis process takes place in which situation, i.e. with or without an inflammation and/or anemia. To consider the most general cases, the upper part of Table A.3 shows the number of feasible t-invariants which include an endocytosis in preerythrocytes and one-nuclear cells split by a present inflammation and anemia. The total sum of invariants is higher than 102 (i.e. the number of considered feasible t-invariants) since some invariants include an endocytosis in both cell types, or an inflammation and anemia process, respectively. Based on the marginal totals, the frequencies can be calculated which are theoretically expected with the assumption that the events are independent of each other, see the lower part of Table A.3. Using these values the Pearson’s chi-square test provides 2 ≈ 0.0272. This provides a p-value of 0.869. That suggests independence of the health status (inflammation or anemia) and the cell type in which the iron from the serum is endocytosed. The next general case considers the number distribution of invariants including inflammation and anemia distinguishing if the

Table A.3 The distribution of feasible t-invariants split by inflammation and anemia as well as the cell types in which the iron from the serum is endocytosed. Cell types Preerythrocyte One-nuclear cell Total sum

Inflammation

Anemia

Total sum

86 87

83 81

169 168

173

164

337

iron is endocytosed by receptor TfR1 or TfR2, for the observed and the theoretically expected frequencies see the upper, and the lower part, respectively, of Table A.4. These values provide 2 ≈ 0.0716. Thus, with a p-value of 0.789 there is no significant reason to assume a dependency between the used receptor and the health state. The next case considers only the endocytosis in preerythrocytes. It is distinguished which receptor endocytosis the iron and if there is an inflammation or an anemia, see Table A.5, where the upper part contains the observed and the lower part the theoretically expected frequencies. Here, the calculations lead to 2 ≈ 0.0673 and a p-value of 0.7953. Therefore, the distribution between TfR1 Table A.4 The distribution of feasible t-invariants split by inflammation and anemia as well as the receptors by which the iron from the serum is endocytosed. Receptors TfR1 TfR2 Total sum

Inflammation

Anemia

Preerythrocyte One-nuclear cell

29,237/337 29,064/337

27,716/337 27,552/337

This gives 2 ≈ 0.0272 and p-value ≈ 0.869. a Theoretically expected frequencies (calculated based on the marginal totals of the observed fequencies under the assumption of independency of the events).

Anemia

Total sum

97 67

91 59

188 126

164

150

314

Receptors a

Inflammation

Anemia

TfR1 TfR2

30,832/314 20,664/314

28,200/314 18,900/314

This gives 2 ≈ 0.0716 and p-value ≈ 0.789. a Theoretically expected frequencies.

Table A.5 The distribution of feasible t-invariants including an endocytosis in preerythrocytes split by inflammation and anemia as well as the receptors by which the iron from the serum is endocytosed. Endocytosis in preerythrocytes Receptors TfR1 TfR2 Total sum

Cell types a

Inflammation

Inflammation

Anemia

Total sum

70 34

69 31

139 65

104

100

204

Receptors a

Inflammation

Anemia

TfR1 TfR2

14,456/204 6760/204

13,900/204 6500/204

This gives 2 ≈ 0.0673 and p-value ≈ 0.7953. a Theoretically expected frequencies.

A. Sackmann et al. / BioSystems 96 (2009) 104–113 Table A.6 The distribution of feasible t-invariants including an endocytosis in one-nuclear cells split by inflammation and anemia as well as the receptors by which the iron from the serum is endocytosed. The p-value 0.772. Endocytosis in one-nuclear cells Receptors TfR1 TfR2 Total sum

Inflammation

Anemia

Total sum

76 37

72 32

148 69

113

104

217

Receptors a

Inflammation

Anemia

TfR1 TfR2

16,724/217 7797/217

15,392/217 7176/217

This gives 2 ≈ 0.0973 and p-value ≈ 0.7551. a Theoretically expected frequencies.

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