### The kinetic model

Ferritin quaternary structure comprises 24 α-helix protein subunits forming a characteristic nanocage that encloses an inner cavity completely isolated from the external medium [13]. Inside this isolated cavity, ferritin stores iron as insoluble *Fe*^{3+} species (ferrihydrite crystallites) by sequestering *Fe*^{2+} from the external medium. When entering the ferritin protein nanocage, ferrous ions pass through any of eight pores formed by three subunits at each three-fold symmetry axis of the protein shell. After this entrance, *Fe*^{2+} is oxidized to *Fe*^{3+} at diiron catalytic sites followed by electron transfer to dioxygen (premineral, diferric oxo) translocation and mineral nucleation inside the internal cavity [13, 35, 36].

Ferritin iron mineralization as a catalytic mechanism (see Figure 7), namely as a ferroxidase (EC 1.16.3.1), can be described applying usual enzymology mathematical tools, even though the process by which ferritins synthesize mineral and release iron is highly complex and not entirely understood [34].

In recent years, stopped-flow experiments with a transient peroxodiFe(III) formed at the ferroxidase active site of ferritin have revealed very unusual kinetic curves, dependent on the iron-to-protein ratio [37]. A mathematical model for the kinetics of the catalytic process has been previously proposed [38]. The model correctly explains the unusual kinetic behavior of the enzyme and the time course of the appearance/disappearance of the reaction intermediates. This kinetic model for catalysis includes two sequential mechanisms, dependent on the amount of iron available to the protein and the concentration of electron acceptors (dioxygen) for the oxidation step, and a new intermediate in the catalytic reaction (a putative hydroperoxodiFe(III) complex). However complex the catalytic mechanism is, the formation of the mineral core, *i.e*. the final product of the enzymatic reaction, can be described more easily considering the ferritin mineralization mechanism as a serial process for a single chemical species (iron) characterized by an initial affinity equilibrium step and subsequent consecutive monomolecular reactions. These conditions imply that the global enzymatic reaction can be described by a Michaelis-Menten-like mechanism with association, dissociation and catalytic processes characterized by an apparent kinetic constant, respectively *k*_{
aso
} , *k*_{
dis
} and *k*_{
cat
} (see Figure 7). In this mechanism, *k*_{
aso
} /*k*_{
dis
} represent the affinity equilibrium constant of the pores of one ferritin protein cage for external *Fe*^{2+} and *k*_{
cat
} encompasses the oxidation, translocation and precipitation rates in the pores. In fact, *k*_{
cat
} takes the value of the rate constant of the global process rate limiting step. Electron transfer steps can be considered fast processes compared to ion translocation and precipitation, given the relative abundance of cytoplasmic electron acceptors (oxygen) in normal conditions. Therefore, *k*_{
cat
} can be considered at least initially as independent from the oxidation step and so the concentration of electron acceptors was not included in the kinetic model.

Though the ferritin kinetic mechanism includes a Michaelis-Menten enzymatic reaction, there is a fundamental difference from most models: the reaction product (diferric oxo) is not released back to the external medium, but it is transferred into the internal cavity of the catalytic protein cage. This implies that the enzyme, *i.e*. ferritin, is modified by the reaction by increasing its iron content. In this way, a population of different enzymes (ferritin protein cages) with different iron contents is created, but each nanocage retains its catalytic activity intact, because the pores remain unblocked (maintaining the same *k*_{
aso
} , *k*_{
dis
} and *k*_{
cat
} independently from the size of the iron mineral core) unless the protein cage becomes full of iron atoms. In this latter case, there is no iron translocation and mineralization, *i.e*. *k*_{
cat
} = 0. Therefore, the iron entrance mechanism can be described by the following kinetic equations defined for each different ferritin species *Fn*_{
i
} containing *i* iron atoms inside its protein nanocage:

F{n}_{i}+Fe\stackrel{{k}_{aso}/{k}_{dis}}{\leftarrow}{C}_{F{n}_{i}Fe}\stackrel{{k}_{cat}}{\to}F{n}_{i}{}_{+1}\forall i=0,\mathrm{..},N-1

(1)

F{n}_{i}+Fe\stackrel{{k}_{aso}/{k}_{dis}}{\leftarrow}{C}_{F{n}_{i}Fe}\forall i=N

(2)

Ferritin can also release iron through nanocage pores, but this process is less understood than iron uptake. It is clear that rates of iron loss will depend on the accessible surface of the mineral core. The process also depends on the presence of reductants and on complex structural changes of the pores between folded and unfolded states that can affect iron release rates [12, 14]. As the mechanism of iron release is not completely understood, a first order kinetic equation was proposed to model the loss of one iron atom from the mineral core as:

F{n}_{i}\stackrel{{k}_{l}{}_{oss,i}}{\to}F{n}_{i-1}+Fe\forall i\ne 0

(3)

Even though *k*_{
loss,i
} is initially defined as a constant, it can be dependent on other variables such as the iron content of the mineral core and the pore availability and folding state. These functional dependencies of *k*_{
loss,i
} will be explored and determined below based on the comparative analysis of model simulations and experimental results.

*In vivo*, ferritin can also be proteolytically degraded by lysosomal or proteasomal degradation. When the ferritin protein cage is degraded to its amino acid components, the iron of the mineral core is reduced and returned to the cLIP [33, 39] and amino acids are recycled back to the cytoplasm. In Caco-2 intestinal cells, ferritin has a half-life of 16 h [26]. Therefore, a first order kinetic equation, characterized by a kinetic constant, *k*_{
d
} , can be proposed for this proteolytic degradation process regardless of its specific mechanism:

F{n}_{i}\stackrel{{k}_{d}}{\to}a{a}_{Pool}+iFe\forall i=0,\mathrm{..},N

(4)

where *aa*_{
Pool
} represents the cytoplasmic amino acid pool. *In vivo*, ferritin subunits can also be synthesized *de novo* from the cytoplasmic amino acid pool. This complex process can also be represented by a first order synthesis rate:

a{a}_{Pool}\stackrel{{k}_{synt}}{\to}F{n}_{0}

(5)

In this work, we are interested in the simulation of the fast stages of the intracellular iron regulation process mediated by ferritin in short periods of time. In these short time spans, the total amount of ferritin can be assumed to be constant, because the time scale of regulation is much slower. This is a standard quasi-steady-state assumption which comes from the way we have separated time scales for synthesis and degradation; in doing so, we have followed usual biochemistry rules as explained in [40]. Hence, to preserve the total amount of ferritin, the amount of degraded and synthesized ferritin must be equal. That is

Therefore, equations (4) and (5) become

F{n}_{i}\stackrel{{k}_{d}}{\to}F{n}_{0}+iFe\forall i

(7)

Notice that according to Eq. (7) there is no change in the amount of *Fn*_{0} due to degradation since degradation and synthesis processes occur at the same rate. Given the order of magnitude of *k*_{
d
} with respect to the other parameters in the model, the effect of removing Eq. (7) is not significant in this scenario, where regulation of ferritin expression levels is not considered. In fact, eliminating this flux resulted in a change of less than 1.5 × 10^{-3}% in all the system's variables (data not shown). However, this terms were kept in the model in order to provide a more general framework, suitable for the inclusion of regulatory factors later on.

Finally, the kinetic model is represented by the following equations:

F{n}_{i}+Fe\stackrel{{k}_{aso}}{\to}{C}_{F{n}_{i}Fe}\forall i=0,\mathrm{..},N

(8)

{C}_{F{n}_{i}Fe}\stackrel{{k}_{dis}}{\to}F{n}_{i}+Fe\forall i=0,\mathrm{..},N

(9)

{C}_{F{n}_{i}Fe}\stackrel{{k}_{cat}}{\to}F{n}_{i+1}\forall i\ne N

(10)

F{n}_{i}\stackrel{{k}_{loss,i}}{\to}F{n}_{i-1}+Fe\forall i\ne 0

(11)

F{n}_{i}\stackrel{{k}_{d}}{\to}F{n}_{0}+iFe\forall i=0,\mathrm{..},N

(12)

### The mathematical model

The kinetic system presented in the previous section has three state variables: *Fn*_{
i
} , {C}_{F{n}_{i}Fe} and *Fe* with *i* = 0, .., *N*. In order to study the dynamics of the kinetic model described by equations (8) to (12) a dynamic model was developed. This model was built performing a mass balance over the system's species and using mass action kinetic relationships. The final model consists of the following system of ordinary differential equations:

*Iron mass balance*

\begin{array}{c}\frac{dFe}{dt}=+{k}_{dis}{\displaystyle \sum _{i=0}^{N}{C}_{F{n}_{i}Fe}}+{k}_{d}{\displaystyle \sum _{i=0}^{N}i}\cdot F{n}_{i}\\ +{\displaystyle \sum _{i=1}^{N}{k}_{loss,i}}\cdot F{n}_{i}-{k}_{aso}\cdot Fe{\displaystyle \sum _{i=0}^{N}F}{n}_{i}\end{array}

(13)

*Ferritin mass balance*

\begin{array}{c}\frac{dF{n}_{0}}{dt}=+{k}_{dis}\cdot {C}_{F{n}_{0}Fe}+{k}_{d}{\displaystyle \sum _{i=1}^{N}F{n}_{i}}\\ +{k}_{loss,1}\cdot F{n}_{1}-{k}_{aso}\cdot Fe\cdot F{n}_{0}\end{array}

(14)

\begin{array}{c}\frac{dF{n}_{i}}{dt}=+{k}_{dis}\cdot {C}_{F{n}_{i}Fe}+{k}_{cat}\cdot {C}_{F{n}_{i-1}Fe}\\ +{k}_{loss,i+1}\cdot F{n}_{i+1}-({k}_{d}+{k}_{aso}\cdot Fe)\xb7F{n}_{i}\\ -{k}_{loss,i}\cdot F{n}_{i}\forall i=1,\mathrm{..},N-1\end{array}

(15)

\begin{array}{c}\frac{dF{n}_{N}}{dt}=+{k}_{dis}\cdot {C}_{F{n}_{N}Fe}+{k}_{cat}\cdot {C}_{F{n}_{N-1}Fe}\\ -({k}_{d}+{k}_{aso}\cdot Fe)\cdot F{n}_{N}-{k}_{loss,N}\cdot F{n}_{N}\end{array}

(16)

*Ferritin-Iron complex mass balance*

\begin{array}{c}\frac{d{C}_{F{n}_{i}Fe}}{dt}=+{k}_{aso}\cdot F{n}_{i}\cdot Fe\\ -({k}_{dis}+{k}_{cat})\cdot {C}_{F{n}_{i}Fe}\forall i=0,\mathrm{..},N-1\end{array}

(17)

\frac{d{C}_{F{n}_{N}Fe}}{dt}=+{k}_{aso}\cdot F{n}_{N}\xb7Fe-{k}_{dis}\cdot {C}_{F{n}_{N}Fe}

(18)

This mathematical model has been conceived to represent the uptake or release of one iron atom at a time, which results in a system of 2*N* + 3 differential equations. Mathematically, the same equation system can be equivalently used to represent the uptake of one iron atom package, *i.e*. a fixed number of iron atoms, at a time. The only difference is the loss in resolution caused by lumping iron in packages instead of treating atoms one by one. A suitable iron package size of 50 iron atoms per package was chosen to solve the differential equation system. Hence, the value of *N* used in the simulations of these work was decreased from 2,500 to 50, and the size of the equation system was lowered from 5,003 to 103 simultaneous equations with no significant loss of resolution with respect to the amount of iron contained in the mineral core.

Since this model does not consider diffusional or spatial distribution effects, this is a lumped component model. General mathematical results for this type of ordinary differential equations show that their steady states strongly and directly depend on the kinetic constant values and relative ratios (see, *e.g*., [41]), particularly on the values of the constants *k*_{
loss,i
} with *i* = 0, .., *N* . Thus, the performance of the system to realistically model the global process depends on an appropriate choice of this set of constants. Certainly, a fundamental part of the construction of our model consisted in proposing a suitable functional dependency of *k*_{
loss,i
} in terms of *i*. Table 1 shows a list of available values in the literature for all those constants that we have assumed to be independent from *i*. For the dependence of *k*_{
loss,i
} in terms of the number *i* of iron packages stored in ferritin, we proposed an expression based on Hill's equation:

{k}_{loss,i}=\{\begin{array}{ll}{\tilde{k}}_{loss}\hfill & \text{if}i=0,\hfill \\ {\tilde{k}}_{loss}\left(1+\frac{\kappa \cdot {i}^{n}}{{\theta}^{n}+{i}^{n}}\right)\hfill & \text{if}i\ne 0.\hfill \end{array}

(19)

where *n* = 1, θ = 1 and κ = 2.4 were chosen in order to recover the behavior of experimental results. A plot of *k*_{
loss,i
} as a function of the number of iron atoms in ferritin (*Fe*_{
pack
} ) is shown in Figure 8. This expression was developed through a mathematical rationalization of biological intuitions about the system. In addition, similar equations have been shown to be good phenomenological approximations for biological rates [42].

It is clear that iron release rates should increase as the surface of the mineral increases and hence the iron release rate will reflect the size of the mineral core in each ferritin protein cage. Iron release from the ferritin mineral requires electron transfer to reduce the ferric to ferrous, hydration of the *Fe-O-Fe* bonds and migration of ferrous iron released from the mineral through the ferritin protein cage to the outside of the ferritin molecule, presumably to be trapped by iron carriers. While much of the molecular details of the process remain unknown it is clear that the folding/unfolding of the pores at the three-fold axis of ferritin protein cages influences the rate of iron removal [12, 14]. Thus iron release rates are a combination of mineral dissolution rates, which reflect the amount of exposed surface of the mineral core, and of the folding state of the pores in the protein cage. Removal of iron in small minerals, estimated at 500 iron atoms for this model, has a different relationship to mineral size than for larger minerals, possibly because with larger minerals, there has been more dehydration in the iron core. Ferritin pore unfolding appears to influence transfer of electrons from the reductants, such as FNH_{2} or DTT. However, nothing is known about the relationship of the protein pore structure and reprotonation of the *Fe-O-Fe* bonds to release iron from the mineral.

A linear relationship with the form {k}_{loss,i}={\tilde{k}}_{loss}\cdot i similar to a first order kinetic with respect to stored iron, proved to be too strong to represent the global iron release phenomenon (in the model simulations, all ferritins loose a fraction of the iron very rapidly). This mathematical expression does not consider the geometric relationship between the mass (volume) of the iron mineral core inside the protein cage and its exposed surface. In fact, for a given number of atoms *i* in the solid core inside a pseudo-spherical cage, it is easy to demonstrate mathematically that the exposed surface available for hydration and dissolution increases roughly as a function of *i*^{2/3}, the value of which is always less than *i*.

The linear relationship does not take into account the fact that iron release from the central core occurs via the three-fold axis pores of the protein cage. Each 24-mer protein cage has eight pores that serve as gates for the release of iron from the mineral core. As the number of available exit channels ("active sites") for a single massive mineral core is fixed, simple kinetic considerations allow prediction of a saturation behavior expected to occur as a function of the amount of available iron atoms able to be released from the mineral core, which limits the global iron release rate from the protein cage. The maximal saturation rate could vary depending on additional factors, such as the folding/unfolding state of the pores, concentration of iron chelators and/or electron transfer to reduce ferric to ferrous ions.

Taken together, the iron release rate depends on (a) the area of exposed mineral core surface, (b) the degree of hydration of the mineral core, (c) the saturation of three-fold axis pores of the ferritin protein cage, (d) the concentration of chelators and (e) the concentration of reducers, in a mechanism wherein each factor affects a different kinetic process in a mostly sequential global mechanism wherein the rate-determining factor will be the slowest one. Taking into account these iron release mechanistic factors, the expression in Eq. (19) was proposed. This function captures the dependence of the iron release rate on the hydrated surface area of the mineral core when *i* is small (small mineral core) and a pore saturation behavior that becomes relevant for larger values of *i* (large mineral core). The maximal saturation rate is a constant that in turn will depend implicitly on the concentration of chelators and reducers and the gating behavior of the pores. This type of kinetic dependence of *k*_{
loss
} as a function of the iron amount in the mineral core (*i*) allowed obtention of simulation results in agreement with the observed experimental results.

### Simulations

Simulations of the model (equations (13) to (18)) were performed in order to investigate the dynamic characteristics of the system. Parameters and initial conditions used in numerical simulations are summarized in Table 1. Briefly, the ODE system was simulated until steady state conditions were satisfied. Those conditions included a maximum number of iterations and a unnoticeable change in the system's variables along two consecutive time spans. As the ODE system was found to present stiff behaviour, a commercially available stiff ODE package was used to perform all simulations.

As shown in Table 1, all simulations were carried out using an initial condition with no *C*_{
FnFe
} complex and empty apoferritin protein cages. The iron available for ferritin mineralization was provided by iron from the surrounding medium, *i.e*. the cLIP. Notice that even when the reported theorical iron capacity of ferritin is *Fn*_{
max
} = 4500 iron atoms per ferritin [43], numerical simulations were performed using a lower value of F{n}_{max}^{eff} = 2500 iron atoms per ferritin, since our experimental results indicated that ferritin precipitates *in vitro* for a ratio *Fe/Fn* > 2500 (data not shown). Finally, the value of {\tilde{k}}_{loss} was adjusted in order to obtain a cLIP corresponding to 5% of the total cell iron at steady state, which is coherent with previously reported values [9].

### Experimental setting: Velocity of sedimentation of ferritins with different iron contents in a sucrose gradient

Commercial horse spleen ferritin (Sigma Chem. Co.) was depleted of iron by treatment with thioglycolic acid [44] and loaded with different amounts of iron as described [45]. After iron loading, ferritins were resolved by gel filtration in a Sephacryl S400 column equilibrated in Hepes 0.15 M, NaCl 0.1 M, pH 7.0. The main peak was collected, its protein content was determined by the bicinchoninic acid (BCA) assay (Pierce, Rockford, Ill) and its iron content by absorbance at 420 nm [44]. To determine their migration properties, ferritins were loaded into a 5 ml 1-25% sucrose density gradient and centrifuged for 2.5 h at 112,000 × g in a Sorvall Combi ultracentrifuge equipped with an AH-850 rotor [45]. The gradients were fractionated in 250 *μ* L aliquots and ferritin content was determined by the BCA assay. A gradient sedimentation curve was constructed by plotting the position of each ferritin fraction in the gradient versus its iron content. This migration standard curve was later used to determine ferritin iron content in cell extracts and to model the iron uptake by ferritin *in vitro* and *in vivo*.