AN IN VIVO INTERMEDIATE FILAMENT ASSEMBLY MODEL

A model is developed to study the in vivo intermediate filament organization in terms of repartition between four different structural states: soluble proteins, particles, short and long filaments. An analysis is conducted, showing that the system has a unique, globally asymptotically stable equilibrium. By means of sensitivity analysis, the influence of parameters on the system is studied. In agreement with biological observations, it is shown that post-translational modifications of intermediate filament proteins resulting in filament solubilization are the main regulators of the intermediate filament organization. A high signalling-dependent solubilization of filaments favours the intermediate filament aggregation in particles.


1.
Introduction.The cytoskeleton is a complex arrangement of structural proteins organized in three different networks: microfilaments, intermediate filaments and microtubules.Each network has specific physical properties and spatial organization, and plays particular roles in the cell.Here, the focus is on the intermediate filament network, which is involved in the stabilization and mechanical resilience of the cell, cell migration, and signal transduction [6,8].
Intermediate filaments are cell type and differentiation stage dependent.They are classified in five different types: Types I-IV are cytoplasmic filaments; Type V, lamins, are nuclear filaments.Intermediate filament proteins share a central rod domain that forms a highly conserved α−helix involved in the formation of coiled-coil structures [11].The central rod domain is flanked by the head and tail domains, which are less conserved and confer specific structural properties to the different types of intermediate filament proteins.Mutations occur at the conserved central rod domain, which modulates the assembly; they then lead to defaults in the organization of intermediate filament networks [1].The head and tail domains contain the sites that can undergo post-translational modifications [16,18]; therefore, the terminal domains have mainly regulatory roles.The α−helical coiled-coil structures, called dimers, laterally associate to form tetramers.The latter are considered as the soluble subunits of intermediate filaments.Depending on the type of intermediate filament proteins, dimers are hetero-or homopolymers.
The in vitro assembly process of cytoplasmic intermediate filaments is initiated by a rapid lateral aggregation of tetramers into unit-length filaments (ULFs) [10]; this aggregation is directly followed by filament elongation, resulting from end-toend annealing of ULFs and filaments [10,13].Evidence seems to indicate that tetramer addition to the filament ends does not contribute significantly to filament elongation [13].Finally, the assembly process is terminated by a compaction event inducing a reduction of the filament diameter [10].
The in vivo assembly process of intermediate filament networks (Types I-IV) is also a multistage process: soluble proteins aggregate rapidly to constitute particles [5].The ultrastructural organization of the particles is still not known, however, it is likely that they consist of filament precursors such as ULF [8].Otherwise the emergence of the objects in next stage of assembly, "squiggles" alias short filaments, that are longer than the restrictions imposed by the diffraction limit of the light microscope, would make no sense (in particular in the light of the in vitro data as published by Kirmse et al [13]).These particles then grow and elongate into short filaments; particles are the precursors of filaments [21,27].Short filaments assemble into longer filaments which are also able to self-interact to create networks [5,29].Filaments are regulated by post-translational modifications of intermediate filament proteins resulting in solubilizations and reorganizations [18,22,25].
The organization of a cytoskeletal network is the main determinant of its function in cells.Defects in the intermediate filament network organization are correlated to different human pathologies such as skin fragility disorders, myopathies, and neurodegenerations [3,15,17].Understanding mechanisms determining the intermediate filament network organization in cells, hence its cytopathological features in diseases, could lead to new therapies.
To the best of our knowledge, so far only a few mathematical models studying the organization of the intermediate filaments were developed.Beil et al [2] investigate, with a non-dynamical approach, the structural organization of cytokeratin networks by fitting stochastic tessellations to graph structures obtained by image segmentation of networks visualized by scanning electron microscopy.Brown et al [4] use stochastic simulations to study the pulsatile motion of neurofilaments in axons.They do not describe the kinetics of neurofilament assembly.Craciun et al [7] develop a deterministic analogue of the model in [4] and obtain comparable results in agreement with experimental data.Portet et al [19] focus on the cytokeratin network organization in a cell in response to its extracellular mechanical environment.A hybrid model governing the soluble pool diffusion and the dynamics of individual filaments is introduced and simulations are conducted.In Portet et al [20], two dynamical models of the in vitro and in vivo intermediate filament assembly are proposed, but the mathematical analysis is limited and no sensitivity analysis is conducted.In Kirmse et al [13], the in vitro model in [20] is extended and its reponses are compared to experimental data to determine the elongation mechanism of a vimentin filament.
The aim of the present work, which extends that of [20] in the in vivo case, is to model the assembly dynamics of in vivo intermediate filaments, • to better understand the steps leading to the organization of the intermediate filament material, • and identify the major reactions acting on the whole process.

IN VIVO INTERMEDIATE FILAMENT ASSEMBLY 3
Here, the organization of the network is described in terms of the repartition of intermediate filament material between the different structural states that this material can be in: soluble proteins, particles, and short and long filaments.
2. Modelling concepts.It is assumed that the simplest intermediate filament structural state is the tetramer, which constitutes the pool of soluble proteins of intermediate filaments.Three other structural states for the intermediate filament material are considered: particles, short filaments and long filaments [9].The dynamics of these four structural states are governed by five reactions: particle formation, particle growth, filament formation, integration and solubilization.
A particle is arranged from the soluble protein pool, particle formation is the aggregation of several soluble proteins to form particles [5].The size of a particle is unknown [8]; a particle can take the form of an octamer (2 tetramers), ULF (8 tetramers), or a bigger structure.The size of a particle is defined here as the number of soluble proteins initially constituting a particle.
Particles grow by addition of soluble proteins [28]; this process is called the growth of particles.As lateral interactions between intermediate filament proteins can occur in 3 different manners [24], particle formation and growth of particles are described with different rates.Here, the particle is assumed to be the sole structural state recruiting from the soluble protein pool.
Particles convert to short filaments, this conversion is called filament formation.Particles are the precursors of filaments [21,27].Particles are considered here as non-filamentous units in the sense that they are not yet components of the filamentous pool and cannot directly interact with filamentous units.Note, however, that the specific shape of objects such as the form of particles or the configuration of the networks are neither described nor addressed by the model.
Short filaments assemble to form longer filaments; short and long filaments also interact.These two mechanisms are described by the same rate; this hypothesis is supported by the fact that there exists a single mode of longitudinal interaction between intermediate filament proteins [24].Based on observations made during in vitro assembly [13], it is also assumed that filament elongation only results from the interaction of filamentous units; soluble protein addition is considered as only playing a significant role in particle formation and growth.Filaments elongate and self-interact to form the network [28].This set of processes (filament elongation and linking) is called integration.Intermediate filament linking does not require any associated proteins, and the integration of filaments to the network results solely from the interactions between long filaments.Consequently, for the sake of simplicity and as the model is expressed in mass, filament linking is not explicitly described in the integration.Here, long filaments are implicitly considered to constitute the networks.
Particles and short and long filaments can disassemble into soluble proteins; this is called solubilization.The solubilization of particles can result from the instability of the particle structure.The solubilization of filaments results from posttranslational modifications of intermediate filament proteins in response to stress, apoptosis or mitosis (e.g.phosphorylation) [14,18,22,23,25].Such signalling events can also cause particle solubilization.
Particle formation, particle growth and integration of filaments are assumed to be biochemical events.Therefore, they are modelled using the Law of Mass Action.On the other hand, the conversion of particles into short filaments (the formation of filaments) is modelled as a structural event, a change of state, because little is known about the process involved.The initial number of soluble proteins involved in the formation of particles can also be interpreted as a structural feature.
3. The model.The model describes the dynamics of the four structural states of the intermediate filament material: soluble protein, particle (precursor of filaments), short filament, and long filament.So four state variables are defined: • S(t), density of soluble proteins at time t, • P (t), density of particles at time t, • F S (t), density of short filaments at time t, and • F L (t), density of long filaments at time t.
The soluble pool is the state variable S. The insoluble pool consists of particles, short and long filaments, P +F S +F L .The filamentous pool is composed of short and long filaments, F S + F L .
Based on the modelling concepts above, the following system is obtained: The constant a represents the initial number of soluble proteins involved in a particle; it is assumed that a ≥ 2. The aggregation rate of soluble proteins to form a particle is α.The constant π is the growth rate of particles.The constant ε represents the filament formation rate.The parameter µ represents the rate of assembly of filaments.The functions κ P (•) and κ F (•) represent the rate of solubilization of the particles and the filamentous pool, respectively.The rates of solubilization can take the form of a constant rate, a time-dependent function, or a concentration-dependent function of signalling proteins.All numbers and rate constants are positive; they are listed in Table 1.System (1) is considered with nonnegative initial conditions such that M 0 := S(0) + P (0) + F S (0) + F L (0) > 0, i.e., there is initially a positive concentration of proteins.
4. Mathematical analysis.Since S + P + F S + F L = 0, it follows that, defining Ω = {(S, P, F S , F L ) : Ω is positively invariant under the flow of system (1), and thus (1) is well-posed, with bounded nonnegative solutions.
The following result is a corollary of Theorem A.
Then system (1) has with respect to Ω the unique globally asymptotically stable equilibrium point with S * the only solution of and F * L given by , where the equilibrium proportion of the filamentous pool F * is Interpretation: When particle and filament solubilization can occur, then starting with any initial structural configuration of the intermediate filament material, the system reaches an equilibrium distribution in which all structural states are present.The equilibrium distribution depends on parameter values.The level of filamentous pool is directly determined by the rate of network solubilization due to signalling.
Interpretation: When there is no solubilization of the filamentous pool (no signalling), then starting with any initial configuration of the intermediate filament material, the system reaches an equilibrium at which all intermediate filament material is integrated in the network.5. Numerical results.System (1) has seven parameters.The values of these parameters are, for most of them, difficult to estimate.The first estimates of in vitro lateral and longitudinal aggregation rates of intermediate filament proteins were published only recently [13].In vivo studies have established only the time scales of the processes involved [27,28].
As the variation of the parameters of system (1) considered with constant rates does not induce any change of stability, but changes the value of the unique positive equilibrium, a sensitivity analysis allows to overcome the limitations resulting from the uncertainty on parameter values, and to better understand how parameters influence the intermediate filament organization.Numerical simulations are then run to investigate the response of system (1) considered with time-dependent rates.
5.1.Constant rates κ F (•) ≡ κ F and κ P (•) ≡ κ P : sensitivity analysis and numerical simulations.Understanding the influence of each reaction on the organization is very important, but characterizing reactions that determine the structural distribution of intermediate filaments is essential.
For instance, the value of the rate of particle growth depends on the molecular affinities of soluble proteins.If a mutation occurs on an intermediate filament gene, the affinity of soluble proteins can be modified [1,15].Hence, the value of the rate π can increase or decrease relative to a typical value.Also, cell signalling (mitosis, apoptosis, or mechanical stimulation) regulates post-translational modifications of intermediate filament proteins resulting in the solubilization of the filamentous pool [14]; this is described in the model by the rate κ F .Sensitivity analysis allows the characterization of any change in structural state concentrations induced by the variation of a given parameter.Hence, influences of each reaction on the different structural states, and as a consequence on the whole intermediate filament organization, can be quantified.
Mathematical procedures used for performing the sensitivity analysis are summarized in Appendix B. To obtain sensitivities of various structural states to parameters, a system of 4 × (7 + 1) equations consisting of ( 13) and ( 14) is numerically integrated with respect to time.Here transient behaviours are not examined, just the effect of parameters on the equilibrium values.To allow comparisons, normalized sensitivity coefficients (15) are used.Results of sensitivity analysis are shown in Figure 1, some illustrations of the dependence of equilibrium proportions on parameters are depicted in Figure 2, and the main conclusions are discussed below.
The initial size of particles, a, does not really change the organization: it mainly acts positively on the soluble pool and negatively on the insoluble pool.Results are not shown in Figure 2, as an increase of a does not induce any fundamental modification of the repartition of the states as the other parameters do.
High lateral aggregation rates, α and π, favour the insoluble pool.These parameters, which can be interpreted as rates of lateral aggregation of soluble proteins [24], affect similarly each structural state.High rates of lateral aggregation can modify the organization of intermediate filaments from a soluble configuration to an arrangement in networks (Figures 2(a A high rate of conversion of particles to short filaments, ε, favours solubility.Particles are the only recruiters of soluble proteins: the faster particles convert to short filaments in comparison to the growth of particles (π), the lower the soluble pool The rate of integration of filaments, µ, regulates the repartition of the filamentous pool.The proportion of filamentous pool stays constant, but the repartition of the filamentous pool between short and long filaments is modified.A high integration rate favours the organization of intermediate filament material in networks (Figure 2(d)).The rate of solubilization of particles, κ P , can affect the organization.An increase in κ P can switch the repartition between the soluble and insoluble pool (Figure 2(e)).
A high signalling-regulated filament solubilization rate, κ F , favours particles.An increase in κ F has opposite effects on filaments integrated in networks and particles, changing the organization of intermediate filaments from a network configuration to a particle configuration (Figure 2(f)).
In biological terms, reactions related to the lateral assembly and disassembly of intermediate filament proteins (α, π, and κ P ), which can be affected by mutations [1], determine the structural organization of the intermediate filament material, and regulate the repartition between soluble and insoluble pools (mainly between soluble proteins and filaments integrated in networks).The signalling-regulated reaction (κ F ) totally determines the type of organization and favours particles [30].

5.2.
Non-constant rate κ F (t): numerical simulations.It is shown above that a main determinant of the intermediate filament organization is the rate of solubilization of filaments.Effects of non-constant rates of filament solubilization are now numerically investigated.
The rate κ F (•) is taken to be a time-dependent function, with a baseline rate of solubilization and peaks resulting from signalling events.Two signalling events are considered here; they differ in duration and magnitude; they can be triggered by a mechanical stimulation or mitosis [14,23].
The non-constant rate of solubilization is defined as follows (bottom of Figure 3) with all parameters positive.Parameters t 1 and t 2 (t 1 < t 2 ) are times at which signalling events take place.Parameters b and c regulate the slope of the functions.Finally, κ F represents the magnitude of the baseline solubilization rate of filaments, κ m is the maximal magnitude of the rate of solubilization induced by the first signalling event, and κ M is the maximal magnitude of the rate of solubilization resulting from the second signalling event (κ F κ m < κ M ).When the rate of filament solubilization κ F is at its baseline level, intermediate filament material organizes in long filaments constituting networks.Interphasic cells have intermediate filaments organized in networks; the soluble pool represents a low proportion of the material, the major part is assembled in filaments [27].
Due to the first signalling event, the magnitude of κ F starts increasing; consequently, the proportion of long filaments drops as they disassemble into soluble proteins that in turn aggregate in particles.Almost instantaneously, these particles convert into short filaments.The increase in the proportion of particles stops when the magnitude of the rate of filament solubilization decreases.For a while, short filaments are the dominant structural state.As the rate κ F returns to its baseline level, the network reorganizes.This first signalling event can result from a mechanical stimulation that locally acts on the cell.The induced solubilization is not dramatic and induces the disassembly-reassembly of about 20% of the long filaments involved in the network; this can be interpreted as a local re-organization of the network in response to the stimulation.
The second signalling event induces a longer and stronger increase of the magnitude of the solubilization rate.Similarly as with the first signal, long filaments first disassemble, inducing an increase of the soluble pool followed by an increase of the proportion of particles and subsequently, an increase of the proportion of short filaments.For a while, particles dominate, and, as the magnitude of the solubilization rate decreases, short filaments reassemble and become the dominant structural state; finally the network reorganizes.In certain cell types, at metaphasis, cells have a reduced filamentous pool, the intermediate filament material is mainly soluble or aggregated into granules and short rod shaped objects [27].This second signal can therefore be interpreted as a mitosis, inducing an almost complete disassembly of the network of the cell before the reformation of networks in daughter cells.Note that similar results are obtained when using a square wave instead of a smooth function such as (2) for the non-constant rate of solubilization (results not shown).

6.
Discussion.A nonlinear model for the dynamics of in vivo intermediate filaments is proposed; the model describes the assembly and the organization of this cytoskeletal component in terms of repartition between different structural states: soluble proteins, particles, short filaments and filaments integrated in networks.The model includes several nonlinearities, one of which is of the form S a .Because of these nonlinearities, the analysis is complicated.However, using mass conservation properties and the theory of asymptotically autonomous systems, it is shown that the system has a unique positive globally asymptotically stable equilibrium.
When signalling events occur (κ F = 0), all structural states of intermediate filaments are present (Theorem 4.1).In the absence of signalling events (κ F = 0), the intermediate filament material is entirely organized in networks (Corollary 4.2).In both cases, due to the global stability, there is no dependence on the initial configurations of the intermediate filament material.Moreover, the proportion of filamentous pool is inversely proportional to the rate of signalling-dependent solubilization of filaments κ F .Qualitative behaviours of the model are in agreement with biological observations [27]: the organization of intermediate filament networks is mainly regulated by the filament solubilization [30], which can be interpreted as a consequence of post-translational modifications.
By means of sensitivity analysis, reactions involved in intermediate filament assembly are studied to identify those that are the most determinant for the organization.The lateral aggregation of soluble proteins and the signalling-dependent solubilization of filaments are found to be the major regulators of this organization.The lateral aggregation of proteins acts on the solubility of the intermediate filament material.The signalling-dependent solubilization of filaments determines the intermediate filament organization and favours aggregation in particles.Furthermore, filament integration only modifies the distribution of the filamentous pool in terms of short and long filaments.This induces changes in the mechanical properties of networks.
More and more diseases have been associated to mutations of intermediate filament genes [17].These mutations are not well understood, but they are known to affect the assembly of filaments and their integration in networks: the cytological signature of these diseases is a misorganization of the intermediate filament material.For example, in certain skeletal and cardiac myopathies, large aggregates of desmin are observed [1].Furthermore, for several neurodegenerative diseases, intermediate filament aggregates also are a pathological hallmark; they seem to be correlated to high phosphorylations [23].The putative effects of mutations on the network organization or the formation mechanismes of aggregates could be studied using models of the type proposed here.
Furthermore, the total length L of intermediate filament material assembled into filaments can be computed from this model.Letting be the linear density of intermediate filaments [10], the total length L can be expressed as The length L is an experimentally observable quantity, which could help identify parameter values of the system, as this was done in vitro in [13].The model has limitations; it does not specify the synthesis of the soluble pool, and motile properties of different structural states of intermediate filament material [9,29] are not described.Depending on the cell type, antero and/or retrograde transports have been observed, with varying speeds for the different structural states of intermediate filaments.Some of those intracellular motions are related to the microtubule network, whereas others are microfilament network dependent [29].Further work will extend the model by taking into account the motile properties of each structural state of the intermediate filament material.A.1.Simplified models.Summing the four equations in system (1), the conservation equation d dt (S + P + F S + F L ) = 0 is obtained.Consequently, there holds that S(t) + P (t) + F S (t) + F L (t) ≡ M 0 , that is, solutions are confined to the hyperplane , the dimensionality of system (1) can be reduced to 3. Considering the proportions S = S M0 , FS = F S M0 and FL = F L M0 of soluble pool, short filaments and long filaments in the cell, respectively, system (1) is now expressed as follows (bars are omitted) where F := F S + F L is the filamentous pool.The dynamics of P (in proportions) is reconstructed from the state of system (3) using the fact that P (t) = 1−S(t)−F (t).This system is called the 3-dimensional system in proportions.Note that since for system (1), S +P +F S +F L ≡ M 0 , the dynamics of (1) can be entirely reconstructed from the dynamics of (3) by multiplying variables of (3), as well as P = 1 − S − F , by M 0 .The two systems are therefore equivalent.The dynamics of the filamentous pool F is obtained by summing equations (3b) and (3c), giving Consequently, system (3) can be reduced to and the dynamics of P is deduced from ) has nonnegative solutions bounded above by 1, summing to 1 if the dynamics of P (t) is also considered.Since system (4) is obtained by linear combination of variables of (1), the problem ( 4) is also well-posed, with solutions bounded above by 1, summing to 1 if the dynamics of P (t) is also considered.A.2. Existence and stability of equilibria for the two-dimensional system (4).First, the existence stability of equilibria of system (4) is considered.Let Ω = {(S, P, F S , F L ) : S + P + F S + F L = 1}.
Theorem A.1.Suppose that κ P (•) ≡ κ P ∈ R + and κ F (•) ≡ κ F ∈ R + .Then system (4) has with respect to Ω a unique globally asymptotically stable equilibrium where S * is the only solution of Proof.Existence and uniqueness of the equilibrium E * .
To find the equilibria of system (4), consider its nullclines; an equilibrium point (S * , F * ) is solution to the following system, From equation (5b), it is obtained that Substituting ( 6) in (5a) implies that Thus, the positive roots S * of ( 7) are the S components of the equilibrium solutions of (5).Rewriting (7) as implies that the roots S * lie at the intersections in R 2 + of Φ and Ψ.The polynomial Ψ is an increasing function on R + such that Ψ(0) = 0. Φ is a concave up parabola such that Φ(0) = κ F (ε + κ P ) > 0 and with 2 positive roots, 1 and ε + κ P πM 0 . Φ decreases on 0, πM 0 + ε + κ P 2πM 0 , increases on Since state variables in system (4) are in proportions, a biologically relevant equilibrium must satisfy the condition S ≤ 1.As a consequence, S * is unique and satisfies 0 < S * < min 1, ε + κ P πM 0 , and system (4) has a unique interior equilibrium Stability of E * .The Jacobian of system ( 4) at E * is Begin with the conclusion.In those regions in parameter space where the Jacobian matrix J E * has a negative trace, it follows from Bendixson's criterion that there are no non-constant periodic solutions in the positive quadrant.It already has been shown that solutions of (4) are bounded.As the equilibrium E * is unique, it follows from the Poincaré-Bendixson theorem that all trajectories limit to the equilibrium E * , i.e. that the equilibrium E * is globally asymptotically stable with respect to Ω.So now the sign of the trace of J E * is studied.Rewrite tr(J E * ) as the difference between two polynomials in S * , tr(J E * ) = P 1 (S * ) − P 2 (S * ), where The sign of tr(J E * ) then depends on the relative positions of the graphs of P 1 and P 2 .The following observations can be made.First, P 1 is an increasing function whose graph is a straight line with negative y−intercept, and a positive S * −intercept denoted S int , Therefore, P 1 is negative for all S * < S int .On the other hand, P 2 is an increasing function with P 2 (0) = 0, so P 2 is positive for all S * > 0.
At this point, the result that S * is globally asymptotically stable has been shown everywhere except in the case where S * is between S int and κ P +ε πM0 , with S int < κ P +ε πM0 , that is, But this inequality is equivalent to πM 0 − κ P + κ F < 0, that is, πM 0 < κ P − κ F .However, the current case corresponds to κ P + κ F + ε < πM 0 .Using both inequalities, there must hold that κ F + ε < −κ F , a contradiction.Therefore the case S int < (κ P + ε)/(πM 0 ) is irrelevant.It follows that tr(J E * ) < 0, implying that E * is globally asymptotically stable.
A.3.Existence and stability of equilibria for the three-dimensional system (3).In the system (3) at equilibrium, the variables S, F and P take their equilibrium values S * and F * obtained in Theorem A.1, and P * = 1 − S * − F * , respectively.Since S and F converge to S * and F * , respectively, it follows that (3) is an asymptotically autonomous system, with limiting system In (10), the dynamics of S and F , which reduce to S(t) ≡ S * and F (t) ≡ F * , are omitted.The following result holds.
Theorem A.2. Suppose that κ P (•) ≡ κ P ∈ R + and κ F (•) ≡ κ F ∈ R + \ {0}.Then system (10) has with respect to Ω a unique globally asymptotically stable equilibrium and Proof.The value of the equilibrium (F S , F L ) = (F * S , F * L ) is readily established.Also, at an arbitrary point (F S , F L ), the Jacobian matrix takes the form and has eigenvalues −κ F and −κ F −µM 0 (F S +F L ), so that the equilibrium (F * S , F * L ) is locally asymptotically stable.Furthermore, the Jacobian has trace −(F S + F L )M 0 µ − 2κ F < 0, so that Bendixson's criterion applies, implying, since solutions are bounded, that the equilibrium (F * S , F * L ) is globally asymptotically stable for the limit system (10).
2. Therefore, system (1) can be transformed into a lower dimensional system, involving S, F S and F L , with P = M 0 − (S + F S + F L ).Going to proportions, this is formulated as the 3-dimensional system (3).Systems (1) and ( 3) are equivalent, and therefore (3) is considered.3. System (4), in the variables (S, F ), is then deduced from (3) by considering the compounded variable F = F S + F L .4. System (4) has a unique, globally asymptotically stable equilibrium point (S * , F * ) (Theorem A.1). 5. System (3) is then considered as a system in which S(t) and F (t) are time dependent quantities (and not obtained as solutions of a differential equation).Thus, (3) is of the form x = f (t, x), with f (t, x) → g(x) as t → ∞, where g has S = S * and F = F * .In this new system, called the limiting system of "(3) seen as a non-autonomous system", the equation for S is trivial, so only ( 10) is studied.6.For (10), it is shown in Theorem A.2 that there is a unique globally asymptotically stable equilibrium point (F * S , F * L ).So it remains to show that the behaviour of system (3) in proportions can indeed be derived from the information obtained on S, F S and F L .Note that the method that is used in the proof of Theorem A.1 to establish the existence of S * involves F S and F L only has a sum, i.e., as F = F S + F L .The method is thus unchanged when considering F S and F L rather than F , and the conclusion still holds that there can be only one equilibrium value of S * such that 0 < S * < min 1, ε+κ P πM0 .It is then easy to check that the equilibrium obtained from (3b) and (3c) with F S + F L substituted to F in (3b), is the same to the one obtained for (10), εκ F (1 − S * ) κ F (ε + κ F ) + M 0 εµ(1 − S * ) and which are the values given by ( 11) and (12) in which ε(1 − S * ) ε + κ F is substituted to F * .Thus (3) also has a unique equilibrium.
From the theory of asymptotically autonomous differential equations, using for example Theorem 4.2 and Corollary 4.3 in [26], since the limit system (10) has the unique equilibrium point (S * , F * S , F * L ), it follows that every forward orbit of (3) converges to (S * , F * S , F * L ).As a consequence, the following result holds.
Theorem A.3.The 3-dimensional system in proportions (3) has with respect to Ω the globally asymptotically stable equilibrium point (S, F S , F L ) = (S * , F * S , F * L ), with S * given as in Theorem A.1, F * S given by (11) and F * L given by (12).From this, the value of P * can be deduced.Since P (t) = 1 − S(t) − F (t), it follows that The equilibrium values of S, P , F S and F L in (1) are then deduced by multiplying all terms by M 0 .This equilibrium is globally asymptotically stable for system (1). )-2(b)).

Figure 1 .
Figure 1.Normalized sensitivity coefficients of the structural states to the seven reactions estimated using (15): (a) soluble proteins, (b) particles, (c) filaments and (d) filaments integrated in networks.For each structural state, computations were repeated for 4096 sets of parameters; results are displayed as boxplots for each parameter.The box shows the interquartile range.The median is indicated as the dividing line in the box.The length of the whiskers is specified as 1.5 times the interquartile range.All data with values beyond the ends of the whiskers are outliers, they are displayed with a + sign.

Appendix A .
Proof of Theorem 4.1.

Table 1 .
Parameters in the model.
3, Appendix A. The existence and uniqueness of S * is obtained in Theorem A.1, Appendix A. Theorem 4.1.Suppose that κ P