Abstract�It is generally accepted that aging of the

vascular system plays an important role in cardiovascular disease (CVD). Recent experimental findings have indicated the involvement of the cytokine transforming growth factor-β1BBB (TGF-β1) in these vascular aging processes. This cytokine is, after binding to a cell receptor, associated with numerous cellular processes, including formation of the extracellular matrix (ECM, the biomolecular network that surrounds the cell). We implemented TGF-β1 signaling and its effect on ECM formation with piecewise-linear differential equations (PLDEs), which have several advantageous properties over traditional continuous modeling. Aging of the system was simulated as a reduction in cell sensitivity for TGF-β1. The model predicted a disturbed ECM balance during aging, which corresponds well to findings from the literature. The outcome of this hybrid approach was satisfactory and, therefore, we will continue modeling of various aging-related pathways with PLDEs in future research.

Keywords�Biomolecular pathway, differential equations,

hybrid system, system analysis1

I. INTRODUCTION In general, biological aging is the process of physical deterioration that accompanies advancing age, making the organism more susceptible to a whole range of diseases. In particular cardiovascular disease (CVD) is of our interest, because it is currently the main cause of death in the USA and Europe. Despite its important relation to CVD, the main mechanism of vascular aging has not been elucidated yet by the various proposed aging theories. However, our recent experimental findings together with those from others [1] suggest the implication of transforming growth factor-β1 (TGF-β1) pathway in vascular wall aging. Moreover, many cellular processes have been imputed to this cytokine [2]. At older age, genes that inhibit intracellular TGF-β1 signaling were shown to be upregulated, which would reduce the sensitivity of the cell receptor for TGF-β1. The effect of TGF-β1 includes the formation and maintenance of the extracellular matrix (ECM), a meshwork of biological macromolecules that surrounds each cell. Disturbances of the TGF-β1 pathway could lead to malfunctioning of ECM formation/degradation and would, subsequently, result in a disturbed vascular ECM. Fig. 1 TTTThis work was funded by Senter (Dutch Ministry of Economic affairs, no. TSGE1028) and Unilever Research and Development Vlaardingen, The Netherlands.

shows the interactions between both pathways. Biological systems are traditionally modeled as a set of continuous differential equations and require quantitative parameter values, which are hard to obtain in practice. In addition, system analysis of these differential equations is often hampered by biological nonlinearities, e.g. sigmoid functions in signaling pathways. We implemented a model of TGF-β1 and ECM with piecewise-linear differential equations (PLDEs), where the sigmoid functions were replaced by discontinuous step functions. This generated a set of equations that combined a continuous (the degradation of biomolecules) and discrete (the step functions) nature, i.e. a hybrid system. The hybrid system paradigm can also be useful to describe and analyze regulatory networks operating in both space and time, such as TGF-β1. Cellular space is compartmentalized and/or can be assumed to be segmented. Regulatory signals trigger proteins to move to another compartment where they become (in)active or change function. This can be described as switch-like dynamics while a lumped parameter structure can be maintained. Applications include various biological topics [4], [5]. We employed a special implementation of PLDEs, which required only qualitative parameter values to do qualitative simulations, i.e. information on whether a parameter is larger or smaller than the others [6]. Our goal was to design a qualitative piecewise-linear model of TGF-β1 signaling and the ECM formation and degradation balance to investigate the effects of aging on both processes. This is to our knowledge the first application of PLDEs and qualitative simulation in a biomedically relevant case, i.e. bioprocesses involved in vascular aging.

Fig. 1. Interactions between the TGF-β1 pathway and ECM

formation/degradation balance. Five concentrations take part in this process: x1 = insoluble ECM proteins, x2 = proteolysis fragments, x3 = protease, x4 = activated TGF-β1 receptor and x5 = TGF-β1 transcription

products. The interactions are graphed according to the symbol conventions for molecular interaction maps [3]: solid arrow =

stoichiometric conversion, open arrow = stimulation, open circle = enzymatic stimulation, bar = inhibition and crossed out circle =

degradation products.

Analysis of the transforming growth factor-β1 pathway and extracellular matrix formation as a hybrid system

M. W. J. M. Musters1, N. A. W. van Riel1,2

1Department of Electrical Engineering and 2Department of Biomedical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands

II. METHODOLOGY A mathematical model of the ECM was obtained that considered the ECM in balance with proteolysis fragments, controlled by the enzymes protease and transglutaminase [7]. The proteolysis fragments stimulated the formation of protease, which created a positive feedback; the system showed multistationary behavior as a result [8]. TGF-β1 is a cytokine that binds to membrane receptors and initiates the activation of SMAD proteins [2], which translocate into the cell nucleus to stimulate a range of transcription products, including the ones that result in ECM formation. Furthermore, ECM was formed externally in the original model [7], but we assumed that mainly TGF-β1 induced ECM formation. The total model of the ECM balance with TGF-β1 signaling and its interactions is shown in Fig. 1. A. Piecewise-linear modeling Systems composed of steep sigmoid curves, as in regulated gene activity, can be approximated by step functions, for which different modeling methods exist [9]. We applied PLDEs [10], [11] to describe the dynamics of our model, because of several advantageous properties [11]. An introduction to PLDEs is listed below; the work of Hidde de Jong gives a more detailed mathematical description [6], [12]. The general form of the state equations is defined as

( ) ( )i i i ix f x g x x= −! , (1)

with 1[ ] Tnx x x= … and each protein concentration ix , encoded by gene i, is synthesized and degraded at a rate ( )if x and ( )i ig x x , respectively. Both ( )if x and ( )ig x are functions of rate parameters ( iκ for synthesis, iγ for

degradation) and step functions s+ and/or s− : 1,

( , )0,

i ii i

i i

xs x

xθ

θθ

+ >⎧= ⎨ <⎩

, ( , ) 1 ( , )i i i is x s xθ θ− += − , (2)

where iθ represents a threshold concentration ( iθ > 0), at which ix is synthesized or degraded. In addition, possible multiple thresholds per ix have to be ordered and equilibrium inequalities have to be defined, according to their biological function, as will be clarified by the following example. Two concentrations, 4x and 5x , were isolated from Fig. 1 to describe the TGF-β1 pathway alone. Fig. 2 shows the corresponding phase space box. Assuming that 3x is always large enough, the PLDEs of this system are stated as

where 14κ ( 1

4γ and 24γ ) and 1

5κ ( 15γ ) represent the synthesis

(degradation) rate parameters for 4x and 5x , respectively.

Two thresholds of x5 are present: 35θ and 4

5θ are the thresholds for x4 formation and inhibition, respectively. The assumption is made that x4 formation occurs before x4 inhibition and hence

3 45 5 50 maxθ θ< < < , (5)

is the threshold order, with 5max the maximum concentration of x5. The system can be divided in different domains D, which are all thresholds (switching domains) and regions between these thresholds (regulatory domains). For each domain, a target equilibrium exists, towards which x4 and x5 converge. From (3) and (4), one obtains five target equilibria (0,0), (0, 1

515 γκ / ), ( 1

414 γκ / ,0), ( )/( 1

414

14 γγκ + ,0)

and ( )/( 14

14

14 γγκ + , 1

515 γκ / ). The positions of all possible

target equilibria relative to the thresholds are defined by the equilibrium inequalities that determine the level of gene expression. In Fig. 2, the target equilibrium for domain D3 is

3( )Dφ with the value ( 14

14 γκ / , 0). The corresponding

equilibrium inequality was chosen to be

414

14

14 max<< γκθ / , (6)

otherwise threshold 14θ could not be reached and,

consequently, x5 would only degrade. B. Model of the ECM balance and TGF-β1 signaling A hybrid system was generated from Fig. 1. The five PLDEs are

( ) ( )x f x g x x= −! , 1 2 3 4 5 Tx x x x x x= ⎡ ⎤⎣ ⎦ , (7) where

1 2

3 4 1

5 1

insoluble ECM proteins, proteolysis fragments, protease, activated TGF-β receptor, TGF-β transcription products,

x xx xx

= == =

=

(3) ( )1 3 1 2 44 4 5 5 4 4 5 5 4

1 1 15 5 4 4 5 5

( , ) ( , ) ,

( , ) ,

x s x s x x

x s x x

κ θ γ γ θ

κ θ γ

+ +

+

= − +

= −

!

!

(4)

Fig. 2. Phase space box of the isolated TGF-β1 pathway. The system

comprised 15 domains: 9 switching and 6 regulatory domains. The third domain D3 is shown here. The arrows show how the trajectories of the solutions in this domain converge to the target equilibrium φ(D3). The

position of this target equilibrium was defined by the equilibrium inequalities.

1 1 2 11 2 2 1 5 5

1 1 12 1 1 3 31 3 23 2 2 5 51 4 34 3 3 5 5

1 15 4 4

( , ) ( , )

( , ) ( , )( ) ( , ) ( , )

( , ) ( , )

( , )

s x s x

s x s xf x s x s x

s x s x

s x

κ θ κ θ

κ θ θ

κ θ θ

κ θ θ

κ θ

+ +

+ +

+ −

+ +

+

⎡ ⎤+⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

,

1 2 1 11 1 1 1 3 3

1 2 1 3 2 22 2 2 2 2 2 2 3 3

1 2 33 3 3 31 2 44 4 5 5

15

( , ) ( , )

( , ) ( , ) ( , )( ) ( , )

( , )

s x s x

s x s x s xg x s x

s x

γ γ θ θ

γ γ θ γ θ θ

γ γ θ

γ γ θ

γ

+ +

+ + +

+

+

⎡ ⎤+⎢ ⎥

+ +⎢ ⎥⎢ ⎥

= +⎢ ⎥⎢ ⎥+⎢ ⎥⎢ ⎥⎣ ⎦

.

Its threshold inequalities are 11 10 maxθ< < , 2 3 1

2 2 2 20 maxθ θ θ< < < < , 1 4 2 33 3 3 3 30 maxθ θ θ θ< < < < < , 1

4 40 maxθ< < , 1 3 4 25 5 5 5 50 maxθ θ θ θ< < < < < ,

(8)

and the equilibrium inequalities become 1 1 1 1 1 2 11 1 1 1 1 1 1 11 2 1 2 1 2 11 1 1 1 1 1 1 1

1 1 2 1 1 2 1 2 11 1 1 1 1 1 1 1 1 1

/ max ,0 /( ) ,

/ max ,0 /( ) ,

( ) / max ,0 ( ) /( ) ,

θ κ γ κ γ γ θ

θ κ γ κ γ γ θ

θ κ κ γ κ κ γ γ θ

< < < + <

< < < + <

< + < < + + <1 1 1 1 1 1 22 2 2 2 2 2 2 2 2

1 1 3 2 1 1 2 3 22 2 2 2 2 2 2 2 2

/ max , /( ) max ,

0 /( ) ,0 /( ) ,

θ κ γ θ κ γ γ

κ γ γ θ κ γ γ γ θ

< < < + <

< + < < + + <

3 1 1 1 1 2 13 3 3 3 3 3 3 3/ max ,0 /( ) ,θ κ γ κ γ γ θ< < < + <

1 1 14 4 4 4/ max ,θ κ γ< < 1 1 1 2

4 4 4 4 4/( ) maxθ κ γ γ< + < , 2 1 15 5 5 5/ max .θ κ γ< <

(9)

Aging of TGF-β1 signaling was simulated by changing the equilibrium inequality 1 1 1 2

4 4 4 4 4/( ) maxθ κ γ γ< + < into 1 1 2 14 4 4 40 /( )κ γ γ θ< + < . This reflects the strong inhibition of

the TGF-β1 receptor that outclasses its synthesis, so threshold 1

4θ could not be reached. The Genetic Network Analyzer (GNA) [12] was used to simulate the PLDEs. To reduce the number of modes1, we carried out all simulations with the following initial conditions, based on physiological considerations:

Analysis without restrictions on the initial conditions resulted in a few additional equilibrium modes and cycles. Furthermore, the corresponding attraction domains2 were small compared to the attraction domains of the dominant

1 mode = unique combination of concentration levels (in this case) 2 attraction domain = domain set from which an equilibrium mode or cycle could be reached

equilibrium modes and cycles. The combination of only a few modes or cycles and a small attraction domain are in practice less likely to occur.

III. RESULTS

Equilibrium modes are presented in Table I. In both

situations (normal and aging), one unfavorable stable mode (Ia) could be reached, with low ECM density and too low TGF-β1 activity to trigger other TGF-β1-controlled processes in the cell (x1, x2, x3, x4, and x5 are low). Eventually, all modes of the system, 3739 in total, could converge to this stable mode. The unstable equilibrium modes (Ib) had smaller attraction domains and were saddle points in the cycles displayed in Table II.

Additionally, the model contained multiple cycles in both simulations (Table II). These were divided into five different types, viz. IIa: oscillating ECM (x1, x2, x3), low TGF-β1 receptor (x4) and transcription activity (x5); IIb: oscillating ECM, high TGF-β1 receptor and low transcription activity; IIc: oscillating ECM and oscillating TGF-β1 (x3, x4, x5); IId: low ECM density (x1) and oscillating TGF-β1; IIe: high ECM density and oscillating TGF-β1 (Table II). In the normal situation, four different cycle types existed, of which three with an attraction domain of more than 200 modes that covered the more probable cycles. The cycle type with low ECM and an oscillating TGF-β1 pathway (IId) was feasible, but only for one cycle with a small attraction domain. Aging resulted in extra cycles with a low and high density ECM (IId and IIe). Their attraction domains were small, but contained multiple cycles, i.e. a higher probability that this mode was reached.

1 2 31 1 1 2 2 3 3 3

1 14 4 4 5 5

max ,0 , max ,

max ,0 .

x x x

x x

θ θ θ

θ θ

< ≤ ≤ < < ≤

< ≤ ≤ < (10)

TABLE II CYCLIC BEHAVIOR OF THE MODEL

# of cycles Cycle type

Normal Aging

IIa x1, x2, x3 oscillating; x4, x5 low 2a 2a

IIb x1, x2, x3 oscillating; x4 high; x5 low 2a 2a

IIc x1, x2, x3, x4, x5 oscillating 1a 1a

IId x1 low; x2 high; x3, x4, x5 oscillating 1b 6b

IIe x1 high; x2 low; x3, x4, x5 oscillating 0 5b Attraction domain is alarger than 200, or bsmaller than 100 modes

TABLE I EQUILIBRIUM MODES OF THE MODEL

# of equilibrium modes Equilibrium modes

Normal Aging

Ia Stable: x1, x2, x3, x4, x5 low 1 1

Ib Unstable 10 7

IV. DISCUSSION The hybrid system of TGF-β1 signaling and ECM balance showed multiple cycles and equilibrium modes under normal and aging conditions. Based on the range of the attraction domains, four different steady-state types were identified in the normal situation (Table I and II), namely a stable mode with no ECM and TGF-β1 activity (Ia); two cycles with a normal ECM balance, but with insufficient TGF-β1 transcription products (IIa and IIb); a cycle with an active ECM balance and TGF-β1 signaling (IIc). The latter cycle is suggested as present under healthy conditions, because both the ECM balance and TGF-β1 pathway are then functioning properly. In concordance with experimental findings, we simulated the aging of the TGF-β1 pathway as a reduction in TGF-β1 receptor activity by changing its corresponding equilibrium inequality. This led to the formation of additional cycles (IId and IIe), all with adverse consequences for the ECM balance. Likewise, the chance to come in these unfavorable modes increased due to the combination of an increase in cycles and extra attraction domains that do not exist under normal conditions. On the whole, it seems that reduced sensitivity of the TGF-β1 pathway could lead to a low or high ECM density (x1). Such alterations in ECM density are consistent with observations of the aging vascular system [13]. The modeling of ECM as a set of PLDEs is criticizable, because no steep sigmoids were present in the original model [7]. However, comparison of the continuous ECM model with our piecewise-linear approximation showed the same qualitative behavior, like bistability and oscillations (unpublished results). In addition, extensive numerical simulations of the continuous model were performed, replacing the Michaelis-Menten equations by sigmoid functions with varying steepness. Results showed no significant difference in qualitative equilibrium modes between both methods, which was another indication that our piecewise-linear approach could be justified for this specific case. Most thresholds in the system were ordered according to their biological function, a few were assumptions. To test the possible effects, we changed the threshold order. This did not affect the overall simulation results importantly, except if thresholds involved in the degradation of a biomolecule were set smaller than synthesis thresholds; the system would contain no cycles and automatically converge to the stable mode with low ECM and TGF-β1 activity (Ia).

V. CONCLUSION

In this paper, a piecewise-linear model was presented that combined ECM formation and TGF-β1 signaling. The goal of this hybrid model was to get a better understanding of the vascular aging process, which was simulated as a reduced sensitivity to TGF-β1 stimulation. Based on our

findings, it appears that elimination of TGF-β1 stimulation can get the delicate ECM formation/degradation cycle out of balance, which would eventually result in undesired side effects, such as a high- or low-density ECM. This is consistent with characteristic features of CVD [13]. Piecewise-linear modeling and qualitative simulation were demonstrated to be helpful in analyzing TGF-β1 signaling and the ECM balance. We will continue to apply this method on various other aging pathways in future research to gain a better understanding of this process.

ACKNOWLEDGMENT

We would like to thank Hidde de Jong for providing GNA (TUTUTUhttp://bacillus.inrialpes.fr/gna/UUUTTT) and Hugues Berry for his help regarding the ECM model.

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