Research Article Finding Solvable Units of Variables in Nonlinear ODEs of ECM Degradation Pathway Network

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1 Hindawi Computational and Mathematial Methods in Mediine Volume 2017, Artile ID , 15 pages Researh Artile Finding Solvale Units of Variales in Nonlinear ODEs of ECM Degradation Pathway Network Shuji Kawasaki, 1 Dhisa Minerva, 2 Keiko Itano, 2 and Takashi Suzuki 2 1 Iwate University, Morioka , Japan 2 Osaka University, Toyonaka , Japan Correspondene should e addressed to Shuji Kawasaki; shuji1113@gmail.om Reeived 14 Deemer 2016; Revised 28 Marh 2017; Aepted 2 April 2017; Pulished 30 May 2017 Aademi Editor: Chung-Min Liao Copyright 2017 Shuji Kawasaki et al. This is an open aess artile distriuted under the Creative Commons Attriution Liense, whih permits unrestrited use, distriution, and reprodution in any medium, provided the original work is properly ited. We onsider ordinary differential equation (ODE) model for a pathway network that arises in extraellular matrix (ECM) degradation. For solving the ODEs, we propose applying the mass onservation law (MCL), together with a stoihiometry alled douling rule, to them. Then it leads to extrating new units of variales in the ODEs that an e solved expliitly, at least in priniple. The simulation results for the ODE solutions show that the numerial solutions are indeed in good aord with theoretial solutions and satisfy the MALs. 1. Introdution Differential equations are a useful method for modeling dynamis of reation pathways in ells. They an e used to formulate iohemial kinetis, that is, the interative dynamis of systems onsisting of proteins, enzymes, produts, and other omponents in terms of their transitive onentrations. The iohemial kinetis of a system desries the reations through the mass ation law (MAL); this results in a system of first-order nonlinear ordinary differential equations (ODEs). In general, the nonlinear ODEs annot e solved expliitly, and so they are usually studied y numerial simulations. For the modeling of suh a system, we propose an idea of using the mass onservation laws (MCLs) together with the MALs. That is, the variales in the ODEs otained y the MALs are grouped into new units of variales to onstitute new MALs, aording to loal alaning relations of inflows and outflows. It is important here that these new MCLs hold for all t 0. As a result, the nonlinear ODEs presented y the new units of variales turns out to e a ompletely integrale system; that is, the ODEs an e solved expliitly, at least in priniple. We apply this idea to analyze the kinetis of the moleule onentrationdynamisinanerellinvasiontotheecmy a matrix metalloproteinase alled MT1-MMP. If we an find appropriate MCLs for alaning relations in the kinetis, then suh relations turn out to exhiit linear relations etween ODE variales that are valid for all t 0.Also,thenonlinear ODEs with the new unit of variales are ompletely solvale expliitly. That is, these nonlinear ODEs of new unit of variales an e solved as a group. Thesegroupsmayoften orrespond to network motifs, that is, loal funtions; they suggest a undle of meaningful omponents in a omplex pathway network (PWN). As the simulation results show, the numerial solutions are in good aord with the theoretial solutions, indiating our modeling y the new unit of variales is right. We thus larify how to look at the PWN model of the ECM degradation with the unit variales. In fat, we indiate that the relevant system of the PWN an e grouped into several units of original variales and, with the unit variales, the system of ODEs an e solved expliitly. Taking linear omination of ODEs for the grouping has existed so far, uthangingoriginalodesintonew,solvaleonesythe linear omination has not een onsidered so far. This is a speial stoihiometry. In addition, in order to enjoy suh grouping method, ertain reformulation inluding douling rule is neessary as in the following, whih has not een onsidered properly so far.

2 2 Computational and Mathematial Methods in Mediine X 1 (a) Tale 1: List of assoiations with a-b. Assoiation with a-b Asso. onst. +X 2 () X 4 (a) k 1 +X 5 () X 7 (a) k 1 +X 8 () X 9 (a) k 1 +X 10 () X 11 (a) 2k 1 +X 11 (a) X 12 (aa) k 1 Tale 2: List of assoiations/dissoiations with B-C. Assoiation/dissoiation with B-C X 2 () X 4 (a) X 3 () Asso. onst. Disso. onst. l 1 =0 Disso. onst. +X 3 () X 5 () k 2 l 2 +X 6 () X 8 () 2k 2 l 2 +X 8 () X 10 () k 2 2l 2 +X 9 (a) X 11 (a) k 2 l 2 +X 3 () X 7 (a) k 2 l 2 +X 6 () X 9 (a) 2k 2 l 2 +X 8 () X 11 (a) k 2 l 2 +X 9 (a) X 12 (aa) k 2 2l 2 Tale 3: List of assoiations/dissoiations with C-C. Assoiation/dissoiation with C-C Asso. onst. Disso. onst. +X 3 () X 6 () 2k 3 2l 3 +X 5 () X 8 () k 3 l 3 +X 7 (a) X 9 (a) k 3 l 3 2. Elementary Reation Proesses That Redue to Solvale Seond-Order Riati Equations In this setion, we will show how the ODEs for the elementary reation proesses an e solved y redution to the wellknown Riati equations of a new unit of variales. In PWN, in addition to the simple dimerization of monomers, there are polymerizations of higher omplexes. For those omplexes that are modified from relevant monomers, we will assume that the assoiation and dissoiationonstantsarethesameasthoseforthemonomer assoiations or dimer dissoiations, respetively. For simpliity, we will show the four asi forms of elementary reations as in the following. The assoiation dissoiation onstants are here denoted y k and l,respetively. (i) Assoiation of two different moleules/dissoiation of their produt: A+B k AB. (1) l (ii) Assoiation of the same moleules/dissoiation of their produt: A+A k AA. (2) l (iii) Assoiation with a modified moleule/dissoiation of their produt: A+AB k ABA or AAB. (3) l (iv) Assoiation with a homodimer/dissoiation of their produt: A+BB k ABB. (4) l X 5 () X 7 (a) +X 3 () X 8 () k 3 l 3 +X 5 () X 10 () 2k 3 2l 3 +X 7 (a) X 11 (a) k 3 l 3 +X 3 () X 9 (a) k 3 l 3 +X 5 () X 11 (a) k 3 l 3 +X 7 (a) X 12 (aa) 2k 3 2l 3 Note that AB + B k ABB is the same as reation (iii). l We now show that the ODEs for the MALs for reations (i) (iv) an e solved expliitly. First, the MALs for reation (i) are d [A] = k[a][b] +l[p], The organization of the paper is as follows. Setion 2 disusses four elementary reation proesses that are modeled y seond-order nonlinear ODEs that an e solved expliitly. In Setion 3, the ECM degradation mehanism is introdued, and we present one of the key onepts ehind the hoie of a good model for ODE kinetis, the douling rule in stoihiometry. In Setion 4, as an appliation of elementary reation proesses, we onsider solving the ODE system that arises from the PWN of moleule onentration dynamis in ECM degradation. We list the tales ited in Setion 4 and present the expliit solutions to the ODE system in Tales 1 7 and Appendix A, respetively. d [B] = k[a][b] +l[p], d [P] =k[a][b] l[p]. The resulting system of ODEs an e solved expliitly: (5) implies the MCLs d ([A] + [P]) = [A] + [P] I A, [B] + [P] I B, d ([B] + [P]) =0 (5) (6a)

3 Computational and Mathematial Methods in Mediine 3 Tale4:Massonservationlawsofa,,and. Items MCL d(x a 1 +X 4 +X 7 +X 9 +X 11 +2X 12 ) 0 0 d(x 2 +X 4 +X 5 +X 7 +X 8 +X 9 +2X 10 +2X 11 +2X 12 ) 0 0 d(x 3 +X 5 +2X 6 +X 7 +2X 8 +2X 9 +2X 10 +2X 11 +2X 12 ) 0 0 Tale 5: Reation laws of (a, ), (, ),and(, ). (a, ): X 1 +(X 2 + X 5 + X 8 +2X 10 + X 11 ) k 1 (X 4 + X 7 + X 9 + X 11 +2X 12 ) l 1 0 dx {(E 1 ) 1 = k 1 X 1 (X 2 + X 5 + X 8 +2X 10 + X 11 ) { d(x (E ) 2 + X 5 + X 8 +2X 10 + X 11 ) = k { 1 X 1 (X 2 + X 5 + X 8 +2X 10 + X 11 ). (, ): (X 2 + X 4 )+(X 3 +2X 6 + X 8 + X 9 ) k 2 (X 5 +X 7 +X 8 +X 9 +2X 10 +2X 11 +2X 12 ) {(E 24 ) {(E 3689 ) { l 2 d(x 2 + X 4 ) = k 2 (X 2 + X 4 )(X 3 +2X 6 + X 8 + X 9 )+l 2 (X 5 +X 7 +X 8 +X 9 +2X 10 +2X 11 +2X 12 ) d(x 3 +2X 6 + X 8 + X 9 ) = k 2 (X 2 + X 4 )(X 3 +2X 6 + X 8 + X 9 )+l 2 (X 5 +X 7 +X 8 +X 9 +2X 10 +2X 11 +2X 12 ). (, ): (X 3 + X 5 + X 7 )+(X 3 +X 5 +X 7 ) k 3 (2X 6 + X 8 + X 9 + X 8 +2X 10 + X 11 + X 9 +X 11 +2X 12 ) l 3 d(x (E 357 ) 3 +X 5 +X 7 ) = 2k 3 (X 3 +X 5 +X 7 ) 2 +2l 3 (X 6 +X 8 +X 9 +X 10 +X 11 +X 12 ) Variale X 1 Tale 6: Proesses of otaining solutions X 1 X 12. Method Single Riati equation ξ A ompanion of X 1 ξ 24 Single Riati equation ξ 3689 A ompanion of ξ 24 X 2 Method of variation of oeffiient X 4 X 4 =ξ 24 X 2 ξ 357 Single Riati equation X 3,X 5 Method of variation of oeffiient X 7 X 7 =ξ 357 (X 3 +X 5 ) X 6, X 8, X 10 Method of variation of oeffiient X 9, X 11, X 12 y (41) d ([A] [B]) =0 [A] [B] =I A I B (6) for some positive onstants I A and I B, whih are typially the initial values of [A] + [P] and [B] + [P], respetively.then, sustituting [A] =I A [P], [B] =I B [P] (7) Tale 7: Initial values and reation onstants used in the simulation. Variales Values a [M] [M] [M] k [M 1 s 1 ] l 1 0 [M 1 s 1 ] k [M 1 s 1 ] l [M 1 s 1 ] k [M 1 s 1 ] l [M 1 s 1 ] into the third equation in (5), we otain an equation in only the variale [P]: d [P] =k(i A [P])(I B [P]) l[p] =k[p] 2 (8) {k(i A +I B )+l}[p] +ki A I B k([p] z 1 )([P] z 2 ), where the last expression is the fatorization of the quadrati polynomial in the seond line. If z 1 =z 2,thisleadsto 1 1 d [P] ( ) = k(z [P] z 2 [P] z 1 1 z 2 ), (9)

4 4 Computational and Mathematial Methods in Mediine or ([P] z 1 )/([P] z 2 )=Ce k(z 1 z 2 )t, and hene the expliit solution [P] t = z 1 Cz 2 e k(z 1 z 2 )t 1 Ce k(z 1 z 2 )t for t 0.Here,z 1,z 2 are z 1,z 2 =z 2 + z 1 z 2 1 Ce k(z 1 z 2 )t, C= [P] 0 z 1 [P] 0 z 2 = {k(i A +I B )+l}± {k(i A +I B )+l} 2 4k 2 I A I B. 2k (10) (11) If z 1 and z 2 are distint and z 1 z 2 >0so that lim t [P] t = z 1 in (10), then we find that [P] t is monotonially dereasing and nonnegative. Solutions to [A] and [B] an e found in a similar manner y using (7). Although the situation is slightly different, a similar alulation was presented in ([1], Setion 6.5). Seond, for reation (ii), the MAL yields kineti equations of the form d [A] = 2k[A] 2 +2l[P], (12) d [P] =k[a] 2 l[p]. In the first equation of (12), the oeffiient of 2 stems from the fat that, in the reation A+A P, two moleules of A are onsumed; similarly, in the dissoiation, two moleules of A are produed. This is alled a doule rate of onsumption/reprodution. The ODEs (12) imply the MCL d ([A] +2[P]) =0 (13) [A] +2[P] I A for a positive onstant I A.Then,sustituting2[P] = I A [A] into the first equation of (12), we have d [A] = 2k[A] 2 l[a] +li A 2k([A] ζ 1 )([A] ζ 2 ), [A] t = ζ 1 Cζ 2 e 2k(ζ 1 ζ 2 )t 1 Ce 2k(ζ 1 ζ 2 )t for t 0.Hereζ 1,ζ 2 are =ζ 2 + ζ 1 ζ 2 1 Ce 2k(ζ 1 ζ 2 )t, C= [A] 0 ζ 1 [A] 0 ζ 2 (14) ζ 1,ζ 2 = l ± l2 +8klI A. (15) 4k Sine ζ 1 ζ 2 >0,wehavelim t [A] t =ζ 1.Asinase(i),the solution is monotonially dereasing and nonnegative. The solution to [P] an also e otained y the MCL (13). For reation (iii), we have the MALs and MCL as shown in the following: d [A] d [AB] d [P] d ([A] + [P]) d ([A] [AB]) = k[a][ab] +l[p], = k[a][ab] +l[p], =k[a][ab] l[p], (16) =0, (17a) =0. (17) TheODEs(16)anesolvedinamannersimilartothat used for reations (i) and (ii). In fat, the assoiation of A-BA redues to reation (i), and the assoiation of A-AB redues to reation (ii). Finally, for reation (iv), we have the following MALs and the MCL: d [A] d [BB] d [ABB] d ([A] [BB]) d ([A] + [ABB]) = 2k[A][BB] +l[abb], = 2k[A][BB] +l[abb], =2k[A][BB] l[abb], =0, =0. (17) (18) The oeffiient of 2 in (17) arises eause a moleule of A may ind to either of the two B moleules. The solution to these ODEs (17) isthesameasthatforreation(i)utwithk replaed y 2k. This is alled a doule hane of assoiation. These elementary reation proesses (i) (iv) will e used elow to solve the ODE system for the kinetis of ECM degradation. The given ODE system is grouped into several supathways, and in the reformulation ased on these groups, we will find that the new variales redue the equations as groups to those of the elementary reation proesses. 3. Appliation to the Model of ECM Degradation y MT1-MMP In this setion, we show how the elementary reation proesses in the previous setion an e applied to a prolem in ell iology, that of extraellular matrix (ECM) degradation y memrane type 1 matrix metalloproteinase (MT1-MMP), whih is a step in the progression of aner ECM Degradation Mehanism. As is well known, in order for the proliferated aner ells to metastasize from a primary lesion, they first invade and degrade the ECM and then

5 Computational and Mathematial Methods in Mediine 5 Pro-MMP2 MT1-MMP TIMP2 MMP2 (ative) TK P Y Y P TK PEX a Figure 1: Mehanism of ativation of a (MMP2) y (TIMP2) and (MT1-MMP). migrate. The aner ells serete MMP, and its dimer MMP2 is then ativated y MT1-MMP; the MMP2 then degrades the ECM. After the ECM is degraded, other enzymes, inluding MT1-MMP, egin degrading the interstitium eyond the ECM. Therefore, mathematial or iologial eluidation of the ativation proess of MMP2 is important for finding alinialureordevelopingdrugs.below,wewilldenote MMP2, TIMP2, and MT1-MMP y a,,and,respetively. Sato et al. [2] have revealed experimentally the mehanism of the ativation of MMP2. The steps of the ativation senario of MMP2, as a ell iologial model, proeed as follows (see Figure 1): (0) Although (MT1-MMP) is an ECM degradation enzyme as well, it is usually inativated y (TIMP2) immediately after vesile transportation to the ell memranes. However, ativates a (MMP2), whih is another ECM degradation enzyme, as follows. (1) The moleules that penetrate the ell memrane form homodimers on the memrane. (2) On one of the in the dimer --, the heterodimer a (pro-mmp2, TIMP2) is oupled to produe the tetramer a; alternatively,a- mayeoupledwith the heterotrimer - to produe the tetramer. Thus, a is assoiated with via. (3) One of the in a that is not oupled with a uts the onnetion etween a and. (4) a thatwasutandreleaseheneomestheativated one. This proess is thus the assoiations/dissoiations of the three moleules a,, and, and a is the key moleule in the ativation of a. In order to produe ertain amount of ativated a (MMP2), there must e suffiient. However,if there is too muh, then the remaining vaant site of atends to assoiate with -,whih results in thefat that a or aa will e produed preferentially to a, and an Binding of a (MMP2) Assoiating only with (TIMP2) y a single inding site Binding of (TIMP2) Assoiating with a (MMP2) or (MT1-MMP) y eah of the single inding sites, respetively Binding of (MT1-MMP) Assoiating with (TIMP2) or (MT1-MMP) y eah of the single inding sites, respetively Figure 2: Binding sites of a (MMP2), (TIMP2), and (MT1- MMP). insignifiant amount of a will e ativated. See Hoshino et al.[3]anhenwatanaeetal.[4]. The moleular inding rules that follow from this senario are that a and do not ouple with eah other, ut with, and may form the homodimer (see Figure 2). Although it is diffiult to deny the possiility of other inding patterns, to the est of our knowledge, there is no evidene of suh patterns in the literature. Therefore, we will assume the following inding patterns: (1) a hasasiteofindingonlywith. (2) has a site of inding with a and another site of inding with. (3) has a site of inding with and another site of inding with itself. (4) There are no other inding patterns. By inspetion, we see that there appear to e nine omplexes (dimers to hexamer) otainale from the monomers a a

6 6 Computational and Mathematial Methods in Mediine Hexamer Pentamer aa a The moleules a and do not dissoiate after they are one assoiated. Thus we set l 1 0. (II) Assoiation/dissoiation of B-C (assoiation onstant k 2 ;dissoiationonstantl 2 ): Tetramer Trimer Dimer a a a (-) B + C k 2 l 2 BC. (20) (III) Assoiation/dissoiation of C-C (assoiation onstant k 3 ;dissoiationonstantl 3 ): Monomer a (-) C + C k 3 l 3 CC. (21) k 1 k 2, l 2 k 3, l 3 Figure 3: Pathway network of assoiation/dissoiation pairs of omplexes of a (MMP2), (TIMP2), and (MT1-MMP). Here, arrows indiating only assoiation are depited. a,, and. The PWN of assoiation or dissoiation of these omplexes are shown in Figure 3. Thus, a quantitative model of ECM degradation an e ased on the kinetis of assoiations/dissoiations of the three proteins (a: MMP2; : TIMP2; and : MT1-MMP) and their nine omplexes, for a total of twelve ompounds. The PWNs of the twelve moleules are depited in Figure 3. A quantitative model was first onsidered y Karagiannis and Popel [5], in whih the general interative ehavior of the moleules a,, and was investigated through simulation in the ontext of type-i ollagen proteolysis. Further, their ehavior in more realisti situations was onsidered y Hoshino et al. [3] and then y Watanae et al. [4] as the mathematial interpretation of the ell iologial model in [2]. Inthequantitativemodelsin[3,4],thedetailedell iologial ehaviors of these enzymes (a,, and ) were studied. In partiular, they found that MT1-MMP has two fluoresene reovery time onstants, 29 [s] and 259 [s], and, in FRAP experiments, they determined that the reovery is not due to lateral diffusions ut to the insertion of a new memrane. As a result, they verified y simulation that the inativation of MT1-MMP y TIMP2 proeeds very quikly (halving time is 4.5 [s]); therefore, a very rapid turnover of MT1-MMP must e ourring at the loation of the ECM degradation. We onsider a mathematial treatment of the ODE kinetis with a more omplete theoretial analysis of suh quantitative studies; see (E1) (E12). Now, aording to the inding rules disussed aove, we have the following three assoiations or dissoiations in the PWN: (I) Assoiation of a-b (assoiation onstant k 1 ): (a-) a+b k 1 ab. (19) l 1 0 For reations (I) (III), there are twelve moleules in the PWN in Figure 3: a,,, a,,, a,, a,, a, and aa. We denote their onentrations as X 1 =[a], X 2 = [], X 3 =[],andsoon.thethreeassoiation/dissoiation groupsarepresentedintales1,2,and3.wemakethe following asi assumptions: (A1) For the assoiation/dissoiation of the modified moleules Aa and B, we use the same assoiation/dissoiation onstants as those used for the elementary reation a-.the sameisassumedforb- C and C-C. (A2) The initial onentrations for X 4,...,X 12 are all 0: X i (0) = 0, i=4,...,12. In order to otain the orret ODEs, it is neessary to extrat the appropriate douling rules for the reation oeffiients in the PWN. We will explain it in the next susetion Pathway Network Dynamis and Douling Rules. From Figure 3 and Tales 1 3, we an write down the following twelve seond-order nonlinear ODEs: dx 1 dx 2 dx 3 dx 4 dx 5 = k 1 X 1 (X 2 +X 5 +X 8 +2X 10 +X 11 ). = k 1 X 1 X 2 k 2 X 2 (X 3 +2X 6 +X 8 +X 9 ) +l 2 (X 5 +X 8 +2X 10 +X 11 ). = k 2 X 3 (X 2 +X 4 ) 2k 3 X 3 (X 3 +X 5 +X 7 ) +l 2 (X 5 +X 7 )+l 3 (2X 6 +X 8 +X 9 ). =k 1 X 1 X 2 k 2 X 4 (X 3 +2X 6 +X 8 +X 9 ) +l 2 (X 7 +X 9 +X 11 +2X 12 ). = k 1 X 1 X 5 +k 2 X 2 X 3 2k 3 X 5 (X 3 +X 5 +X 7 ) l 2 X 5 +l 3 (X 8 +2X 10 +X 11 ). (E1) (E2) (E3) (E4) (E5)

7 Computational and Mathematial Methods in Mediine 7 dx 6 =k 3 X 2 3 2k 2X 6 (X 2 +X 4 )+l 2 (X 8 +X 9 ) l 3 X 6. (E6) dx 7 =k 1 X 1 X 5 +k 2 X 3 X 4 2k 3 X 7 (X 3 +X 5 +X 7 ) l 2 X 7 +l 3 (X 9 +X 11 +2X 12 ). (E7) dx 8 = k 1 X 1 X 8 +2k 2 X 2 X 6 k 2 X 8 (X 2 +X 4 ) +2k 3 X 3 X 5 +l 2 ( X 8 +2X 10 +X 11 ) (E8) Doule hane of assoiation at a time Doule hane of dissoiation at a time Figure 4: Douling rules in the assoiation/dissoiation of - and -C. l 3 X 8. dx 9 dx 10 dx 11 dx 12 =k 1 X 1 X 8 k 2 X 9 (X 2 +X 4 )+2k 2 X 4 X 6 +2k 3 X 3 X 7 +l 2 ( X 9 +X 11 +2X 12 ) l 3 X 9. = 2k 1 X 1 X 10 +k 2 X 2 X 8 +k 3 X 2 5 2l 2X 10 l 3 X 10. =k 1 X 1 (2X 10 X 11 )+k 2 X 2 X 9 +k 2 X 4 X 8 +2k 3 X 5 X 7 2l 2 X 11 l 3 X 11. =k 1 X 1 X 11 +k 2 X 4 X 9 +k 3 X 2 7 2l 2X 12 l 3 X 12. (E9) (E10) (E11) (E12) Note that ODEs (E1) (E12) inorporate douling rules. The oeffiient of 2 on (1) k 3 X 2 3 in (E3), (2) k 3 X 2 5 in (E5), (3) k 3 X 2 7 in (E7) orresponds to the doule rate of onsumption/reprodution in reation (ii). See Figure 4. The oeffiient of 2 on (1) k 3 X 3 (X 5 +X 7 ) in (E3), (2) k 3 X 5 (X 3 +X 7 ) in (E5), (3) k 3 X 7 (X 3 +X 5 ) in (E7), (4) k 3 X 3 X 5 in (E8), (5) k 3 X 3 X 7 in (E9), (6) k 3 X 5 X 7 in (E11), whih orresponds to C-C assoiation or dissoiation reations in Tale 3, will e explained in Setion 4.3. The remaining oeffiients of 2 in the ODEs are due to the doule hane of assoiation/dissoiation in reation (iv), as disussed in Setion 2. What is the rationale for the aove douling rules? It is the ingeniously well-formed MCLs and reation laws, whih areshowninthenextsetion. 4. Analyzing the PWN Dynamis 4.1. Kinetis of Those Reations Containing a-b. Aording to Tale 1, the reations related to a-b are summarized as X 1 +(X 2 + X 5 + X 8 +2X 10 + X 11 ) k 1 l 1 0 (X 4 + X 7 + X 9 + X 11 +2X 12 ). (22) Looking at X 1 and the produts (the right hand side) of this reation, we infer from the flux alane of onsumptionprodution that we may have d(x 1 + X 4 + X 7 + X 9 + X 11 +2X 12 ) 0. (23) By taking summation (E1) + (E4) + (E7) + (E9) + (E11) +2 (E12) of the ODEs, we find that (23) is indeed true. Here, X 12 (aa) has a oeffiient of 2 eause the derease in X 1 is equal to the inrease in (X 4 + X 7 + X 9 + X 11 +2X 12 ),sine the omplex aa onsumes two a moleules, unlike the others: a, a, a, a.equation(23)isanmclofform (6a) or (17a). Equation (23) may e onsidered as an MCL for a 0,sine (X 1 + X 4 + X 7 + X 9 + X 11 +2X 12 ) (t) onst. X 1 (0) =a 0 (24)

8 8 Computational and Mathematial Methods in Mediine y assumption (A2); thus, a loal alane holds: a 0 X 1 (t) (X 4 + X 7 + X 9 + X 11 +2X 12 )(t) for all t 0. From (22), we also have d( X 1 + X 2 + X 5 + X 8 +2X 10 + X 11 ) 0, (25) whih may e onsidered to represent an MCL for a 0 0, sine ( X 1 + X 2 + X 5 + X 8 +2X 10 + X 11 ) (t) onst. X 1 (0) +X 2 (0) = a (26) Here, X 10 has a oeffiient of 2 eause the omplex has two - sites forindingwitha, whih implies a doule hane of assoiation, as in reation (iv). Equation (26) is an MCL of form (6) or (17). Note that (E1) and (25) suggest that X 1 and (X 2 +X 5 +X 8 + 2X 10 + X 11 ) form a pair of a-b kinetis, (E 1 ) and (E ), as displayed in Tale 5. The two ODEs an e solved expliitly using the method for reation (i) in Setion 2; details are shown in Appendix A Kinetis of Those Reations Containing B-C. Next, we onsider the kinetis of assoiations/dissoiations with B-C involved in the ODE system. As an e seen in Tale 2, the reations related to B-C are summarized as (X 2 + X 4 )+(X 3 +2X 6 + X 8 + X 9 ) k 2 l 2 (X 5 + X 8 +2X 10 + X 11 )+(X 7 + X 9 + X 11 +2X 12 ). (27) In the manner similar to that used for the Aa-B kinetis, we infer, from the flux alane of onsumption-prodution, that we may have d(x 2 + X 5 + X 8 +2X 10 + X 11 + X 4 + X 7 + X 9 + X 11 +2X 12 ) =0; that is, d(x 2 +X 4 +X 5 +X 7 +X 8 +X 9 +2X 10 +2X 11 +2X 12 ) =0. (28) (29) By taking summation (E1) + (E4) + (E7) + (E9) + (E11) +2 (E12) of the ODEs, we find that (28) is indeed valid. Equation (28) is an MCL of form (6a) or (17a). Equstion (29) may e viewed as an MCL of 0,sine From (27), we also have d( X 2 X 4 + X 3 +2X 6 + X 8 + X 9 ) =0, (31) whih is onfirmed y taking the summation (E2) (E4) + (E3) + 2 (E6) + (E8) + (E9) oftheodes.thisisanmclof form (6) or (17). Equation (31) represents an MCL of 0 0, sine ( X 2 X 4 + X 3 +2X 6 + X 8 + X 9 ) (t) onst. X 2 (0) +X 3 (0) = (32) Here, X 6 has a oeffiient of 2 eause it has two - sitesfor inding with a or a. The MCL (31) implies that ξ 24 (t) (X 2 + X 4 )(t) and ξ 3689 (t) (X 3 +2X 6 + X 8 + X 9 )(t) are governed y thesameode,uptotheinitialvalues.fromtheodes (E1) (E12), it follows that the two variales form a pair of B-C kinetis, (E 24 ) and (E 3689 ), as in Tale 4. The variales ξ 24 (t) and ξ 3689 (t) an e solved expliitly; details are shown in Appendix A.2. Now, we find that eah of X 2 (t) and X 4 (t) an e solved expliitly, as well. This is eause the right-hand side of (E2) ontains only known terms, exept for X 2 : dx 2 = [k 1 X 1 (t) +k 2 ξ 24 (t) +l 2 k 2 ( 0 0 )] X 2 (t) +l 2 (X 1 (t) a ), (E2) y (26) and (32). Hene X 2 (t) anealulatedythemethod of variation of oeffiients. See Appendix A.3. In this way, X 4 (t) aneotainedyx 4 (t) = ξ 24 (t) X 2 (t). 4.3.KinetisofThoseReationsContainingC-C. We finally onsider the kinetis of assoiations/dissoiations of C- C involved in the ODE system. As shown in Tale 2, the reations related to C-C an e summarized as (X 3 + X 5 + X 7 )+(X 3 +X 5 +X 7 ) k 3 l 3 (2X 6 +X 8 +X 9 +X 8 +2X 10 +X 11 +X 9 +X 11 +2X 12 ). (33) (X 2 +X 4 +X 5 +X 7 +X 8 +X 9 +2X 10 +2X 11 +2X 12 ) (t) onst. X 2 (0) = 0. (30) In a manner similar to that used previously, we infer from thefluxalaneofonsumption-produtionthat d(x 3 +2X 6 +X 8 +X 9 +X 5 +X 8 +2X 10 +X 11 +X 7 +X 9 +X 11 +2X 12 ) =0; (34)

9 Computational and Mathematial Methods in Mediine 9 that is, d(x 3 +X 5 +2X 6 +X 7 +2X 8 +2X 9 +2X 10 +2X 11 +2X 12 ) =0. (35) By taking the summation (E3) + (E5) +2 (E6) + (E7) +2 (E8) +2 (E9) +2 (E10) +2 (E11) +2 (E12) of the ODEs, we find that (35) is indeed valid. Equation (34) is an MCLofform(6a)or(17a).Equation(35)mayeviewedas an MCL of 0,sine (X 3 +X 5 +2X 6 +X 7 +2X 8 +2X 9 +2X 10 +2X 11 +2X 12 ) (t) onst. X 3 (0) = 0. (36) From (E3), (E5), and(e7), wehavetheode(e 357 ),whih is displayed in Tale 5, and in whih we have used (30) and (32). The oeffiient of 2 on k 3 X 3 (X 3 +X 5 +X 7 ) 2 in (E 357 ) is aresultofthesummationoftheodes(e3), (E5), and(e7). But then, where do the oeffiients of 2 on k 3 (X 3 +X 5 +X 7 ) in (E3), onk 3 X 5 (X 3 +X 5 +X 7 ) in (E5), andonk 3 X 7 (X 3 + X 5 +X 7 ) in (E7) ome from? They stem from the following stoihiometry. For those reations in Tale 3 that orrespond to the reation (ii) in Setion 2, X 3 () +X 3 () X 6 (), X 5 () +X 5 () X 10 (), X 7 (a) +X 7 (a) X 12 (aa), (37) the reason is the same as in (12); that is, it is aused y the doule rate of onsumption/reprodution. Hene, we must ount these three reations with the assoiation/dissoiation oeffiients on 2k 3 and 2l 3 in (E3), (E5), and(e7). Forthe reation X 3 ()+X 5 () X 8 () orresponding to reation (iii), oth X 3 () +X 5 () X 8 (), X 5 () +X 3 () X 8 () (38) must e ounted. The first reation in (38) is the one ontained in (E3), and the seond reation is in(e5). In the first reation, the sujet is X 3 (), anditassoiateswith the ojet X 5 (). Wewillnamethismoleule = 1.In theseondreation,thesujet is X 5 (), anditassoiates with the ojet X 3 (). Wewillnamethismoleule = 2. In other words, a moleule of (=ojet) isassoiatedwith amoleuleof (=sujet). In general, = 1 and 2 are different moleules. The first reation must e ounted as a reation of X 3 (). Also, seen as a reation of X 5 (), the seond reation must also e ounted. Both reations take plae at the same time. The assoiation/dissoiation oeffiients are k 3 and l 3, respetively, for oth reations. However, now onsider, for example, how X 3 () is onsumed douly y the amount of 1 and 2. In Tale 3, other ominations of this kind are X 5 () +X 7 (a) X 11 (a), X 7 (a) +X 5 () X 11 (a), X 3 () +X 7 (a) X 9 (a), X 7 (a) +X 3 () X 9 (a). (39) The ontriution of the reations in (38) and (39) are therefore douled. Comining the douling of the reations in (38), we otain the third reation in Tale 5, whih auses the term 2k 3 (X 3 +X 5 +X 7 ) 2 in E 357 and the ODEs (E3), (E5),and(E7). Finally, in onsidering X 3 (), the dissoiation term l 3 (X 8 +X 9 ) need not e douled, sine only one moleule of is produed from a moleule of : -; also,this dissoiation is not involved in (E5). By the aove arguments for the douling rules, we have otained ingeniously well-formed MCLs and reation laws, as shown in Tale 4, respetively. This may suggest that our douling rules are justified. Also, the aove argument of C-C kinetis may imply the following: suppose that, for assoiation of moleules, p 1,p 2,...,p n, say, some ominations of oupling p i -p j (1 i, j n), are possile and others not. Then, it does not hold that d(x 1 + +X n ) = 2k(X 1 + +X n ) 2 + dissoiation term (40) in general. However, if all of p i has a ommon inding ite, that is, all p i are of the form -C, then we do have (40). In the latter ase, the douling rule holds for (X 1 + +X n ) as a unit, asif(p 1,p 2,...,p n ) are a single moleule with inding ite -, and the elementary douling rule (12) holds with [A] in (12) replaed y (X 1 + +X n ) Solutions to Remaining Variales. Solutions to the remaining variales an also e otained. X 3 and X 5 an e found y the method of variation of parameters, as in the ase of X 2. X 7 an e found y X 7 (t) = ξ 357 (t) (X 3 +X 5 )(t),where ξ 357 (t) is the solution to (X 3 +X 5 +X 7 )(t). By applying the solution ξ 357 to (E3),weotainthesolutiontoX 3 (t) as dx 3 = [(k 2 l 3 )ξ 24 +(2k 3 l 2 )ξ 357 +l 2 +l 3 ]X 3 (t) +l 2 ξ 357 +l 3 ξ 24 l 3 ( 0 0 ) (E3) y (30) and (32); note that ξ 24 and ξ 357 have een otained already. Similarly, X 6, X 8,andX 10 analsoefoundythe method of variation of oeffiient.

10 10 Computational and Mathematial Methods in Mediine Supply Feedak ξ 689 X 3 Feedforward ξ ξ 3689 X 1 a k 1 X 7 ξ Prodution only Feedforward X 7 + ξ 357 Prodution ξ 689 ξ X 4 + ξ24 k 2 ξ ξ Prodution only ξ 357 k 3 ξ Prodution only ξ X 2 Feedforward ξ X 5 Prodution Prodution X 5 ξ81011 ξ Supply Feedak Figure 5: Looking at PWN with unit variales. All of X 1 to X 8 and X 10 have thus een otained in priniple. The remaining ones, X 9, X 11,andX 12,ane otained y solving X 9 (t) =ξ 3689 (t) (X 3 +2X 6 +X 8 ) (t), X 11 (t) =X 1 (t) (X 2 +X 5 +X 8 +2X 10 ) (t) a 0 + 0, X 12 (t) = 1 2 [a 0 (X 1 +X 4 +X 7 +X 9 +X 11 ) (t)] (41) (see (A.8), (26), and (24), resp.). The proesses for otaining thesesolutionsarelistedintale Biomedial Impliation. As desried in Setion 3.1, an appropriate amount of X 2 () (TIMP2) is neessary to otain a sustantial amount of the key moleule X 9 (a). Therefore, development and mediation of suh drugs that inhiit TIMP2 outside the ell memrane may e onsidered to e effetive. Similarly, drugs that inhiit MT1-MMP inside the ell memrane might e onsidered to e useful as well. In addition, upon eing larified the model ehaviors we notie the most important pathways in the PWN. They are three parts produing ξ (assoiation parts just efore the threelueoxesinfigure5).thepwnmayesaioe suh a devie that ontinues to produe ξ 91112, as explained in the following. In fat, the key moleule X 9 (a) is thus produed, through ξ 91112, ontinuously as far as possile. The initial onentration a 0 is dispersed in the network, to X 4 (a), X 7 (a), X 9 (a), X 11 (a),andx 12 (aa), aording to the first MCL in Tale 4. Likewise, 0 and 0 are dispersed in the network, aording to latter two MCLs in Tale 4, respetively. The units of variales orrespond to oth the MCLs in Tale 4 and reations in Tale 5. For example, the first MCL in Tale 4 means that the onsumption of X 1 and prodution of X 4 (a)+x 7 (a)+x 9 (a)+x 11 (a)+2x 12 are of the same rate in the reation (a, ) in Tale 5. Thus, the initial mass ontinues to exist in the PWN with its omponents hanging to dimer, trimer, tetramer,...: at every moment, suh dimer, trimer,..., and hexamer ontained in the unit variales oexist. This is true for all the initial masses a 0, 0,and 0.Bythis,inthereations y unit variales, (1) X 1 versus ξ = X 2 + X 5 + X 8 +2X 10 + X 11, (2) ξ 24 = X 2 + X 4 versus ξ 3689 = X 3 +2X 6 + X 8 + X 9, (3) ξ 357 = X 3 + X 5 + X 7 versus ξ 357 = X 3 + X 5 + X 7, all possile intermediates of different level oligomers, towards X 9, are produed simultaneously at every moment. That is, thepwnsystemissettledsoastoontinuetoproduex 9 as muh as possile at every moment.

11 Computational and Mathematial Methods in Mediine t t 0.4 X 1 (t), numerial X 1 (t), theoretial Figure 6: X 1 : simulation result and theoretial solution. X 1 (t) (X 4 +X 7 +X 9 +X 11 +2X 12 )(t) Sum of the two Figure 8: ξ : simulation result and theoretial solution and MCL (24) a 0 X 1 (t) (X 2 +X 5 +X 8 +2X 10 +X 11 )(t) Sum of the two Figure 7: ξ : simulation result and theoretial solution and MCL (26). t X 2 +X 4 (t), numerial X 2 +X 4 (t), theoretial Figure 9: ξ 24 : simulation result and theoretial solution. t In addition, the node that produes X 9 (a) is not single. There are three suh nodes in the PWN, eah in reation of (a, ), (, ),and(, ).AsinFigure5,X 9 is produed through the prodution of ξ =X 9 +X 11 +2X 12.Here,thisξ is only produed and no more reused y feedak or additional reations. Therefore, we may say that the PWN is suh a system that produes X 9 (through ξ )inthetripleway(three prodution nodes) and ontinues the prodution as muh as possile. This kind of onsideration has eome possile eause the PWN model is analyzed through the unit variales On the Simulation Results. We present the time ourse plots of the theoretial and numerial solutions in Figures 6 12.Thetheoretialsolutionsaoveareingoodagreement with our simulation results. In Figures 7, 8, 10, and 12, the MCLs (26), (24), (32), and (34), respetively, are onfirmed; the solutions are also onfirmed. The values of reation onstants and initial values used in the simulations are listed in Tale 7. The reation onstants are the same as those used in Watanae et al. [4], while the initial values are set as follows: the initial values used in [4] are a 0 = 0 = 0 = [M] ased on experimental measurement; if the initial values are adopted as they stand, then the simulation auses a time-delay; that is, the responses eome muh slower. Therefore, for the sake of onveniene in simulation, we take the initial values with the order of In addition, in order to show the relationship of the ehaviors of solutions and initial values in a more effetive way, we set as a 0 = [M]. For example, in Figure 6, we an see that X 1 ( ) = a 0 0 indeed (see (A.1)). Also, in Figure 8, the MCL an e seen visually. It should e noted that the simulation results otained in Watanae et al. [4] and hene in this paper are ased on

12 12 Computational and Mathematial Methods in Mediine t t 0 (X 2 +X 4 )(t) (X 3 +X 6 +2X 8 +X 9 )(t) Sum of the two Figure 10: ξ 3689 : simulation result and theoretial solution and MCL (32). (X 3 +X 6 +2X 8 +X 9 )(t) (X 7 +X 9 +X 11 +2X 12 )(t) (X 5 +X 8 +2X 10 +X 11 )(t) Sum of the three Figure 12: ξ ξ ξ : simulation result and theoretial solution and MCL (36) X 3 +X 5 +X 7 (t), numerial X 3 +X 5 +X 7 (t), theoretial Figure 11: ξ 357 : simulation result and theoretial solution. t X 9 (t) Model without douling rule Model with douling rule t Figure13:TimeourseofX 9 (t): omparison of the models with or without the douling rule. atual aner data. The study in this paper provides the exat solutions to the ODE model and hene a mathematially rigorousfoundationisaddeothesimulationresults. Conerningthequestionthatifthereisathresholdofthe key moleule onentration that leads to ECM degradation, we do not know the exat threshold at least presently. It may e onsidered that, aording to the amount of the key moleule, ECM degradation is promoted weakly or strongly. However, as for the unit variale ξ 3689, it is indiated in Watanae et al. [4] that, at around 0 = 100 [nm], transient peak of ξ 3689 eomes maximal and the ECM degradation is promoted therey the most. See Figures S3, S4, and S5 in [4]. Figure 13 presents the time ourses of a simulation result of the key moleule X 9 forthemodelwithandwithoutthe douling rule. Here, the initial values of the onentration are a 0 = [M], 0 = [M], and 0 = [M]. By modifying the model with the douling rule, the peak at aout t = 0.25 was inreased y approximately 1.5 times. Atually, the authors did not egin with the idea of the douling rules in Setion 3.2. The only douling rule we had exploited was the one orresponding to the general asi model in (12). Thus the only oeffiients of 2 were on k 3 X 2 3 and l 3 X 6 in (E3), k 3 X 2 5 and l 3X 10 in (E5), k 3 X 2 7 and l 3X 12 in (E7), and l 2 X 11 in (E11). The solutions to this first version of the model and the modified version in (E1) (E12) are different, ut they have the same initial and asymptoti ehaviors. In Figure13,weshowasanexamplethetimeseriesofthekey moleule X 9 (t). Figure 14 shows a plot of X 9 ( ) versus X 2 (0). As desried in Setion 3.1, the key moleule of the ECM degradation is a (onentration X 9 ), and the amount of a that is produed is influened primarily y 0.Ifthereis

13 Computational and Mathematial Methods in Mediine 13 X 9 ( ) log 10 X 2 (0) Model without douling rule Model with douling rule Figure 14: X 9 ( ) versus X 2 (0): omparison of the models with or without the douling rule. too muh or too little 0,aninsignifiantamountofa will e produed. As shown in the figure, in oth models, X 9 ( ) has a peak at around 0 = [M]. The peak inreases when the model was modified. 5. Conluding Remarks We presented a method for solving a nonlinear ODE system from ell iology. With the setting in this paper, the ODEs eome a ompletely integrale system so that they an e solved expliitly. The key idea was to use the MCL to otain linear relations of the ODE variales that would e valid for all t 0. The previous analysis of suh ODE systems was primarily ased on the alane of flux at equilirium (t = ) or y omputer simulation (for t < ). The theoretial solutions to the ODEs are in omplete agreement with those otained y simulation. We gained several new insights in analyzing and modeling the PWNs: in the MALs, a oeffiient of 2 must e attahed sometimes to appropriate moleules. Determining all of the douling laws is ruial for otaining an integrale systemofodes.wehaveeluidatedsomesituationsinwhih the douling law is required. We also note that the total PWN an e grouped into several loal PWNs in whih loal linear relationships hold. This may suggest a way to otain an appropriate view of the omposition of network motifs in systems iology. In a future study, we will try to determine a suffiient ondition suh that the given ODE system is ompletely integrale. It may e desirale to larify under whih onditions the ODE system framework is suffiiently lose to a realisti PDEsystem;thiswouldeusefulforthedevelopmentof therapies or drugs. Appendix A. Expliit Solutions to the Unit Variales The solutions to X 1 X 12 and related materials are presented as follows. A.1. Solution to X 1 : Single Riati Equation. Theorem A.1. One has the following solution to (E 1 ): a 0 (a 0 0 ) a 0 0 e, k 1(a 0 0 )t (a 0 > 0 ), { 0 X 1 (t) = 1+k 1 0 t, (a 0 = 0 ), a { 0 ( 0 a 0 )e k 1( 0 a 0 )t, { 0 a 0 e k 1( 0 a 0 )t (a 0 < 0 ). Corollary A.2. X 1 (t) is dereasing monotonially: X 1 (0) =a 0 X 1 ( ) = { a 0 0, (a 0 > 0 ), { 0, (a { 0 0 ). (A.1) (A.2) Thus X 1 (t) is nonnegative as well. Let us denote ξ (t) (X 2 + X 5 + X 8 +2X 10 + X 11 )(t). Corollary A.3. Onehasthefollowingsolutionto(E ): 0 (a 0 0 )e k 1(a 0 0 )t, a 0 0 e k 1(a 0 0 )t (a 0 > 0 ), ξ (t) = { 0 1+k 1 0 t, (a 0 = 0 ), { 0 ( 0 a 0 ) { 0 a 0 e, (a k 1( 0 a 0 )t 0 < 0 ). Thus ξ (t) is dereasing monotonially: and is nonnegative. ξ (0) = 0 ξ ( ) = { 0, (a 0 0 ), { { 0 a 0, (a 0 < 0 ) A.2. Solution to X 2 +X 4 : Single Riati Equation Theorem A.4. Onehasthefollowingsolutionto(E 24 ): ξ 24 (t) (X 2 + X 4 ) (t) = z 1 C 24 z 2 e k 2(z 1 z 2 )t, 1 C 24 e k 2(z 1 z 2 )t where C 24 = 0 z 1 0 z 2, (A.3) (A.4) (A.5) z 1,z 2 = {k 2 ( 0 0 ) l 2 }± {k 2 ( 0 0 ) l 2 } 2 (A.6) +4k 2 l k 2

14 14 Computational and Mathematial Methods in Mediine We note that 0 0 >z 1 >0>z 2.Sine(X 2 + X 4 )(t) = z 2 +(z 1 z 2 )/(1 C 24 e k 2(z 1 z 2 )t ),wehavethefollowing. Corollary A.5. ξ 24 (t) is dereasing monotonially: ξ 24 (0) = 0 ξ 24 ( ) =z 1. Thus ξ 24 (t) is nonnegative as well. Corollary A.6. Onehasthefollowingsolutionto(E 3689 ): (A.7) Thus ξ 3689 (t) is dereasing monotonially: and is nonnegative. ξ 3689 (0) = 0 ξ 3689 ( ) =z (A.9) The solutions ξ 24 (t) and ξ 3689 (t) are thus otained irrespetive of 0 > 0 or 0 < 0. Eah of the solutions to X 2 (t) and X 4 (t), however, does depend on a 0 > 0, a 0 = 0 or a 0 < 0 as seen in the following. ξ 3689 (t) (X 3 +2X 6 + X 8 + X 9 ) (t) = z 1 C 24 z 2 e k 2(z 1 z 2 )t 1 C 24 e k 2(z 1 z 2 )t 0 + 0, t 0. (A.8) A.3. Solution to X 2 and Thus X 4 :MethodofVariationof Constant. Let E 1 = e k 1( 0 a )t 0, E 2 = e k 2(z 1 z )t 2,andE 3 = e (k 2z 1 +l )t 2. Theorem A.7. One has X 2 (t) { = C 1,1 (a 0 0 )E 1 E 3 (1 C 1,1 E 1 )(1 C 24 E 2 ) [(1 C 24)+l 2 { 1 E 3 C 24 (1 E 2 E 3 ) }], (a k 2 ζ 1 +l 2 k 2 (z 1 z 2 +ζ 1 )+l 0 > 0 ), 2 0 (1 + k 1 0 t) (1 C 24 E 2 ) [(1 C 24)E 3 +l 2 C 24 {(1 E 3 1+k 1 0 t )κ 2 1 E 2 [k 2 (z 1 z 2 ζ 1 ) l 2 ] 1 E 1 2 (1 + k 1 0 t) [k 2 (z 1 z 2 ζ 1 ) l 2 ] 2 }], (a 0 = 0 ), { 0 a 0 {(1 C 1,3 E 1 )(1 C 24 E 2 ) [(1 C 24)E 3 +l 2 ( 1 E 3 + E 3 E 2 )], (a k 2 z 1 +l 2 k 2 z 2 +l 0 < 0 ), 2 (A.10) respetively. Here κ 2 >0is given y κ 2 =η(k 2 ζ 1 +l 2 )+η(k 2 (z 1 z 2 ζ 1 ) l 2 ), with η (x) =x 1 +k 1 0 x 2. (A.11) The solutions turn out to e nonnegative in either of the three ases. Now X 4 is given y X 4 (t) = ξ 24 (t) X 2 (t), although we do not write down the solution here eause it is ompliated like X 2. A.4. Solution to X 3 +X 5 +X 7 : Single Riati Equation. The solution to (E 357 ) is otained, similarly as ξ 24. ξ 357 (t) (X 3 +X 5 +X 7 ) (t) where = η 1 η 2 C 357 e 2k 3(η 1 η 2 )t, C 1 C 357 e 2k 3(η 1 η 2 )t 357 = 0 η 1, 0 η 2 (A.12) η 1,η 2 = l 3 ± l3 2 +8k 3l 3 0, (A.13) 4k 3 respetively. Properties of positivity and monotone dereasing of ξ 357 are valid as well. The solutions to other variales, X 3, X 5, X 7, X 6, X 8, X 10, X 9, X 11,andX 12, are too ompliated to write down and omitted. Conflits of Interest The authors delare that they have no onflits of interest. Aknowledgments The first and the last authors are supported y JSPS KAK- ENHI Grant-in-Aid for Exploratory Researh, no The last author is supported also y JST-CREST and KAK- ENHI Grant-in-Aid for Sientifi Researh on Innovative Areas,no.16H06576.Theauthorswouldliketoexpresstheir appreiation. Referenes [1] P. Atkins and J. de Paula, Physial Chemistry for the Life Sienes, Oxford University Press, 2nd edition, [2] H.Sato,T.Takino,Y.Okadaetal., Amatrixmetalloproteinase expressed on the surfae of invasive tumour ells, Nature, vol. 370, no. 6484, pp , [3] D. Hoshino, N. Koshikawa, T. Suzuki et al., Estalishment and validation of omputational model for MT1-MMP dependent ECM degradation and intervention strategies, PLoS Computational Biology,vol.8,no.4,ArtileIDe ,2012. [4] A.Watanae,D.Hosino,N.Koshikawa,M.Seiki,T.Suzuki,and K. Ihikawa, Critial role of transient ativity of MT1-MMP

15 Computational and Mathematial Methods in Mediine 15 for ECM degradation in invadopodia, PLoS Computational Biology,vol.9,no.5,ArtileIDe ,2013. [5] E. D. Karagiannis and A. S. Popel, A theoretial model of type I ollagen proteolysis y matrix metalloproteinase (MMP) 2 and memrane type 1 MMP in the presene of tissue inhiitor of metalloproteinase 2, The Journal of Biologial Chemistry, vol. 279,no.37,pp ,2004.

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