MATHEMATICAL ANALYSIS OF A MODULAR NETWORK COORDINATING THE CELL CYCLE AND APOPTOSIS. Gheorghe Craciun. Baltazar Aguda.

Transcription

1 MATHEMATICAL BIOSCIENCES AND ENGINEERING Volume, Number, Xxxx XXXX pp. MATHEMATICAL ANALYSIS OF A MODULAR NETWORK COORDINATING THE CELL CYCLE AND APOPTOSIS Gheorghe Crciun Mthemticl Biosciences Institute, Ohio Stte University 25 W. 8th Avenue, Columbus, OH 432 Bltzr Agud Systems Biology Group, Bioinformtics Institute 3 Biopolis Street, #7- Mtrix, Singpore 3867 Avner Friedmn Mthemticl Biosciences Institute, Ohio Stte University 25 W. 8th Avenue, Columbus, OH 432 Abstrct. The cell-division cycle nd poptosis re key cellulr processes deregulted during crcinogenesis. Recent work of Agud nd Algr suggests modulr orgniztion of regultory moleculr pthwys linking the cellulr processes of division nd poptosis. We crry out detiled mthemticl nlysis of the Agud-Algr model to unrvel the dynmics implicit in the model structure. In ddition, we further explore model prmeters tht control the bifurctions corresponding to the forementioned cellulr stte trnsitions. We show tht this simple model predicts interesting behvior, such s hysteretic oscilltions nd different conditions in which poptosis is triggered.. Introduction. It is now widely ccepted mong cncer reserchers tht the cell-division cycle nd the cell-deth progrm (clled poptosis) re key cellulr processes deregulted during crcinogenesis. Enhnced cell prolifertion cn be due to incresed cell cycling, inhibited poptosis, or both. Recently, Agud nd Algr [] reviewed nd nlyzed the regultory moleculr pthwys linking the initition of the cell cycle nd poptosis. These uthors suggested modulr orgniztion of the complex networks nd presented corresponding kinetic model for the cellulr stte trnsitions from quiescence (nondividing) to cell cycling, nd eventully to poptosis s the strength of extrcellulr signling increses. This kinetic model, however, ws not nlyzed in detil in the originl pper. Here, we extend the mthemticl nlysis to show tht this simple model predicts other interesting behvior, such s hysteretic oscilltions nd different conditions in which poptosis is triggered. The modulr structure of the Agud-Algr model is shown in Figure. The lines tht end with indicte tht one module inhibits the other, nd the lines tht end with indicte tht one module ctivtes the other. The modules involved correspond to intrcellulr signling, cell cycle, poptosis, nd control node of trnscription fctors tht stimulte the expression of cell cycle nd poptosis genes. 2 Mthemtics Subject Clssifiction. 92C37. Key words nd phrses. cellulr processes models, moleculr pthwys, cell cycle, poptosis.

2 2 G. CRACIUN, B. AGUDA AND A. FRIEDMAN The key trnscription fctors re suggested to be the proteins E2F- nd c-myc. The primry components of the cell-cycle module re the enzymes clled cyclindependent kinses, nd the key plyers in the poptosis module re the cspses. An exmple of signling pthwy tht impinges on the ctivtion of E2F- is the Rs/MAPK pthwy which is cscde of ctivtion involving protein kinses [5]. This signling pthwy hs been shown in [5] to possess n ultrsensitive switch-like behvior, s depicted in the topmost box of Figure. Figure. Modulr structure of the model. Agud nd Algr [] reviewed the literture tht points to the proteins E2F- nd c-myc s the key trnscription fctors in this control node. Figure 3 in ppendix B gives summry of the importnt pthwys implicting E2F- nd c- Myc, nd offers detils nd justifiction of the modulr structure shown in Figure (Fig. 3 is Fig. 3 of []). The interction lbeled in Figure represents the fct tht ctivted cyclindependent kinses (e.g., CDK2) needed for entry into the cell cycle would further enhnce the trnscriptionl ctivity of E2F- vi phosphoryltion of the retinoblstom (Rb) protein. The interction lbeled b in Figure represents the suggested negtive feedbck between the Rb/E2F pthwy nd the Rs signling pthwy ([6]). The interction lbeled b involves the negtive regultion of cyclin A/CDK2 towrd E2F- through the phosphoryltion of the ltter s prtner, the DP proteins, which re required for binding DNA. Finlly, the interction lbeled c includes the cell-cycle-enhnced poptosis vi the indirect regultion of cspses by cyclin-dependent kinses (e.g., ctivtion of cspse by D-type cyclin, s reported by Mendelsohn et l. [7]). More detils nd justifiction of these modulemodule interctions re discussed in []. 2. Description of the model. A kinetic implementtion of the modulr model in Figure is shown in Figure 2. The mening of the nottions used in Figure 2 is

3 MODULAR NETWORK COORDINATING CELL CYCLE AND APOPTOSIS 3 Figure 2. Mechnistic implementtion of the model. The solid rrows indicte input into or output from node; dotted rrows indicte influence of node on process, without loss of mss. s follows: = signling molecule S 2 = ctive signling protein G 2 = control node of trnscription fctors = ctive cell cycle mrker (for initition of S phse) S 2 = ctive poptosis mrker We will use the sme species symbols for their corresponding concentrtions. Then, ccording to either mss-ction or Michelis-Menten kinetics, we get the following system of differentil equtions: d dt ds 2 dt dg 2 dt d dt da 2 dt = ε (k s k sd k sd2 ), () = ( ε 2 k S v ) ms 2, K M + S K Mr + S 2 (2) = ε 3 (k 2 S 2 + k 2 k m2 G 2 k m2 G 2 ), (3) ( = ε k3 G 2 C v ) m3, (4) K M3 + C K Mr3 + = ε 2 ( k4 G 2 A K M4 + A + k 4 A K M4 + A v m4a 2 K Mr4 + A 2 ). (5)

4 4 G. CRACIUN, B. AGUDA AND A. FRIEDMAN Except for the time-scling fctor ε, equtions () (5) re identicl to those used in []. The cyclic rections shown in Figure 2 nmely, those between the pirs of species (S, S 2 ), (C, ), nd (A, A 2 ) imply tht the totl concentrtions of ech pir of species is constnt. Of course, the corresponding moleculr species re, in generl, not constnt inside the cell; however, the primry purpose of this simplifiction is merely to crete n ultrsensitive switch (see the two bottom boxes of Fig. ) tht cn be modeled using the zeroth-order ultrsensitivity property of cyclic enzyme rections when the Michelis constnts re close to zero [4]. The individul rection steps in these cyclic rections re ssumed to possess the Michelis-Menten type of kinetics; tht is, the rection rte hs the form vx/(k M + X), where X is the substrte (rectnt), v is the mximum rection rte, nd K M is the Michelis constnt. The mximum rte v is itself proportionl to the totl enzyme concentrtion ctlyzing the step; for exmple, for the conversion of S to S 2, v is proportionl to, since the ltter is the ctlyst for this step. In () nd (3) the interctions tke plce by mss-ction lw nd degrdtion. For exmple, the term k m2 G 2 in (3) is due to degrdtion promoted by, nd the term k m2 G 2 is due to intrinsic degrdtion. We include the vrious powers of ε to ccount for the fct tht the vrious processes hve different time scles. For negtive powers of ε, the smller the vlue of ε, the fster the process pproches equilibrium. For positive powers, smll ε mens tht the evolving vrible (e.g. A 2 ) chnges very slowly. Thus, for exmple, in the subsystem (2)-(4), the vribles S 2, G 2, re fst evolving, nd G 2 nd S 2 evolve fster thn, s suggested by the prmeter vlues in []. We fix S + S 2 =, C + =, A + A 2 =. We tke the prmeter reference vlues (PRVs) to be the following: k s =.2, k sd =., k sd2 =.5, k = k 3 =, v m =, K M = K Mr =.2, k 2 =., k 2 =., k m2 =, k m2 =., v m3 =., K M3 =.2, K Mr3 =.2, k 4 =, k 4 =, v m4 =, K M4 =.2, K Mr4 =., ε =.. For these PRVs, the time scle is such tht one cell cycle typiclly tkes 5 time units. Except for ε, ll other prmeters re of the sme order of mgnitude s those chosen in []. Our im is to show tht the model is sufficient to ccount for the following experimentl observtions (see [2, 3, ]): Assertion A. If the signl rte k s is smll then the cell is quiescent; if k s is of intermedite size then the cell is cycling; nd if k s is lrge then the cell goes into poptosis. We ssume tht there exists threshold on the poptosis mrker beyond which poptosis occurs irreversibly; without loss of generlity, we suppose tht this threshold is 2. (The prmeters were chosen so tht the ultrsensitive switch occurs round this vlue of A 2 ). Note tht, for smll ε, the vrible G 2 reches equilibrium much fster thn, S 2,, nd A 2. Thus we my suppose tht, in the time it tkes the vrible G 2 to rech equilibrium, the chnge in the other vribles is negligible. Therefore, we my equte the right-hnd side of (3) to zero nd determine G 2 s function of S 2 nd. Similr fst-slow vrible considertions pply to the remining equtions, nd they help us determine some rte prmeters in our computtions s we work with the full dynmicl system. The computtionl results tht will be described here for ε = hve been vlidted lso for smller vlues of ε, such s ε = 2, ε = 3, nd ε = 4.

5 MODULAR NETWORK COORDINATING CELL CYCLE AND APOPTOSIS 5 3. Hysteresis loop for signl/cycle interply. We first consider only the dynmics of S 2 nd G 2 which re the fstest vribles in the system () (4) for smll ε. Bsiclly, the S 2 -G 2 interction represents the ctivtion of trnscription fctors tht involves fst protein-protein interctions. The other vribles (,, nd A 2 ) re ll considered slow, becuse they re ssumed to represent the dynmics of much lrger regultory networks. We equte the right-hnd side of (3) to zero, nd we solve for G 2 s function of S 2 nd. Then we equte the right-hnd side of (2) to zero nd solve for S 2 s function of. Note tht lthough in generl this lst eqution hs two solutions, only the one given by the Goldbeter-Koshlnd function is biologiclly relevnt (the other solution would led to negtive vlue for S, see [4, 9]). Then (4) reduces to the eqution d = Ψ(, ). (6) dt More detils bout the computtion of Ψ re given in ppendix A. Equtions () nd (6) cpture the dynmics of () (4) restricted to the mnifold obtined by mking the right-hnd sides of (2) nd (3) equl to zero, nd they represent good pproximtion of the dynmics of () (4) for smll ε. The S-shped -nullcline in Figure 3 is the curve Ψ(, ) = ; tht is, for ech fixed, we compute by solving (2), (3), nd (4) t equilibrium. The nullcline nullcline Figure 3. The projection of the trjectory on the (, ) plne t PRVs. upper nd lower brnches of the S-shped curve re stble, nd the middle brnch is unstble. This suggests tht if the nullcline of () intersects only the middle brnch, then s increses to some criticl vlue 2, there is n upwrd jump (which cn be interpreted s hving crossed the checkpoint for cell-cycle entry); nd when the signl is decresed to some criticl vlue, the cell returns to its initil (post-mitotic) phse.

6 6 G. CRACIUN, B. AGUDA AND A. FRIEDMAN.9.8 S 2 G 2 A t Figure 4. Solutions of () (5) s explicit functions of time t PRVs. We shll now describe our computtionl results for the full system ()-(5). Any choice of initil vlues will become negligible fter some time; for simplicity we tke initil vlues = S 2 = G 2 = = A 2 =. First, for the PRVs, we get the results in Figures 3 nd 4. The curve spnned by dots in Figure 3 is the projection of the solution on the (, ) plne. Figure 4 shows the periodic behvior of ll the components of the solution of () (5) fter some time (pprox. time units), so the effect of the choice of initil vlues becomes negligible. Note tht A 2 < 2, so the cell does not go into poptosis. The nullcline of () is determined by the eqution = k s k sd + k sd2. (7) This -nullcline vries with the prmeters k s, k sd, nd k sd2. In prticulr, s we vry the signl input rte k s (nd keep ll the other prmeters fixed), the -nullcline chnges, but the -nullcline does not. If, roughly, k s <.65, nd ll other prmeters re t PRVs then the two nullclines intersect below the lower knee of the S-shped -nullcline, nd the cell is quiescent (see Fig. 5). In this cse the cell gets stuck t the intersection of the nullclines of nd before reching the lower knee of the S-shped curve. This is quiescent cell, with A 2 < 2 (no poptosis). If k s >.25 then the two nullclines intersect bove the higher knee of the S- shped -nullcline (see Figure 6). The vrible jumps up (i.e., the cell crosses the checkpoint for cell cycle entry), but it does not complete the cycle: it gets stuck t the intersection of the two nullclines. In this cse A 2 > 2 fter bout 35 time units, i.e., the the cell enters poptosis.

7 MODULAR NETWORK COORDINATING CELL CYCLE AND APOPTOSIS S 2 G 2 A b t Figure 5. The cse k s =.2: () reduced phseplne (b) explicit functions of time S 2 G 2 A b t Figure 6. The cse k s =.3: () reduced phseplne, (b) explicit functions of time. For k s =.65 the two nullclines intersect just bove the lower knee of the -nullcline (see Fig. 7), nd for k s =.25 they intersect just below the higher knee (see Fig. 8). In both cses the system exhibits periodic oscilltions. As shown in Figures 7b nd 8b, for k s just bove the quiescent limit, the system spends lot of time in the inctive stte (i.e., ), nd for k s just below the poptotic limit the system spends lot of time in the ctive stte (i.e., ). As we cn see in the bifurction digrm in Figure 9, these oscilltions re born out of Hopf bifurctions. Awy from the point B, ech verticl line intersects the ner-rectngulr thick curve t two points, the distnce between them being the mplitude of stble periodic solution. Ner B, the thick curve ctully bulges slightly leftwrd, so tht ech verticl line intersects it t four points. The distnce between the two nerest points is the mplitude of n unstble periodic solution, nd the distnce between the two frther points is the mplitude of stble periodic solution. As shown in Figure 3, these oscilltions re hysteretic in the sense tht they involve periodic switching between the upper nd lower brnches of the -nullcline. Strictly speking, these oscilltions cnnot be interpreted s the CDK oscilltions in the cell cycle. However, it is interesting to note tht the hysteretic oscilltions, s illustrted in Figure 3, which rise from our nlysis of the Agud-Algr model,

8 8 G. CRACIUN, B. AGUDA AND A. FRIEDMAN S 2 G 2 A b t Figure 7. The cse where k s is ner the quiescent limit: () reduced phseplne (b) explicit functions of time S 2 G 2 A b t Figure 8. The cse where k s is ner the poptotic limit: () reduced phseplne (b) explicit functions of time. resemble the hysteretic oscilltions of biologicl control systems which occur, for exmple, in the yest cell cycle (see [9] for review). For our purposes, we consider entry into the cell cycle when is switched on. 4. The role of other prmeters. Consider the prmeter k 2, relted to the input provided by the cell-cycle mrker to the control node G 2. This prmeter mesures the depth of the hysteretic loop of the -nullcline. If we increse k 2 from its reference vlue. to k 2 =.2, we get deeper -nullcline, s we see by compring the -nullclines from section 3 with the -nullcline in Figure. On the other hnd, if we decrese k 2, then the depth of the S shpe decreses. In prticulr, if k 2 =, then the -nullcline loses its S shpe, so we cnnot get hysteretic loop (see Fig. ). In this cse the solution, insted of being periodic, converges to some fixed point t the intersection of the nullclines. We shll now consider the cse where k 2 =.2. If we keep ll the other prmeters t their reference vlues then, s seen in Figure, the cycle begins its upwrd jump, but does not come down to completion, since the intersection between the two nullclines is now on the upper brnch of the -nullcline. Therefore, the cell goes into poptosis (A 2 > 2 ) fter some time. However, if k s is decresed

9 MODULAR NETWORK COORDINATING CELL CYCLE AND APOPTOSIS 9 Figure 9. Bifurction digrm of (verticl xis) with respect to k s (horizontl xis). The vlues of ll other prmeters re given by the PRVs. The Hopf bifurction t A is supercriticl, nd the Hopf bifurction t B is subcriticl. The thin solid lines represent stble stedy sttes, nd the thin dshed line represents unstble stedy sttes b.5. Figure. k 2 =.2: () for k s t its reference vlue; (b) for k s =., smller thn its reference vlue. (from the reference vlue k s =.2 to k s =.), then the cell goes into cycling (with A 2 < 2 ) s shown in Figure b. The bove considertions suggest tht the ssertion A in section 2 is vlid only for k 2 restricted to some intervl k 2 < k 2 < k 2, which includes the points. nd.2. If k 2 becomes very lrge then the two nullclines will lwys intersect long the stble upper brnch of the S-shped nullcline, nd the trjectory will get stuck t tht stble fixed point. In this cse we cn only hve quiescence or poptosis (i.e., there cn be no cycling for ny vlues of k s nd k sd2 ). Finlly, let us look t the interply between the prmeters k sd, k sd2, nd k s. The three possible modes of the cell re summrized in the digrms of Figure 2. These digrms re bsed on the observtion tht the prmeter vlues for

10 G. CRACIUN, B. AGUDA AND A. FRIEDMAN S 2 G 2 A b t Figure. The cse k 2 = : () reduced phseplne (b) explicit functions of time. 5 x 3 7 x poptosis 5 k s poptosis k s 2.5 cycling 4 cycling 2 O q O quiescent quiescent k sd2 b k sd2 Figure 2. The mode digrm: () k sd =. (b) k sd =.3. The specil points re the PRVs for k s nd k sd2. which the -nullcline psses through the knees of the -nullcline re very good pproximtions of the prmeter vlues where the Hopf bifurctions occur. The eqution of the lower line in Figure 2 is determined by the condition tht the -nullcline psses exctly through the lower knee of the S-shped curve (i.e., the lower knee of the -nullcline). Similrly, the eqution of the higher line in Figure 2 is determined by the condition tht the -nullcline psses exctly through the higher knee of the -nullcline. If k sd is very smll (e.g., k sd =., its reference vlue) then it is roughly the quotient k s /k sd2 tht determines the mode (see Figure 2). As we increse k sd both lines re rised, but the lower line is rised fster thn the higher line. Hence, the cycling region shifts upwrd nd t the sme time decreses (see Figure 2b), nd new region ppers between the two lines. For exmple, if ll prmeters re set t their reference vlues, except for k sd =.3, then the cell is quiescent s in Figure 5, insted of cycling s in Figures 8 nd Conclusions. It hs been shown experimentlly (see [2, 3, ]) tht there is n optimum rnge of trnscriptionl ctivities for E2F- nd c-myc for cells to proliferte. If the intensity of growth fctor signling is too smll, cells re quiescent nd do not cycle. However, if signls re too lrge then the poptosis progrm is

11 MODULAR NETWORK COORDINATING CELL CYCLE AND APOPTOSIS triggered; this cn be interpreted s n importnt fil-sfe mechnism to prevent uncontrolled cell prolifertion s it occurs in cncer. A simple model developed in [] estblished the bove biologicl observtion for certin rnge of prmeters. In this pper, we crried out full nlysis of the model for wide rnge of prmeters. We hve shown tht the positive feedbck loop (shown s in Fig., nd represented by the prmeter k 2 in eqution (3)) is required for n S-shped - nullcline. This nullcline geometry is essentil for switch-like behvior between the vrious cellulr sttes, prticulrly from quiescence to cell cycling nd/or poptosis. The biochemicl bsis of the loop is well estblished (see []); the trnscriptionl ctivities of the fctors E2F- nd c-myc (represented by G 2 in the model) re incresed by CDK2 (represented by in the model) vi inhibition of the retinoblstom protein (Rb) which inhibits the G 2 fctors. Our nlysis suggests tht the prmeters involved in this positive feedbck loop re the key prmeters for controlling the cellulr stte trnsitions. Our simultions on the effect of signl strengths re consistent with experimentl reports on the induction of poptosis (see []). The cross-shped phse digrm in Figure 2b is n interesting result. The digrm shows tht the series of cellulr stte trnsitions depends on the mgnitude of the negtive feedbck loop between nd the signl (s mesured by the vlue of k sd2 ). For exmple, if the signl rte k s is incresed while keeping k sd2 =.5, then the system goes from quiescence to cycling, nd eventully to poptosis. But for k sd2 below.2, sy k sd2 =., the sme increse in k s will not give rise to n intermedite cell-cycling stte. The region denoted by q- in Figure 2b is region where the cell could either be quiescent or poptotic depending on the cell s history. If the cell is previously quiescent nd the signl increses to the q- region, then the cell remins quiescent; however, further increse in the signl will send the cell directly into poptosis (without going through the intermediry cell cycle phse). Thus, we predict tht: Assertion B. If the negtive feedbck loop between the cell cycle nd signling pthwys tht impinge on the E2F-/c-Myc node is very wek (nmely, k sd2 is smll) then for lrge enough signl degrdtion rte (k sd ), the cell will bypss the cycling phse, s it goes (with incresing signl input rte k s ) from quiescent phse to poptosis. As we hve mentioned in section 3, the periodic oscilltions generted by the model my not correspond to experimentlly observed CDK oscilltions, but they resemble the oscilltions of biologicl control systems which occur, for exmple, in the yest cell cycle (see [9]). Using different model, consisting lso of five ODEs, Obeyesekere et l. [8] exhibited bifurction digrm similr to Figure 7, but for different vribles; tht model does not include poptosis. It would be interesting to extend our model to more relistic network tht includes key components of the cell-cycle module, such s Rb, cyclin E/CDK2, Cdc25A nd p27kip (see []); such n extension might enble one to test our ssertion B experimentlly. It would lso be interesting to include in the model detils of the poptosis module tht incorporte positive feedbck loop between executioner cspses nd cytochrome c relese, which ws lso reviewed in []. Such model would involve lrger system of ODEs.

12 2 G. CRACIUN, B. AGUDA AND A. FRIEDMAN Acknowledgments. Crciun nd Friedmn were supported by the Ntionl Science Foundtion upon Agreement No. 25. Friedmn ws lso supported by the Ntionl Science Foundtion Grnt DMS Appendix A. Here re the detils of computing the expression of Ψ(, ) in eqution (6). First, by equting the right-hnd side of eqution (3) to zero, we get G 2 = k 2S 2 + k 2 k m2 + k m2. (8) Next, by equting the right-hnd side of eqution (2) to zero nd solving for S 2, we get S 2 = ( k + k K Mr + v m K M + v m ± (k k 2 2 K Mr 2k v m K M 2k v m + k 2 2 K 2 Mr + 2v m K M k K Mr 2v m k K Mr + v 2 mk 2 M + 2v 2 mk M + v 2 m) 2 )/(2vm 2k ). As we mentioned in section 3, of the two solutions for S 2, only the one with the minus sign is biologiclly relevnt (nd corresponds to the Goldbeter-Koshlnd function). We substitute this expression of S 2 into eqution (8) nd then substitute the resulting expression for G 2 into eqution (4). We get Ψ(, ) = ε (k 3 (k 2 ( k + k K Mr + v m K M + v m (k k 2 2 K Mr 2k v m K M 2k v m + k 2 2 K 2 Mr +2v m K M k K Mr 2v m k K Mr + v 2 mk 2 M + 2v 2 mk M +v 2 m) 2 )/(2vm 2k ) + k 2 )( ))/((K M3 + )(k m2 + k m2 )) (v m3 )/(K Mr3 + ). Appendix B. Figure 3 presents summry of the importnt pthwys relevnt to our model nd offers some detils nd justifiction of the modulr structure shown in Figure. Figure 3 is reproduces Figure 3 of [], where more detiled discussion of these pthwys is given. REFERENCES [] Agud B. D., nd Algr C. K. (23). A structurl nlysis of the qulittive networks regulting the cell cycle nd poptosis. Cell Cycle 2: [2] Blgosklonny M. V. (999). A node between prolifertion, poptosis nd growth rrest. BioEssys 2: [3] Evn G. I., nd Vousden K. H. (2). Prolifertion, cell cycle nd poptosis in cncer. Nture 4: [4] Goldbeter A., nd Koshlnd D. E. Jr. (98). An mplified sensitivity rising from covlent modifiction in biologicl systems. Proc. Ntl. Acd. Sci. USA 78: [5] Hung C. Y. F., nd Ferrell J. E. Jr. (996). Ultrsensitivity in the mitogen-ctivted protein kinse cscde. Proc. Ntl. Acd. Sci. USA 93: [6] Lee K. Y., Ldh M. H., McMhon C., nd Ewen M. E. (999). The retinoblstom protein is linked to the ctivtion of Rs. Mol. Cell. Biol. 9: [7] Mendelsohn A. R., Hmer J. D., Wng Z. B., nd Brent R. (22). Cyclin D3 ctivtes Cspse 2, connecting cell prolifertion with cell deth. Proc. Ntl. Acd. Sci. USA 99: [8] Obeyesekere, M. N., Zimmermn, S. O., Tecrro, E. S., nd Auchmuty, G. (999). A Model of Cell Cycle Behvior Dominted by Kinetics of Pthwy Stimulted by Growth Fctors. Bull. Mth. Biol. 6:

13 MODULAR NETWORK COORDINATING CELL CYCLE AND APOPTOSIS 3 Figure 3. Pthwys linking the trnscription fctors c-myc nd E2F- with the initition of the cell cycle nd poptosis. Dshed lines re trnscriptionl processes; solid lines re post-trnsltionl processes. Arrows signify ctivtion, while hmmerheds signify inhibition. [9] Tyson J. J., Chen K. C., nd Novk B. (23). Sniffers, buzzers, toggles nd blinkers: Dynmics of regultory nd signling pthwys in the cell. Curr. Opin. Cell Biol. 5:22 23 [] Ymski L., Jcks T., Bronson R., Goillot E., Hrlow E., nd Dyson N. J. (996). Tumor induction nd tissue trophy in mice lcking E2F-. Cell 85: E-mil ddress: gcrciun@mbi.ohio-stte.edu E-mil ddress: bdgud@gmil.com E-mil ddress: friedmn@mbi.ohio-stte.edu