AN ANALYI OF A FRACTAL KINETIC CURE OF AAGEAU by John Maloney and Jack Hedel Departent of Matheatcs Unversty of Nebraska at Oaha Oaha, Nebraska 688 Eal addresses: aloney@unoaha.edu, jhedel@unoaha.edu
Runnng Head Fractal Knetcs Curve Analyss
Abstract The fractal knetcs curve g g s e p p =, K f derved by avageau [6], s analyzed to show that the paraeters K g and g are not unquely deterned, f, s e gven four approprately stuated data ponts ( p, ), =,,,. Coparson s ade to an alternate fractal Mchaels-Menten equaton derved by Lopez-Quntela and Casado, J. []. Key words: fractal knetcs, bochecal knetcs, data analyss, paraeter unqueness
. Introducton The Mchaels-Menten equaton p = + K has long been the standard fraework for bochecal knetcs [], descrbng the reacton of a substrate on a free dp enzye E to for a product P. Here p = and and K are eprcally deterned paraeters. It s dt convenent to rewrte ths equaton n the for F HG K I F HG p KJ F = H G I K J p I KJ. () If the reacton s allosterc (cooperatve) so that n olecules of bnd to E, the knetcs s descrbed by the Hll equaton [5] p = n K + n whch can be wrtten F HG K n I F HG p KJ F = H G I K J p I KJ. () Recently, avageau [6] has shown that both the Mchaels-Menten and Hll equatons are specal cases of a ore general fractal power law echans whch produces the knetc equaton F HG K f I g F HG I KJ s e p p KJ F = H G I K J. g () Clearly, () ncludes () and () as specal cases where g and g s e can be nterpreted as generalzed cooperatvty exponents. In fact, g e = reduces () to the Hll equaton ().
Note that equaton () has two paraeters and K, equaton () has three paraeters, K whle equaton () has four paraeters K g and g. Recently, the authors [] have ade a data fttng analyss of equatons () and (). For the Mchaels-Menten equaton () let dp, and d p, be two ponts on the hyperbolc (concave down) response curve descrbed by (). Then these two peces of data unquely and K deterne the paraeters., f, s e d p,, and n For the Hll equaton () three data ponts,, and ether deterne the three paraeters K and n unquely or there ay be no soluton for the three paraeters. It depends on a precsely, deterned but coplcated relatonshp between the three data ponts []. d p dp, The purpose of ths paper s to carry out a slar analyss for the ore general equaton () wth four paraeters. It turns out that there wll always be two, nfntely any, or no solutons for the four paraeters gven four data ponts. That s, t s possble to have a sngle unque solutons for the four paraeters regardless of the specfed data. Before begnnng the analyss of (), another fractal generalzaton of the Mchaels-Menten equaton should be entoned. By assung that the reacton rate constants K are scale dependent, D K = A where D s the fractal denson of the scag varable, Lopez-Quntela and Casado [] were led to the equaton p = K eff eff D + () whch s dstnct fro equatons () and () above. The data-fttng proble for () has been analyzed n []. It turns out that for ths three-paraeter equaton there can also be ether one, two, or no solutons dependng on the algnent of the data ponts n a coplcated way. However, a unque soluton exsts only f the three data ponts le on a sngle specal curve separatng the regon of no solutons fro the regons of two solutons (Curve () n fgure 7 of []). 5
Thus, the two papers, [] and [6], propose dfferent odels for fractal reacton knetcs. In one odel there exsts a sngle data curve producng a unque soluton for the paraeters, whch s not robust. In the other odel, there s never a unque soluton for the syste paraeters. Thus data analyss of the two odels uncovers shortcongs n each one and rases the queston as to whether ether s an adequate odel of realty. Ths s a theoretcal analyss, assung only the nal aount of data s avalable to deterne the paraeters of the odel. uch an analyss s a frst step n evaluatng the practcal utlty of a odel. It s clearly portant to know whether or not the odel paraeters are unquely deterned by data even f only n a theoretcal settng. Then the probles of paraeter estaton and senstvty analyss can be carred out later f the odel s appled to specfc sets of data.. Results We now turn to the analyss of equaton (). For ease of notaton we use nstead of. p Theore Gven four data ponts (, ), =,,, where 0 < < < < and 0 < < < <, then there ether exsts one, two, nfntely any, or no solutons for, K f, gs and ge where and K f are assued to be postve. Proof ubsttute (, ), =,,, nto () for (, ) and take logarths of both sdes to obtan ( K ) ( ) ge ( ) g = g. olvng for K f gves: s f e g e K f = + + ( ). gs gs ubtractng the frst two of these four equatons (that s for = and ) eates the K f ter and gves: g + + = gs gs e 0. 6
larly usng = and gves:: g + + = gs gs e 0. These two equatons can be solved for geand gs to gve: g e = + and g s =. + K g and g f, e, s (, ) Thus, are all expressed n ters of and the data ponts. nce we haven t yet used ( = and ) g e + + = 0 we can substtute nto ths the expressons for g e and gs gs g s to obtan: + + F I + = 0. HG KJ Ths equaton can be rewrtten as: 7
( ) ( ) ( ) ( ) α + α + α + α = 0 or ( ) ( ) ( ) ( ) α α α α = (5) where: and α + α + α + α = 0. α = + + ( ) ( ) ( ) α = + + ( ) ( ) ( ) α = + + ( ) ( ) ( ) α = + + ( ) ( ) ( ) We note that represents the axu velocty n (), (), and (). Then wth > for =,,, and usng the fact that the α = α α α, we set ( ) h and note that (5) s equvalent to fndng a such that α α = α = α = α = 0 n whch case there are nfntely any solutons. It s easy to show that: α h ( ) =. We also note that h ( ) f and only f h ( ) Q ( ) ( )( )( )( ) h ( ) = α. 8
where Q b g s a quadratc n whose coeffcents are cobnatons of the s and α s. nce on ts doan (, ) and assung ( ), l h ( ) = h + = o r 0, and h / ( ) h b g> 0 h( ) l h ( ) = =, then h changes sgn at ost twce, and the graph of h b g ( ) has one of the fors: wth h ( + ) = and for h ( + ) = 0 fgure It s thus clear that MAPLE): h b g ntersects the e fgure y = ether once, twce, or no tes and the theore s proved. b g To proceed further, we need to know the coeffcent of the squared ter n Q. It s (found by ( ) ( ) ( ) α = α + α + α 9
= ( ) + + + ( ) + + + ( ) + + ( ) It turns out that the queston of the exstence of a unque soluton of (5), e a unque such that h =, can be resolved copletely usng only α, the coeffcent of the quadratc ter n Q. It s clear. ( ) fro fgure that when ( ) h+ = there s a unque soluton f α > 0, and ether two solutons or no soluton α <. In a slar anner t s clear fro fgure that when ( ) f 0 and ether two solutons or no soluton f α > 0. Of course, the expresson for α nvolves and sple and useful geoetrc nterpretaton. h+ = 0, there s a unque soluton f α < 0 k α, α, α. It turns out that the sgn of each of the α has a Lea 0, ( 0 ),( 0 α > = < ) f and only f (, ) les above, (on), (below) the e fro ( ), to (, ). Proof uppose that ( ) ( ) + ( ) 0= α = + Thus or = + +. = 0
= + whch s the e y = x + b wth y = and x= between (, ) and (, ). establshed. nce we have dvded by a postve nuber, the two nequaltes follow drectly. In a copletely analogous anner the followng geoetrc nterpretaton of α, α, and α can also be Lea 0, ( 0 ),( 0 α > = < ) f and only f (, ) les above, (on), (below) the e fro (, ),. To ( ) Lea 0, ( 0 ),( 0 α > = < ) f and only f (, ) les above, (on), (below) the e fro (, ),. to ( ) Lea 5 0, ( 0 ),( 0 α > = < ) f and only f (, ) les above, (on), (below) the e fro(, ) to (, ). We are now able to establsh the an result of ths paper. Theore There s never a unque set of paraeters for the equaton (). Otherwse stated, equaton (5), ( ), h = never has a unque soluton. Proof larly a We adopt the notaton ( + +,,, ), for exaple, to ndcate that α > 0, α < 0, α > 0, and α < 0. + or replaced by a zero ndcates that the correspondng coponent s zero. Frst suppose that α α α 0 + + > so that ( ) h+ = and fgure apples. Thus also α < 0 snce α+ α + α + α = 0. We now have to consder varous cases for the sgn possbltes for α, α, and α < 0 ). α (wth +. nce α+ α + α > 0 and therefore α > α α > 0, then Case (,,, )
( ) ( ) ( ) α = α + α + α < and (5) has ether two solutons or no soluton. 0 Case (,,, ) +. Ths case lke any others can be ruled out by usng the geoetrc nterpretaton of the provded by the leas. In ths case consder the ponts and es: α 's The frst nus sgn eans that (, ust be below L, a contradcton. Case ( +,,, ). Ths case s ruled out n a copletely slar anner to case. ) Case ( ++,,,. Ths case s ruled out n a copletely slar anner to case. ) Case 5 ( + +,,, ). Ths case s ruled out ether by the sae reasonng as case or by the sae reasonng of case. Case 6 ( ++,,, ). In a slar anner to case, consder the ponts and es
The frst nus sgn says that L s below L but the last nus sgn says that L s below L, a contradcton. Case 7 ( +++,,, ). Here the sae approach as n case leads to a contradcton. Case 8 If any one the frst three α 's s zero, consstent wth α+ α + α > 0, the very sae arguents as above stll apply. If any two of the frst threeα 's are zero then all α 's are zero (usng the geoetrc arguent) and therefore α = 0 and h = whch gves an nfnty of solutons. We now suppose that α+ α + α = 0 = α and agan consder several sub cases. Case 9 α > 0. Usng the geoetrc approach we consder the ponts and es:
Thus we also have α > 0 and hence α < 0, a contradcton. α <. In a copletely slar anner we fnd that α < 0, α > 0, also a contradcton. Case 0 0 = = = = and h ( ) Case α = 0. Agan, by the geoetrc arguent we fnd that α α α α 0 nfnty of solutons. ( ) 0 = has an Ths dsposes of all cases wth α+ α + α 0. If α+ α + α < 0 we have h + = (fgure ). lar arguents to the above show that there can't be a unque soluton here ether. The theore s proved.. Conclusons. The lack of a unque soluton for the paraeters, K, g, and gven any four regular (e onotoncally arranged ) data ponts (, (), ay be defcent as a odel of realty. ) f s, I =,,,, suggest that avageau's fractal knetcs odel, equaton g e
References [] Hedel, Jack and Maloney, John, When can sgodal data be ft to a Hll curve?, J. Australan Math. oc., eres B, Appled Math (999),8-9. [] Hedel, Jack and Maloney, John, An analyss of a fractal Mchaels-Menten curve, J. Australan Math. oc., eres B, Appled Math to appear. [] Lopez-Quntela, M.A. and Casado, J., Revson of the ethodology n enzye knetcs: a fractal approach, J. Theor. Bology 9 (989), 9-9. [] Mchaels, L. and Menten, M.L., De knetk der nvrtnwrkung, Boche. Z. 9 (9), -69. [5] Monod, J., Wyan, J. and Changeux, J.P., On the nature of allosterc transtons: a plausble odel, J. Mol. Bol. (965), 88-8. [6] avageau, Mchael A., Mchaels-Menten echans reconsdered: plcatons of fractal knetcs, J. Theo. Bol. 76 (995), 5-. [7] ot, E. O., Canoncal Nonear Modeg, an Nostrand Renhold, N. Y., 99. 5