Institute of Biochemistry and Biophysics of the Polish Academy of Sciences, Warszawa, Poland

Transcription

1 Supplmtary Matrials Vol., No 2/ o-li at: Rgular papr Thortical mol xplaiig th rlatioship tw th molcular mass a th activatio rgy of th zym rval y a larg-scal aalysis of ioiformatics ata iotr H. awlowski a iotr Zilkiwicz Istitut of Biochmistry a Biophysics of th olish Acamy of Scics, Warszawa, ola A gral pc of th zym catalytic rat o its mass was rval wh a statistical aalysis of 175 rcors from th M ataas was prform. Th stimat activatio rgy of th catalytic procss crass asymptotically with th zym molcular mass icras. Th propos thortical mol postulats th xistc of a itrmiat complx of th zym a th partig prouct. It allows for th xplaatio of th iscovr mass-rgy rlatioship, as a ffct of th al zym prouct itractios urig complx issociatio. Fitt paramtrs of th mol sm to i agrmt with thos wily accpt for th va r Waals rgy of molcular itractios. Thir valus also agr with th pictur of th hyrog oig i th catalytic procss a suggst that surfac walk ca th favoral way of th prouct partur. Ky wors: turovr, catalysis, zym, molcular mass, statistics Rciv: 18 March, 21; rvis: 2 May, 21; accpt: 1 Ju, 21; availal o-li: 14 Ju, 21 -mail: piotrp@i.waw.pl

2 rgy, Appix 1 Accorig to q. 2 a q. 24 th al rgy of zym-prouct itractios, ca calculat as 2, (A.1.1) whr appropriat compots of th sum ar tak for th i -th atom of zym a th j -th atom of prouct, sparat y th istac. Othr symols ot:, a asolut valu of miimal rgy of va r Waals itractios, a Waals miimum., th itratomic istac of th va r Assum i th Thortical mol, th homogity of zym a prouct, a quality of Loo atomic itractios allow for th rplacmt of th aov, poit-lik, charactristics of th systm y cotiuous os, a th appropriat sum y th oul-volum itgral 2. (A.1.2) I th aov formula, a ar ifiitsimal lmts of th zym () a th prouct () volum, rspctivly. a ar th sitis of atoms. aramtrs a ar costat valus of th ix os. Varial is th istac tw giv poits of th zym a th prouct.

3 For a sphrical prouct of raius R, rlativly small compar to th zym charactristic imsio, w ca glct th chag i th cosir istac rlat to th chag of th atio i th prouct. For this raso, w may trat istacs from iffrt poits of th zym to iffrt poits of th prouct as masur oly to a chos sigl poit of th lattr, for xampl, its ctr. As a rsult, itgratio isi prouct coms trivial, a las to th formula 8 R, (A.1.) whr w itrouc, which ots th istac from a giv poit of th zym to th prouct ctr. For assum gomtry of th mol (Fig. 4), a for th half-immrs prouct isplacmt from th zym ctr, th asist way to calculat th fi itgral (A.1.) is to us Cartsia cooriats x, y, z, with origi at th ctr of th prouct, th y axis orit prpicularly towars th zym oy, a th z axis goig through th prouct a zym ctrs. Ths cooriats aitioally wr rlat to th st of sphrical varials,, (Fig. 8). I th propos mol, th poit (,, ou-stat prouct. ) rprsts th positio of th Now, for R R, wh itgratig usig th sphrical varials, o otais: 8 R si( ) δθ R max 1 4 δρ δ, (A.1.4)

4 whr th uppr limit of th raial itgratio max is max R 2 2 si 2 ( ) cos( ). (A.1.5) Wh o calculats th azimuthal a raial itgrals i formula A.1.4, o gts: 2 8 R 1 1 si( ) δθ R si ( ) cos( ) R. (A.1.) I th aov xprssio, th xpasio of th fuctio i th curly rackts i a Maclauri sris up to th powr of 2 (with rspct to ) allows for approximat polar itgratio. Th R ( ) R 5, (A.1.7) whr th rgy of al itractios i th zym ctr, ( ), quals

5 2 1 R 1 1 () 9 R R. (A.1.8) If th gomtric istac tw th ou-stat sit a th sit whr th prouct is crossig th arrir for its rlas, quals, th rspctiv iffrc i rgy is R (A.1.9) R 5 Appix 2 Assumig that th ma umr of os of a crtai typ which hav to rok urig th cosir procss is, th chag i th rgy of al itractios ca writt as:, (A.2.1) whr is a sigl-o rgy. Th, th staar viatio of th iscuss rgy ca simply calculat as:. (A.2.2) Assumig oisso istriutio of th umr of os, o may put

6 . (A.2.) Thus, (A.2.4) Fially,. (A.2.5) Th st of otai quatios scris th rlatios tw microscopic a macroscopic paramtrs of th systm. W hav 2. (A.2.)