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Electrophoresis of interacting proteins II. A simple method of simulation
Bioehimica et Biophysica Acta. 336 (1974) 213-221 © Elsevier Scientific Publishing C o m p a n y , A m s t e r d a m - Printed in The N e t h e r l a n d s BBA 36635
ELECTROPHORESIS OF IN:?ERACTING PROTEINS I1. A S I M P L E M E T H O D O F S I M U L A T I O N
H. N I J H U I S a n d T. A. J. P A Y E N S
Department of Biochemistry. Agrictd;arai Unirerslty Wagenbtgen, and Netherlands htstitute for Dairy Research. Erie (The Netherl~'.ds) (Received A u g u s t 22rid, 1973)
SUMMARY
G o a d ' s method for the numerical solution ,:)f the conservation-of-mass equation of a system of migrating and simultaneously interacting macromolecules is simplified by taking into account the velocity flu:~es between adjacent boxes only. The error introduced in this way is used to a d v a r t a g e to simulate the effect of the diffusionai flux. It is shown that this procedure is e~sentially the same as the countercurrent analog method developed by Bethune and Kegeles to account for the effect of diffusion on transport patterns of interacting proteins. The adjustment of the simulated diffusion cgefficients to their actual values is discussed and the results of different calculations are compared. It is shown that diffusional spreading has only a secondary influence on the development of a reaction boundary.
INTRODUCTION
Transport techniques such as eleetrophoresis, ultracentQifugation and gel filtration can be used to advantage in studying biopolymer interactions [1-6]. Several methods have been proposed to solve the conservation-of-mass equation for a system of interacting biopolymers in transport experiments. Gilbert [7] and Gilbert and Jenkins [1 ] have presented analyt;, al expressions for ':he concentration and the concentration gradient in p o l y m e r i z i n g ~ n d complex forming systems neglecting the effect of diffusion on the spreading of a boundary. The usefulness of this approach resides in the fact ~hat in.prolonged experiments the contr:bution of diffusioqpoften can be neglected when compared to the spreading caused by the differentiaVmigration of the different polymer species. It is realized, howe~er, that in experiments of finite duration a n d / o r with the occurrence of small peaks ()r shoulders, it may be impt>rtant tc assess the blurring effect of the diffusion on the saape of a boundary. This has led a n u m b e r of authors to solve the complete conservation-of-mass equation by numerical or simulati( n methods [2, 8, 9]. C a n n apd G o a d [2] solved the complete conservation°of-mass equation by considering the contributions of the indivudual species to the fluxes due to velocity
214 and diffusional transport, In this approach the b o u n d a r y is divided into a large number of segments and the flux at a particular segment edge is calculated from the average concentrations in a n u m b e r of neighbouring boxes. Equilibrium between the individual species is re-established after each transport cycle. Actually the flux at a particular edge is related to the average concentration in all boxes. G o a d showed that if the velocity flux is restricted to the transfer of material from one box to the next a diffusion-like error is introduced. This author therefore preferred to suppress this error by taking into account the contribution of the average concentrations o f several boxes to the flux and to introduce the diffusional flux separately. This is certainly a most accurate procedure, but we shall demonstrate below that for practical purposes the diffusion-like error can be used to simulate the diffusional flux, which leads to a drastic reduction of the computations. The method of simulation presented in this paper is (as stated above) a simplification of the C a n n - G o a d procedure, it will be shown that the method also carries a close relationship to the countercurrent simulation approach of Bethune and Kegeles [10, i I ]. Application of the method to the simu;ation of the anomalies observed during the electrophoresis of complex forming c~- and r-casein has been presented in Part I of this series [12]. THEORY AND DISCUSSION
Let us consider the following system of interacting biopolymers A and B: iA~ ~A~ jA, + B~AIB
(i----2,3...p) [ (j== ! , 2 . . . q) J
(I)
The complex formation between a~- and//-casein at the temperature of free electrophoresis [12] and the iW,eraction between R N A and virus coat protein [13] offer among others [3] excellent examples of such a system. The computation of the equilibrium concentrations of the various species requires the solution of the following set of equations: gi
. l AI/AI
(i =~ 2.3 ..... p)
Lj
AjB/A~. B
~_-_-
Z~ A i +
P
(j ::- 1,2 .....q)
(2) (3]
#
I=l
~' (t -- ;.j)A~B
(4)
./=1 q
==~ B + Z' ,;.jAjB J=l
(5)
In tk'.se equations K~ and Lj represent the equilibrium constants for the self-association of A and for the complex formation between A and B. respectively, /~ and 1~ stand for the constituent concentration (w/v) of A and B and ;.j is defined as •~j ~
Mu/(jMa
-+-
Mn)
with MA and Mn the m o n o m e r molecular weights of A and B.
215 Eqns 2-5 lead to the following expression for the m o n o m e r concentration~ of Component A: P
q
q
( 2~ K, AI)( 27 ~L,A~) 4- 2~ lIB - (A 4- B)2j }L~AJ,I ---- 0 l=l
(6)
j=O
j=O
with K1 =- Lo = i. Eqn 6 is a polynomial in the m o n o m e r concentration At, the graphical appearance of which is shown in Fig. I. In the computation of the equilibrium concentration, At is found by N e w t o n - R a p h s o n iteration [14]. The constituent concentration ~ , being the highest possible value of Al, appears to be a natural ~tarting value for the iteration. Actually, however, the n u m b e r of iteration steps can be reduced considerably by estimating the starting value of At in a particular box from its concentration in the preceding one [i 5].
/
/
/
/
/
/
/
•
/
/
/~
/
Fig. I. Schematic plot o f the p o l y n o m i a l (Eqn 6) used for the c o m p u t a t i o n o f the m o n o m e r concent r a t i o n (A,) o f C o m p o n e n t A.
The simulation of the free elcctrophorcsis of a system of interacting biopolymers is achieved by subdividing the electrophoretic channel into a large number of small boxes of equal length, lx. The development of the reaction boundaries is brought about by alternate rounds of velocity transport of the indiv,dual species from one box to the next during an interval of t i m e . lt, followed by re-equilibration according to Eqns 2-5, On the ascending side velocities are taken relative to the slowest species (i.e. C o m p o n e n t B) and the ratio dx/,it is chosen such that the fastest component just reaches the end of a box, therefore VA -- Va -=
(7)
216 The complexes, having intermediate mobilities [12], penetrate the boxes only partially. Similarly on the descending side the velocities are c h a n g e d o f sign a n d taken relative to the c o m p o n e n t with the highest absolute value of the electrophoretic mobility (i.e. C o m p o n e n t A). The source p r e g r a m for the simulation, a flow scheme o f which is presented in Fig. 2, was written in A L G O L - 6 0 . All c o m p u t a t i o n s have been carried o u t on the CDC-3200 digital c o m p u t e r complex o f the Agricultural University.
I
o,,,, I
Fig. 2. Flow diagram for the simulation of reaction boundaries of complex forming proteins in transport experiments. It is worthwh,~le to analyse the above simulation p r o c e d u r e s o m e w h a t further. With the (n -~- I)th transfer, the change o f mass in Box n u m b e r r d u e to C o m p o n e n t i is given by ..iml .
• +~
~
m,.,+~
--
~
m,..
i
:.L(m~_~.,-
i
m,~,)
(8)
where fl == Vf
zlt//Jx
a n d VI is the relative velocity of Species i.
(9)
217 Obviously, f r o m o u r choice o f reference o f the relative velocities a n d of d t / A x ( E q n 7), we have 0 < 3q < I. T h e mass transfer o f each species from one box to the next is thus seen to be d e t e r m i n e d by a c o n s t a n t fraction Ji o f the mass difference between adjacent boxes. This transfer is therefore f o u n d to be identical with the mass transfer taking place in the e o u n t e r c u r r e n t distributional process. I n the latter case we h a v e j i -~ Pi/(Pt • ~ I ), where Pi is the p a r t i t i o n coefficient o f the Species i [16]. As m e n t i o n e d in the Introduction, B e t h u n e a n d Kegeles [9-11] have use the mass distribution achieved in a cont i n u o u s c o u n t e r c u r r e n t process to a c c o u n t for the effect o f the diffusional spreading on the t r a n s p o r t p a t t e r n o f interacting proteins. It is worth noting, however, that the c o u n t e r c u r r e n t a n a l o g involves m o r e c o m p u t a t i o n s than the present m e t h o d of simulation, since it requires n o t only the c o m p u t a t i o n of the partition equilibrium but also those o f the chemical equilibria in both the upper a n d lower layer o f each counterc u r r e n t tube. F o l l o w i n g Bethune [9] we thus find for the mass distribution of Species i after a sufficiently large n u m b e r o f transfer cycles n :
6Ci/6n~-½./i(i
(lO)
--fOb2Cl/brZ--fl3Cl/br
where r is box n u m b e r a n d Ct is the c o n c e n t r a t i o n of Species i in g/dl. Similarly, for the mixture o f interacting biopolymers w'c have
h._,rC,/bn :: t)2{.~7½.~(I --.f~)C~ }liar 2 -- ,~..~f~C~/hr i
i
( I 1)
i
If now, following again Bethune and Kegeles, we draw the following analogies: a ,--, t a n d r *-, x, then Eqns l0 a n d ! l show the formal analogy with the complete conservation-of-mass e q u a t i o n in t r a n s p o r t experiments in which the distance is expressed in units o f length , I x a n d t i m e in units tit. As a consequence the simulated diffusion coefficient o f C o m p o n e n t i, expressed in cm2/s, becomes Di* -- ½ji(l - J i ) (/lx)z//]t
(12)
In the mixture o f b i o p o l y m e r s the actual diffusional flux is defined by 18]
Jo =~ B.U(i~Ci/,~x)
(13)
i
w h e r e / ~ is the gradient averaged diffusion coefficient (14)
D ~-~ _v[O, ~Cdax]iZ[OCi/i~x]
:nd Dl is the true diffusion coefficient o f C o m p o n e n t i. in a c c o r d a n c e with Eqn 12 the simulated gradient averaged diffusion coefficient, expressed in em2/s, b e c o m e s D- = [....E{½f~(i -- fObC,/br}/2;{bC,/&}l(Ax)2/At I
1
(15)
218 The adjustment o f the simulated to the actual diffusion coefficient now d e m a n d s that /5, ~ /5
(16)
Eqns 14--16 and 7 show that d x and At are fixed by the values o f the relative velocities, the diffusion coefficients DI and the concentration gradients ~Ct/~x. The latter will be discussed below. The number of transfer cycles, n, which is necessary for the simulation o f a particular reaction boundary, finally follows as the ratio of the time of the experiment a n d / I t . Conversely in the simulation o f a given experiment an arbitrary choice o f n determines together with the duration of the experiment an arbitrary /It. The correspemding fx then follows from Eqn 7 and the diffusional spreading operative in this simulation from Eqn 15. It is obvious that in general then /5* will not e q u a l / ) . We are now to able to elaborate upon the simulation of the electrophoresis of complex forming a~l- and fl-casein dealt with in the previous paper [12]. Since we have no a priori knowledge o f the concentration gradients ~C~/~x existing in the reaction boundaries, we must start with a comparison of the simulated patterns coraputed for arbitrary numbers of transfer. Typical results of the simulation of the ascending electrophoretic pattern of a I:l mixtures of ~¢~- and fl-casein for n : 27, 46, 73 and a duration of the experiment o f 4800 s are presented in Figs 3a-3c, from which it is seen, that the general appearance of the pattern is not affected substantially by the number of transfers. More importantly, as is seen from Table I, also the maximum and minimum gradient mobilities and the concentration changes over the reaction boundary do not change appreciably with n. This result, of course, is in agreement with Gilbert and Jenkins" conclusion [I] that in prolonged experiments the effect of diffusional spreading on a boundary is negligible. In other words: the proper choice of the number of transfers is o f minor importance for a correct simulation of the percentages and mobilities found from experimental electrophoretic patterns. The calculations underlying Table i and Fig. 3 also yield the concentration gradients necessary for comparison of the averaged true and simulated diffusion coeflicienls defined by Eqns 14 and 15. To this end v.e have approximated the various ,~C~/Oxby the total concentration change of the Species i over the boundary. The true value o f !3 is then estimated as follows. From existing data [[7. t8] we find the diffusion coefficient, D~. of monomeric ¢~- and fi-casein, corrected for the t e m p e r a t u r e of free eleclrophoresis, to be 3.2.10 -~ cm2/s *. The diffusion coetticients of the polymers or complexes containing i subunits are then calculated from that of the monoo mers by the Stokes-Einstein relation: D~=-- D ~ ' i ~
(17)
A similar calculation is performed for the estimation of the simulated diffusion coefficient, /5", according to Eqn 15. The comparison of t h e / 5 a n d / 5 * estimated in this way for n 27. 46 and 73 is given in Table !!, from which it may be noted that the value o f the diffusion coeffi" The molecular parameters of monomeric +,~- and fl-casein are almost the same [17. 18].
219 ,
.
.,
.
!
.
.
il
i
i.
~ i
!i
~ ~
,!~.,
,
X K K M ~ K ~ K W ~
~
;
" !
ii
:
XX
K ~ M X K K M
~~ NNK ~~N N ~ N~K X ~ N ~N K M I NKKKKKNN~ K KN M~ N K~MMNNKNKX N~KNNN~K~ ~NXK~K~N~N~ N W N ~ N M ~ N
ascending
ascendin9
a
b
I
C
ascending Fig. 3. Simulation of the ascending elcctrophoretic react!on boundaries for a 1:1 mixtu;'e of ,t,=- and /~-¢,asein. (a) (top left) Number of transfers 27. (b) (top right) Number of transfers 4 6 (¢) Number of
220 TABLE 1 COMPARING VELOCITIES AND PERCENTAGES OF THE ASCENDING ELECTROP H O R E T I C B O U N D A R I E S O F C O M P L E X F O R M I N G ~1- A N D /%CASEIN S I M U L A T E D FOR DIFFERENT NUMBERS OF TRANSFER Number of Number of Velocities* transfers boxes (IO -s el-n/s)
27
25 40 60
46 73
1.96 1.94 1.94
!.64 1.60 1.58
P ~ c e n t a g es*° At - A2 ..... ,4s 1.09 1.09 1.07
0.30 0.31 0.31
0.24 0.22 0.22
0.47 0.47 0.46
" Velocities corresponding to the maxil.numn and minimum gradients o f the reaction boundaries. *" At relative area of pure ~t-easein; ,'lz and As. relative areas under bimnoda| peak.
cient, D, is hardly affected by this change of n. The simulated diffusion coefficients/~*, however, are found to be inversely proportional to ~, as expected from the theory presented above. Actually Table Ii indicates that the most accurate number of transfers should be about 35, cot responding to an averaged simulated diffusion coefficient of 2.55.10 -~ cm2]s. Similar considerations hold for the simulation of descending electrophoretic or ultracentrifugal patterns. T A B L E li COMPARING SIMULATED AND TRUE GRADIENT AVERAGED DIFFUSION COEFF I C I E N T S P E R T A I N I N G T O T H E C O M P U T A T I O N S P R E S E N T E D IN T A B L E 1 Nqmnbgr of . It {s) transfers
27 4~! 7?.
17"P.8 104.3 658
lx (r'mn)
277.3 162.8 102.6
tX)z F : - IOT. . . . . It (cmnZ,'s) Eqn. [ 5
/3. tO~ (cl.nZ/s)
3.286 1.958 1,241
2.527 2.534 2.536
Eqn 14
The above analysis shows the usefulness of the simplified procedure for the simulation or the reaction boundaries observed in the electrophoresis (or ultracentrifugationj of interacting proteins, i n agreement with the conclusions arrived at by Cilbert and Jenkins [I ] the value of the average simulated diffusion coefficient thereby ai~pears to be of minor importance for the reproduction of the experimental peak areas and mobilities. The adjustment of the simulated to true diffusion coefficient can easily be achieved, however, by carrying out a preliminary computation with an arbitrary number of transfers. From this the true average diffusion coefficient, /5, is estimated as exposed above. The most realistic number of transfers then follows from the equalization of the tl ue and simulated average diffusion coefficients { Eqn 16). Bethune [9] has presented an alternative procedure for the adjustment of the simulated to the true diffusion coefficient for the case o f discrete self-association. His treatment can also be extended to complex formations of the kind A q- B -~ C, as will be shown elsewhere (Nijhuis, thesis to be published).
221 ACKNOWLEDGEMENTS T h i s s t u d y w a s s p o n s o r e d by t h e N e t h e r l a n d s F o u n d a t i o n for C h e m i c a l Res e a r c h ( S . O . N . ) , w i t h f i na nc i a l ai d f r o m t he N e t h e r l a n d s O r g a n i z a t i o n f o r the Adv a n c e m e n t o f P u r e R e s e a r c h ( Z . W . O . ) . W e a r e grateful t o the s t a f f o f A g r i c u l t u r a l U n i v e r s i t y C o m p u t e r D e p a r t m e n t f o r c o m p u t i n g facilities. We arc m u c h i ndept ed t o D r H . J. V r e e m a n , N e t h e r l a n d s I n s t i t u t e f o r D a i r y R e s e a r c h , Ede, f o r critical remarks. REFERENCES 1 Gilbert, G. A. and Jenkins, R. C. LI. (1959) Proc. Royal Soc. London, Ser. A, 253, 420~37 2 Cann, J. R. and Goad, W. B. (1970) Interacting Macromolecules, Academic Press, New York 3 Nichol, L W., Bethune, J. L., Kegeles, G. and Hess, E. L. (1964) in The Protein~ (Neurath, H., ¢cl.), 2nd edn, Vol. 2, pp. 305- 403, Academic Press, New York 4 Ackers, (3. K. (1970) Advances in Protein Chemistry, Vol. 24, pp. 343-446, Academic Press, New York 5 Nichol, L. W. and Winzor, D. J. (1964) J. Phys. Chem. 68, 2455-2463 6 Winzor, D. J. and Scheraga, H. A. (1963) Biochemistry 2, 1263-1267 7 Gilbert, G. A. (1959) Proc. Royal Soc. London Set. A., 250, 377-388 8 Cox, D. J. (1971) Arch. Biochern. Biophys. 146, 181-195 ') Bethune, J. L. (1970) J. Phys. Chem. 74. 3837-3845 l, ~ liethune. J. L. and Kggelcs, G. (I961) J. Phys. Chem. 65, 1755-1760 li Belhune, J. L. and Kegeles, G. (1961) J. Phys. Chem. 65, 1761-1764 12 Payens, T. A, J. and Nijhuis, H. (1974) Biochim. Biophys. Acta 336, 201-212 13 Durham, A. C. H., Finch, J. T. and Klug, A. (1971) Nat. New Biol. 229, 37-42 14 Margenau, H. and Murphy, G. M. (I943) The Mathematics of Physics and Chemistry, 14th edn, pp. 475-476, D. van Nostrand Company, Inc., New York 15 Nijhuis, H. (1974), Thesis Agricultural University Wageningen, H. Veenman and Zn, Wageningen 16 Craig, L. C. and Craig, D. (1950) in Technique of Organic Chemistry (Weissberger, A, ed.), Vot. I11, pp. 171-311, lnterscien¢¢, New Y o r k
17 Schmidt, D. G. (1970) Biochim. Biophys. Acta 207, 130--138 18 Noelken0 M. and Reibst¢in, M. (1968) Arch. Biochem. Biophys. 123, 397-402
ELECTROPHORESIS OF IN:?ERACTING PROTEINS I1. A S I M P L E M E T H O D O F S I M U L A T I O N
H. N I J H U I S a n d T. A. J. P A Y E N S
Department of Biochemistry. Agrictd;arai Unirerslty Wagenbtgen, and Netherlands htstitute for Dairy Research. Erie (The Netherl~'.ds) (Received A u g u s t 22rid, 1973)
SUMMARY
G o a d ' s method for the numerical solution ,:)f the conservation-of-mass equation of a system of migrating and simultaneously interacting macromolecules is simplified by taking into account the velocity flu:~es between adjacent boxes only. The error introduced in this way is used to a d v a r t a g e to simulate the effect of the diffusionai flux. It is shown that this procedure is e~sentially the same as the countercurrent analog method developed by Bethune and Kegeles to account for the effect of diffusion on transport patterns of interacting proteins. The adjustment of the simulated diffusion cgefficients to their actual values is discussed and the results of different calculations are compared. It is shown that diffusional spreading has only a secondary influence on the development of a reaction boundary.
INTRODUCTION
Transport techniques such as eleetrophoresis, ultracentQifugation and gel filtration can be used to advantage in studying biopolymer interactions [1-6]. Several methods have been proposed to solve the conservation-of-mass equation for a system of interacting biopolymers in transport experiments. Gilbert [7] and Gilbert and Jenkins [1 ] have presented analyt;, al expressions for ':he concentration and the concentration gradient in p o l y m e r i z i n g ~ n d complex forming systems neglecting the effect of diffusion on the spreading of a boundary. The usefulness of this approach resides in the fact ~hat in.prolonged experiments the contr:bution of diffusioqpoften can be neglected when compared to the spreading caused by the differentiaVmigration of the different polymer species. It is realized, howe~er, that in experiments of finite duration a n d / o r with the occurrence of small peaks ()r shoulders, it may be impt>rtant tc assess the blurring effect of the diffusion on the saape of a boundary. This has led a n u m b e r of authors to solve the complete conservation-of-mass equation by numerical or simulati( n methods [2, 8, 9]. C a n n apd G o a d [2] solved the complete conservation°of-mass equation by considering the contributions of the indivudual species to the fluxes due to velocity
214 and diffusional transport, In this approach the b o u n d a r y is divided into a large number of segments and the flux at a particular segment edge is calculated from the average concentrations in a n u m b e r of neighbouring boxes. Equilibrium between the individual species is re-established after each transport cycle. Actually the flux at a particular edge is related to the average concentration in all boxes. G o a d showed that if the velocity flux is restricted to the transfer of material from one box to the next a diffusion-like error is introduced. This author therefore preferred to suppress this error by taking into account the contribution of the average concentrations o f several boxes to the flux and to introduce the diffusional flux separately. This is certainly a most accurate procedure, but we shall demonstrate below that for practical purposes the diffusion-like error can be used to simulate the diffusional flux, which leads to a drastic reduction of the computations. The method of simulation presented in this paper is (as stated above) a simplification of the C a n n - G o a d procedure, it will be shown that the method also carries a close relationship to the countercurrent simulation approach of Bethune and Kegeles [10, i I ]. Application of the method to the simu;ation of the anomalies observed during the electrophoresis of complex forming c~- and r-casein has been presented in Part I of this series [12]. THEORY AND DISCUSSION
Let us consider the following system of interacting biopolymers A and B: iA~ ~A~ jA, + B~AIB
(i----2,3...p) [ (j== ! , 2 . . . q) J
(I)
The complex formation between a~- and//-casein at the temperature of free electrophoresis [12] and the iW,eraction between R N A and virus coat protein [13] offer among others [3] excellent examples of such a system. The computation of the equilibrium concentrations of the various species requires the solution of the following set of equations: gi
. l AI/AI
(i =~ 2.3 ..... p)
Lj
AjB/A~. B
~_-_-
Z~ A i +
P
(j ::- 1,2 .....q)
(2) (3]
#
I=l
~' (t -- ;.j)A~B
(4)
./=1 q
==~ B + Z' ,;.jAjB J=l
(5)
In tk'.se equations K~ and Lj represent the equilibrium constants for the self-association of A and for the complex formation between A and B. respectively, /~ and 1~ stand for the constituent concentration (w/v) of A and B and ;.j is defined as •~j ~
Mu/(jMa
-+-
Mn)
with MA and Mn the m o n o m e r molecular weights of A and B.
215 Eqns 2-5 lead to the following expression for the m o n o m e r concentration~ of Component A: P
q
q
( 2~ K, AI)( 27 ~L,A~) 4- 2~ lIB - (A 4- B)2j }L~AJ,I ---- 0 l=l
(6)
j=O
j=O
with K1 =- Lo = i. Eqn 6 is a polynomial in the m o n o m e r concentration At, the graphical appearance of which is shown in Fig. I. In the computation of the equilibrium concentration, At is found by N e w t o n - R a p h s o n iteration [14]. The constituent concentration ~ , being the highest possible value of Al, appears to be a natural ~tarting value for the iteration. Actually, however, the n u m b e r of iteration steps can be reduced considerably by estimating the starting value of At in a particular box from its concentration in the preceding one [i 5].
/
/
/
/
/
/
/
•
/
/
/~
/
Fig. I. Schematic plot o f the p o l y n o m i a l (Eqn 6) used for the c o m p u t a t i o n o f the m o n o m e r concent r a t i o n (A,) o f C o m p o n e n t A.
The simulation of the free elcctrophorcsis of a system of interacting biopolymers is achieved by subdividing the electrophoretic channel into a large number of small boxes of equal length, lx. The development of the reaction boundaries is brought about by alternate rounds of velocity transport of the indiv,dual species from one box to the next during an interval of t i m e . lt, followed by re-equilibration according to Eqns 2-5, On the ascending side velocities are taken relative to the slowest species (i.e. C o m p o n e n t B) and the ratio dx/,it is chosen such that the fastest component just reaches the end of a box, therefore VA -- Va -=
(7)
216 The complexes, having intermediate mobilities [12], penetrate the boxes only partially. Similarly on the descending side the velocities are c h a n g e d o f sign a n d taken relative to the c o m p o n e n t with the highest absolute value of the electrophoretic mobility (i.e. C o m p o n e n t A). The source p r e g r a m for the simulation, a flow scheme o f which is presented in Fig. 2, was written in A L G O L - 6 0 . All c o m p u t a t i o n s have been carried o u t on the CDC-3200 digital c o m p u t e r complex o f the Agricultural University.
I
o,,,, I
Fig. 2. Flow diagram for the simulation of reaction boundaries of complex forming proteins in transport experiments. It is worthwh,~le to analyse the above simulation p r o c e d u r e s o m e w h a t further. With the (n -~- I)th transfer, the change o f mass in Box n u m b e r r d u e to C o m p o n e n t i is given by ..iml .
• +~
~
m,.,+~
--
~
m,..
i
:.L(m~_~.,-
i
m,~,)
(8)
where fl == Vf
zlt//Jx
a n d VI is the relative velocity of Species i.
(9)
217 Obviously, f r o m o u r choice o f reference o f the relative velocities a n d of d t / A x ( E q n 7), we have 0 < 3q < I. T h e mass transfer o f each species from one box to the next is thus seen to be d e t e r m i n e d by a c o n s t a n t fraction Ji o f the mass difference between adjacent boxes. This transfer is therefore f o u n d to be identical with the mass transfer taking place in the e o u n t e r c u r r e n t distributional process. I n the latter case we h a v e j i -~ Pi/(Pt • ~ I ), where Pi is the p a r t i t i o n coefficient o f the Species i [16]. As m e n t i o n e d in the Introduction, B e t h u n e a n d Kegeles [9-11] have use the mass distribution achieved in a cont i n u o u s c o u n t e r c u r r e n t process to a c c o u n t for the effect o f the diffusional spreading on the t r a n s p o r t p a t t e r n o f interacting proteins. It is worth noting, however, that the c o u n t e r c u r r e n t a n a l o g involves m o r e c o m p u t a t i o n s than the present m e t h o d of simulation, since it requires n o t only the c o m p u t a t i o n of the partition equilibrium but also those o f the chemical equilibria in both the upper a n d lower layer o f each counterc u r r e n t tube. F o l l o w i n g Bethune [9] we thus find for the mass distribution of Species i after a sufficiently large n u m b e r o f transfer cycles n :
6Ci/6n~-½./i(i
(lO)
--fOb2Cl/brZ--fl3Cl/br
where r is box n u m b e r a n d Ct is the c o n c e n t r a t i o n of Species i in g/dl. Similarly, for the mixture o f interacting biopolymers w'c have
h._,rC,/bn :: t)2{.~7½.~(I --.f~)C~ }liar 2 -- ,~..~f~C~/hr i
i
( I 1)
i
If now, following again Bethune and Kegeles, we draw the following analogies: a ,--, t a n d r *-, x, then Eqns l0 a n d ! l show the formal analogy with the complete conservation-of-mass e q u a t i o n in t r a n s p o r t experiments in which the distance is expressed in units o f length , I x a n d t i m e in units tit. As a consequence the simulated diffusion coefficient o f C o m p o n e n t i, expressed in cm2/s, becomes Di* -- ½ji(l - J i ) (/lx)z//]t
(12)
In the mixture o f b i o p o l y m e r s the actual diffusional flux is defined by 18]
Jo =~ B.U(i~Ci/,~x)
(13)
i
w h e r e / ~ is the gradient averaged diffusion coefficient (14)
D ~-~ _v[O, ~Cdax]iZ[OCi/i~x]
:nd Dl is the true diffusion coefficient o f C o m p o n e n t i. in a c c o r d a n c e with Eqn 12 the simulated gradient averaged diffusion coefficient, expressed in em2/s, b e c o m e s D- = [....E{½f~(i -- fObC,/br}/2;{bC,/&}l(Ax)2/At I
1
(15)
218 The adjustment o f the simulated to the actual diffusion coefficient now d e m a n d s that /5, ~ /5
(16)
Eqns 14--16 and 7 show that d x and At are fixed by the values o f the relative velocities, the diffusion coefficients DI and the concentration gradients ~Ct/~x. The latter will be discussed below. The number of transfer cycles, n, which is necessary for the simulation o f a particular reaction boundary, finally follows as the ratio of the time of the experiment a n d / I t . Conversely in the simulation o f a given experiment an arbitrary choice o f n determines together with the duration of the experiment an arbitrary /It. The correspemding fx then follows from Eqn 7 and the diffusional spreading operative in this simulation from Eqn 15. It is obvious that in general then /5* will not e q u a l / ) . We are now to able to elaborate upon the simulation of the electrophoresis of complex forming a~l- and fl-casein dealt with in the previous paper [12]. Since we have no a priori knowledge o f the concentration gradients ~C~/~x existing in the reaction boundaries, we must start with a comparison of the simulated patterns coraputed for arbitrary numbers of transfer. Typical results of the simulation of the ascending electrophoretic pattern of a I:l mixtures of ~¢~- and fl-casein for n : 27, 46, 73 and a duration of the experiment o f 4800 s are presented in Figs 3a-3c, from which it is seen, that the general appearance of the pattern is not affected substantially by the number of transfers. More importantly, as is seen from Table I, also the maximum and minimum gradient mobilities and the concentration changes over the reaction boundary do not change appreciably with n. This result, of course, is in agreement with Gilbert and Jenkins" conclusion [I] that in prolonged experiments the effect of diffusional spreading on a boundary is negligible. In other words: the proper choice of the number of transfers is o f minor importance for a correct simulation of the percentages and mobilities found from experimental electrophoretic patterns. The calculations underlying Table i and Fig. 3 also yield the concentration gradients necessary for comparison of the averaged true and simulated diffusion coeflicienls defined by Eqns 14 and 15. To this end v.e have approximated the various ,~C~/Oxby the total concentration change of the Species i over the boundary. The true value o f !3 is then estimated as follows. From existing data [[7. t8] we find the diffusion coefficient, D~. of monomeric ¢~- and fi-casein, corrected for the t e m p e r a t u r e of free eleclrophoresis, to be 3.2.10 -~ cm2/s *. The diffusion coetticients of the polymers or complexes containing i subunits are then calculated from that of the monoo mers by the Stokes-Einstein relation: D~=-- D ~ ' i ~
(17)
A similar calculation is performed for the estimation of the simulated diffusion coefficient, /5", according to Eqn 15. The comparison of t h e / 5 a n d / 5 * estimated in this way for n 27. 46 and 73 is given in Table !!, from which it may be noted that the value o f the diffusion coeffi" The molecular parameters of monomeric +,~- and fl-casein are almost the same [17. 18].
219 ,
.
.,
.
!
.
.
il
i
i.
~ i
!i
~ ~
,!~.,
,
X K K M ~ K ~ K W ~
~
;
" !
ii
:
XX
K ~ M X K K M
~~ NNK ~~N N ~ N~K X ~ N ~N K M I NKKKKKNN~ K KN M~ N K~MMNNKNKX N~KNNN~K~ ~NXK~K~N~N~ N W N ~ N M ~ N
ascending
ascendin9
a
b
I
C
ascending Fig. 3. Simulation of the ascending elcctrophoretic react!on boundaries for a 1:1 mixtu;'e of ,t,=- and /~-¢,asein. (a) (top left) Number of transfers 27. (b) (top right) Number of transfers 4 6 (¢) Number of
220 TABLE 1 COMPARING VELOCITIES AND PERCENTAGES OF THE ASCENDING ELECTROP H O R E T I C B O U N D A R I E S O F C O M P L E X F O R M I N G ~1- A N D /%CASEIN S I M U L A T E D FOR DIFFERENT NUMBERS OF TRANSFER Number of Number of Velocities* transfers boxes (IO -s el-n/s)
27
25 40 60
46 73
1.96 1.94 1.94
!.64 1.60 1.58
P ~ c e n t a g es*° At - A2 ..... ,4s 1.09 1.09 1.07
0.30 0.31 0.31
0.24 0.22 0.22
0.47 0.47 0.46
" Velocities corresponding to the maxil.numn and minimum gradients o f the reaction boundaries. *" At relative area of pure ~t-easein; ,'lz and As. relative areas under bimnoda| peak.
cient, D, is hardly affected by this change of n. The simulated diffusion coefficients/~*, however, are found to be inversely proportional to ~, as expected from the theory presented above. Actually Table Ii indicates that the most accurate number of transfers should be about 35, cot responding to an averaged simulated diffusion coefficient of 2.55.10 -~ cm2]s. Similar considerations hold for the simulation of descending electrophoretic or ultracentrifugal patterns. T A B L E li COMPARING SIMULATED AND TRUE GRADIENT AVERAGED DIFFUSION COEFF I C I E N T S P E R T A I N I N G T O T H E C O M P U T A T I O N S P R E S E N T E D IN T A B L E 1 Nqmnbgr of . It {s) transfers
27 4~! 7?.
17"P.8 104.3 658
lx (r'mn)
277.3 162.8 102.6
tX)z F : - IOT. . . . . It (cmnZ,'s) Eqn. [ 5
/3. tO~ (cl.nZ/s)
3.286 1.958 1,241
2.527 2.534 2.536
Eqn 14
The above analysis shows the usefulness of the simplified procedure for the simulation or the reaction boundaries observed in the electrophoresis (or ultracentrifugationj of interacting proteins, i n agreement with the conclusions arrived at by Cilbert and Jenkins [I ] the value of the average simulated diffusion coefficient thereby ai~pears to be of minor importance for the reproduction of the experimental peak areas and mobilities. The adjustment of the simulated to true diffusion coefficient can easily be achieved, however, by carrying out a preliminary computation with an arbitrary number of transfers. From this the true average diffusion coefficient, /5, is estimated as exposed above. The most realistic number of transfers then follows from the equalization of the tl ue and simulated average diffusion coefficients { Eqn 16). Bethune [9] has presented an alternative procedure for the adjustment of the simulated to the true diffusion coefficient for the case o f discrete self-association. His treatment can also be extended to complex formations of the kind A q- B -~ C, as will be shown elsewhere (Nijhuis, thesis to be published).
221 ACKNOWLEDGEMENTS T h i s s t u d y w a s s p o n s o r e d by t h e N e t h e r l a n d s F o u n d a t i o n for C h e m i c a l Res e a r c h ( S . O . N . ) , w i t h f i na nc i a l ai d f r o m t he N e t h e r l a n d s O r g a n i z a t i o n f o r the Adv a n c e m e n t o f P u r e R e s e a r c h ( Z . W . O . ) . W e a r e grateful t o the s t a f f o f A g r i c u l t u r a l U n i v e r s i t y C o m p u t e r D e p a r t m e n t f o r c o m p u t i n g facilities. We arc m u c h i ndept ed t o D r H . J. V r e e m a n , N e t h e r l a n d s I n s t i t u t e f o r D a i r y R e s e a r c h , Ede, f o r critical remarks. REFERENCES 1 Gilbert, G. A. and Jenkins, R. C. LI. (1959) Proc. Royal Soc. London, Ser. A, 253, 420~37 2 Cann, J. R. and Goad, W. B. (1970) Interacting Macromolecules, Academic Press, New York 3 Nichol, L W., Bethune, J. L., Kegeles, G. and Hess, E. L. (1964) in The Protein~ (Neurath, H., ¢cl.), 2nd edn, Vol. 2, pp. 305- 403, Academic Press, New York 4 Ackers, (3. K. (1970) Advances in Protein Chemistry, Vol. 24, pp. 343-446, Academic Press, New York 5 Nichol, L. W. and Winzor, D. J. (1964) J. Phys. Chem. 68, 2455-2463 6 Winzor, D. J. and Scheraga, H. A. (1963) Biochemistry 2, 1263-1267 7 Gilbert, G. A. (1959) Proc. Royal Soc. London Set. A., 250, 377-388 8 Cox, D. J. (1971) Arch. Biochern. Biophys. 146, 181-195 ') Bethune, J. L. (1970) J. Phys. Chem. 74. 3837-3845 l, ~ liethune. J. L. and Kggelcs, G. (I961) J. Phys. Chem. 65, 1755-1760 li Belhune, J. L. and Kegeles, G. (1961) J. Phys. Chem. 65, 1761-1764 12 Payens, T. A, J. and Nijhuis, H. (1974) Biochim. Biophys. Acta 336, 201-212 13 Durham, A. C. H., Finch, J. T. and Klug, A. (1971) Nat. New Biol. 229, 37-42 14 Margenau, H. and Murphy, G. M. (I943) The Mathematics of Physics and Chemistry, 14th edn, pp. 475-476, D. van Nostrand Company, Inc., New York 15 Nijhuis, H. (1974), Thesis Agricultural University Wageningen, H. Veenman and Zn, Wageningen 16 Craig, L. C. and Craig, D. (1950) in Technique of Organic Chemistry (Weissberger, A, ed.), Vot. I11, pp. 171-311, lnterscien¢¢, New Y o r k
17 Schmidt, D. G. (1970) Biochim. Biophys. Acta 207, 130--138 18 Noelken0 M. and Reibst¢in, M. (1968) Arch. Biochem. Biophys. 123, 397-402