BULLI~TIN O~F

~ATH~MATIC~ BmPHYSICS VOLUMV. 31, 1969

APPLICATION OF MOBILITIES IN ELECTROMAGNETIC FIELDS TO THE DETERMINATION OF KINETIC PARAMETERS

[ ] R. A. MTT.T.S, Department of Mathematics, University College of Townsville, Queensland 4812, Australia

Changes in eleetrophoretie and/or eleetromagnetophoretie mobilities during the course of biochemical reactions are related to first order rate constants of those reactions. By linearization of the mobility/reaction-coordinate relations, a method for the determina- tion of rate constants is suggested, an assessment being made of some likely advantages and limitations of this approach to kinetic problems.

1. Introduction. Certain reactions involving biological materials have been studied by measuring electrophoretie mobilities after different reaction-times (Brinton and Lauffer, 1959). An extensive s tudy of this kind was carried out by Fischer and Lauffer (1949) on the reaction between formaldehyde and tobacco mosaic virus by stopping reaction with the aid of specially prepared buffer, thus lowering the pH to the isoelectric point of the virus and bringing about precipitation. The mean electrophoretic mobility of a given precipitate was found to depend on the reaction-time prior to precipitation, and functions describing this dependence yielded estimates of rate constants based on pro- posed reaction mechanisms.

To date no such studies have been conducted with the aid of electromagneto- phoresis but, in view of the fact tha t an additional mobility is determined by this technique, it would seem to provide a much more sensitive approach to

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564 R.A. MILLS

reaction mechanisms. In what follows, therefore, both electrophoretic and e!ectromagnetophoretic mobilities will be considered.

2. Mobilities from Boundary Velocities. In an earlier paper (Mills, 1968) it was indicated that particles suspended in conducting fluids subjected to mu- tually orthogonal electric and magnetic fields will acquire velocities deter- mined by diffusive, viscous and electromagnetic forces. I f a system contain- ing two layers, one of suspension and the other of pure conducting fluid, is placed in crossed electric and magnetic fields, the boundary between the layers is found to move with a velocity v given, to a first approximation, by

V = U1E + U2aE • B (1) where

U1 = electrophoretic mobility U2 = electromagnetophoretic mobility

a = conductivity of fluid E = electric field strength B -- magnetic induction.

Equat ion (1) is most accurate when the suspension is dilute and only small fields are applied. For simplicity it will be assumed that the mobilities U1 and U2, as well as the conductivity a, may be treated as scalars. One may then express boundary velocities in terms of the current density j by introducing the relation

j = (2)

whence v = (U1/a)j + U2j • B. (3)

Equat ion (3) enables one to estimate the mobilities U1 and U 2 from observed velocity components in the directions of j and j x B.

3. Relations Between Mobilities and Reaction-Times. Biochemical reac- tions are often very difficult to understand because of the presence of several types of active site on the reacting species. Relations derived in this section are intended to show how mobilities may be used to investigate reaction mech- anisms at these sites by assessing how far each type of reaction has proceeded. In addition to elucidating biochemical reaction mechanisms, it is also fore- seeable that mobility data will provide a useful means for the determination of rates and mechanisms of other biological processes. An indication of this is seen in the work of Mayhew (]966) who has measured electrophoretic mobili- ties at successive stages of the mitotic cycle and has used his data to suggest

M O B I L I T I E S I N E L E C T R O M A G ~ E T I C F I E L D S 565

corresponding changes in cell chemistry. For the present, however, only first-order biochemical reactions will be considered.

As a preliminary to the derivation of mobility/reaction-time relations, a number of quantities must be defined. In a given system, let there be p types of active site initially present on each particle of the biochemical species and let aj be the overall fraction of sites type j deactivated after a reaction- t ime t. A p-dimensional vector space Vp may be constructed from the ordered p-tuples ~ = (al, a2 . . . . . %) and progress of the various reactions described by the trajectory in V~ traced out by a as reaction-time increases. V~ will be termed reaction space, and corresponding to a given point ~ of Vp there will be a unique ordered pair U = (U1, U2). The latter may be considered the image point of ~ under the mappings

U, = Ut(a) i = 1,2 (4)

and will be regarded as a point in a new spade M which will be called mobility

space. The above formulation will now be used to derive mobility/reaction-co-

ordinate relations by first expanding the Ui as Taylor series in al, a~ . . . . . % about some point r = (a*, a* . . . . . a*) of Vv, tha t is

p

U,(~) = U,(a*) + ~ a,i(a s - a'~) Sffil

+ - - + . . . ( i - - 1 , 2 ) ( 5 ) lffil =

where

% \~aj]r b,,, = ~ a s 3a----~J.."'" ( i - - 1, 2). (6)

Finally the trajectories in V~ and M will be parameterized by means of the reaction4ime. To do this it will be assumed tha t in all cases an excess of reagent is added to the biochemical species so tha t reaction at each active site of the latter may be regarded as irreversible and of first order. Under these conditions it follows that there exist constants k s > 0 such tha t

da---! = kj(1 - aj) (j = 1, 2, . ,p) . (7) dt ""

Integration of (7) subject to the initial conditions a s = 0 (j = 1, 2 . . . . ,p) when t = 0 gives

a s = 1 --e-kJ t ( j = 1,2 . . . . . p). (8)

566 R.A. MILLS

On substitution of (8) in (5), expressions for the Ut are obtained in terms of the rate constants kj and reaction-times t and t* corresponding to a and a* respectively. Thus writing U~(t) for the mobility Ut considered as a function of the reaction-time t, one obtains the approximate relation

P

Udt) - U,(t*) = A t - ~ as, e-kJ t (i = 1,2) (9) i = 1

where P

= e-~ , ~* (i 1 , 2 ) (lO) A f ~ a~j = j = l

by truncating (5) after the terms containing %. The value of t* has not yet been specified, but to give the expression (9) its greatest accuracy, a value near the mid-point of the experimental reaction-times should be chosen in each case so that deviations from linearity in the mobility/reaction-coordinate relations (5), that is truncation errors, are minimized.

4. Es t ima t ion o f _Rate Constants. Except for the A t, mobility differences Udt) - U~(t*) in (9) are the sums of exponential functions of time. Parsons (1968) has presented an interesting method for estimating constants in such expressions without using measurements taken over long intervals of time (as in the conventional procedure). This is clearly advantageous here because the truncation errors discussed earlier increase as I t - t* I increases. At the same time, the above author's finite-difference technique allows one to evaluate the number of constants tha t may be determined from any given set of data. The method is most useful when observations are made at equally spaced intervals of time, otherwise interpolation procedures are necessary. But in the present ease this stipulation does not introduce any further difficulty.

Estimates of the A t may be obtained from (9) since

lim U~(t) - U , ( t * ) = A t ( i = 1, 2) , ( 1 1 ) t--~ cO

tha t is, equation (9) may be rewritten

P

= e-kit Ud~)- Udt) ~1% (12)

When the a~j and which is of the form required for computational purposes. k s have been estimated, substitution in equation (10) yields second estimates of the A t and the calculations may then be repeated until satisfactory agree- ment between successive estimates of the A t is achieved.

MOBILITIES IN ELECTROMAGNETIC FIELDS 567

5. Linearization of Equations. In truncating equations (5), it has been assumed that the mobility/reaction-coordinate relations are almost linear. The validity of this assumption must be examined if reliable estimates of rate constants are to be obtained from (9). Unfortunately no data are available to test this in the case of electromagnetophoresis, but electrophoretic experi- ments suggest that approximately linear relations hold over sufficiently small regions of reaction space. Thus a change of 17 per cent in the degree of sulphonation of sodium polyvinyl-sulphonate has been shown to give rise to a change of electrophoretic mobility which, to within the limits of experimental error, is also equal to 17 per cent (Nagasawa, Soda and Kagawa, 1958). A more extensive s tudy of this type reported by Noda, Nagasawa and Ota (1964) indicates that a linear relationship may be useful over small regions of I/v but not over the entire course of a reaction. Further experimental evidence is afforded by the mobility data of Fischer and Lauffer (1949) which follow very closely an equation similar to (9).

In addition to this experimental justification, it is also known that at low transport velocities and small electric field strengths a particle moving with velocity v through a viscous fluid experiences a retarding force approximately equal to fv, where f is the frictional coefficient. Further an electrical field of strength E exerts an accelerating force approximately equal to QE on a particle of charge Q. In steady electrophoretic motion, these two forces will be equal in magnitude and hence, from the relation

v = U1E (13) it follows that

Uz = Q/f. (14)

Thus, provided the coefficient f is constant throughout reaction, U1 will vary linearly with the charge Q which, in turn, is a linear function of the reaction coordinates. I t is assumed that similar linear relations hold for electromagneto- phoresis.

6. Conclusions. The use of electrophoretie mobilities in studying biochemi- cal kinetics as discussed by Brinton and Lauffer (1959) possesses, in certain cases, several distinct advantages over other analytical techniques. Thus the relative concentrations of several components produced by reaction may be determined from optical measurements during electrophoresis, whereas chemi- cal analysis may only yield an average composition for the whole sample. In addition, electrophoresis succeeds in some instances where analysis is made difficult due to the breakdown of molecular structure in the presence of cer- tain reagents. Finally, transport processes tend to be very sensitive to small

568 R.A. MILLS

changes in the propert ies of mater ials such as thei r electric charge. The present t r e a t m e n t is a general izat ion to the e lectromagnet ic case and is inten- ded to serve as a guide in the de te rmina t ion of ra te constants f rom more exten-

sive mobi l i ty data . I n using this method, however, i t is necessary to stop react ion before meas-

ur ing t r anspor t coefficients. This m a y be difficult in some cases, and i t is for this reason t ha t the me thod is no t sui table for the s t u d y of fast reactions. Others (see, for example, Gilbert and Jenkins , 1959; Bak and K au m an , 1959) have s tudied a l ternat ive approaches in which react ion and t r anspor t occur s imul taneously and have ob ta ined first order ra te constants for cer ta in fast ionic react ions (Bak, 1963) b y this means. However , in te rp re ta t ion of the results of such exper iments m a y be difficult and the s i tua t ion is fu r the r com- pl icated by the t empera tu re -dependence of mobilities, heats of react ion being

a possible source of difficulty. This research was suppor ted in pa r t by a Commonweal th Pos tg radua te

Award, for which the au thor would like to express his thanks .

LITERATURE

Bak, T. A. and W. G. Kauman. i959. "Theory of Electro--ion." Trans. Ear. Sot., 55, 1109-1121.

1963. Contributions to the Theory of Chemical Kinetics. New York : Benjamin. Brinton, C. C. (Jr.) and M. A. Lauffer. 1959. In Electrophoresie, M. Bier, ed. Now

York: Academic Press. Fischer, M. A. and M. A. Lauffer. 1949. "The Reaction of Tobacco Mosaic Virus with

Formaldehyde. I I Kinetics." J. Amer. Chem. Soc., 71, 3800-3804. Gilbert, G. A. and 1%. C. L. Jenkins. 1959. "Sedimentation and Electrophoresis of

Interacting Substances. II Asymptotic Boundary Shape for Two Substances Inter- acting Reversibly." Proc. Roy. Soc., A253, 420-437.

Mayhew, E. 1966. "Cellular Electrophoretic Mobility and the Mitotic Cycle." J. Gen. Physiol., 49, 717-725.

Mills, R . A . 1968. "A Macroscopic Formulation of Electromagnetophoresis." Bu/l. Math. Biophysics, 30, 309-318.

Nagasawa, M., A. Soda and I. Kagawa. 1958. "Electrophoresis of Polyelectrolyte in Salt Solutions." J . Polymer. Science, S1, 439-451.

Noda, I., M. Nagasawa and M. Ota. 1964. "Electrophoresis of a Polyelectrolyte in Solutions of High Ionic Strength." J. Amer. Chem. Soc., 86, 5075-5079.

Parsons, D .H. 1968. "Biological Problems Involving Sums of Exponential Functions of Time: A Mathematical Analysis that Reduces Experimental Time." Math. Bio Sciences, 2, 123-128.

RECEIVED 1-17-69