Eleclrophoresis 1990,11,281-289

Steady state electrolysis and isoelectric focusing

28 1

Original papers Rolf Hagedorn Gunter Fuhr

Steady state electrolysis and isoelectric focusing

Department of Biology, HumboldtUniversity of Berlin

The properties of pH gradients formed by stationary electrolysis of weak mobile or fixed electrolytes are analyzed. The model uses the appropriate balance equations and those of chemical equilibria. It is shown how the equation of the current density has to be modified for considering that fraction of current that is associated with the diffusion of neutral buffer molecules within a pH gradient. Furthermore it is shown that the pH gradients themselves give rise to water production within the gradient and that essential properties of the steady state are related to chemical reactions between the electrolyte constituents. The differential equations describing the gradients of the concentration of a given component, the pH, conductivity and potential are explicitly formulated in relation to those reactions. The equations are solved numerically and the significance of the results for isoelectric focusing is discussed. The experimental conditions to reach shallow and smooth pH gradients exhibiting sufficient ionic strength are formulated.

1 Introduction In the 1960’s, Svensson deduced the properties of useful isoelectric focusing (IEF) systems by a theoretical analysis of steady state electrolysis [ 1-31. Unfortunately, the shape ofpH gradients formed by electrolysis of a simple salt are unsuitable for IEF since at sufficiently high field strength the anions of the salt are completely separated from the cations by an undesirable zone of pure water. According to Svensson it is possible to improve the properties of the pH gradients by the use of carrier ampholytes. If such ampholytes are added to the initial salt solution they will be focused in the region of water that would be developed in their absence. In this way the former water region is filled by the added ampholytes and pH gradients useful for IEF are generated. This method became feasible in practice with the synthesis of suitable carrier ampholytes by Vesterberg in 1969 [4]. Without doubt Svensson’s concept of carrier ampholytes laid the theoretical foundations of IEF and his theories have brought great success in practice. However, experimental investigations have shown that some problems of practical and theoretical interest remain. These problems are: (i) During IEF a progressive pH gradient decay, incompatible with the assumption of a steady state, is observed. (ii) It was experimentally found that the prolonged electrolysis of mixtures of weak electrolytes (acids and/or bases) results in a more or less stepwise pH gradient [5- 121 with electrolytes aligned along the electric field in the order of increasing pK values. At first glance such a result appears self-evident. However, according to Rilbe’s (Svensson’s) theory [ 131 such an aligment should at best be of a transient nature. The movement of a weak acid towards the acidic region of the anode may result in a total loss of charge. This stops the electric migration of the given acid but not its diffusion and thus a mixture of more than two weak electrolytes (one acid and one base) may never reach equilibrium conditions [ 121. (iii) If during an IEF experiment a steady state is reached, all components (except H ‘and OH-) should be at rest and the question arises how the pH gradient can be stable with Correspondence: Dr. R. Hagedorn, Humboldt-Universitat zu Berlin, Sektion Biologie, InvalidenstraDe 43, DDR-1040 Berlin, German Democratic Republic Abbreviations: IEF, isoelectric focusing; Tris, tris(hydroxymethy1)aminomethane 0VCH Verlagsgesellschaft mbH, D-6940 Weinheim, 1990

time: The relation between the currents carried by the protons and hydroxyde ions varies with the pH. Consequently, each of both currents is divergent. This, in turn, seems to exclude a stationary concentration of protons. In the last few years several theoretical treatments of steady state electrolysis and related processes have been published [ 14-29]. The major theoretical treatments ofthe generation of pH gradients are the steady state formulation of Palusinski et al. [ 19, 201 and the transient model of the Bier group [25, 261. Although the basic concept of these theories includes the consideration of chemical reactions between the electrolyte subspecies, the fact has been neglected so far that just by those reactions the diffusion of originally uncharged buffer subspecies produces an electric current. At the steady state this current is counterbalanced by the ordinary current arising from the electromigrational and diffusional fluxes of the ionic subspecies. The theoretical model elaborated in this paper considers the additional current derived from the diffusion of uncharged species. Since the steady state distributions of strong and weak electrolytes are distinguished mainly by this term, it explains the stepwise pH gradients generated on prolonged electrolysis of nonamphoteric buffer mixtures. Furthermore in our model we consider the water production occurring within pH gradients as a result ofchemical reactions between the protons and hydroxide ions. Starting from the law of mass conservation, the appropriate equations (pH gradient, charge distribution, water formation, field strength and conductivity) are deduced with the aim of addressing the above questions and to give practical guidelines for the generation of optimum and stationary pH gradients.

2 Theory of steady state electrolysis 2.1 Characterization of the electrolytic steady state

If ci denotes the total concentration of a given buffer component the general balance equation may be written as

where the source term a,ci/at, (in mol/(m3s), is the rate of production and (vici) is the mass flux resulting from the 0173-0835/90/0404-0281$2.50/0

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translocation of ciwith drift velocity vi. The flux may also beexpressed in terms of the current densityji(A/m2) 130, 3 11

ing concentration gradient and an associated pseudo-force of diffusion. At the steady state both forces balanceeach other and the mean drift velocity of the ions becomes zero. Conseji = F Qi (vi ci) ( 2 ) quently, the N a ions can no any longer transfer charge. The where F is Faraday's constant (96 500 C/mol) and Qi is the statement that the conductivity and the transference number mean number of elementary charges per buffer particle as in- of the N a ions is finite is not in conflict with avanishing current troduced by Svensson [2l density. Both the transference number and conductivity are c; - c; only proportional to the current in homogeneous systems Qi = (3 [ 321. In which manner stationary distributed electrolytes may Ci increase the conductivity of the solution is discussed below. The total concentration of an ampholyte is ci = c: t c;

+ c:-

t cp

(4)

where c:- is the concentration of the zwitterionic subspecies, c9 that of the undissociated subspecies, and c: (c;) the concentration of ionic subspecies. If the concentrations of the appropriate subspecies are taken as zero this relation holds also for monovalent electrolytes. Since the equilibrium reaction between undissociated electrolytes and its ionic subspecies cannot alter its total number we get

(i

* H, OH)

(5)

It therefore follows (Eq. 1) that the flux of buffer components and consequently also the ratioj,/Q,(Eq. 2 ) has to be constant at the steady state where a c,/ a t = 0. Furthermore, since the electrodes are neither sinks nor sources of buffer particles (decomposition at the electrodes is excluded) it follows that vi ci = 0 and j , = 0 (i H, OH) (6a, b)

*

It is important to note that Eqs. (5) and (6) cannot be applied to protons and hydroxide ions which exhi bit divergent fluxes. Hence the source terms a, H / a t , a,Ol-I/at cannot vanish. Eq. (6b) leads to the consequence that the whole steady state current is only maintained by the protons and hydroxide ions j = 2 j , = j H tioH =const. (7) Combining this with the conservation law (1) and Eq. (2) we obtain

a r H - arOH at at

(81

(ii) At the steady state the mass flux is either constant (infinite systems, systems with buffer exchange) or zero (finite systems as considered in this work). The statement of a constant flux requires that the ratio jJQiis also constant. Here the quantity Qi is a p H dependent term. The equilibrium in a solution containing a weak undissociated electrolyte and its ionic form may be represented as:

HA = H + t ABOH =+B+ t OHwhere A denotes an acid and B a base. Ifthegiven electrolyteis an ampholyte, then both dissociable groups belong to the same organic ampholyte radical. Taking into account that the given electrolytes may also be ampholytes we define as degree of ionization:

.+

c; t c;c? t c-; = 1 and a;=Ci

Ci

Then, as usual:

where K , and K b are the dissociation constants of the associated reactions (lo), the subscripts a and b referring to acids and bases, respectively. By using Eq. (1 2) the quantity Q (see Eq. 3) is given as:

Qj = (u: -a;) = f(pH)

(13)

Since the dissociation constants for strong electrolytes tend to Since both ion species react to water ( a ,H / a t = - a H,O/a t ) infinity this equation may also be applied to these electrolytes. it can easily be shown (Eqs. 1,2,6,7) that water formation is By using this equation the condition of electroneutrality may given by be written as:

(9) The above steady state conditions are discussed below. (i) At the steady state the current densities of the buffer components (not H, OH) have to vanish. At first glance, the assumption that electrolytes which have achieved a steady state distribution do not contribute to the current seems to be in conflict with accepted theory. Svensson [2] has shown that ampholytes also exhibit finite conductivity in the vicinity of their isoelectric point furthermore, it is known that buffer components which have achieved a stationary distribution also exhibit a finite transference number and finally, it is known that the presence of carrier ampholytes within a pH gradient increases conductivity. In order to discuss these problems we consider an NaOH solution. Two kinds of forces are acting on the N a ions: the electric force which has the direction of the field and the pseudo-force of diffusion with a direction opposite to the concentration gradient of the ions. As long as no concentration gradients exist, the Na ions are transferred toward the cathode. This process in turn creates acorrespond-

Since (jJQi) has to be constant, Eq. (13) shows that the assumption of a constant current density ji is not justified in case of weak electrolytes. (iii) Even at the steady state protons react with hydroxide ions, presuming a p H gradient exists. In general the result is a local formation of water. Water formation/consumption was already predicted by Planck [311 and Kohlrausch [331. (iv) T o complete the discussion we consider the current densitiesj; The current of a given buffer component is commonly expressed as the sum of the contributions of the associated subspecies. ii=-

2 {.nk(Fz~CkV~+RTzkVCk)}

(15)

where the subscript k (for once) numbers the subspecies. R is the electrophoretic mobility, m2/(V s), vQ,is the potential gradient, R is the molar gas constant, 8:3 145 J/(K mol) and T the temperature (in K). The valences zk are constant integers

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(1, 0, - 1, . . .).It seems plausible that the uncharged subspecies do not contribute to the current ( z k = 0). However, repeating Planck's derivations [ 30, 3 11 it becomes evident that not the valences but the mean charge Qi (Eq. 3) has to be used in Eq. ( 14).

where ci denotes the total concentration of the buffer component and Q,its average charge (with sign). Even in the excellent work ofBier et al. [26] anequationsimilartoEq.(l6)wasonly used for proteins to describe the current. The other components were considered by Eq. (1 5). The main difference between both equations is due to the fact that the valences are constants whereas Qiis a pH-dependent quantity. The use of the valences neglects the fact that even buffer molecules may become charged while diffusing within a pH gradient. Keeping in mind that protolysis occurs in a much shorter time than electrophoretic and diffusional transport, the buffer particles may be described via their average charge. Compared with Eq. (1 5) the electromigrational term in Eq. (16) is decreased by factor Ql. This factor takes into account that the translation of one elementary charge requires the translocation of l/Ql buffer particles, due to the time domain of protolysis. O n the other hand, both diffusional terms differ by the term RTQ,c,vQl which is not considered by Eq. (15). This term reflects that diffusion occurs within a pH gradient. While diffusing, the average charge of a buffer particle is altered. This is equivalent to a contribution to the current density. The balance equation (1) may also be written for the subspecies of a buffer component. In the case of a weak base we have two such equations: one for the undissociated buffer molecules and a second for the ions. Contrary to the above, more rigorous derivation for the total concentration now the source terms cannot be neglected. Considering a base fromthesteady statecondition ac/at=O(seeEq. 5)itfollows that a c'/a t = -8 P l a t . This condition couples both balance equations for the subspecies. Hence it follows at the steady state by eliminating the source terms that

A(k+covo) =O

(i*H,OH)

283

This follows by differentiating the dissociation product of water [HI [OH] = Kw. Relating the current density to the potential gradient we define:

where the quantity 8(S/m) in contrast to thecommon conductivity (a) is denoted as apparent conductivity. The difference between both is that only the apparent conductivity, but not the common conductivity, pertains to the fraction of current also produced by diffusion. Now the above equations may be solved for the quantities of interest:

v @=

F

{s,

= [HI

ts2}is,

In order to arrive at equations that may be surveyed we have introduced the abbreviations: S, = H

+ OH t 2

ci (uf (1 -uT) + a; ( I -a;))

Fi B

S2 =

2F

Qivci; S3 =

c Q:cj 13

(17)

Here covoisthe finite flux of diffusing uncharged buffer molecules. This shows that ji cannot be a constant quantity and that, furthermore, Fcovomay be considered as a current density devoted to the fact that diffusion occurs within a pH gradient.

For the sake of convenience we have used the summation indices B, F and W concerning moveable buffer constituents (nonamphoteric and amphoteric components (B)), fixed charges (F) and protons/hydroxide ions (W), respectively. Immobilized charges are treated just as moveable charges except that their electrophoretic mobilities are zero.

2.2 Calculation of the pH gradient created by stationary electrolysis The above equations give a complete but implicit description of the steady state. To deduce the explicit equations we need the following additional equations:

3 Results and discussion 3.1 Experimental results In order to exemplify the theoretical predictions some experiments with strong and weak electrolytes have been carried out. Stationary p H gradients were created by prolonged

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electrolysis (24 h) of weak or strong electrolytes in a water cooled vertical density gradient column (see Fig. 1). After the column was filled with electrolyte, the field was applied. The p H gradients were measured by displacing the sucrose gradient with a dense sucrose solution. The p H of the eluate was continuously recorded. Fig. 2 shows the pH gradients obtained by electrolysis of 0.01 mol/L NaOH (Fig. 2a) and 0.02 mol/L Tris (Fig. 2b) at a field strength of 1500 V/m. The gradient was calculated by numerical integration of Eq. (2 1); the following quantities were used: C ~ H =3.6 lO-'rn*/V s; C ~ O H = -2.1. lo-' m2/V s; p H at the cathode = 12.3. In the case of NaOH the length of the actual pH gradient (pH > 7) was measured to be 3.9 cm. Fig. 3 shows the pH gradient formed by electrolysis of an electrolyte containing both NaOH and Tris. The correspondence between the measured and calculated p H profile may at least be considered a semiquantitative verification of the theoretical results. Differences in the initial slopes are mainly due to the pH measurement. The diameter of the spherical, pH sensitive electrode range was about 0.8 cm and the volume of the unstirred pH measuring chamber about 1.5 mL. Hence the sharp pH increment around pH 7 (predicted by the model) could not be registered. Figure 1 . Schemeofthe90mLdensitygradientcolumn.Thelowerplatinum electrode is enclosed by agas-escapetube (t), tightened at its lower end with 3.2 pH Gradient formation acrylamidegel(g).Thedistancebetween both platinum electrodesequals 20 cm. The column was cooled by a water tube (w). Thermoconvection was Petterson [51 demonstrated that acidic pH gradients between suppressed by a linear sucrose gradient (0-1 mol/L).

0

X

Figure 2. pH Gradients formed after 24 h electrolysis. (a) 0.01 M NaOH solution. Measured gradient: open circles; calculated gradient: full line. (b) 0.02 M Tris-solution. Measured gradient: closed circles; calculated gradient: solidline. Field strength: 1500V/m,finalcurrentdensity:0.5 Aim'.

"7 X Figure3. pH Gradient afterelectrolysisofanNaOH/Trismixture(CNaOH = 0.01 rnol/L; CTns= 0.04 mol/L). Experimental conditions as in Fig. 2. Measured gradient: circles; calculated gradient: solid line. For the calculation we have used pK, of the Tris-buffer = 14-8.08 = 5.92.

p H 1 and 3 could be formed by electrolysis of a mixture of six acids and two ampholytes. These pH gradients increased stepwise. In a series of further publications 16-1 1I Petterson's results were confirmed and extended. They indicate that (i) pH gradients created by electrolysis are in principle suited for IEF, and (ii) that the experimental conditions leadjng to proper pH gradients are hardly to be found by empirical variations. Although the poor properties of pH gradients formed so far by weak buffers have given this technique a poor reputation 1341, it seems to us that it still has potential. One of the most successful attempts to generate p H gradients by buffer mixtures was by [ 111. Unfortunately, the majority of buffer components as used by these authors (47 components) were ampholeric buffers with properties unsuited for pH gradient generation [341.Thus, in acertain sense, conventionalIEF with carrier ampholytes was substituted by IEF with unsuitable carrier ampholytes and some added weak electrolytes. Other investigators have successfully used weak electrolytes in order to improve the properties of carrier ampholyte generated pH gradients r351. In contrast to these studies we restrict the following discussion to pH gradients formed by electrolysis of pure nonamphoteric buffers. In order to interpret the steady state we have calculated the distributions of the buffer species corresponding to the pH gradient of Fig. 3 (see Fig. 4). According to Fig. 4 (lower part) the slope of the buffer molecule concentration (solid 1ine)does not vanish and thus the molecules will still keep diffusing from the right to the left side even at the steady state. The concentration gradient of the charged buffer component (dotted line) is unable to stop the migration of the ions (at the right hand side of the maximum concentration of weak ions the ion gradient even favors electromigration). Thus at the steady state a translocation of ions will also occur. But a molecule starting from a position around X2 will reach a position around X 1 in a more or less ionized form (due to the dissociation of OH) and likewise an ion starting from position X 1

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285

could participate in charge transfer. This indicates in our opinion an indirect mechanism. (ii) By fractionation of the appropriate pH gradients it was shown that the conductivity is increased by a factor corresponding to the transference numbers of the electrolytes (e.g. ampholytes) present within the steady state gradient. But note that such a fractionation eliminates the former distribution of the electrolytes. The conductivity within the fractions is different from that of the appropriate gradient regions. This was previously reported to be true [361.

3.3 General propertiesofpH gradientsgeneratedby steady state electrolysis 3.3.1 Conductivity The general conductivity course is determined by the factor F(& H - no") (see Eq. 22); note that &,His negative). Due to this factor the conductivity has a minimum near pH 7 and increases toward the electrodes (see Fig. 8a, dashed line). (a) Conductivity of immobilized pH gradients: If the gradient is generated only by fixed charges then the conductivity is H ) the remaining factors in Eq. equal to F(S2"H - S ~ ~ H Oand (22) cancel out. In this case only the protons and hydroxide ions contribute to the conductivity. As compared with both other techniques, the immobilized pH gradients exhibit the lowest conductivities 1371. Figure 4. Upper panel: pH gradient corresponding to Fig. 3. Lower panel: Distribution of Na ions: dashed curve; Tris ions: dotted curve; Tris molecules: solid curve. For further details see Section 3.2.

will reach X2 with reduced charge, due to the association of OH. The diffusion of originally uncharged weak electrolyte molecules within a p H gradient in general produces a current density due to changes in the degree of ionization. These changes reflect chemical reactions (see Eqs. 10a, b). The migration of ions is also connected with changes in the degree of ionization. The steady state is reached if both processes compensate each other and thus no chemical net reactions occur (only the protons and hydroxide ions undergo net reactions). This is the stabilizing principle of stepwise p H gradients. An interesting secondary effect concerns the O H ions. If aparticular buffer particle would run from X 1to X 2 and back from X 2 t o X I , then about oneOHionistransferredfromX2toXl due to association/dissociation of OH. This indicates why the presence of stationary distributed buffer particles can affect the current that is associated with the transfer of only protons and hydroxide ions according to Eq. (7). A similar mechanism should also be possible in the case of ampholytic buffer components but not in the case of p H gradients produced by immobilized charges. Hence the later should exhibit the lowest conductivities. Furthermore this mechanism indicates that the slope of a p H gradient (i.e. its extention) and its conductivity are to a certain degree inversely proportional quantities. T o finish the discussion we want to highlight the following two problems. (i) Since the mass flux of a given buffer component vanishes at the steady state (the physical arrangement is a finite system with no buffer exchange) it may be hard to explain in which manner stationary distributed buffer molecules

(b) Conductivity of p H gradients established by amphoteric and nonamphoteric buffers: By an appropriate transformation of Eq. (22) the following criteria may be deduced. (i) An increase of conductivity (as compared with the immobilized p H gradients) is only possible at those positions of the pH gradient where buffer anions and buffer cations coexist. This presumption is automaticaly performed if the buffer is composed of amphoteric components: then each particle carries both positive and negative charges. In the case of nonamphoteric buffer mixtures the coexistence of both buffer anions and cations has to be secured by a suitable buffer composition. (ii) A given buffer component may only contribute to conductivity within those p H ranges where its mean charge (Q3 is different from zero: In the case of amphoteric systems the ampholytes are focused around pH points corresponding to their isoelectric points. Since at these points the Qitends to be zero each ampholyte contributes to conductivity only in the vicinity of its isoelectric point and this contribution is proportional to ci(at- a;)2.In the second case (pH gradients established by aciddbases) each component contributes to conductivity according to its degree of ionization (either cia:* or cia7). As compared with ampholytes these contributions are higher. For this reason such systems should exhibit the highest conductivities. (iii) A further practical consequence of analyzing Eq. (22) concerns the possibility of increasing the apparent conductivity by using more concentrated buffers: Above a relatively low buffer concentration (ionic strength > a further increase of conductivity cannot be achieved by using the same buffer with higher concentrations.

3.3.2 Conductivity and pH gradient extention It can been shown that factors increasing the conductivity at the same time decrease the extention of a p H gradient. If a given gradient spans a certain p H interval, then the gradient is

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the more extended the lower its conductivity. This inverse relation leads to the following practical consequence that, especially in the case of buffer generated pH gradients, a remarkable increase of conductivity, by a factor of 10-100 and higher, should be possible. Such a drastic increase is paid for by a reduction of pH gradient extension. However, if pH gradients of small extension may be used for special, e.g. preparative applications, then buffer generated pH gradients should exhibit the highest conductivities, provided the buffers are composed by relatively strong electrolytes (with regard to the given pH range). 3.3.3 Buffering capacity In the case of immobilized pH gradients the buffering capacity depends on the concentrations and the pK values of the immobilized acidic and basic monomers. By an appropriate choice of fixed electrolytes high buffering capacities are possible. In the case of ampholyte-gener,ated p H gradients the criterion for a high buffering capacity was given by Svensson. A given ampholyte exhibits maximum buffering capacity if at pH = p 1 both a+and a- have the value 1/2 (see Eq. 12). This and the ampholyte concentration determines the buffering capacity. In the case of buffer-generated pH gradients we have to distinguish two cases. (i) Extended pH gradients: Figure 5 shows the pH gradient obtained by electrolysis of a solution containing only a weak base (pK = 6, note that Kb is assigned to the reaction (lob). The extention of such a pH gradient is mainly determined by the choice ofthe weak electrolyte. Above pH = (pK +3)thegradientisextended(pH range b, at afield strength of 1500 V/m the slope is about 4 cm per p H unit). Below pH = (pK + 1) the gradient extention is minimal (about 0.1 cm per pH unit). The buffering capacity of such a base-generated gradient is maximal around pH = 14 pK k0.5 (where a equals 0.5). As can be seen from Fig. 5 there is no overlapping between pH ranges of sufficient buffering capacity (range a) and sufficient gradient extention (range b). For this reason extended gradients that are generated by monovalent weak electrolytes exhibit no buffering capacity in practice. This disadvantage should be easily compensated for by the addition of fixed buffering ions. Since the p H gradient is already determined by the weak electrolytes, a uniform distribution of suited buffering ions is sufficient for this purpose. (ii) pH

PH

13 12 11

.

I0

’

. .

9 . 8 7 .- - - - - - - - - - - - - - - - - - - -

I

I I I I

$.

r

gradients with small extention: In this case by an appropriate choice of buffer composition a maximal buffering capacity is possible. A criterion for the selection of suited buffer constituents is given in the following equations: pK, ~ p H - ( 1 . . . l S ) , p K b x 14 - p H + ( I . . . 1.5)

where pKbdenotesthe pK of a base and pK,that of an acid. According to this the buffer has to be composed by pairs of acids and bases. Each of those pairs has to meet the above criteria. These criteria secure both a sufficient buffering capacity and increased conductivity. 3.4 pH Gradient decay

We now discuss the p H gradient decay in systems as used in IEF. Experimentally one observes two phenomena, namely the (cathodic) drift of the whole gradient and/or the so-called “plateau” effect 112,391. The reason for the drift seems to be electroosmosis [ 121. This effect produces a unidirectional water flow throughout the system. The later effect, the socalled “plateau” effect, apparently requires a bidirectional flow [121. Recently [27-291 both effects were explained as follows: (i) It was found (27,281 that “focusing occurs in two stages, a relatively rapid stage during which the ampholytes migrate to their (preliminary) isoelectric position, and a much slower stabilizing phase during which the final steady state is achieved. This later phase provides an explanation for. . . the plateau effect”. Accordingly, the formation of a pH plateau could be interpreted as a transient phase in approaching the steady state. However, by a proper choice of buffer systems, linear pH gradients should be obtained, even at the steady state (seeFig. 8). (ii)In [291it was emphasized that steady state considerations have to include the total distance between the electrodes, i.e. the electrolyte composition has to be chosen such that the buffer components of interest finally become located within the separation chamber (and not within the electrolyte reservoirs). Both the plateau effect and the gradient drift were discussed in terms of a migration of buffer components out of the separation chamber into the adjacent electrolyte reservoirs. Such losses may be prevented either by (i) isolating the focusing space from the adjacent electrolyte chambers using ion exchange membranes; or by (ii) an optimum concentration of analyte/catholyte which will locate the carrier ampholytes within the focusing space [291. In this case the strongest acidic/basic buffer components operate as spacers which isolate the remaining buffer species from the electrodes. Whereas this later procedure is flexible and generally applicable, the use of ion exchange membranes may be problematic in special cases: (i) It is known that the net charge of the membrane matrices may stimulate electroosmosis [381. (ii) The current across the ion exchange membranes is carried primarily by hydrogen and hydroxydeions [291,i.e. only these ions will enter the separation space. Therefore, an appropriate formation of water will occur even before the steady state is reached. The associated hydrodynamic pressure on the membranes may become critical. (iii) Whereas such membranes may effectively control the passage of ions they can barely prevent the diffusion of low molecular weight buffer subspecies. Hence such membranes should not be used in the case of pH gradients generated by monovalent acids/bases.

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Steady state electrolysis and isoelectric focusing

-

H,O I - d

mol

0.1

0.05

7 0

7

1

X Fignre6. Dashedline: IdealpHcourseinanIEF system. Solidline: development of a “plateau” after prolonged duration of I E F [ 121. For further explanation see Section 3.4.

Ql

0

C

X

14

PH

Figure 7. Water formation within a pH gradient. If the slope of the pH gradient is 10unitdm and the steady state current density 1 mA/cm*, then a volume of one liter of pH 7 electrolyte would produce about 0.1 mol water per day. If this process leads to pH discontinuities in the neutral region a drastical increase of water formation may be expected.

In order to predispose our further discussion we consider the conventional system for performing IEF. Such a system consists of an anolyte and a catholyte and a zone between both electrolytes filled by carrier arnpholytes. The obtainable ideal pH course is shown in Fig. 6 (dashed line). The slope of the pH gradient is positive, that is, near pH 7.1 the water production has a maximum (Fig. 7) and mobile ampholytes that are present in this region are bilaterally washed out by the water stream. After a certain time a plateau will be formed (Fig. 6, solid line). Now the slope of the pH gradient around the neutral point increases and thus the pH gradient decay will be progressive with duration of the focusing process. Regarding the stability of pH gradients it is desirable that the pH gradient should posses a constant slope. Thus the pH of the anolyte should correspond to the most acidic pH of the given pH gradient and that of the catholyte to the most basic pH. In addition, the buffering capacity of the gradient must be high

C 1

1

0 0

0

x

X

X

0

X

Figure 8 . pH Gradient formed by electrolysis of a mixture of two weak acids and two weak bases. (a) pH Gradient: dashed curve; conductivity: dotted curve. Corresponding concentrations ofthe neutral (superscript o), and charged subspecies (superscript + and -, respectively). (b) Weakestacid(pK=8).(c)Stronger acid(pK = 6.8).(d) Weakest base (pK= 8). (e) Stronger base ( p K = 6.8). The ionic strength at pH 7 was 10 mol/L.

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Electrophoresis 199O,II, 281-289

R. Hagedorn and G. Fuhr

enough in order to compensate for pH changes that would gradients, it should easily be possible to calculate the approotherwise be induced by the focused proteins. These state- priate steady state concentrations of weak acids (bases). ments are independent ofthe particular method of pH gradient formation. The conditions for pH gradient decay are discussed below. If the charge distributions that determine the steady 4 Concluding remarks state pH gradient would be fixed in any way, the water flow could not result in a pH gradient destruction. At the steady Steady state electrolysis has beenreinvestigated. It was shown state in free solution the charges are “fixed” by the balance that the consideration of chemical reactions occurring in the between electric forces and diffusional (]pseudo-) forces. Ad- interior of the pH gradients is necessary for the explanation ditional forces, as may arise by the superimposed water flow, of stationary conditions. By considerating these reactions it would alter the shape of the steady state pH gradient. Since the seems to be possible to explain some properties of pH graadditional forces are small, these changes may be neglected, dients that have remained in doubt so far (see Section 1). The provided that a lateral loss of charged components generating derivations presented allow for a more accurate calculation of the pH gradient is prevented. InFigs. (8b--c)thedistributionof pH gradients for IEF. This concerns especially pH gradients the buffer subspecies of an ideal electrolytic system is shown; generated by weak aciddbases. Hitherto, it has been assumed the associated pH gradient and its conductivity is shown in that in free solution a stable system could contain only two Fig. 8a. These distributions would not essentially be altered by components [231 which should limit its applicability. The the superimposed water flow, that is, the ideal system is practical significance of the theoretical results for IEF is disalready stationary. However, the real system may be con- cussed. Unfortunatelly, the equations of interest cannot be structed in such a way that electrolyte is also present behind solved analytically. The optimum concentrations of the single the electrodes (see Fig. 9). electrolyte components have to be estimated by computer simulation. At present the program is written in BASIC. After To reach the steady state it is necessary that the concentra- this paper has been published, the program will be rewritten in tions of all components in this additional electrolyte region the TURBO-PASCAL-4 language and will become available correspond to their surface concentraticins at the electrodes. upon request (see Section 6: Appendix). According to Fig. 8 all acids are present at the cathode surface and all bases at the anode surface such that at the steady state Received May 12, 1989 all acids and bases have to be present in the appropriate regions behind the electrodes in concentrations corresponding to their surface concentration at the electrodes. Since the buf- References fer components have to be homogeneously distributed within I11 Svensson, H., Acta Chem. Scand. 1961,15,325-341. these regions it seems to be advisable to isolate them by 121 Svensson, H., Acta Chem. Scand. 1962,16,456-466. dialysis membranes and to stir within the isolated compart131 Svensson, H., Arch. Biochem. Biophys. Suppl. I , 1962, 132-138. ments. Furthermore, large compartments would provide for a [41 Vesterberg, O., Acta Chem. Scand. 1969,23,2653-2666. compensation of buffer decomposition due to reactions at the [51 Petterson, E., Acta Chem. Scand. 1969,23, 2631-2635. electrodes. 161 Nguyen, N. Y. and Chrambach, A., Anal. Biochem. 1976, 74, In general, it isdesirabletoconstructtheelectrolytic chambers in such a way that (i) either the space behind the electrodes becomes as small as possible (in this way lateral losses may be reduced; however, losses due to a buffer decomposition at the electrodes are not compensated), or that (ii) the buffer composition within these regions corresponds already at the beginning to the final steady state composition. Whereas this condition (appropriate composition of anolyte and catholyte) can hardly be realized in the case of carrier arnpholyte created pH

Po h

0

P

Figure 9. Schematic presentation of an electrolytic system containing lateral electrolyte compartments. The dotted lines indicate in which manner the buffer concentrations have to be continued in order to reach steady state.

145-153. 171 Nguyen, N. Y., Rodbard, D., Svendsen, P. J. and Chrarnbach, A., Anal. Biochem. 1977, 77, 39-55. 181 Nguyen, N. Y., Salokangus, A. and Chrambach, A,, Anal. Biochem. 1977,78,287-294. [91 Nguyen, N. Y. and Chrambach, A., Anal. Biochem. 1977, 79, 462-469. [ 101 Caspers,M. L. andchrambach, A.,Anal.Biochem. 1977,81,28-39. 11 11 Couno, C. B. and Chapo, G. A., Electrophoresis 1982,3, 65-75. I121 Righetti, P. G., in: Righetti, P. G., Van Oss, J. W. and Vanderhoff, J. W. (Eds.), Electrokinetic Separation Methods, Elsevier, North Holland Biochemical Press, Amsterdam 1979, pp. 389-441. 113 I Rilbe, H., in: Catsimpoolas, N. (Ed.),IsoelectricFocusing, Academic Press, New York 1976, pp. 13-52. 1141 Rilbe, H., Electrophoresis 1982,3, 332-336. [ 151 Thormann, W., Mosher, R. A. and Bier, M., Electrophoresis 1985,6, 78-8 1. [161 Tsai, A,, Mosher, R. A. and Bier, M., Electrophoresis 1986, 7, 481-491. 171 Thormann, W. andMosher,R. A.,Electrophoresis 1985,6,4 13-4 18. I181 Hjelmeland, L. M. and Chrambach, A., Electrophoresis 1983, 4, 20-26. [ 191 Palusinski, 0. A,, Bier, M. and Saville, D. A,, Biophys. Chem. 1981, 14,389-397. 1201 Bier, M., Mosher, R. A. and Palusinski, 0. A.,J. Chromatogr. 1981, 211,313-335. L211 Mosher,R. A.andThorrnann,W.,Electrophoresis1986,7,395-400. [221 Mosher, R. A,, Bier, M. and Righetti, P. G., Elecfrophoresis 1986,7, 56-66. 1231 Mosher, R. A,, Thormann, W., Graham, A. and Bier, M., Electrophoresis 1985,6, 545-551. [241 Shimao, K., Electrophoresis 1987,8, 14-19.

Electrophoresis in an expanded stationary boundary

Electrophoresis 1990,l l 289-292

1251 Bier, M., Palusinski, 0.A., Mosher, R. A. and Saville, D. A,, Science 1983,219,1281-1287. I261 Mosher, R. A., Dewey, D.,Thormann, W.,Saville,D. A. andBier,M., Anal. Chem. 1989,61,362-366. 1271 Thormann, W., Mosher, R. A. and Bier, M., J. Chromatogr. 1986, 351, 17-29. [281 Mosher, R. A., Thormann, W. and Bier, M., J. Chromatogr. 1986, 351,31-38. [29l Mosher, R. A., Thormann, W. and Bier, M., J. Chrornatogr. 1988, 436, 191-204. 1301 Planck, M.,Ann. Phys. 1890,39, 161-186. I311 Planck, M., Ann. Phys. 1890,40,561-576.

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[321 Alberty, R. A . , J . Am. Chem. SOC.1950, 72,2361-2367. 1331 Kohlrausch, F., Ann. Phys. 1897,62, 209-239. 1341 Righetti, P. G., Fazio, M. andTonani, C.,J. Chromatogr. 1988,440, 367-377. [351 Gill, P., Electrophoresis 1985,6, 282-286. 1361 Murel, A., Vilde, S., Kongas, A. and Kirret, O., Electrophoresis 1984, 5, 139-142. 1371 Bjellqvist, B., and Ek, K., Application Note 321, 1982, LKB-Produkter AB, Bromma, Sweden. [381 Rilbe, H., US Patent 4 217 193, 1980. [391 Miles, L. E. M., Simmons. J . E. and Chrambach, A., Anal.Biochem. 1972,49, 109-117.

6 Appendix: Structure of the computer program Themain program (inTURBO PASCAL)callsfor thefollowing procedure: eingabe (* input of the initial values at x = 0: *> (* mobile components: c Lp K , iP, *> (* fixed charges: pK, c, = c(x> *I (* pH, current density *I; integration (* integration and calculation of eqs. 20-25 *I ; ausgabe (* printing of the initial values; drawing of the curves *);

The procedure integration operates as follows: repeat 1st-step: (* calculation of S1-S5 (eqs. 25) 2 n d s t e p : (* integration of eqs. 21, 24 3 r d ~ t e p :repeat approximation o f n e w - p H until electroneutralityis-reached; 4th-step: (* calculation of the remaining quantities: (* eqs. 20; 22; 23 until finalpHisieached or ionicstrength = 2 .

*>; *>;

*)

*>;

Further instructions provide for the control and regulation of the step width.

Budin M. Michov Department of Biochemistry, Academy of Medicine, Sofia

Electrophoresis in an expanded stationary boundary A method of electrophoresis in an expanded stationary boundary is described, in which a pH gradient formed in a single buffer is used. The degrees of ionization and the concentrations of the electrolytes have values at which the polyions stack and resolve simultaneously in a polyacrylamide gel. T o illustrate the method, the resolving ofserum proteins in a pH gradient formed by Tris - glycinate buffer is considered.

1 Introduction A disc electrophoresis buffer system [ 1-51 consists of buffers of different composition and pH values. These buffers are necessary for stacking the sample in a stacking gel as a result of the Kohlrausch effect [6], after which the polyions are resolved in a resolving gel. To accomplish the Kohlrausch effect, the leading ion (the ion which moves before the fastest polyion) and the trailing ion (the ion which follows the slowest polyion) must migrate with the same velocity, thus forming a moving boundary which stacks the polyions when it crosses them. In the methods of Ornstein Ill, Davis [2], and Allen [7, Correspondence: Dr. Budin M. Michov, Ph. D., lnstitut fur Lebensmitteltechnologie und Analytische Chemie, Technische Universitat Miinchen, 8050 Freising-Weihenstephan, Federal Republic of Germany

Abbreviations: TEMED, N,N,N’,N’-tetramethylethylenediamine; Tris, Tris-(hydroxymethy1)aminomethane 0VCH Verlagsgesellschaft mbH, D-6940 Weinheim, 1990

81, and also in the theoretical models of Jovin [3, 41 and Chrambach et al. L51, the leading and trailing ions are different. In contrast to this we proposed 191 that the leading and trailing ions be the same - for example glycinate ion alone. As we proved, the same leading and trailing ion forms a boundary of changing electric field strength, which is designated as the stationary boundary [9-111. When polyions cross this boundary, they arrange themselves according to their effective mobilities in homogeneous polyion zones. We now attempt to analyze the stationary boundary when expanded along the resolving gel. Ifwe imagine this boundary as being divided into many parts, we can assume that every part resembles a separate boundary, with the result that every boundary part will stack certain polyions, after which they will separate from each other in accordance with their electric charge and mass against the increasing strength ofthe expand0 I73-0835/90/0404-0289%2.50/0