Etectrophoressis 1991, 12, 693-703
Fabrizio C. Celentano’ Elisabetta Gianazza2 Pier Giorgio Righetti’ ‘Department of Biology, University of Milano and Department of Biomathematics, UCLA School of Medicine ’Department of Biomedical Sciences and Technologies, University of Milano
Computational approach to IPGs
693
On the computational approach to immobilized pH gradients The unified treatment for computing the pH of complex mixtures of mono- and polyprotic buffers, including ampholytes, as utilized in the gradient simulation program PGS, is presented. Its ability to compute pH, buffering power and ionic strength is shown by discussing a few simulations. The problems arising in the automatic formulation of optimal mixtures are presented, as well as the merits and limits of several target functions utilized in such optimizations. It is shown that no universal target function exists and that a proper optimization method should account for the fact that more than one formulation is possible for a given pH range.
1 Introduction 1.1 Early simulations on isoelectric focusing in carrier ampholytes There exist several approaches to the study of pH gradient formation in isoelectric focusing in carrier ampholytes (CAIEF). Perhaps the earliest model was that proposed by Almgren [I]. By assuming Gaussian CA distribution around the pl, equal diffusion coefficients, electric mobilities and relative concentrations and evenly spaced plvalues of CA-buffers along the pH scale, Almgren was able to simulate conductivity, pH and distribution profiles, and concluded that at least 30 CA species/pH unit were needed for the production of a stepless pH gradient. Similar steady-state profiles were also proposed by Cann [2] and by Schafer-Nielsen [3] (see Figs. 1.9 and 1.10 in this last article). The most extensive modeling proposed to date appears to be that performed by Bier’s group [4-61: their program allows the simulations of all four modes of electrophoresis (moving boundary, zone electrophoresis, isotachophoresis, and IEF. Generally speaking, for characterizing one such electrophoretic system, one needs a relationship (called the velocity relation) describing the steady-state, plus the three fundamental laws, i.e. electroneutrality, Ohm’s and mass conservation laws. In all these cases, the simulation allows prediction of the ways molecules (and macromolecules) approach the final steady-state and what particular distribution profile they assume once attaining such a situation. It was shown that, with some modifications, such a model will predict also the steady-state distribution profiles of amphoteric compounds in immobilized pH gradients (IPGs) [71. In a related field, Hadded and Cowie [8] have described a computer-assisted optimization of eluent concentration and pH in ion chromatography. This procedure provides for systematic selection of eluent concentration (at a fixed pH value) or eluent pH (at a fixed eluent concentration) in ion chromatography. Poncelet eta/. [9] computed, in complex chemical and biochemical systems at equilibrium, pH (or
Correspondence: Prof. P. G. Righetti, University of Milano, Via Celoria 2, 1-20133 Milano, Italy Abbreviations: B, buffering power; CA, carrier ampholytes; CV@), coefficient of variation of b; DMlmr the greatest deviation, either maximal positive or maximal negative, from linearity; IEF, isoelectric focusing; IPG, immobilized pH gradients; MGS, monoprotic electrolyte gradient simulation; PGS, polyelectrolyte gradient simulation; SD, standard deviation; 2-D, two-dimensional.
0VCH Verlagsgesellschaft mbH, D-6940 Weinheim, 1991
the concentration of any chemical species), partitions between acidic and basic forms, global charge, molar mean charges, ionic strength and molar mean contribution to the ionic strength. In this last case, activity coefficients were introduced, in order to correct the dissociation constant for ionic strength. However, “when the number of acid-base systems is large (more than three) the system of equations becomes very difficult to handle”. Sillero and Ribeiro [lo] have recently published a procedure for calculating the p1 of proteins: this is in fact an algorithm for estimating a fixed pH value given a mixture of weak acidic and basic groups. Their progam has been applied to proteins bearing up to 10 acidic and 10 basic groups. 1.2 Monoprotic electrolyte and polyelectrolyte gradient simulation: A chronicle of events In the last few years, we have developed simulation programs whose aim is to model and optimize mixtures of buffers for generating linear pH gradients. Although of general applicability, they parallel in their history the evolution of the concepts and applications of IPGs, started a decade ago by our group together with Dr. Bjellqvist’s and Dr. Gorg’s teams [Ill. IPGs were born as a continuation of conventional CA-IEF and originally were mostly meant to extend the fractionation capability of the existing technique to extremely narrow pH ranges. In fact, in the cornerstone publication [ 111, only modified Henderson-Hasselbach’s equations were given for computing narrow to ultranarrow linear pH gradients, extending to a maximum of 1 to 1.2 pH units. Such equations were based on the use of only two species (the acrylamido acids and bases used to generate and stabilize the pH gradient within the IPG matrix), one of them buffering and the other titrating within the desired pH range. However, stability and reproducibility as afforded by IPGs seemed appealing features for wide-range formulations, including a larger number of buffering Immobilines. This arrangement in turn required such complex calculations that automatic computation became a must for both the simulation of the gradient steps (pH, ionic strength and buffering power p ) and the optimization of their composition in order to give the most linear pH course. Our first simulation program [I21 (MGS, or monoprotic electrolyte gradient simulation) was operating by the end of 1982. A first approach to the formation of extended pH gradients was through the sequential elution of the buffering species of increasing pKfrom a multichamber (actually, a 5-chamber) mixer [12], developed for gradient elution as 0173-0835/91/1010-0693 $3.50+.25/0
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Electrophoresis 1991, 12, 693-703
F. C . Celentano, E. Gianazza and P. G. Righetti
first proposed by Peterson and Sober [13]. This laborious procedure was soon abandoned in favor of classical twovessel gradient mixing [14, 151, for which we theoretically studied the conditions of gradient linearity, as a function of pKdistribution of the buffers, and of the titrants [15].Along this line in 1984 we published the formulations for a series of wide immobilized pH gradients (spanning 2-6 pH units within the pH 4-10 range), optimized in terms of gradient linearity, the standard deviation from linearity within the stated pH extremes not exceeding 1% of the range span [16]. In this paperwe compared two approaches to the generation of extended pH gradients: (a) in one case each Immobiline had the same concentration in both vessels of the mixer; (ii) in the other, different concentrations of buffering species could be present in each chamber. In the latter case the apparent pK of a given buffer may be shifted in such a way that the buffering power across the pH range is smoothed and the conditions for linearity better approached. Gradient linearity, however, is not the optimal solution in quite a few cases: e.g., in developing two-dimensional (2-D) maps, best resolution in the focusing dimension would clearly be obtained by nonlinear pH gradients, following the relative abundance of isoelectric proteins along the pH scale. Thus, we also calculated wide, nonlinear IPG recipes for use in 2-D maps and in most cases requiring analysis of highly heterogeneous samples [17]. As at that time we had given extended recipes including also two commercially unavailable components (a strong acid and a strong base) we were forced by colleagues to publish a new set of formulations, optimized in the absence of such titrants [18].During that period, other types of focusing in nonamphoteric buffers were introduced: Chrambach’s ’arrested stack’ [191, Rilbe’s ’steady-state rheoelectrolysis’ [20] and Bier’s ’physically’ immobilized pH gradients [21]. With MGS we could resimulate their data and find the merits and limits of each alternative route [22]. By 1986 IPG at pH extremes was first tried. Thus our computer program was expanded to include the effects of H+ and OH- (negligible within the 4-10 pH range) on p , ionic strength and pH profile [23].As a provocative concept, we introduced the idea of water as two Immobiline species, one with pK=-1.74 [H’], the other with pK= 15.75 [OH-]. The first compounds to be focused in the pH 3-4 interval were dansylated amino acids [24].In the case of very acidic and very basic intervals, the IPG behavior was modeled also [7] by modifying the general program for steady-state IEF, as developd by Palusinski et al. [4] and by Bier et al. [5]. As chemicals with more acidic and more basic pK became available, extended formulations including pH extremes were recently computed: the widest pH range that could be formulated is a pH 2.5-11 interval, spanning 8.5 pH units [25, 261. In 1988 we started a program on the characterization of existing Immobilines [27,28] and on the synthesis of new species (a pK3.1 [29], a pK6.6 and a pK7.4 [30], a pK8.05 [31] and a pK 10.3 [26]).At this point the number of useful buffers for IPGs had considerably expanded and MGS, which was limited to mixtures of no more than 10 species (later 12) could no longer handle this new complexity. In addition, oligoprotic acrylamido buffers have been deLrribed [32], and these species too were not included in our simula-
tion program. These facts forced us to develop a brand new program, polyelectrolyte gradient simulation (PGS) for IEF and chromatography.
2 pH Calculation in complex mixtures We summarize in the following the derivation of the expressions utilized for computing a pH gradient, as utilized in the PGS program, accounting for mono and polyprotic species, as well as for polyampholytes, treated separately in different preceding papers [12, 15,33,34]. The pH of a solution of an ampholyte i having nAi acid groups and nBibasic ones is obtained from the electroneutrality condition
where nAl and n,, are the numbes of the acid and basic groups, alA,and a,,, the degrees of dissociation of the j-th acid or basic group, zAJand zBJthe charges of the zwitterion and [CTJthe total concentration of the ampholyte. A similar equation holds also for a mixture of one polyprotic acid and one polyprotic base, but in this case the concentrations of the two may be different, while for monoprotic species the upper and lower values of the summation index are both 1. If follows that for a mixture of m acids and I bases, summing for all the species, the electroneutrality condition becomes m
I
“Ai
where [A:] and [BT] are the total concentrations of the i-th acid or base. For ampholytes the total concentrations corresponding to the acid and basic groups of such species will be equal. It follows that the above equation can be utilized for computing the pH of any mixture of mono and polyprotic species, given a general expression for the degrees of dissociation. This is found by considering the generic dissociation steps of the i-th polyacid Ai or polybase B,: A(h-l)-
Ak: i
h-tHt .h
whence the concentrations of the anion having 11 LliLiigc\ and of the cation with n,,-h+l charges, can be computed. Obviously, for monoprotic species nA,= nBI= h = 1. One obtains: A?-’)A:= k: (34 IH+I
Electrophoresis 1991, 12, 693-703
Computational approach to lPGs
After the dissociation of 1,2, ..J..., nAIprotons, the concentrations of the anions from acid A, can be computed from Eq. (3a), recursively substituting [A:] into [A;], the latter into [A12-],and so on, up to [A:*'-]. An identical recursive substitution is followed for the bases. The procedure may be visualized using a linear graph where the nodes represent the concentrations of the anions whose values are obtained multiplying the value of the preceding node by the value of the interposed branch. The latter value, in effect, is a (constant) transfer function transforming each node value into the following, as depicted below for an acid:
[A01
[A:, 1
0
>O
k,'/[H+]
[A:-]
k;/[H+]
The substitution of the Eqs. (6) and (5) into Eq. (2), finally gives the equation allowing computation of the pH of any mixture of mono and polyprotic buffers. In effect, for monoprotic species, the dissociation coefficients become CLA~ =KAi/(KAi
t H+); u B ~= H+/(KB~ t H+)
and Eq. (2) reduces to m
1
(7)
[A:-] >o
s o . . .o
695
...
#+'/ [H']
One obtains, for h =j:
(44
The total concentrations of the acid [AT] and of the base [B:] are obtained summing the Eqs. (4) for all t h e j values between one and nA, or nB,and adding the concentration of the remaining undissociated acid or base:
identical to Eq. (5) in [15] with the addition of the hydrolytic products of water, not directly accounted for in the MGS program. The buffering power p of the mixture, an important quantity both for the gel behavior and the optimization of its and pBI composition, is the sum of all the contributions PAL of all the anions and cations (or different groups of the ampholytes) plus that of water. For each of them
and, for the expression of these derivatives, the reader is referred to the previous publications [15,33-351.
3 Examples of simulations with polyprotic buffers
The dissociation coetlicient for the ion with charge j, i.e the ratio between the ion concentration and the total concentration of the considered species, is now obtained by dividing each one of the Eqs. (4) by the corresponding Eq. ( 5 ) :
a i i=
[H+]j
j-1
hkh
Earlier simulations, and their comparison with experiments, have shown the satisfactory accuracy of our computing methods for both MGS [12, 141 and PGS [34-351, as long as the Immobiline concentrations are low enough to keep the activity effects negligible. We have also observed that the best results are obtained when the pKs of the buffers are evenly distributed along the gradient. As this is seldom the case, we present in the following a few simulations, showing the importance of filling up the ,,holes" in the pK distribution along the gradient. If citric acid (pK, 2.9, pK2 4.1, pK, 5.8) is dissolved at the same concentration in both chambers of a gradient mixer (1 :1ratio) and titrated between pH 3 (close to pK,,,,,) and pH 6.8 (above pK,,,), the gradient of Fig. 1A is obtained. The pH gradient is linearbetween pK, and pK, and there is a sudden change of slope in proximity of the p minimum (step #28). If one adds imidazole (pK6.9) to the mixture (in a 1:l ratio with citric acid, thus extending the pK range of the mixture to cover the whole gradient span), the linear portion of the gradient extends up to pH 6, but the slope then flattens (too much buffering power) (Fig. lB).The addition of a third component with intermediate pK (itaconic acid, pK, 3.85, pK2 5.49, thus filling "pK holes" in the mixture, should improve the situation. In effect, with our simulation program, we can find by trial and error the proper ratio of the three components (54:20:30 citric acid : itaconic acid : imidazole) allowing a smooth pH gradient in the whole titration interval to be obtained (Fig. 1C).
696
-
PH 7 -
-n-
CITRIC K I D 1:l
0'
dmBn
6 -
Electrophoresis 1991, 12, 693-703
F. C. Celentano, E. Gianazza and P. G. Righetti
I
=
I
D,
225
m
8
0'
*
R B
m h
p
/ *
0
185 34
-
32
-
165 145 125 185
38
- 85 65
- 45
28
-
e PH
5
ie
ze
15
25
30
35
CITR:ITAC:IMID (54:28:3m
48 p
-0-
STEP
.* I
- 345
* I
S8
,
18
- 325 - 385 - 285 - 265
GI
'
- 46 -1 z45 225
'I
J ,**m I
a el'
5
18
15
- 285 44 I185
- 165
c
8
-
28
25
38
35
Figure 1. Simulation of the behavior of a simple oligo acid (citric acid). In all cases citric acid (54 mM) is titrated at pH3.0 and pH 6.8 and the linear mixing ofthese limiting solutions is done linearly, as with a two-vessel gradient mixer. Forty steps are computed. (A) The citric acid molarity is kept constant in the two chambers (1:l ratio). (B) Each chamber contains a mixture of 54 mM citric acid and 54 mM imidazole (imid). ( C ) Each chamber contains a mixture of 54 mM citric acid, 20 mM itaconic acid and 30 mM imidazole. Note in this last case the good pH gradient linearity and the smoothing of the p power. I , ionic strength, p buffering power.
Ekctniphoresis 1991. 12,693-703
PH
Computational approach to IPGs
-
697
TEPA 1 : l
1°/** * * * *
t
fl.**
A@-
445
-** I I I I I I I I I
0
5
5
0
PH
** * + L
5
d , ~ ~ 1 1 1 , 1 1 1 1 1 , l ~ 1 I I 1 , I , , , , , , , , , , , , , , , , , , , , , , , , f ~ *
10
15
10
15
-
20
20
25
25
4
339
39
30
35
40
45
50
55
60
STEP
30
35
40
45
50
55
68
STEP
TEPA - TETA
-
EDA
** *
I
0
Figure 2. Simulation of the behavior of a simple oligobase (tetraethylenepentamine,TEPA) and ofits mixture with other oligobases. In all cases TEPA (60 mM) is titrated at pH 4 and pH 10 and the mixing of these limiting solutions is made linearly, as in a two-vessel gradient mixer. Sixty steps are computed. (A) The TEPA molarity is kept constant in the two chambers (1:l ratio).(B)The molarityratioisaltered in the two chambers, so that the acidic reservoir (pH 4) contains a 2.2 higher molarity than the basic one (pH 10). (C) Each chamber contains a mixture of 60 mM TEPA, 20 mM tetraethylene tetramine (TETA) and 20 mM ethylenediamine (EDA). Note in this last case the smoothing of the sigmoidal transition around pH 6 and the filling-in of the p power minimum.l,ionic strength;p,bufferingpower.
698
Electrophoresis 1991, 12, 693-703
F. C . Celentano, E. Gianazza and P. G. Righetti
An oligobase, tetraethylenepentamine (TEPA, pK, 2.7, pK2 4.3, pK, 7.9, pK, 9.1, pK, 9.9) titrated between pH 4 and pH 10 is a particularly interesting case: in fact, TEPA was originally proposed as one of the oligoamines forming the backbone of carrier ampholytes [36,37]. As shown in Fig. 2A, such oligoamines alone cannot form a linear pH gradient. Due to the lack of suitable pKvalues, there is a steep sigmoidal transition around pH 6, corresponding to a deep minimum of p power. If the concentration ratio between the two chambers is shifted from 1: 1 to 2.2: 1, the /? power of the acidic region is increased (the p minimum moves from fraction 12 to 25) and the linear portion of the gradient can be extended, but there remains the sigmoidal transition between the two segments ofthe curve. When adding ethylendiamine (EDA, pK, 7.0, pK, 10) and triethylenetetramine (pK, 3.3, pK26.7, pK, 9.2 and pK, 9.9) and modelling the mixture with our simulation program, much better pH and /? profiles can be obtained (Fig. 2C): the sigmoidal transition has been substantially reduced and the minimum of p power has been increased from less than 10 to more than 20% of its maximum value. The optimal concentration ratio between the three components corresponds to TEPA : TETA : EDA= 3 : 1 : 1 (Fig. 2C).
4 Optimization: An “experimental” approach 4.1 The problem of the target function Gradient optimization, that is the determination of the composition of the solutions in the two mixer’s chambers allowing the best approximation of the required gradient shape to be obtained, requires the definition of a target function to be minimized. Such a function of the mixture compositions in both vessels should accurately represent the gradient deviation from the ideal shape, in order to play the same role as the sum of squared deviations, in linear and nonlinear regression. In this paper we refer only to linear gradients because the optimization approach will be the same with every gradient shape. In fact, from the mathematical point of view, we have anyhow a nonlinear optimization problem, as the function describing the gradient shape is intrinsically nonlinear. Optimization cannot be treated as a regression problem. The latter, in fact, deals with the optimal fitting of a linear or nonlinear function to randomly scattered experimental points, while optimization requires the comparison of points sampled from two continuous functions, i.e. the actual and the required gradients. It follows that the deviations to be minimized are not normally distributed and independent, as required in regression, but highly correlated. The choice of the target function thus becomes nontrivial, and different functions usually lead to sensibly different results, depending on the system’s behavior and on how the functions to be compared are sampled (for instance, the mean deviation from linearity depends on the number of samples). The target function for the evaluation of the comparison should thus closely reflect the behavior of the considered system. In fact, the purpose of a target function is twofold: (i) to depict the linearity of the gradient course, and (ii) to describe the overall congruity of the pH range, i.e. the correspondence between expected and found. We have first sug-
gested keeping constant the buffering power p of the mixture, a welcome property also for other reasons, byminimizing its variation coefficient CV@) [14, 151, but stressed the fact that a constant /3 leads to linearity if and only if the mixture concentration (rigorously the concentration of each species in the mixture) is constant. It follows that such a target function does not work at its best when the species concentrations are different in the two vessels. We currently minimize CV(/?)only when using the “same concentrations” procedure [16].Other target functions have been used, and are still under test, with the “different concentrations” procedure. The first is the standard deviation, SD,not the plain sum of squares, in order to lessen the aforementioned drawbacks by taking a square root. By referring the deviations to the ideal linear gradient shape [15], both the linearity and congruity issues are addressed. It should be noticed that the same occurs with nonlinear gradients, by computing the deviations from their ideal shape. In this case, obviously, we must substitute nonlinearity for linearity. A second interesting target function is the same maximum absolute deviation DMlm,the greatest between the maximal positive deviation (DM)and the absolute value of the maximal negative one (DJ.This may be a good target when its minimization is not obtained at the expense of many small oscillations, introducing a kind of noise in the gradient. Another possible target comes from the observation that the best gradients are often almost symmetric around their center. It follows that the asymmetry to be minimized can be measured by the absolute value of the mean, D,,,or by the algebraic sum of the two maximal deviations A = 1 D M + D, 1. In a survey of the published simulations, the value of A corresponding to the optimized formulation was found in all cases very low. It should be noticed that also the sums of any two of the above targets can be assumed as a new target function. Fig. 3 compares the course of these target functions when optimizing a pH 5.5-8.5 gradient obtained with a single triprotic base whose pKs are 6,7 and 7.5, as the ratio of the buffer concentration in the acidic (A) and basic chamber (B) is
I \
0.1 -..,
0.0 0.50
4 0.75
1 .oo
SD
0-0
~
1.25
A : B concentration ratio
Figure 3. Optimization, in terms of linearity, of the pH gradient 5.5-8.5 obtained from the linear mixing of two limiting solutions (A = acidic, pH 5.5; B=basic,pH 8.5) containing different and variable concentrations of a triprotic base with pK1 6, pK2 7, pK3 7.5. The deviation from linearity is maximum deviation plotted in different ways: standard deviation (SD), towards negative (D,) or positive ( D M )values, average deviation (Day),as a function of the buffer concentration ratio between the acidic (A) and basic (B) chambers.
Ekcirophoresis 1991, / 2 , 693-703
Computational approach to IPGs
increased from 0.6 to 1.1. SD gives a shallow parabolic curve, whereas D,, is a steeper V-shaped function. The minima of the two curves are close to one another but do not coincide. Still close to these minima the curves representing D, and D, tangle one another, where A (not shown) gets to its minimum: zero. Both limbs ofthis cross are parabolic, but over a short distance may be approximated to straight lines. Their intersection is thus quickly found graphically or analytically: each simulation gives in fact two “experimental” points, D, and D, and two runs suffice to find two couples of points. The steepest and most informative function could be used to get close to the target, and the absolute minimum then searched while resorting to a more accurate if shallower function. The A function appears most useful for this first step within the simple simulations allowing for just one degree of freedom, but the presence of N variables would require the solution of an algebraic system of degree N+1. It should be anyhow kept in mind that the above functions may have several minima in systems with more degrees od freedom. Figure 4 shows the course of the minima of the above target functions when the same pH 5.5-8.5 gradient is obtained with bases having pK, 6 and pK, 7, as before, while pK, varies between 7.1 and 10.5.It appears that the different targets have different minima, except when the pKs are equally spaced and symmetrically disposed around the center of the gradient span. The interesting feature is that the observation above can be generalized, as one usually finds that the minima of the different target functions are, as could be expected, seldom identical, although, often, rather similar. Figure 4 also shows that a nonmonotonous relationship exists between the pKs of the buffer and its concentration for optimal linearity. Moreover, for monoprotic species the course of the gradient only depends on the pK value and not on the buffer dissociation as an acid or as a base (not shown). I
I 0-0
0 .c e
1 .oo --
A-AD
DM,., av
O-OSD
699
4.2 Exploration of the composition hyperspace
As a consequence of the results, and of the doubts, exposed in the preceding section, we have undertaken the systematic investigation of a panel of simple buffer mixtures, whose composition is varied stepwise; at each step, the pH gradient course arising from their linear mixing is simulated and a number of target functions are computed. By analyzing the structure of the multidimensional spaces representing different target functions as a function of composition of the mixtures in the two mixer’s vessels, we have sought: (i) general features in the configuration of such spaces that might help in devising an efficient optimization algorithm; (ii) the easiest target for the optimization procedure. We restricted ourselves to simple buffer systems because the number of experimental points to be computed dramatically increases with the number of the allowed degrees of freedom; in addition the graphical representation of multidimensional spaces soon becomes impracticable. Thus, the highest number of independent variables (buffer concentrations) in our tests is 3; these are drawn along the Cartesian axes, while the fourth, dependent, variable (the target function) is rendered with a color shade, using the UNIRAS graphic package installed on the Unisys 2200 mainframe of the University computing consortium CILEA. When dealing with less degrees of freedom, the same package allows the drawing of contour lines. In detail, the buffer systems that we are studying are: (i) with 1 degree of freedom: 1 diprotic or triprotic buffer, whose concentration is changed between the two chambers; (ii) with 2 degrees of freedom: 3 monoprotic buffers, with constant concentrations in the two chambers; (iii) with 3 degrees of freedom: 2 monoprotic buffers, with varying concentrations in both chambers. In all cases, the dissociating groups whose concentration is taken as the independent variable behave in the system as buffers, while strong acids or bases titrate the solutions to the desired pH extremes. Within each experimental series, the concentration of one of the buffers in one (cases i and iii) or in both chambers (case ii) is kept constant through0ut.A single (hyper)space is explored, i.e. only those solutions corresponding to a given pH range, or a given pair of pH extremes, are considered.
Figure 4. Comparison of the behavior of different target functions in the
MGS or PGS may enter any (hyper)space point corresponding to a predefined concentrations vector. Our selection for the present set of simulations greatly simplifies the spatial exploration. It should be stressed, however, that, in comparison with the MGS current optimization procedure, the present exploration runs overestimate one of the reference parameters, i.e. congruity vs. linearity, and emphasize two points along the gradient rather than evaluating the curve as a whole.
optimization, in terms of linearity, of gradients in the pH 5.5-8.5 range, obtained by linear mixing of different solutions of a triprotic base with pK, 6 , p K ~7and pK3 varying from 7.1 to 10.S.The optimization is obtained varying the base concentration in the acidic chamber A, whilst the solution in the basic chamberB has a constant concentration (10 mM).The minima of the targets are plotted as a function of the difference between the last two pKs. SD,standard deviation from linearity; DMlm),maximum absolute deviation in the positive or negative directions; D,,,absolutevalue of the average deviation. Note that there is a continuum of solutions allowing to cast optimized gradients (obviously of different quality), having the same pH span, and that the same A/B ratios can be used with solutions containing different species at different concentrations.
Figure 5 shows two sets of results concerning a pH 5.5-8.5 gradient obtained using monoprotic bases. In the first case the gradient is obtained using a pK 6.5 and a pK 8 species, the former at constant concentration in the acidic chamber and the latter at constant concentration in the basic chamber of the mixer. The other case shows a symmetric pK distribution, with a pK6 base constant in the first chamber and a pK 7.5 base constant in the second (basic) chamber. In both cases the behavior ofSD, of DM,,, and A appears rather different. The first function shows a single minimum and a
.-a c
0.50-
0.25 -I
i
700
Electrophoresis 1991, 12, 693-703
F. C. Celentano, E. Gianazza and P. G. Righetti
rather flat appearance, while DMlm (maximum deviation in the graph) is steeper and shows two minima.The sum of the two still shows two well defined minima (the distance between contour lines is 0.025). Still steeper, but more complicated is the behavior of A (delta), showing several minima of comparable depth.
Also interesting is the symmetry of behavior ofthe two mixtures. In effect, as the protolytic products of water play no role above pH 4 or below pH 10, by rotating the axes corresponding to the two chambers identical results are obtained, in consequence of the symmetric pK distribution along the gradient.This fact agrees with the finding that the
SD pK 6/1 = 1
; pK
7.5/2 = 1.5
pK 6 / 1
; pK
7.5/2
= 1.5
pK 6 / 1
Max. Dev. ; pK 7.5/2 = 1.5
= 1
pK 7.5/1
delta pK 6/1
=
1
SD
+
= 1
max dev ; pK 7.5/2 = 1.5
Figure 5. Contour maps of different target functions showing the minima predicting the optimal composition of the solutions to be linearly mixed for
casting a p H 5.5-8.5 gradient.These solutions contain two monoprotic bases with pKs6.5 and 8 (seriesA‘-D’) or6and7.5 series (A”-D”), titrated to the limiting pH values with a strong acid. In the A’-B‘ series pK6.5 is kept at constant concentration in the acid chamber (l),where pK8 is allowed to vary, whilst its concentration is varied in the basic (2) chamber, where pK8 is constant.An identical design is utilized for the simulations ofthe A - D s e r i e s . The degrees offreedom of the system are thus only two.The behavior of standard deviation (SD)is depicted by arather shallow surface with asingle minimum. The absolute maximum deviation from linearity DM,,, (Max. Dev.) shows steeper slopes and still steeper appears the algebraic sum of the maximum positive (DM)and negative (D,) deviations A (delta). These functions show more than one minimum because are representative of one single point of the gradient, whose position varies with varying composition.The shape of SD+ DM,,,(SD+Max. Dev.) is most influenced by the steep behavior ofthe 1atter.Note that,as the pKdistribution and the composition ofthe two mixtures series are symmetrical in respect to the gradient extremes, the two series turn out to be identical when the axes pertaining to the two chambers are exchanged.
Compulational approach to IPCs
Electrophoresis 1991, 12, 693-703
functions plotted in Fig. 4 are not monotonous. There is a continuum of different ways of casting the same gradient, obviously by using different species. These results, to be further extended, confirm that the gradient optimization is not a simple task. Although exceedingly good gradients can be obtained by trial and error, unsophisticated optimization procedures may miss the goal or, at least, a complete description of the system. We finally must point out that running the same program on computers with different CPUs (Z80, 8086, 80286, 80386, with or without mathematical coprocessor) may lead to different results as well as when the same program is
pK 6.5/1
compiled with different compilers (Microsoft FORTRAN v. 3 and v. 4,Lahey FORTRAN 77, Pascal). The differences inthe composition of the mixtures, due to numerical noise, are usually negligible when the target function is steep in the composition hyperspace, but may grow up to 10% when the function is shallow in the optimal region. Fortunately enough, such a situation indicates that the buffering power is high, and thus that the pH dependence on composition is not critical. FORTRAN, although challenged by C, still appears the most reliable language for numerical computation, but our experience indicates that the accuracy of the results obtained with PCs sometimes needs to be checked with a well proven mainframe.
M a x . Dev.
SD =
1.5 ; pK 8/2
70 1
pK 6.5/1 = 1.5
= 1
;
pK 8/2
=
1
3.4 3.2 3.0
2.8 2.6 2.4 2.2 (u
\2.0
u? 1.8 co 1.6
Y n1.4 1.2 1 .o
.a .6 .4
.2
.O
pK 6.5/1
delta = 1.5
;
pK 8/2
3.4
3.2 3.0 2.8 2.6 2.4
N
2.2
\2.0
uW? 1.8 Y
1.6
ni.4
1.2 1 .o
.E .6
.4 .2
.o
.O
.5
1.0
1.5
2.0
2.5
3.0
3.5
=
1
.O
.5
1.0
1.5
SD
pK 6.5/1
2.0
+
=
2.5
3.0
3.5
m a x dev
1.5 ; pK 8/2 = 1
702
Elecrrophoresis 1991, I2, 693-703
F. C . Celentano, E. Gianazza and P. G. Righetti
5 Practical concluding remarks
Transferring into the practical laboratory life the above theoretical considerations is surely not a painless task, as the message does not bounce off the pages so brightly and clearly as required. As the main application of pH gradients, and the need to accurately forecast their behavior, is in the field of the IPG techniques, we summarize in the following a few guidelines drawn from our experience in casting IPG gels. (i) There is a fundamental way of smoothing a pH gradient: whenever possible, arrange for mixtures with many, evenly and symmetrically placed pKs and then use the “equal concentrations” method, minimizing CV(p). The “different concentrations” technique is better when the above conditions are not met, but then a different target function should be used. (ii) For creating linear extended pH intervals, it is necessary to use strong titrants, which will carry the pH values to the required extremes without contributing to the /? power of the system (for that to occur the pKof the titrants should be at least 2 pH units outside the extremes of the interval). (iii) The supporting density gradient interferes with the linear dispensing of the pH gradient (see Fig. 1 in [12]). Thus the densitygradient should be kept to a minimum.We have shown that, although a glycerol density gradient of 0-200/0 is recommended, a 0-100/0 gradient is more than adequate to stabilize the liquid during gel casting [38]. (iv) An average /? power of 3 meq L-’pH-’ is more than adequate to support and stabilize a pH gradient in IPG (as well as in CA-IEF). As an example, a 10 mg/mL solution of hemoglobin, at the PI, has a /? power of only 1 peq L-’pH-’. Often, well-functioning IPG gradients can be obtained with a p of only 1 meq L-’ pH-’. (v) In such dilute gels as normally used for IPGs (see point iv) no corrections for the activity coefficients is needed. But if a gel contains more than 20 mM total Immobilines, or polyprotic species, then such corrections will have to be introduced. (vi) Whenever possible avoid the use of oligo- and polyprotic buffers. As shown in Figs. 1 and 2, the optimization of mixtures of these compounds is difficult. Monoprotic species should always be preferred, when available. In addition, oligoprotic species could bind to proteins and be quite difficult to remove [39]. Ourprogram could be easilymodified to compute the pZs of proteins and to calculate and optimize pH gradients in ionexchange chromatography. The first problem is like calculating a fixed pH value, given a concoction of 7-8 different buffers of known pK, but we doubt there is any meaning in this. What about the micro-dissociation constants of ionizable amino acids, once incorporated in the polypeptide chain? The second requires the knowledge of the retention of different ions on the column. Our program will be made available after the optimization problem is fully understood and a reliable method will be implemented.
This work has been supported by grants from Agenzia Spaziale Italiana and CNR, Progetto Finalizzato Chimica Fine II (P. G.R.)and progetto Finalizzato Biotecnologie e Biostrumentazione (EC.C and P.G.R.). The authors are indebted to Dr. Monica Boccato of the Consorzio Interuniversitario Lombardo Elaborazione Automatica, (CILEA)for writing the hyperspaces visualization programs and for her skillful and patient help in plotting the data, and to Mr. Carlo Tonani, who played a main role in the implementation of PGS v.1 during the preparation of his doctoral thesis. Received August 10, 1990
6 References Almgren, M., Chem. Scripta 1971, 1, 69-75. Cann, J. R., in: Righetti, P. G., van Oss, C. J. and Vanderhoff, J. W. (Eds.), Electrokinetic Separation Methods, Elsevier, Amsterdam 1979, pp. 369-387. Schafer-Nielsen, C., in: Dunn, M. J. (Ed.), Gel Electrophoresis of Proteins, Wright, Bristol 1986, pp. 1-36. Palusinski, 0. A., Allgyer, T. T., Mosher, R. A., Bier, M. and Saville, D . , Biophys. Chem. 1981,13, 193-203. Bier,M.,Mosher,R.A. and Palusinski, O.A.,J. Chromatogr. 1981,211, 313-323. Bier,M.,Palusinski, O.A.,Mosher, R.A. and Saville, D.,Science 1983, 221, 1281-1287. Mosher, R. A,, Bier, M. and Righetti, P. G., Electrophoresis 1986, 7, 59-66. Haddad, P. R. and Cowie, C. E., J. Chromatogr. 1984,303,321-330. Poncelet, D., Pauss, A,, Naveau,H., Frere, J. M. and Nyns,E. J.,Anal. Biochem. 1985,150,421-428. Sillero, A. and Ribeiro, J. M., Anal. Biochem. 1989, 179,319-325. Bjellqvist, B., Ek, K., Righetti, P. G., Gianazza, E., Gorg, A,, Westermeier, R. and Postel, W., J. Biochern. Biophys. Methods 1982, 6, 3 17-33 9. Dossi, G., Celentano, F. C., Gianazza, E. and Righetti, P. G., J. Biochem. Biophys. Methods 1983, 7, 123-142. Peterson, E. A. and Sober, H. A. in:Alexander, P. and Block, R. J. (Eds.), Analytical Methods of Protein Chemistry, Pergamon Press, London 1960, pp. 88-102. Gianazza, E., Dossi, G., Celentano, F. C. and Righetti, P. G., J. Biochem. Biophys. Methods 1983,8, 109-133. Celentano, F. C.,Gianazza,E.,Dossi,G.and Righetti,P. G., Chemom. Intell. Lab. Systems 1987, I , 349-358. Gianazza, E., Celentano, F. C., Dossi, G., Bjellqvist, B. and Righetti, P. G., Electrophoresi~ 1984. 5, 88-97. Gianazza, E., Giacon, P., Sahlin, B. and Righetti, P. G., Electrophoresis 1985, 6, 53-56. Gianazza, E., Astrua-Testori, S . and Righetti, P. G., EIectrophoresis 1985, 6, 113-117. Chrambach, A. and H.jelmeland, L. M. in: Hirai, H. (Ed.). . ,. Electrophoresis ’83, De Gruyter, Berlin 1984, pp. 81-97. [20] Rilbe, H., J. Chromatogr., 193-205. [21] Bier, M., Mosher, R. A., Thormann, W. and Graham, A. in: Hirai, H. (Ed.), Electrophoresis ’83, De Gruyter, Berlin 1984, pp. 99-107. [22] Righetti, P. G. and Gianazza, E., J. Chromatogr. 1985, 334,7l-82. [23] Righetti, P. G., Gianazza, E. and Celentano, F. C., J. Chromatogr. 1986,356,9-14. [24] Bianchi-Bosisio, A , , Righetti, P. G. , Egen,N. B. and Bier, M., Electrophoresis 1986, 7, 128-133. [25] Gianazza, E., Celentano, F., Magenes, S., Ettori, C. and Righetti, P. G., Electrophoresis 1989, 10, 806-808. [26] Gelfi, C.,Bossi,M. L. Bjellqvist,B. and Righetti,P. G.,J. Biochem. Biophys. Methods 1987, I S , 41-48. [27] Chiari,M., Casale, E., Santaniello, E. and Righetti,P. G., Theor. Appl. Electrophoresis 1989, I , 92-102. [28] Chiari, M., Casale, E., Santaniello,E. and Righetti, P. G., Theor. Appl. Electrophoresis 1989, I , 103-107. [29] Righetti,P. G.,Chiari,M., Sinha,P.K.and Santaniello,E.,J.Biochem. Biophys. Methods 1988, 16, 185-192.
Electrophoresis 1991,
12, 703-709
Computer program MOL MASS
[30] Chiari, M.,Righetti, P. G., Ferraboschi, P.,Jain,T. and Shorr,R.,Electrophoresis 1990, 11, 617-620. [31] Chiari, M., Pagani, L., Righetti, P.G., Jain, T., Shorr, R. and Rabilloud, T., J.Biochem. Biophys. Methods 1990,21, 165-172. [32] Charlionet, R., Sesboul, R. and Davrinche, C., Electrophoresis 1984, 5, 176-178. [33] Celentano, F. C., Tonani, C., Fazio, M., Gianazza, E. and Righetti, P. G., J. Biochem. Biophys. Methods 1988, 16, 109-128. [34] Righetti, P. G., Fazio, M., Tonani, C., Gianazza, E., and Celentano, F. C., J. Biochem. Biopliys. Methods 1988, 16, 129-140.
Gunter M. Rothe Hans Weidmann Institut fur Allgemeine Botanik, Johannes Gutenberg-Universitat, Mainz
1351 Celentano, F. C., Gianazza, E., Tonani, C. and Righetti, P. G . in: Schafer-Nielsen, C. (Ed.), Electrophoresis '88, VCH Weinheim 1988, pp. 15-27. [36] Vesterberg, O., Acta Chem. Scand. 1969, 23, 2653-2666. [37] Righetti,P. G., Pagani, M. and Gianazza, E.,J. Chromatogr. 1975,109, 341-356. [38] Bossi, M. L., Bossi, O., Gelti, C. and Righetti, P. G., J. Biochem. Biophys. Methods 1988, 16, 171-182. [39] Righetti, P. G., fsoe(ectric Focusing, Theory, Merhodology and Applications, Elsevier, Amsterdam 1983.
Computer-aided calculation of the molecular size of nondenatured proteins in pore-gradient gel electrophoresis A computer program written in Turbo C is described, which uses the two-step mathematical procedure published recently (Rothe, G. M., Electrophoresis 1988, 9, 307-316) to evaluate the molecular mass, Stokes'radius, spherical radius, and frictional coefficient of nondenatured proteins. The program runs on any IBM-PC or 100% compatible IBM-PC, provided the disk operating system MS-DOS or PCDOS 3.0 or later has been installed. Functions that are permanently in use are accessible by menu. Storage and loading of data from disk and help instructions can be called by use of function keys. The program provides several tables into which inserted and calculated data is automatically integrated. Each table can be printed out, provided a printer with IBM character set is connected to the computer.
1 Introduction The high band resolution of polyacrylamide gel electrophoresis [l,21 can be improved by using gradient gels instead of homogeneous gels [3-81. The average pore radius of gradient gels decreases with increasing gel concentrations, i.e. in the direction of the moving proteins. This results in a sharpening of migrating protein bands and a separation of proteins over a larger molecular mass range than in homogeneous polyacrylamide gels [3-81. Since the polyacrylamide matrix causes no electroendosmosis, pore gradient gels have also been used to estimate the size of nondenatured [7-261 and sodium dodecyl sulfate (SDS)denatured proteins [14,27-34].To enable the calculation of the molecular mass of proteins from data obtained by poregradient gel electrophoresis (PGGE), both one-step [lo171 and two-step mathematical procedures [18-261 have been described. It was shown only recently, however, that the reliability of one-step procedures is practically limited to SDS-PGGE [9,26,27] and to electrophoresis in the presence of the nondenaturing detergent cetyltrimethylammonium bromide CTAB polyacrylamide gel electrophoresis [ 171 where all proteins under investigation attain the ~~
Correspondence: Prof. Dr. Gunter M. Rothe, Institut fur Allgemeine Botanik, Johannes Gutenberg-Universitat, Saarstr. 21, W-6500 Mainz, Germany Abbreviation: PGGE, pore-gradient gel electrophoresis
0VCH Verlagsgesellschaft mbH, D-6940 Weinheim,
703
1991
same charge per unit mass. Without the anionic detergent CTAB, which can only be applied at acid pH values, reliable size estimations of native proteins afford a time-dependent version of PGGE [35].This means that before a calibration line can be set up to calculate the molecular mass of sample proteins, the time-dependent migration distances or migration velocities of both a set of marker proteins and the sample proteins must be analyzed [9,25,26]. Migration velocities are either expressed as (i) relative mobilities (retarded mobility divided by free mobility) [lo, 19, 211 or (ii) as apparent mobilities [lo, 25,261. So-called Ferguson plots were also applied to describe the relative mobilities of proteins in polyacrylamide gels although their original use was to describe protein mobilities in starch gels [ 191.The validity of this simple relation, however, is lost at polyacrylamide concentrations above 15%T [36] and as a result it cannot be applied to linear gradient gels in the range of 4-30%T [9, 25, 26, 361. Even in gradient gel electrophoresis the migration velocity of nondenatured proteins is strongly influenced by their net native charge [35]. Therefore, it is likely that a sample protein migrates faster or slower than an equally sized marker protein [35].This fact implies that a charge-independent protein parameter must be found first before the molecular size of a nondenatured protein can be calculated from PGGE data [35,36]. Such a parameter is represented by the maximum migration distance (D,,,), i.e., the distance a protein travels in pore gradient gel electrophoresis until its mi0173-0835/91/1010-0703 $3.50+.25/0
Fabrizio C. Celentano’ Elisabetta Gianazza2 Pier Giorgio Righetti’ ‘Department of Biology, University of Milano and Department of Biomathematics, UCLA School of Medicine ’Department of Biomedical Sciences and Technologies, University of Milano
Computational approach to IPGs
693
On the computational approach to immobilized pH gradients The unified treatment for computing the pH of complex mixtures of mono- and polyprotic buffers, including ampholytes, as utilized in the gradient simulation program PGS, is presented. Its ability to compute pH, buffering power and ionic strength is shown by discussing a few simulations. The problems arising in the automatic formulation of optimal mixtures are presented, as well as the merits and limits of several target functions utilized in such optimizations. It is shown that no universal target function exists and that a proper optimization method should account for the fact that more than one formulation is possible for a given pH range.
1 Introduction 1.1 Early simulations on isoelectric focusing in carrier ampholytes There exist several approaches to the study of pH gradient formation in isoelectric focusing in carrier ampholytes (CAIEF). Perhaps the earliest model was that proposed by Almgren [I]. By assuming Gaussian CA distribution around the pl, equal diffusion coefficients, electric mobilities and relative concentrations and evenly spaced plvalues of CA-buffers along the pH scale, Almgren was able to simulate conductivity, pH and distribution profiles, and concluded that at least 30 CA species/pH unit were needed for the production of a stepless pH gradient. Similar steady-state profiles were also proposed by Cann [2] and by Schafer-Nielsen [3] (see Figs. 1.9 and 1.10 in this last article). The most extensive modeling proposed to date appears to be that performed by Bier’s group [4-61: their program allows the simulations of all four modes of electrophoresis (moving boundary, zone electrophoresis, isotachophoresis, and IEF. Generally speaking, for characterizing one such electrophoretic system, one needs a relationship (called the velocity relation) describing the steady-state, plus the three fundamental laws, i.e. electroneutrality, Ohm’s and mass conservation laws. In all these cases, the simulation allows prediction of the ways molecules (and macromolecules) approach the final steady-state and what particular distribution profile they assume once attaining such a situation. It was shown that, with some modifications, such a model will predict also the steady-state distribution profiles of amphoteric compounds in immobilized pH gradients (IPGs) [71. In a related field, Hadded and Cowie [8] have described a computer-assisted optimization of eluent concentration and pH in ion chromatography. This procedure provides for systematic selection of eluent concentration (at a fixed pH value) or eluent pH (at a fixed eluent concentration) in ion chromatography. Poncelet eta/. [9] computed, in complex chemical and biochemical systems at equilibrium, pH (or
Correspondence: Prof. P. G. Righetti, University of Milano, Via Celoria 2, 1-20133 Milano, Italy Abbreviations: B, buffering power; CA, carrier ampholytes; CV@), coefficient of variation of b; DMlmr the greatest deviation, either maximal positive or maximal negative, from linearity; IEF, isoelectric focusing; IPG, immobilized pH gradients; MGS, monoprotic electrolyte gradient simulation; PGS, polyelectrolyte gradient simulation; SD, standard deviation; 2-D, two-dimensional.
0VCH Verlagsgesellschaft mbH, D-6940 Weinheim, 1991
the concentration of any chemical species), partitions between acidic and basic forms, global charge, molar mean charges, ionic strength and molar mean contribution to the ionic strength. In this last case, activity coefficients were introduced, in order to correct the dissociation constant for ionic strength. However, “when the number of acid-base systems is large (more than three) the system of equations becomes very difficult to handle”. Sillero and Ribeiro [lo] have recently published a procedure for calculating the p1 of proteins: this is in fact an algorithm for estimating a fixed pH value given a mixture of weak acidic and basic groups. Their progam has been applied to proteins bearing up to 10 acidic and 10 basic groups. 1.2 Monoprotic electrolyte and polyelectrolyte gradient simulation: A chronicle of events In the last few years, we have developed simulation programs whose aim is to model and optimize mixtures of buffers for generating linear pH gradients. Although of general applicability, they parallel in their history the evolution of the concepts and applications of IPGs, started a decade ago by our group together with Dr. Bjellqvist’s and Dr. Gorg’s teams [Ill. IPGs were born as a continuation of conventional CA-IEF and originally were mostly meant to extend the fractionation capability of the existing technique to extremely narrow pH ranges. In fact, in the cornerstone publication [ 111, only modified Henderson-Hasselbach’s equations were given for computing narrow to ultranarrow linear pH gradients, extending to a maximum of 1 to 1.2 pH units. Such equations were based on the use of only two species (the acrylamido acids and bases used to generate and stabilize the pH gradient within the IPG matrix), one of them buffering and the other titrating within the desired pH range. However, stability and reproducibility as afforded by IPGs seemed appealing features for wide-range formulations, including a larger number of buffering Immobilines. This arrangement in turn required such complex calculations that automatic computation became a must for both the simulation of the gradient steps (pH, ionic strength and buffering power p ) and the optimization of their composition in order to give the most linear pH course. Our first simulation program [I21 (MGS, or monoprotic electrolyte gradient simulation) was operating by the end of 1982. A first approach to the formation of extended pH gradients was through the sequential elution of the buffering species of increasing pKfrom a multichamber (actually, a 5-chamber) mixer [12], developed for gradient elution as 0173-0835/91/1010-0693 $3.50+.25/0
694
Electrophoresis 1991, 12, 693-703
F. C . Celentano, E. Gianazza and P. G. Righetti
first proposed by Peterson and Sober [13]. This laborious procedure was soon abandoned in favor of classical twovessel gradient mixing [14, 151, for which we theoretically studied the conditions of gradient linearity, as a function of pKdistribution of the buffers, and of the titrants [15].Along this line in 1984 we published the formulations for a series of wide immobilized pH gradients (spanning 2-6 pH units within the pH 4-10 range), optimized in terms of gradient linearity, the standard deviation from linearity within the stated pH extremes not exceeding 1% of the range span [16]. In this paperwe compared two approaches to the generation of extended pH gradients: (a) in one case each Immobiline had the same concentration in both vessels of the mixer; (ii) in the other, different concentrations of buffering species could be present in each chamber. In the latter case the apparent pK of a given buffer may be shifted in such a way that the buffering power across the pH range is smoothed and the conditions for linearity better approached. Gradient linearity, however, is not the optimal solution in quite a few cases: e.g., in developing two-dimensional (2-D) maps, best resolution in the focusing dimension would clearly be obtained by nonlinear pH gradients, following the relative abundance of isoelectric proteins along the pH scale. Thus, we also calculated wide, nonlinear IPG recipes for use in 2-D maps and in most cases requiring analysis of highly heterogeneous samples [17]. As at that time we had given extended recipes including also two commercially unavailable components (a strong acid and a strong base) we were forced by colleagues to publish a new set of formulations, optimized in the absence of such titrants [18].During that period, other types of focusing in nonamphoteric buffers were introduced: Chrambach’s ’arrested stack’ [191, Rilbe’s ’steady-state rheoelectrolysis’ [20] and Bier’s ’physically’ immobilized pH gradients [21]. With MGS we could resimulate their data and find the merits and limits of each alternative route [22]. By 1986 IPG at pH extremes was first tried. Thus our computer program was expanded to include the effects of H+ and OH- (negligible within the 4-10 pH range) on p , ionic strength and pH profile [23].As a provocative concept, we introduced the idea of water as two Immobiline species, one with pK=-1.74 [H’], the other with pK= 15.75 [OH-]. The first compounds to be focused in the pH 3-4 interval were dansylated amino acids [24].In the case of very acidic and very basic intervals, the IPG behavior was modeled also [7] by modifying the general program for steady-state IEF, as developd by Palusinski et al. [4] and by Bier et al. [5]. As chemicals with more acidic and more basic pK became available, extended formulations including pH extremes were recently computed: the widest pH range that could be formulated is a pH 2.5-11 interval, spanning 8.5 pH units [25, 261. In 1988 we started a program on the characterization of existing Immobilines [27,28] and on the synthesis of new species (a pK3.1 [29], a pK6.6 and a pK7.4 [30], a pK8.05 [31] and a pK 10.3 [26]).At this point the number of useful buffers for IPGs had considerably expanded and MGS, which was limited to mixtures of no more than 10 species (later 12) could no longer handle this new complexity. In addition, oligoprotic acrylamido buffers have been deLrribed [32], and these species too were not included in our simula-
tion program. These facts forced us to develop a brand new program, polyelectrolyte gradient simulation (PGS) for IEF and chromatography.
2 pH Calculation in complex mixtures We summarize in the following the derivation of the expressions utilized for computing a pH gradient, as utilized in the PGS program, accounting for mono and polyprotic species, as well as for polyampholytes, treated separately in different preceding papers [12, 15,33,34]. The pH of a solution of an ampholyte i having nAi acid groups and nBibasic ones is obtained from the electroneutrality condition
where nAl and n,, are the numbes of the acid and basic groups, alA,and a,,, the degrees of dissociation of the j-th acid or basic group, zAJand zBJthe charges of the zwitterion and [CTJthe total concentration of the ampholyte. A similar equation holds also for a mixture of one polyprotic acid and one polyprotic base, but in this case the concentrations of the two may be different, while for monoprotic species the upper and lower values of the summation index are both 1. If follows that for a mixture of m acids and I bases, summing for all the species, the electroneutrality condition becomes m
I
“Ai
where [A:] and [BT] are the total concentrations of the i-th acid or base. For ampholytes the total concentrations corresponding to the acid and basic groups of such species will be equal. It follows that the above equation can be utilized for computing the pH of any mixture of mono and polyprotic species, given a general expression for the degrees of dissociation. This is found by considering the generic dissociation steps of the i-th polyacid Ai or polybase B,: A(h-l)-
Ak: i
h-tHt .h
whence the concentrations of the anion having 11 LliLiigc\ and of the cation with n,,-h+l charges, can be computed. Obviously, for monoprotic species nA,= nBI= h = 1. One obtains: A?-’)A:= k: (34 IH+I
Electrophoresis 1991, 12, 693-703
Computational approach to lPGs
After the dissociation of 1,2, ..J..., nAIprotons, the concentrations of the anions from acid A, can be computed from Eq. (3a), recursively substituting [A:] into [A;], the latter into [A12-],and so on, up to [A:*'-]. An identical recursive substitution is followed for the bases. The procedure may be visualized using a linear graph where the nodes represent the concentrations of the anions whose values are obtained multiplying the value of the preceding node by the value of the interposed branch. The latter value, in effect, is a (constant) transfer function transforming each node value into the following, as depicted below for an acid:
[A01
[A:, 1
0
>O
k,'/[H+]
[A:-]
k;/[H+]
The substitution of the Eqs. (6) and (5) into Eq. (2), finally gives the equation allowing computation of the pH of any mixture of mono and polyprotic buffers. In effect, for monoprotic species, the dissociation coefficients become CLA~ =KAi/(KAi
t H+); u B ~= H+/(KB~ t H+)
and Eq. (2) reduces to m
1
(7)
[A:-] >o
s o . . .o
695
...
#+'/ [H']
One obtains, for h =j:
(44
The total concentrations of the acid [AT] and of the base [B:] are obtained summing the Eqs. (4) for all t h e j values between one and nA, or nB,and adding the concentration of the remaining undissociated acid or base:
identical to Eq. (5) in [15] with the addition of the hydrolytic products of water, not directly accounted for in the MGS program. The buffering power p of the mixture, an important quantity both for the gel behavior and the optimization of its and pBI composition, is the sum of all the contributions PAL of all the anions and cations (or different groups of the ampholytes) plus that of water. For each of them
and, for the expression of these derivatives, the reader is referred to the previous publications [15,33-351.
3 Examples of simulations with polyprotic buffers
The dissociation coetlicient for the ion with charge j, i.e the ratio between the ion concentration and the total concentration of the considered species, is now obtained by dividing each one of the Eqs. (4) by the corresponding Eq. ( 5 ) :
a i i=
[H+]j
j-1
hkh
Earlier simulations, and their comparison with experiments, have shown the satisfactory accuracy of our computing methods for both MGS [12, 141 and PGS [34-351, as long as the Immobiline concentrations are low enough to keep the activity effects negligible. We have also observed that the best results are obtained when the pKs of the buffers are evenly distributed along the gradient. As this is seldom the case, we present in the following a few simulations, showing the importance of filling up the ,,holes" in the pK distribution along the gradient. If citric acid (pK, 2.9, pK2 4.1, pK, 5.8) is dissolved at the same concentration in both chambers of a gradient mixer (1 :1ratio) and titrated between pH 3 (close to pK,,,,,) and pH 6.8 (above pK,,,), the gradient of Fig. 1A is obtained. The pH gradient is linearbetween pK, and pK, and there is a sudden change of slope in proximity of the p minimum (step #28). If one adds imidazole (pK6.9) to the mixture (in a 1:l ratio with citric acid, thus extending the pK range of the mixture to cover the whole gradient span), the linear portion of the gradient extends up to pH 6, but the slope then flattens (too much buffering power) (Fig. lB).The addition of a third component with intermediate pK (itaconic acid, pK, 3.85, pK2 5.49, thus filling "pK holes" in the mixture, should improve the situation. In effect, with our simulation program, we can find by trial and error the proper ratio of the three components (54:20:30 citric acid : itaconic acid : imidazole) allowing a smooth pH gradient in the whole titration interval to be obtained (Fig. 1C).
696
-
PH 7 -
-n-
CITRIC K I D 1:l
0'
dmBn
6 -
Electrophoresis 1991, 12, 693-703
F. C. Celentano, E. Gianazza and P. G. Righetti
I
=
I
D,
225
m
8
0'
*
R B
m h
p
/ *
0
185 34
-
32
-
165 145 125 185
38
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- 45
28
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15
25
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GI
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38
35
Figure 1. Simulation of the behavior of a simple oligo acid (citric acid). In all cases citric acid (54 mM) is titrated at pH3.0 and pH 6.8 and the linear mixing ofthese limiting solutions is done linearly, as with a two-vessel gradient mixer. Forty steps are computed. (A) The citric acid molarity is kept constant in the two chambers (1:l ratio). (B) Each chamber contains a mixture of 54 mM citric acid and 54 mM imidazole (imid). ( C ) Each chamber contains a mixture of 54 mM citric acid, 20 mM itaconic acid and 30 mM imidazole. Note in this last case the good pH gradient linearity and the smoothing of the p power. I , ionic strength, p buffering power.
Ekctniphoresis 1991. 12,693-703
PH
Computational approach to IPGs
-
697
TEPA 1 : l
1°/** * * * *
t
fl.**
A@-
445
-** I I I I I I I I I
0
5
5
0
PH
** * + L
5
d , ~ ~ 1 1 1 , 1 1 1 1 1 , l ~ 1 I I 1 , I , , , , , , , , , , , , , , , , , , , , , , , , f ~ *
10
15
10
15
-
20
20
25
25
4
339
39
30
35
40
45
50
55
60
STEP
30
35
40
45
50
55
68
STEP
TEPA - TETA
-
EDA
** *
I
0
Figure 2. Simulation of the behavior of a simple oligobase (tetraethylenepentamine,TEPA) and ofits mixture with other oligobases. In all cases TEPA (60 mM) is titrated at pH 4 and pH 10 and the mixing of these limiting solutions is made linearly, as in a two-vessel gradient mixer. Sixty steps are computed. (A) The TEPA molarity is kept constant in the two chambers (1:l ratio).(B)The molarityratioisaltered in the two chambers, so that the acidic reservoir (pH 4) contains a 2.2 higher molarity than the basic one (pH 10). (C) Each chamber contains a mixture of 60 mM TEPA, 20 mM tetraethylene tetramine (TETA) and 20 mM ethylenediamine (EDA). Note in this last case the smoothing of the sigmoidal transition around pH 6 and the filling-in of the p power minimum.l,ionic strength;p,bufferingpower.
698
Electrophoresis 1991, 12, 693-703
F. C . Celentano, E. Gianazza and P. G. Righetti
An oligobase, tetraethylenepentamine (TEPA, pK, 2.7, pK2 4.3, pK, 7.9, pK, 9.1, pK, 9.9) titrated between pH 4 and pH 10 is a particularly interesting case: in fact, TEPA was originally proposed as one of the oligoamines forming the backbone of carrier ampholytes [36,37]. As shown in Fig. 2A, such oligoamines alone cannot form a linear pH gradient. Due to the lack of suitable pKvalues, there is a steep sigmoidal transition around pH 6, corresponding to a deep minimum of p power. If the concentration ratio between the two chambers is shifted from 1: 1 to 2.2: 1, the /? power of the acidic region is increased (the p minimum moves from fraction 12 to 25) and the linear portion of the gradient can be extended, but there remains the sigmoidal transition between the two segments ofthe curve. When adding ethylendiamine (EDA, pK, 7.0, pK, 10) and triethylenetetramine (pK, 3.3, pK26.7, pK, 9.2 and pK, 9.9) and modelling the mixture with our simulation program, much better pH and /? profiles can be obtained (Fig. 2C): the sigmoidal transition has been substantially reduced and the minimum of p power has been increased from less than 10 to more than 20% of its maximum value. The optimal concentration ratio between the three components corresponds to TEPA : TETA : EDA= 3 : 1 : 1 (Fig. 2C).
4 Optimization: An “experimental” approach 4.1 The problem of the target function Gradient optimization, that is the determination of the composition of the solutions in the two mixer’s chambers allowing the best approximation of the required gradient shape to be obtained, requires the definition of a target function to be minimized. Such a function of the mixture compositions in both vessels should accurately represent the gradient deviation from the ideal shape, in order to play the same role as the sum of squared deviations, in linear and nonlinear regression. In this paper we refer only to linear gradients because the optimization approach will be the same with every gradient shape. In fact, from the mathematical point of view, we have anyhow a nonlinear optimization problem, as the function describing the gradient shape is intrinsically nonlinear. Optimization cannot be treated as a regression problem. The latter, in fact, deals with the optimal fitting of a linear or nonlinear function to randomly scattered experimental points, while optimization requires the comparison of points sampled from two continuous functions, i.e. the actual and the required gradients. It follows that the deviations to be minimized are not normally distributed and independent, as required in regression, but highly correlated. The choice of the target function thus becomes nontrivial, and different functions usually lead to sensibly different results, depending on the system’s behavior and on how the functions to be compared are sampled (for instance, the mean deviation from linearity depends on the number of samples). The target function for the evaluation of the comparison should thus closely reflect the behavior of the considered system. In fact, the purpose of a target function is twofold: (i) to depict the linearity of the gradient course, and (ii) to describe the overall congruity of the pH range, i.e. the correspondence between expected and found. We have first sug-
gested keeping constant the buffering power p of the mixture, a welcome property also for other reasons, byminimizing its variation coefficient CV@) [14, 151, but stressed the fact that a constant /3 leads to linearity if and only if the mixture concentration (rigorously the concentration of each species in the mixture) is constant. It follows that such a target function does not work at its best when the species concentrations are different in the two vessels. We currently minimize CV(/?)only when using the “same concentrations” procedure [16].Other target functions have been used, and are still under test, with the “different concentrations” procedure. The first is the standard deviation, SD,not the plain sum of squares, in order to lessen the aforementioned drawbacks by taking a square root. By referring the deviations to the ideal linear gradient shape [15], both the linearity and congruity issues are addressed. It should be noticed that the same occurs with nonlinear gradients, by computing the deviations from their ideal shape. In this case, obviously, we must substitute nonlinearity for linearity. A second interesting target function is the same maximum absolute deviation DMlm,the greatest between the maximal positive deviation (DM)and the absolute value of the maximal negative one (DJ.This may be a good target when its minimization is not obtained at the expense of many small oscillations, introducing a kind of noise in the gradient. Another possible target comes from the observation that the best gradients are often almost symmetric around their center. It follows that the asymmetry to be minimized can be measured by the absolute value of the mean, D,,,or by the algebraic sum of the two maximal deviations A = 1 D M + D, 1. In a survey of the published simulations, the value of A corresponding to the optimized formulation was found in all cases very low. It should be noticed that also the sums of any two of the above targets can be assumed as a new target function. Fig. 3 compares the course of these target functions when optimizing a pH 5.5-8.5 gradient obtained with a single triprotic base whose pKs are 6,7 and 7.5, as the ratio of the buffer concentration in the acidic (A) and basic chamber (B) is
I \
0.1 -..,
0.0 0.50
4 0.75
1 .oo
SD
0-0
~
1.25
A : B concentration ratio
Figure 3. Optimization, in terms of linearity, of the pH gradient 5.5-8.5 obtained from the linear mixing of two limiting solutions (A = acidic, pH 5.5; B=basic,pH 8.5) containing different and variable concentrations of a triprotic base with pK1 6, pK2 7, pK3 7.5. The deviation from linearity is maximum deviation plotted in different ways: standard deviation (SD), towards negative (D,) or positive ( D M )values, average deviation (Day),as a function of the buffer concentration ratio between the acidic (A) and basic (B) chambers.
Ekcirophoresis 1991, / 2 , 693-703
Computational approach to IPGs
increased from 0.6 to 1.1. SD gives a shallow parabolic curve, whereas D,, is a steeper V-shaped function. The minima of the two curves are close to one another but do not coincide. Still close to these minima the curves representing D, and D, tangle one another, where A (not shown) gets to its minimum: zero. Both limbs ofthis cross are parabolic, but over a short distance may be approximated to straight lines. Their intersection is thus quickly found graphically or analytically: each simulation gives in fact two “experimental” points, D, and D, and two runs suffice to find two couples of points. The steepest and most informative function could be used to get close to the target, and the absolute minimum then searched while resorting to a more accurate if shallower function. The A function appears most useful for this first step within the simple simulations allowing for just one degree of freedom, but the presence of N variables would require the solution of an algebraic system of degree N+1. It should be anyhow kept in mind that the above functions may have several minima in systems with more degrees od freedom. Figure 4 shows the course of the minima of the above target functions when the same pH 5.5-8.5 gradient is obtained with bases having pK, 6 and pK, 7, as before, while pK, varies between 7.1 and 10.5.It appears that the different targets have different minima, except when the pKs are equally spaced and symmetrically disposed around the center of the gradient span. The interesting feature is that the observation above can be generalized, as one usually finds that the minima of the different target functions are, as could be expected, seldom identical, although, often, rather similar. Figure 4 also shows that a nonmonotonous relationship exists between the pKs of the buffer and its concentration for optimal linearity. Moreover, for monoprotic species the course of the gradient only depends on the pK value and not on the buffer dissociation as an acid or as a base (not shown). I
I 0-0
0 .c e
1 .oo --
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699
4.2 Exploration of the composition hyperspace
As a consequence of the results, and of the doubts, exposed in the preceding section, we have undertaken the systematic investigation of a panel of simple buffer mixtures, whose composition is varied stepwise; at each step, the pH gradient course arising from their linear mixing is simulated and a number of target functions are computed. By analyzing the structure of the multidimensional spaces representing different target functions as a function of composition of the mixtures in the two mixer’s vessels, we have sought: (i) general features in the configuration of such spaces that might help in devising an efficient optimization algorithm; (ii) the easiest target for the optimization procedure. We restricted ourselves to simple buffer systems because the number of experimental points to be computed dramatically increases with the number of the allowed degrees of freedom; in addition the graphical representation of multidimensional spaces soon becomes impracticable. Thus, the highest number of independent variables (buffer concentrations) in our tests is 3; these are drawn along the Cartesian axes, while the fourth, dependent, variable (the target function) is rendered with a color shade, using the UNIRAS graphic package installed on the Unisys 2200 mainframe of the University computing consortium CILEA. When dealing with less degrees of freedom, the same package allows the drawing of contour lines. In detail, the buffer systems that we are studying are: (i) with 1 degree of freedom: 1 diprotic or triprotic buffer, whose concentration is changed between the two chambers; (ii) with 2 degrees of freedom: 3 monoprotic buffers, with constant concentrations in the two chambers; (iii) with 3 degrees of freedom: 2 monoprotic buffers, with varying concentrations in both chambers. In all cases, the dissociating groups whose concentration is taken as the independent variable behave in the system as buffers, while strong acids or bases titrate the solutions to the desired pH extremes. Within each experimental series, the concentration of one of the buffers in one (cases i and iii) or in both chambers (case ii) is kept constant through0ut.A single (hyper)space is explored, i.e. only those solutions corresponding to a given pH range, or a given pair of pH extremes, are considered.
Figure 4. Comparison of the behavior of different target functions in the
MGS or PGS may enter any (hyper)space point corresponding to a predefined concentrations vector. Our selection for the present set of simulations greatly simplifies the spatial exploration. It should be stressed, however, that, in comparison with the MGS current optimization procedure, the present exploration runs overestimate one of the reference parameters, i.e. congruity vs. linearity, and emphasize two points along the gradient rather than evaluating the curve as a whole.
optimization, in terms of linearity, of gradients in the pH 5.5-8.5 range, obtained by linear mixing of different solutions of a triprotic base with pK, 6 , p K ~7and pK3 varying from 7.1 to 10.S.The optimization is obtained varying the base concentration in the acidic chamber A, whilst the solution in the basic chamberB has a constant concentration (10 mM).The minima of the targets are plotted as a function of the difference between the last two pKs. SD,standard deviation from linearity; DMlm),maximum absolute deviation in the positive or negative directions; D,,,absolutevalue of the average deviation. Note that there is a continuum of solutions allowing to cast optimized gradients (obviously of different quality), having the same pH span, and that the same A/B ratios can be used with solutions containing different species at different concentrations.
Figure 5 shows two sets of results concerning a pH 5.5-8.5 gradient obtained using monoprotic bases. In the first case the gradient is obtained using a pK 6.5 and a pK 8 species, the former at constant concentration in the acidic chamber and the latter at constant concentration in the basic chamber of the mixer. The other case shows a symmetric pK distribution, with a pK6 base constant in the first chamber and a pK 7.5 base constant in the second (basic) chamber. In both cases the behavior ofSD, of DM,,, and A appears rather different. The first function shows a single minimum and a
.-a c
0.50-
0.25 -I
i
700
Electrophoresis 1991, 12, 693-703
F. C. Celentano, E. Gianazza and P. G. Righetti
rather flat appearance, while DMlm (maximum deviation in the graph) is steeper and shows two minima.The sum of the two still shows two well defined minima (the distance between contour lines is 0.025). Still steeper, but more complicated is the behavior of A (delta), showing several minima of comparable depth.
Also interesting is the symmetry of behavior ofthe two mixtures. In effect, as the protolytic products of water play no role above pH 4 or below pH 10, by rotating the axes corresponding to the two chambers identical results are obtained, in consequence of the symmetric pK distribution along the gradient.This fact agrees with the finding that the
SD pK 6/1 = 1
; pK
7.5/2 = 1.5
pK 6 / 1
; pK
7.5/2
= 1.5
pK 6 / 1
Max. Dev. ; pK 7.5/2 = 1.5
= 1
pK 7.5/1
delta pK 6/1
=
1
SD
+
= 1
max dev ; pK 7.5/2 = 1.5
Figure 5. Contour maps of different target functions showing the minima predicting the optimal composition of the solutions to be linearly mixed for
casting a p H 5.5-8.5 gradient.These solutions contain two monoprotic bases with pKs6.5 and 8 (seriesA‘-D’) or6and7.5 series (A”-D”), titrated to the limiting pH values with a strong acid. In the A’-B‘ series pK6.5 is kept at constant concentration in the acid chamber (l),where pK8 is allowed to vary, whilst its concentration is varied in the basic (2) chamber, where pK8 is constant.An identical design is utilized for the simulations ofthe A - D s e r i e s . The degrees offreedom of the system are thus only two.The behavior of standard deviation (SD)is depicted by arather shallow surface with asingle minimum. The absolute maximum deviation from linearity DM,,, (Max. Dev.) shows steeper slopes and still steeper appears the algebraic sum of the maximum positive (DM)and negative (D,) deviations A (delta). These functions show more than one minimum because are representative of one single point of the gradient, whose position varies with varying composition.The shape of SD+ DM,,,(SD+Max. Dev.) is most influenced by the steep behavior ofthe 1atter.Note that,as the pKdistribution and the composition ofthe two mixtures series are symmetrical in respect to the gradient extremes, the two series turn out to be identical when the axes pertaining to the two chambers are exchanged.
Compulational approach to IPCs
Electrophoresis 1991, 12, 693-703
functions plotted in Fig. 4 are not monotonous. There is a continuum of different ways of casting the same gradient, obviously by using different species. These results, to be further extended, confirm that the gradient optimization is not a simple task. Although exceedingly good gradients can be obtained by trial and error, unsophisticated optimization procedures may miss the goal or, at least, a complete description of the system. We finally must point out that running the same program on computers with different CPUs (Z80, 8086, 80286, 80386, with or without mathematical coprocessor) may lead to different results as well as when the same program is
pK 6.5/1
compiled with different compilers (Microsoft FORTRAN v. 3 and v. 4,Lahey FORTRAN 77, Pascal). The differences inthe composition of the mixtures, due to numerical noise, are usually negligible when the target function is steep in the composition hyperspace, but may grow up to 10% when the function is shallow in the optimal region. Fortunately enough, such a situation indicates that the buffering power is high, and thus that the pH dependence on composition is not critical. FORTRAN, although challenged by C, still appears the most reliable language for numerical computation, but our experience indicates that the accuracy of the results obtained with PCs sometimes needs to be checked with a well proven mainframe.
M a x . Dev.
SD =
1.5 ; pK 8/2
70 1
pK 6.5/1 = 1.5
= 1
;
pK 8/2
=
1
3.4 3.2 3.0
2.8 2.6 2.4 2.2 (u
\2.0
u? 1.8 co 1.6
Y n1.4 1.2 1 .o
.a .6 .4
.2
.O
pK 6.5/1
delta = 1.5
;
pK 8/2
3.4
3.2 3.0 2.8 2.6 2.4
N
2.2
\2.0
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1.6
ni.4
1.2 1 .o
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.4 .2
.o
.O
.5
1.0
1.5
2.0
2.5
3.0
3.5
=
1
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.5
1.0
1.5
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pK 6.5/1
2.0
+
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1.5 ; pK 8/2 = 1
702
Elecrrophoresis 1991, I2, 693-703
F. C . Celentano, E. Gianazza and P. G. Righetti
5 Practical concluding remarks
Transferring into the practical laboratory life the above theoretical considerations is surely not a painless task, as the message does not bounce off the pages so brightly and clearly as required. As the main application of pH gradients, and the need to accurately forecast their behavior, is in the field of the IPG techniques, we summarize in the following a few guidelines drawn from our experience in casting IPG gels. (i) There is a fundamental way of smoothing a pH gradient: whenever possible, arrange for mixtures with many, evenly and symmetrically placed pKs and then use the “equal concentrations” method, minimizing CV(p). The “different concentrations” technique is better when the above conditions are not met, but then a different target function should be used. (ii) For creating linear extended pH intervals, it is necessary to use strong titrants, which will carry the pH values to the required extremes without contributing to the /? power of the system (for that to occur the pKof the titrants should be at least 2 pH units outside the extremes of the interval). (iii) The supporting density gradient interferes with the linear dispensing of the pH gradient (see Fig. 1 in [12]). Thus the densitygradient should be kept to a minimum.We have shown that, although a glycerol density gradient of 0-200/0 is recommended, a 0-100/0 gradient is more than adequate to stabilize the liquid during gel casting [38]. (iv) An average /? power of 3 meq L-’pH-’ is more than adequate to support and stabilize a pH gradient in IPG (as well as in CA-IEF). As an example, a 10 mg/mL solution of hemoglobin, at the PI, has a /? power of only 1 peq L-’pH-’. Often, well-functioning IPG gradients can be obtained with a p of only 1 meq L-’ pH-’. (v) In such dilute gels as normally used for IPGs (see point iv) no corrections for the activity coefficients is needed. But if a gel contains more than 20 mM total Immobilines, or polyprotic species, then such corrections will have to be introduced. (vi) Whenever possible avoid the use of oligo- and polyprotic buffers. As shown in Figs. 1 and 2, the optimization of mixtures of these compounds is difficult. Monoprotic species should always be preferred, when available. In addition, oligoprotic species could bind to proteins and be quite difficult to remove [39]. Ourprogram could be easilymodified to compute the pZs of proteins and to calculate and optimize pH gradients in ionexchange chromatography. The first problem is like calculating a fixed pH value, given a concoction of 7-8 different buffers of known pK, but we doubt there is any meaning in this. What about the micro-dissociation constants of ionizable amino acids, once incorporated in the polypeptide chain? The second requires the knowledge of the retention of different ions on the column. Our program will be made available after the optimization problem is fully understood and a reliable method will be implemented.
This work has been supported by grants from Agenzia Spaziale Italiana and CNR, Progetto Finalizzato Chimica Fine II (P. G.R.)and progetto Finalizzato Biotecnologie e Biostrumentazione (EC.C and P.G.R.). The authors are indebted to Dr. Monica Boccato of the Consorzio Interuniversitario Lombardo Elaborazione Automatica, (CILEA)for writing the hyperspaces visualization programs and for her skillful and patient help in plotting the data, and to Mr. Carlo Tonani, who played a main role in the implementation of PGS v.1 during the preparation of his doctoral thesis. Received August 10, 1990
6 References Almgren, M., Chem. Scripta 1971, 1, 69-75. Cann, J. R., in: Righetti, P. G., van Oss, C. J. and Vanderhoff, J. W. (Eds.), Electrokinetic Separation Methods, Elsevier, Amsterdam 1979, pp. 369-387. Schafer-Nielsen, C., in: Dunn, M. J. (Ed.), Gel Electrophoresis of Proteins, Wright, Bristol 1986, pp. 1-36. Palusinski, 0. A., Allgyer, T. T., Mosher, R. A., Bier, M. and Saville, D . , Biophys. Chem. 1981,13, 193-203. Bier,M.,Mosher,R.A. and Palusinski, O.A.,J. Chromatogr. 1981,211, 313-323. Bier,M.,Palusinski, O.A.,Mosher, R.A. and Saville, D.,Science 1983, 221, 1281-1287. Mosher, R. A,, Bier, M. and Righetti, P. G., Electrophoresis 1986, 7, 59-66. Haddad, P. R. and Cowie, C. E., J. Chromatogr. 1984,303,321-330. Poncelet, D., Pauss, A,, Naveau,H., Frere, J. M. and Nyns,E. J.,Anal. Biochem. 1985,150,421-428. Sillero, A. and Ribeiro, J. M., Anal. Biochem. 1989, 179,319-325. Bjellqvist, B., Ek, K., Righetti, P. G., Gianazza, E., Gorg, A,, Westermeier, R. and Postel, W., J. Biochern. Biophys. Methods 1982, 6, 3 17-33 9. Dossi, G., Celentano, F. C., Gianazza, E. and Righetti, P. G., J. Biochem. Biophys. Methods 1983, 7, 123-142. Peterson, E. A. and Sober, H. A. in:Alexander, P. and Block, R. J. (Eds.), Analytical Methods of Protein Chemistry, Pergamon Press, London 1960, pp. 88-102. Gianazza, E., Dossi, G., Celentano, F. C. and Righetti, P. G., J. Biochem. Biophys. Methods 1983,8, 109-133. Celentano, F. C.,Gianazza,E.,Dossi,G.and Righetti,P. G., Chemom. Intell. Lab. Systems 1987, I , 349-358. Gianazza, E., Celentano, F. C., Dossi, G., Bjellqvist, B. and Righetti, P. G., Electrophoresi~ 1984. 5, 88-97. Gianazza, E., Giacon, P., Sahlin, B. and Righetti, P. G., Electrophoresis 1985, 6, 53-56. Gianazza, E., Astrua-Testori, S . and Righetti, P. G., EIectrophoresis 1985, 6, 113-117. Chrambach, A. and H.jelmeland, L. M. in: Hirai, H. (Ed.). . ,. Electrophoresis ’83, De Gruyter, Berlin 1984, pp. 81-97. [20] Rilbe, H., J. Chromatogr., 193-205. [21] Bier, M., Mosher, R. A., Thormann, W. and Graham, A. in: Hirai, H. (Ed.), Electrophoresis ’83, De Gruyter, Berlin 1984, pp. 99-107. [22] Righetti, P. G. and Gianazza, E., J. Chromatogr. 1985, 334,7l-82. [23] Righetti, P. G., Gianazza, E. and Celentano, F. C., J. Chromatogr. 1986,356,9-14. [24] Bianchi-Bosisio, A , , Righetti, P. G. , Egen,N. B. and Bier, M., Electrophoresis 1986, 7, 128-133. [25] Gianazza, E., Celentano, F., Magenes, S., Ettori, C. and Righetti, P. G., Electrophoresis 1989, 10, 806-808. [26] Gelfi, C.,Bossi,M. L. Bjellqvist,B. and Righetti,P. G.,J. Biochem. Biophys. Methods 1987, I S , 41-48. [27] Chiari,M., Casale, E., Santaniello, E. and Righetti,P. G., Theor. Appl. Electrophoresis 1989, I , 92-102. [28] Chiari, M., Casale, E., Santaniello,E. and Righetti, P. G., Theor. Appl. Electrophoresis 1989, I , 103-107. [29] Righetti,P. G.,Chiari,M., Sinha,P.K.and Santaniello,E.,J.Biochem. Biophys. Methods 1988, 16, 185-192.
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Computer program MOL MASS
[30] Chiari, M.,Righetti, P. G., Ferraboschi, P.,Jain,T. and Shorr,R.,Electrophoresis 1990, 11, 617-620. [31] Chiari, M., Pagani, L., Righetti, P.G., Jain, T., Shorr, R. and Rabilloud, T., J.Biochem. Biophys. Methods 1990,21, 165-172. [32] Charlionet, R., Sesboul, R. and Davrinche, C., Electrophoresis 1984, 5, 176-178. [33] Celentano, F. C., Tonani, C., Fazio, M., Gianazza, E. and Righetti, P. G., J. Biochem. Biophys. Methods 1988, 16, 109-128. [34] Righetti, P. G., Fazio, M., Tonani, C., Gianazza, E., and Celentano, F. C., J. Biochem. Biopliys. Methods 1988, 16, 129-140.
Gunter M. Rothe Hans Weidmann Institut fur Allgemeine Botanik, Johannes Gutenberg-Universitat, Mainz
1351 Celentano, F. C., Gianazza, E., Tonani, C. and Righetti, P. G . in: Schafer-Nielsen, C. (Ed.), Electrophoresis '88, VCH Weinheim 1988, pp. 15-27. [36] Vesterberg, O., Acta Chem. Scand. 1969, 23, 2653-2666. [37] Righetti,P. G., Pagani, M. and Gianazza, E.,J. Chromatogr. 1975,109, 341-356. [38] Bossi, M. L., Bossi, O., Gelti, C. and Righetti, P. G., J. Biochem. Biophys. Methods 1988, 16, 171-182. [39] Righetti, P. G., fsoe(ectric Focusing, Theory, Merhodology and Applications, Elsevier, Amsterdam 1983.
Computer-aided calculation of the molecular size of nondenatured proteins in pore-gradient gel electrophoresis A computer program written in Turbo C is described, which uses the two-step mathematical procedure published recently (Rothe, G. M., Electrophoresis 1988, 9, 307-316) to evaluate the molecular mass, Stokes'radius, spherical radius, and frictional coefficient of nondenatured proteins. The program runs on any IBM-PC or 100% compatible IBM-PC, provided the disk operating system MS-DOS or PCDOS 3.0 or later has been installed. Functions that are permanently in use are accessible by menu. Storage and loading of data from disk and help instructions can be called by use of function keys. The program provides several tables into which inserted and calculated data is automatically integrated. Each table can be printed out, provided a printer with IBM character set is connected to the computer.
1 Introduction The high band resolution of polyacrylamide gel electrophoresis [l,21 can be improved by using gradient gels instead of homogeneous gels [3-81. The average pore radius of gradient gels decreases with increasing gel concentrations, i.e. in the direction of the moving proteins. This results in a sharpening of migrating protein bands and a separation of proteins over a larger molecular mass range than in homogeneous polyacrylamide gels [3-81. Since the polyacrylamide matrix causes no electroendosmosis, pore gradient gels have also been used to estimate the size of nondenatured [7-261 and sodium dodecyl sulfate (SDS)denatured proteins [14,27-34].To enable the calculation of the molecular mass of proteins from data obtained by poregradient gel electrophoresis (PGGE), both one-step [lo171 and two-step mathematical procedures [18-261 have been described. It was shown only recently, however, that the reliability of one-step procedures is practically limited to SDS-PGGE [9,26,27] and to electrophoresis in the presence of the nondenaturing detergent cetyltrimethylammonium bromide CTAB polyacrylamide gel electrophoresis [ 171 where all proteins under investigation attain the ~~
Correspondence: Prof. Dr. Gunter M. Rothe, Institut fur Allgemeine Botanik, Johannes Gutenberg-Universitat, Saarstr. 21, W-6500 Mainz, Germany Abbreviation: PGGE, pore-gradient gel electrophoresis
0VCH Verlagsgesellschaft mbH, D-6940 Weinheim,
703
1991
same charge per unit mass. Without the anionic detergent CTAB, which can only be applied at acid pH values, reliable size estimations of native proteins afford a time-dependent version of PGGE [35].This means that before a calibration line can be set up to calculate the molecular mass of sample proteins, the time-dependent migration distances or migration velocities of both a set of marker proteins and the sample proteins must be analyzed [9,25,26]. Migration velocities are either expressed as (i) relative mobilities (retarded mobility divided by free mobility) [lo, 19, 211 or (ii) as apparent mobilities [lo, 25,261. So-called Ferguson plots were also applied to describe the relative mobilities of proteins in polyacrylamide gels although their original use was to describe protein mobilities in starch gels [ 191.The validity of this simple relation, however, is lost at polyacrylamide concentrations above 15%T [36] and as a result it cannot be applied to linear gradient gels in the range of 4-30%T [9, 25, 26, 361. Even in gradient gel electrophoresis the migration velocity of nondenatured proteins is strongly influenced by their net native charge [35]. Therefore, it is likely that a sample protein migrates faster or slower than an equally sized marker protein [35].This fact implies that a charge-independent protein parameter must be found first before the molecular size of a nondenatured protein can be calculated from PGGE data [35,36]. Such a parameter is represented by the maximum migration distance (D,,,), i.e., the distance a protein travels in pore gradient gel electrophoresis until its mi0173-0835/91/1010-0703 $3.50+.25/0
