Electrophoresis 1991, 12, 1011-1021

Carlo Tonani Pier Giorgio Righetti Department of Biomedical Sciences and Technologies, University of Milano

IPG simulator: Linear pH gradients

1011

Immobilized pH gradients (IPG) simulator an additional step in pH gradient engineering: 1. ~i~~~~ p~ gradients A new computer program, called immobilized pH gradients (IPG) simulator, is proposed for calculating and optimizing any recipe for use in isoelectric focusing in immobilized pH gradients. Unlike our previous monoprotic electrolyte gradient simulation (MGS) and polyelectrolyte gradient simulation (PGS) programs, based on minimizing CV(P), the present program has as a target function the minimization of the quadratic moment around zero of the residuals (p,). With this algorithm it is possible to formulate IPG recipes which have deviations from linearity well below 1O/o of the given pH interval (a limit set with the previous MGS andPGSprograms),infact,as smallas0.1-0.2%(inpHunits).Thenewsimulatorperforms 2-3 times betterthan the previous ones in the pH4-10 range,and is absolutely necessary when working outside this range, at extreme pH values, where CV(p) cannot work against the buffering power of bulk water, thus generating pH recipes with huge deviations from linearity. In the latter cases, p, performs 10 times better than CV(p). When utilizing strong titrants for extended pH intervals, the “all or none” rule has been discovered: such titrants should always be used in tandem, since omission of one of the two at either the acidic or basic extremes produces strongly distorted pH profiles. Our new, most powerful simulator also contains equations for creating nonlinear gradients, notably: concave and convex exponentials and sigmoidal (see the companion paper: Righetti, P. G. and Tonani, C., Electrophoresis 1991, 12, 1021-1027):

1 Introduction Contrary to isoelectric focusing (IEF) in soluble amphoteric buffers (carrier ampholytes, CA), which seems to be based on the “uncertainty” principle (nothing is known about the several hundred chemicals composing wide-pH mixtures, and the shape and range of the generated pH gradient is never reproducible) [l], immobilized pH gradients (IPG) appear to be an “exact” science: the chemicals are well-defined and the pH gradient can be engineered at whim [2]. This science has been exacting, however; the chemistry of the Immobiline chemicals (the acrylamidobuffers and -titrants grafted to the polyacrylamide matrix) has taken years to develop (see [3] for an update on these compounds) and the calculation of wide-pH recipes has required the development of complex computer algorithms (see [4] for a general survey). The aim of this paper is to present a new, powerful IPG simulation program, based on the experience we have accumulated in the last 8 years in this field. We briefly summarize what has been developed so far and give the reasons for this latest evolution. Our first simulation program (MGS, or monoprotic electrolyte gradient simulation) was in operation by the end of 1982.A first approach to the formation of extended pH gradients was through the sequential elution of buffering species of increasing pKfrom a five-chamber mixer [5].This procedure was soon abandoned in favor Correspondence: Prof. P. G. Righetti, University of Milano, Via Celoria 2, Milano 20133, Italy Abbreviations: /3, buffering power; CA, carrier ampholytes, CV, coefficient of variation; I, ionic strength; IEF, isoelectric focusing; IPG, immobilized pH gradients; MGS, monoprotic electrolyte gradient simulation; PGS, polyeiectrolyte gradient simulation; SD, standard deviation; pz, quadratic moment about zero

0VCH Verlagsgesellschaft mbH, D-6940 Weinheim,

1991

of standard two-vessel gradient mixing [6,7], for which we studied the conditions for gradient linearity as a function of pK distribution of the buffers and of the titrants [7]. We could thus produce formulations for a series of wide IPG (spanning 2-6 pH units within the pH 4-10 range), optimized in terms of gradient linearity [8,9]. In a previous paper [8] we compared two approaches to the generation of extended pH gradients: (i) in one case each buffering Immobiline had the same concentration in both vessels of the mixer; (ii) in the other, different concentrations of buffering ions could be present in each chamber. In the case of two-dimensional maps, however, best resolution in the focusing dimension was 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 two-dimensional maps and in cases requiring analysis of highly heterogeneous samples [101. At the beginning of 1986,we started thinking of expanding the fractionation capability of IPGs: up to that moment, the most extended pH interval described was a pH of 4-10. For this reason, we had not included the dissociation products of water (H’ and OH-) in our simulations, since within the pH 4-10 range their concentration is negligible. At that time, we started focusing dansylated amino acids (which exhibited pls in the pH 3-4 interval) and we realized that there was a strong divergence between calculated and experimental pH gradients; thus, our computer program was expanded to include the effects ofH’ (pK=-1.74) and OH- (pK= 15.75) on p, ionic strength, and pH profile [ll].In fact, simulations were not only limited to acidic, but included also quite basic (pH 10-11) intervals [12]. As chemicals with more acidic and more basic pKs became available, extended formulations including pH extremes were computed: the widest pH range that could be formulated was a pH 2.5-11 interval, spanning 8.5 pH units [13]. * Program available from Fluka Chemie AG, Buchs, Switzerland 0173-0835/91/12 12-1011 $3.50+.25/0

1012

CTonani and P. G. Righetti

In 1988,we started a long-range program on the characterization of existing Immobilines and on the synthesis of new species [3]. At this point the family of “Immobilines” had considerably expanded and our former program (which was limited to mixtures of no more than 10 species, including buffers and titrants) could no longer handle the new generation. These factors forced us to develop a brand new program, PGS (polyelectrolyte gradient simulation) for IEF in IPGs and for chromatography [14, 151. Given this extended know-how, there would appear to be

Electrophoresis 1991, 22, 1011-1021

tions, but by no means does it guarantee that they will be centered on zero. The second requisite can be fulfilled by a parameter that will give an idea of the centrality of a distribution, and this is the data average. SD and average are the principal statistical indexes; however, if we want to take into account not only the dispersion and centrality of data, but also their symmetry and flattening, we may study all these aspects with a single concept, the so-called moment of a distribution (v); there are several types of moments, but we choose here the moments around zero, given by the following relation:

no reason for yet an additional variant. Yet, as also discussed in [4], all the previous programs had some shortcomings, as outlined below: (i) The approach of minimizing the coefficient of variation of the buffering power [CV(@)]to produce linear pH gradients is only a winning strategy if and when the concentration of the Immobiline mixture is constant, i.e. only when the two vessels of the gradient mixer contain the same solution titrated at the two extremes of the pH interval (equal concentration method) [8].Yet, with the presently available 6 Immobiline species, more flexibility in recipe calculation is obtained by the “unequal concentration” method, i.e by using different molarities of the same Immobilines in each vessel. With the latter approach, Righetti etal. [16] proposed minimizing SD(pH), which is the standard deviation along the pH course, by using the steepest descent principle in the search for buffer concentrations allowing for a better linearity of the pH course. (ii) Minimization of CV(P), while working satisfactorily inside the pH range of 4-10, cannot perform properly outside these boundaries, where there is a strong contribution of water to the buffering power; as the latter is represented by two branches of a hyperbole, CV(@)as a target function becomes almost meaningless. (iii) Both our previous programs (MGS and PGS, for mono- and polyelectrolytes) were meant for modelling only linear pH gradients, whereas there are also many applications for nonlinear recipes. All three points above are tightly linked to the problem of the choice of target function, that is to say, the function to be minimized (or in general extremized) in order to attain good optimization. Some possible target functions, as well as their merits and limits, were discussed by Celentano etal. [4].In this paper, and in the companion report, we present a new program, with a new target function, able to guide our optimization module on the grounds of linear as well as nonlinear pH gradients.

2 Theory The target function presented here is not centered on p, but on the residuals from a desired, ideal curve (represented by a function) and the actual curve created by computer modelling. For best results in ideal-actual curve matching, two requisites have to be fulfilled: (i) The residuals must be as small as possible, as they represent the deviation of the computer-generated curve from the expected one; (ii) the residuals must oscillate around zero. The first requisite can be satisfied with a parameter that gives an index of variability (e.g., the standard deviation, SD) and an idea about the dispersion of data, which in our case is represented by the width of oscillations. However, if we consider as a target function the SD of residuals, we see that the optimization module can indeed reduce the oscilla-

that is to say: A=

EX; 7

where xiis the ifhresidual, k is the power and n the data considered. Note that, if k= 1, the moment around 0 is the average of residuals, while for k=2 it is the mean of the squares of residuals. We have tried various moments with different values of k , and we found k = 2 to be the best value for our computer modelling. Note that, by rising to the power of k, one generally obtains an increased numerical value of the moment; this means that high-order moments are extremely sensitive towards the data, so that small data errors can produce unacceptably large errors on moment values. Thus, it was decided to use the quadratic moment around zero of the residuals as a target function of our new program :

cx;

Pl=

-y--

Here the smallest moment thus corresponds to the smallest oscillations around zero.

2.1 The optimization module

The optimization module is written in FORTRAN 77 and is composed of five submodules, each of them producing different kinds of gradients: (i) Linear gradients using p of residuals as target function: This module can create an optimized linear pH gradient, and the function to be matched is the equation of the ideal straight line starting from the initial pH of the gradient (pH,,) to the final pH (pH,,): (4) where n is the total number of fractions. (ii) Linear gradients using SD of residuals as target function: Also this module creates an optimized linear pH gradient, and the function to be matched is the same as in the previous module. Note, however, that this module is not a substitute for the preceding one, but an additional one, since we observed that it is sometimes useful as a preoptimization step, to be performed before using module (i).

Electrophorcvs 1991. 12, 1011-1021

IPG simulator: Linear pH gradients

(iii) Exponential gradients usingp of residuals: This applies to nonlinear, exponential, concave pH gradients; for their shape, these kinds of gradients are useful in developing two-dimensional maps. The function to be matched is that of an exponential curve: y = pH,,*EXP (a*x)

(iv) Logarithmic gradients using 1-1 of residuals: This module can create and optimize nonlinear, logarithmic, convex pH gradients. The function to be matched is that of a logarithmic curve: (7)

where a is (PHfin - PHI”) (8) In (n) (v) Polynomial gradients using ,uof residuals: This module allows the creation of variously shaped pH gradients (e.g. sigmoidal); the function to be matched is apolynomial one, and can be up to the Shdegree: a=

y = A$

+ A , , y + A,X3 + A , y + A,X’ + A ,

chart.The maximum number offractions desired forthe optimization of a gradient is 99; in fact, we observed that to gain a good degree of precision it is necessary to use at least 10 steps per pH unit (e.g., if the creation of a pH 4-11 gradient is desired, at least 60 steps are required).

2.2 About the program (6)

+ a*ln (x)

,8 and r) for each gradient’s fraction and their correlated

(5)

where a is

y = pH,,

1013

(9)

where the coefficients (for a sigmoidal pH 3-1 1 gradient) are : A , = 2.8465366914 A , = 0.1705543657 A, = -0.00743202642 A , = 0.0002091370579 A,, = -0.000002811654986 A , = 0.00000001472126296

The IPG simulator runs on IBM or compatibles supported by the MS DOS operating system; the computer’s hardware may be based on old 8086 coprocessors or new 80286 or 80386 CPUs, but it must be equipped with a mathematical coprocessor.Afloppyof 1.44 Mb or hard disk is required, as well as a 640 kb RAM memory; EGA or VGA graphic cards are supported. This program, developed on an old AT&T PC 6300, was written in two languages: Microsoft QuickBASIC and FORTRAN 77; the modules in which heavy calculations are performed (e.g. optimization) are coded in FORTRAN 77 in order to take advantage of the powerful mathematical libraries of this language, while the editors, the data base and the graphic modules are structured in BASIC.The IPG simulator can handle up to 18 Immobilines plus two strong titrants in asingle mixture; its internal data base can store up to 150 recipes, and the buffer archive also contains up to 150 components and can generate charts from the calculated recipes, plotting up to three parameters at a time; in addition, it is possible to export the simulation results to other programs (statistical, graphics, word processors, etc.). When one runs the program, one enters the main menu, in which several options can be found, each of them leading to one of the following submenus: Buffer editor, Mixture editor, Gradient, Compute, Optimize, Help, Recipe, Delete archive, Plot, Testing gel, Immobilines’ info, Export, Fix beta, Search, Quit. We give here a brief explanation of each option:

Note that these nonlinear pH gradients can be produced in the laboratory without resorting to particular mixing devices (e.g. computer-driven burettes) or unusual experimental conditions (e.g. different volumes in C, and C,, conical or wedge-shaped reservoirs, pumps, etc.), but simply with a standard, two-vessel gradient mixer. As outlined above, the aim of the optimization module is to decrease p 2 ; this is achieved by successive increments (or decrements) of each Immobiline concentration into both vessels. After every change in mixture composition, pz is evaluated and, if the value obtained is lower than the previous one, the change is accepted. When the optimization module is unable to further minimize p2 at a preselected molarity value, a variation of the magnitude of the increments occurs (the width of such increments, either positive or negative, spans from 0.002 to 0.2 mM), and the process will continue to the end; this optimization module allows the use of both the “equal” and “different” concentration approaches; after selecting the Immobilines to be involved into the optimization procedure and choosing the appropriate optimization submodule, the program will give the pH gradient recipe as well as a complete list of parameters (pH,

Buffer Editor:

When pressing the (b) key the Buffer Editor option is entered; here you can create, view, delete or modify the species to be used for pH gradient preparation. Mixture Editor: Here one can prepare and modify any desired mixture; many options are available. By pressing (g) one obtains all the stepGradient: by-step values of pH, p, ionic strength and linear deviation during pH gradient preparation. This option allows you to find pH, p, Compute: ionic strength and the dissociation degrees for each species in a mixture of up to 20 elements. Optimize: The optimize function is able to find the best concentration for each species in the mixture in order to get the best fitting between the ideal and actual shape of the pH gradient. Gives information about the options and Help: features of the IPG simulator.

1014

C . Tonani and P. G . Righetti

Electrophoresis 1991, 1.7, 1011-1021

This option gives all the volumes of components (Immobilines, TEMED, glycerol, etc.) that are involved in pH gradient preparation. Delete archive: Clears and resets mixtures and/or buffer archives. Plot: This function generates charts of the pH gradients, allowing the plotting of up to 3 from a total of 4 parameters: pH, P, ionic strength (I), linear deviation. This option creates a gel map on the Testing gel: screen in which one can evaluate the resolution power of the gradient created and the differences between ideal and actual uH gradients. Recipe:

"[

T 0.2

SD of residuals

A

10-

8--

JJ of residuals

grams (e.g. Microsoft Chart, Lotus 1,2,3 and so on).

B

2 1

BUFFER EDITOR

MIXTURE EDITOR

HELP

FIXING BETA

21

61

b-0.2 81

TESTING GEL

SD of residuals + JJ

'T BUFFER ARCHIVE

41 fraction number

T o.2

MIXTURE ARCHIVE

1

OPTIMIZATION MAIN MODULE

L

I

EXPONENTI

-0.2 1

21

41 fraction number

61

81

MGARITHMIC MODULE

Figure I . Simplified scheme of the structure of the IPG simulator program.

Figure2. Different optimization procedures o n a pH 2.5-1 1 interval with different target functions. (A) Minimization of standard deviation (SD) of the residuals. (B) Minimization of quadratic moments around zero of the residuals (fi2). (C) Two cycles of optimization, first as in (A), followed by (B). Dev, deviation from linearity. Note that, contrary to (A),in (B) and (C) the deviation is a function oscillating around zero.

IPG simulator: Linear pH gradients

Electrophoresis 1991, 12, 1011-1021

2.3 Structure of the program

(P

pH 3-7

I ‘.03T

The program’s structure here depicted (Fig. l),reflects the options of the program itself, and has been greatly simplified.

1015

pH 4-8

0.074 SD of Mean -c--(

Min.

3 Results We give a comparison among the different optimization algorithms. Figure 2A gives the pH profile of an extended IPG interval (pH 2.5-10.5), optimized by using as a target function the minimization of standard deviation of residuals (see point ii in Section 2.1).The pH gradient obtained is linear, the maximum negative deviation being only-0.12 pH units. The deviation from linearity takes an almost sinusoidal shape, but it is all on the negative side, touching zero only at 6 points along the pH gradient. In other words, the pH gradient is linear, but it takes a course which is slightly shifted to the right (negative) side of an ideal pH curve joining the two extreme pHvalues ofthe interval. In Fig. 2B, we see the same optimization of an IPG, pH 2.5-10.5, but using as a target function the minimization of p2 (the quadratic moment around zero of the residuals). While the pH curve apparently seems just as good as the one in Fig. 2A, in reality it follows the ideal shape much better: its deviation from linearity oscillates constantly (in an almost sinusoidal way) around zero, thus drastically cutting (by at least ‘I2) the value of positive (or negative) maximum deviation. The “philosophy” of this optimization algorithm is now quite evident: since it has as a target the constant reduction to zero of all residuals, any time a negative deviation is produced, it is counteracted by a positive one of almost equal intensity. The possibility also exists of subjecting a recipe to successive cycles of optimization. This is illustrated in Fig. 2C: the same pH 2.5-10.5 recipe was optimized first under the target of standard deviation of residuals, followed by a reoptimization by the p2 criterion. The amplitude of the deviations is further reduced (this is particularly evident in the first negative and positive peaks, starting from the acidic side). The deviations cannot be further flattened simply because this mixture is already optimized to the extreme (remember that the cocktail consists of only 8 buffering Immobilines, plus two titrant ones, plus the hydrolytic products of water; see recipe in Table 1). Given these most encouraging findings, we have resimulated all the extended recipes given in the 1984 paper [8]: 5 mixtures of 4 pH units (pH 3-7; pH 4-8; pH 5-9; pH 6-10 and pH 7-11); 4 mixtures of 5 pH units (pH 3-8; pH 4-9; pH 5-10 and pH 6-11); 3 mixtures of 6 pH units (pH 3-9; pH 4-10 and pH 5-11); 2 mixtures of 7 pH units (pH 3-10 and pH 4-1 1); 1mixture of 8 pH units (pH 3-1 1) and finally the most extended, pH 2.5-1 1 interval. All these recipes are given in Table 1 (where C, and C, represent the mixing chamber and reservoir, respectively, of a two-vessel gradient mixer). The “pedigree” of the five mixtures of 4 pH units is shown in Fig. 3: the vertical bars represent (from the left) the maxima of positive (Max.) and negative (Min.) deviations from linearity, their average (Ave.), the median value (Med.), the standard deviation of pH (SD pH), and the standard deviation of mean (SD of mean). All the gradients are extremely linear, in some cases the deviation being only k 0.005 pH units, in the worst case being 3z 0.020. This compares with the data reported in [8]: the pres-

-0.015

I

I

-0.03

-0.03

3 0.03

pH 6-10

pH 5-S

Statistics on Mixtures genesatina 4 pH units width gradients; Max and Min are the maximum positive and negdtive deviations, Ave the average, Med the median and SO the standard deviation of residuals. All values are exprmssed in pH units.

I o.ord

I

1 -0.04 ’igure 3. Statistics on the 5 IPG mixtures extending over 4 pH units. The pH interval is indicated in each panel, starting from the most acidic (pH 3-7) increasing by increments of 1 pH unit. The respective recipes are given in Table 1.

pH 3 - 8

pH 4-9

SDof

-0.03-1

0.03

-0.031 pH 5-10

pH 6-1 1

SD S D o f

-0.015

1

-0.03-L

1

-0.04

1

Figure 4. Statistics on the 4 IPG mixtures extending over 5 pH units. The pH interval is indicated in each panel, starting from the most acidic (pH 3-8) increasing by increments of 1 pH unit. The respective recipes are given in Table 1.All symbols as explained in the bottom left panel of Fig. 3.

1016

C . Tonani and P. G. Righetti

Electrophoresis 1991, 12, 1011-1021

ent deviations often being smaller by a factor of 2-4. Figures 4-6 give the same statistical evaluation of all other ranges, up to the most extended ones. Table I1 summarizes the statistical data on all recipes reported. We have thus far shown that, when optimizing wide pH gradients by the “different concentration’’ approach, our new optimization algorithm, based on p2,performs substantially better than minimizing CV(P), as in the old Dossi etal. program 15-91, just as predicted by Celentano et al. [4]. But we had also stated in

pH 3-9

(2 0 . 0 7

pH 4 - 1 0

the introduction that CV(p) would be quite disastrous outside the pH 4-10 interval, where such a function would have to fight against the buffering power ofwater. Since the latter is represented by two branches of a hyperbole, CV(6) is at a loss because it tries to straighten out a bend that, due to the dramatic rise of fi in bulk water, cannot be smoothed. Even SD(pH), adopted in [16], does not work at its best in this region. This is shown in Fig. 7A: here the pH recipe published by Gianazza etal. in 1989 [13], covering the pH

1

PH 3 - 1 0

Max.

12

Max.

PH 4-11

1

0.01

PH

-0.01

-0.0 pH 5 - 1 1

‘.03T Statistics on mixtures Qenerating 6pH-unit gradients; Max. and Min. are the maximum positive and negative deviations, h e . the average, Med. the median and SD the standard deviation of residuals. All values are expressed in pH units.

0.01

SD

pH

-0.01

-0.0

krgure 6. Statistics on the 2 lPG mixtures extending over J pH units (pH 3-10 and pH 4-11), on 1 IPG recipe of 8 pH units (pH 3-11) and on the most extended (pH 2.5-11) IPG interval. All symbols are explained in the bottom left panel of Fig. 3.

-0.

fraction number 1

21

41

z 12

O I 0.2

10

0.14

8

Figure 7. Comparison between two IPG

PH

:I] -0.

4

-0.

A

1

2

fractionnumber 21

lo

41

pH 2.5-11 recipes obtained with different optimization algorithms. (A) Optimized byminimizing CV@) as in [6].(B) Optimized by the p2 algorithm as in the present article. In both figures the left panel gives the statistics on each gradient (see bottom left panel ofFig.3 for an explanation).The recipe for (B) is in Table 1; for (A) in [13].

Electrophoresis 1991, 12, 1011-1021

IPG simulator: Linear pH gradients

2.5-11 range, and optimized as i n [16], exhibits a huge bump at the acidic extreme and a hump at the basic end. The deviations at these extremes are huge: up to 0.27 of a pH unit. Conversely (see Fig. 7B), by using the ,u2optimization module, even these extreme regions can be flattened out, with a maximum deviation of only 0.03 of a pH unit. Thus it is seen that, in this particular case, the new optimization function can perform up to 10 times betterthan the previously adopted targets.

Mixture # Name Kind LowerpH Higher pH Notes Elements

: : : : : : :

5 Mix 3-11 Lin. Different 3 11 p= 1.5 10

Table 1. Linear IPG recipes spanning 4, 5, 6, 7, 8 and 8.5 pH units Mixture # Name Kind LowerpH HigherpH Notes Elements

:

I

: Mix. 3-7 Lin.

Different : 3 7 : -

: :

Mixture # Name Kind LowerpH HigherpH Notes Elements

/

Mixture # Name Kind LowerpH HigherpH Notes Elements

Ca 0.463 0.581 0.178 3.786 2.911 7.408 0.000

Cb 8.483 1.761 2.093 0.254 2.599 0.000 10.910

: 2

Mixture # Name Kind LowerpH HigherpH Notes Elements

Different 3 : 8 : p= 1.5 : 8

pK3.1 buffer Immobiline 3.6 Immobiline 4.6 Immobiline 6.2 Immobiline 7.0 Immobiline 8.5 pK 1.0 Titrant Immobiline 10.3

Ca 0.445 0.553 0.525 4.210 0.000 0.000 4.908 0.000

Cb 7.961 3.048 2.302 0.036 2.422 1.664 0.000 11.857

: 3

Mix 3-9 Lin. Different 3 9 p= 1.5 9

pK 3.1 buffer Immobiline 3.6 Immobiline 4.6 Immobiline 6.2 Immobiline 7.0 Immobiline 8.5 Immobiline 9.3 pK 1.0 Titrant pK > 12 Titrant

Ca 0.000 1.085 0.483 2.859 0.000 2.582 16.972 23.313 0.000

: Mix. 3-10 Lin.

:

Immobiline 3.6 Immobiline 4.6 Immobiline 6.2 Immobiline 7.0 Immobiline 8.5 pK 1.0 Titrant Immobiline 10.3

Ca 2.872 0.730 3.172 0.527 5.285 6.802 0.000

Cb 6.987 3.873 0.154 2.214 2.369 0.000 8.874

Immobiline 3.6 Immobiline 4.6 Immobiline 6.2 Immobiline 7.0 Immobiline 8.5 Immobiline 9.3 pK 1.0 Titrant pK > 12 Titrant

Ca 1.617 1.516 3.170 0.092 3.185 0.000 4.993 0.000

: 4

: :

5

p = 1.5

: 7

Different 3 10 p = 1.5 10

pK 3.1 buffer Immobiline 3.6 Immobiline 4.6 Immobiline 6.2 Immobiline 7.0 Immobiline 8.5 Immobiline 9.3 Imniobiline 10.3 pK 1.0 Titrant pK > 12 Titrant

Ca 0.897 0.691 0.542 2.485 0.584 3.320 0.000 0.000 6.870 0.000

Mixture # Name Kind LowerpH HigherpH Notes Elements

: 7

: Mix. 4-9 Lin. : Different

: 4 : 9 :

j7= 1.5

: 8

Cb 12.502 0.875 4.213 0.155 2.141 0.989 3.089 0.000 14.880

: 4

: : :

Cb 12.200 2.457 4.954 0.213 1.858 0.750 0.189 1.422 0.000 20.569

: Different

:

: : : :

Ca 0.000 0.859 0.611 1.963 0.414 2.789 2.569 11.231 19.889 0.000

: 6 : Mix: 4-8 Lin.

: Mix. 3-8 Lin.

:

pK3.1 buffer Immobiline 3.6 Immobiline 4.6 Immobiline 6.2 Immobiline 7.0 Immobiline 8.5 Immobiline 9.3 Immobiline 10.3 pK 1.0 Titrant pK > 12 Titrant

p = 1.5 pK 3.1 buffer Immobiline 3.6 Immobline 4.6 Immobiline 6.2 Immobiline 7.0 pK 1.0 Titrant Immobiline 10.3

Mixture # Name Kind Lower pH Higher pH Notes Elements

1017

Cb 13.482 2.667 4.995 0.171 1.965 1.394 1.696 0.204 0.000 20.609

Mixture # Neme Kind LowerpH HigherpH Notes Elements

Cb 0.000

5.259 0.000 2.784 1.666 1.631 0.000 3.540

: 8

: Mix. 4-10 Lin. :

Different

: 4 : 10 : p = 1.5

: 9

Immobiline 3.6 Immobiline 4.6 Immobiline 6.2 Immobiline 7.0 Immobiline 8.5 Immobiline 9.3 Immobiline 10.3 pK 1.0 Titrant pK> 12 Titrant

Ca 2.341 1.336 2.75 6 0.026 3.103 1.909 2.826 8.686 0.000

Cb 2.553 6.926 0.000 2.a74 1.150 1.483 1.196 0.000

8.337

1018 Mixture # Name Kind Lower pH Higher pH Notes Elements

Mixture # Name Kind Lower pH Higher pH Notes Elements

9 Mix. 4-11 Lin. Different 4 11 j7= 1.5 9 Immobiline 3.6 Immobiline 4.6 Immobiline 6.2 Immobiline 7.0 Immobiline 8.5 Immobiline 9.3 Immobiline 10.3 pK 1.0 Titrant p K > 12 Titrant 10 Mix. 5-9 Lin. Different 5 9 p= 1.5 7 Immobiline 4.6 Immobiline 6.2 Immobiline 7.0 Immobiline 8.5 Immobiline 9.3 pK 3.1 buffer pK 3.1 buffer

Mixture # Name Kind Lower pH Higher pH Notes Elements

Mixture # Name Kind Lower pH Higher pH Notes Elements

11 Mix. 5-10 Lin. Different 5 10 p= 1.5 8 Immobiline 4.6 Immobiline 6.2 Immobiline 7.0 Immobiline 8.5 Immobiline 9.3 Immobiline 10.3 pK 3.1 buffer pK > 12 Titrant

Ca 0.690 1.743 2.195 0.197 3.295 2.378 7.171 15.024 0.000

Ca 2.452 3.128 0.131 0.904 1.949 4.137 0.000

Cb 0.000 5.333 0.179 2.511 0.521 0.698 1.546 0.000 6.109

Cb 1.237 0.172 3.667 1.501 1.063 0.000 0.001

Mixture # Name Kind LowerpH HigherpH Notes Elements

: 14 : Mix 6-11 Lin Different : 6 : 11

p=

Immobiline 6.2 Immobiline 7.0 Immobiline 8.5 Immobiline 9.3 Immobiline 10.3 pK 3.1 buffer pK > 12 Titrant Mixture # Name Kind LowerpH HigherpH Notes Elements

:

Ca 2.572 3.024 0.000 0.561 2.417 6.581 10.606 0.000

Ca 1.955 2.418 0.191 2.403 1.616 5.939 11.095 0.000

Cb 0.040 0.000 3.630 0.717 0.906 1.122 0.000 0.825

Ca 1.645 1.516 2.035 3.934 8.312 0.000

Cb 2.486 0.595 0.513 1.375 0.000 0.757

Ca 0.000 0.218 0.568 2.982 0.252 2.773 3.471 8.679 21.721 0.000

Cb 29.096 5.013 6.740 0.484 2.985 1.443 0.941 1.049 0.000 42.051

15

: I : 11

: /3 = 1.5 : 6

Immobiline 7.0 Immobiline 8.5 Immobiline 9.3 Immobiline 10.3 pK 3.1 buffer p K > 12 Titrant Mixture # : 16 Name Mix. 2.5-11 Lin. Kind Different LowerpH : 2.5 HigherpH : 11 Notes : = 2.3 Elements : 10 pK 3.1 buffer Immobiline 3.6 Immobiline 4.6 Immobiline 6.2 Immobiline 7.0 Immobiline 8.5 Immobiline 9.3 Immobiline 10.3 pK 1.0 Titrant pK > 12 Titrant

The following abbreviations and symbols apply: C,, mixing chamber (acidic extreme); Cb,reservoir (basic extreme); 8, average buffering power (in mequiv.L-*pH-'); pK > 12 (for calculation purposes, a pK of 13 is assumed). Elements: total number of Immobilines in a recipe.

4 Discussion 4.1 The new optimization algorithm

6 10

1.5 Immobiline 6.2 Immobiline 7.0 Immobiline 8.5 Immobiline 9.3 Immobiline 10.3 pK3.1 buffer pK3.1 buffer

Cb 1.978 4.053 0.450 0.789 1.035 0.000 0.804

: Mix. 7-11 Lin. Different

Cb 4.146 0.000 4.457 1.562 0.000 2.578 0.000 2.478

13 Mix. 6-10 Lin. Different

7

Ca 2.571 0.189 1.950 1.516 5.230 10.457 0.000

p

p=

p=

1.5

: I

IL

Mix 5-11 Lin. Different 5 11 1.5 8 Immobiline 4.6 Immobiline 6.2 Immobiline 7.0 Immobiline 8.5 Immobiline 9.3 Immobiline 10.3 pK 3.1 buffer pK > 12 Titrant

Mixture # Name Kind Lower pH Higher pH Notes Elements

Electrophoresis 1991, 12, 1011-1021

C.Tonani and P. G. Righetti

Ca 3.099 0.180 1.638 1.987 2.996 8.709 0.000

Cb 1.887 4.415 0.939 1.141 1.300 0.000 1.120

We are aware that there may be countless ways of producing an extended IPG recipe.Yet, if the aim is to generate an IPG mixture as close as possible to a linear shape (and that is often the goal, since interpolation of p i values is then a simple proposition) it would appear that the new IPG simulator here described comes close to this target. In the past, when generating wide IPG recipes, we had allowed 190of the given pH interval as a maximum tolerated deviation from linearity. Thus, in a pH 4-10 gradient (a 6 pH unit in-

Electrophoresis 1991, 12, 1011-1021

IPG simulator: Linear pH gradients

1019

Table 2. Statistical data on the recipes of Table 1”) PH Range

Maximum (+) deviation

Maximum

Error

Average

(-) deviation

(Oh)

o f residuals

3-7 3-8 3-9 3-10 3-1 1 4-8 4-9 4-10 4-11 5-9 5-10 5-11 6-10 6-1 1 7-1 1 2.5-11

0.006 0.015 0.015 0.022 0.025 0.007 0.010 0.014 0.020 0.008 0.014 0.025 0.004 0.028 0.034 0.049

-0.009 -0.012 -0.019 -0.021 -0.030 -0.006 -0.011 -0.010 -0.022 -0.016 -0.014 -0.024 -0.003 -0.009 -0.000 -0.069

0.225 0.300 0.316 0.314 0.375 0.175 0.220 0.233 0.314 0.400 0.280 0.416 0.100 0.560 0.850 0.788

0.000 0.000 0.000 0.000 0.000 0.000 0.000

a) For all formulations,

0.000 0.001 0.000 0.000 0.001 0.000 0.005 0.014 -0.002

SD of mean

Average consumption

0.000 0.000 0.000 -0.001 0.001 0.000 -0.001 0.000 0.000 0.000 0.000 0.001 0.000 0.005 0.017 -0.002

0.004 0.006 0.008 0.011 0.013 0.003 0.005 0.006 0.010 0.006 0.007 0.009 0.002 0.008 0.009 0.024

0.000 0.000 0.001 0.001 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.000 0.001 0.001 0.002

20.713 19.965 43.069 31.286 42.468 21.929 14.726 23.751 25.098 10.171 20.491 16.428 14.705 15.516 11.584 65.253

mM

/lo /

uation is even better: 0.005 as maximum positive and 0.006 as maximum negative deviations. Such deviations are so minute that they are well below the experimental error, i.e. they could not possibly be measured even by the most precise pH meter available today [17] (perhaps only by a differential pH meter).

D

The “philosophy” of the new optimization algorithm is: since it has as a target not only the minimization of the residuals of deviation, but also the “over-imposition” of the experimental with the ideal pH curve, it accomplishes that by having the deviations from linearity constantly oscillating around zero. As clearlyvisible in Fig. 2B, every time the gradient is off the theoretical course by, e.g., a negative deviation, this is immediately counteracted by a positive one, often of the same entity, and these oscillations around “zero deviation” accompany the pH course from beginning to end. When this is taken into account, it is clear that our deviations are even better when compared with the previous program [8]: in the latter, in reality, the pH curve is not overimposed on the ideal line joining the two pH extremes, but is a regression line which could be slightly offset from the theoretical pH curve.

I-0.1

12T\

---0.4

V

_- -0.5

.ti.

-_ -0.6

-0.7

21

SD of residuals

= 1.5

pH 2.5-10.5

2 < 1

Median of residuals

41 fraction number

61

81

Figure 8. Wide pH interval created by omitting one strong titrant Immobiline.ApH2.5-10.5 IPGrange was created byusing only the strongpK1 and omitting the pK> 12 Immobiline. Note the strong, negative deviation from linearity, reflected essentially throughout all the pH gradient.

terval), the maximum deviation from linearity could be 0.06 pH units. Our former program was always able to match these requirements (in fact, in some recipes, we were able to obtain deviations of only0.5 O/o of the stated pH interval), but this had a price. Our previous program was unable to smooth the p power, or minimize SD(pH), while keeping fixed the required starting and end points of the pH interval to be created. Looking at Table 1in [8], one notices that the initial and final pH values often do not match the nominal ones: the simulated values are frequently off by as much as 0.1-0.15 pH units. When we tried to resimulate the given pH interval back to the correct starting and ending pH points, the deviation from linearity increased. On the contrary, with the present optimization algorithm, based on minimizing the quadratic moment around zero of the residuals ( p J , all pH intervals always start and always end with the correct pH values (see Table 1).In addition, our deviations from linearity are in most cases better than 0.25 O/o of the stated pH interval and sometimes are as small as 0.1O/o. If we take e.g. our new pH 7-11 recipe, the maximum positive deviation is only 0.007 and the maximurn negative is 0.006 (both in pH units). Since this is a 4 pH unit interval, this means that these deviations are between 0.1-0.2 O/o of the given interval. If we then take the pH 3-7 span, the sit-

4.2 Behavior at pH extremes The power of the new optimization algorithm can be appreciated even more when examining its behavior in ”forbidden grounds“, i.e. outside the pH 4-10 range, where bulk water begins to buffer appreciably. Here our old programs (both the MGS and PGS) [5-101 are quite ineffective, since at pH extremes bulk water fiercely competes with the grafted Immobilines with its own /3 power, represented by two branches of a hyperbole (see Fig. 2.17 in [2]).Thus the target functions, endeavoring either to minimize CV(l3) or SD(pH), are here helplessly trying to straighten two bends in the pH 2.5-4 and pH 10-11 regions. The results are shown in Fig. 7A: this is the best that could be obtained by Gianazza etal. [13] after extensive trials, when trying to formulate a pH 2.5-1 1 range, the widest possible with the IPG technology. There are huge deviations at the extremes, up to +0.2 in the alkaline region and as much as -0.27 in the acidic region. In addition, the optimization algorithms used, while striving to smooth the ,5 power or minimize SD(pH),

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Electrophoresis 199 I , 12, 1011-102 1

CTonani and P. G . Righetti

lose sight of the original goal, i.e. producing a pH 2.5-1 1 interval. Thus, as stated in the original article [13],while the nominal pH range is 2.5-11, the actual one is pH 2.6710.85.This is also true for a number of other extended recipes: pH 2.5-10.5 (actual pH 2.65-10.38); pH 3-10.5 (actual pH 3.03-10.37). Quite different is the behavior of our new program, based on p2:as shown in Fig. 7B, now the pH 2.511 is essentially a straight line, with maxima in the positive and negative deviations not exceeding 0.030 pH units (i.e. smaller by one order of magnitude). Again, the "philosophy" of our new optimization procedure can be appreciated from Fig. 7B: the deviations from linearityare small, almost sinusoidal oscillations around zero, so that the sum of all positive and negative residuals is essentially zero. 4.3 On the buffering power

There is a price to pay for the above results (Fig. 7B): in order to linearize the pH gradient at pH extremes, the algorithm operates by "flooding" the grounds, i.e. by greatly increasing the concentration of strong titrants at the two extremes. This is necessary, since the system has to counteract the p of water, but it might lead to a too high ionic strength in the recipe, with concomitant poor transfer of proteins in two-dimensional maps from the first to the second dimension. In all past recipes we have, in general, given formulations having a pa"=3. Within the pH 4-10 interval, it is possible to lower the pavto as low as 1,but this is impossible outside this range. In fact, when trying to reformulate the recipe of Fig. 7B (based on p,,=3), we could not bring it below pa,= 2.3 without greatly distorting the pH gradient. However, we have experienced that the most difficult territory of operation is the pH 2.5-3 interval: if one can omit this portion from an extended pH interval, recipes with a much lower consumption of Immobilines and a pa,as low as 1.5 can be obtained. Thus, in many applications, it may pay not to use the widest possible formulation (pH 2.5-11) but a slightly reduced one (pH 3-1 1; note that this last recipe has a barycenter at pH 7). 4.4 On the use of strong titrants We Iearned another fundamental rule when using our new program IPG simulator. This is the "all or none"ru1e: when using strong titrants (the pK 1 and pK > 12 species), in widepH recipes, it is imperative that they are both utilized in a formulation. Omission of one of the two at one extreme is quite deleterious to the shape of the pH gradient: e.g., as shown in Fig. 8, when omitting the pK> 12 in a pH 2.5-10.5 formulation, the pH gradient curve becomes erratic and marked negative deviations are generated all along the pH interval. It is impossible to correct this by computer modelling: the rectification, however, is automatic when the strong alkaline titrant is reintroduced. The same applies, of course, when omitting the pK 1 species. Thus, when using a strong titrant at one extreme, the only way to balance it out is to introduce another strong titrant at the other extreme. In other words, one cannot expect to reach extreme pH values, while maintaining a good linearity of the pH gradient, only by use of buffering Immobilines: e.g., since the lowest and highest of the buffering species have a pK of 3.1 and 10.3, respectively, to titrate these species "outside" their pK values (i.e. for extending the pH gradient below pH 3.1 and above pH 10.3) one has to use strong titrants. Note that, al-

though not explicitly stated, this concept can be traced back to our earlier simulations; as shown in Fig. 6 of [6], when the pK of a fictitious acid was progressively increased from 0.5 to 3, the course of a pH 3.5-9.5 interval was progressively altered and finally totally ruined (in the case of the pK 3.0 compound).

4.5 On the flexibility of the IPG simulator Our previous MGS module required extensive computer time (extended recipes often demanded overnight calculations) and extensive operator know-how (the program was interactive, so the operator could enter it at any time during the calculation with an educated guess). The new PGS also required extensive computer time, since the equations introduced were complex [14, 151. For this reason we have abandoned equations for polyprotic in favor of monoprotic buffers (see [4] for a general explanation). In the present version, the IPG simulator is versatile and easy to use: no educated guesses are expected from the operator and, if so desired, the optimization can be done fully automatically, without intervention from the user. All that is needed is to fix the paydesired for a given pH interval and specify, as initial input, the Immobiline species required and its molarity. For simplicity we often start with the same molarity: e.g. 2 mM per Immobiline in each vessel of the gradient mixer. Computation proceeds automatically to the desired recipe with the minimal p2 value. As for the time of calculation: with an IBM 386,20 MHz clock (definitely not among the fastest available today) and a mathematical coprocessor, computing, even of complex mixtures, can take as little as 10 min to do. PGR is supported by grants from Agenzia Spaziale Italiana (ASI, Roma) and by Progetti Finalizzati Biotecnologie e Biostrumentazione and FATMA (CNR, Roma). We have enjoyed, overtheyears, collaboration with ProjI;: Celentano and Dr. E. Gianazza,for the development of ourprevious MGS and PGS programs. To both of them, our thanks for stimulating discussions and criticism.

Note added: As originally submitted, all the recipes in Table 1 had been calculated with a pay= 3 (mequiv. L-' pH-'). Since, when using such recipes, we experienced both ion binding and a strong electroosmotic flow (producing severely distorted two-dimensional maps) we have recalculated allrecipeswithapa,=1.5 (except forrecipeNo. 16,inwhich pa"could not be lowered below 2.3). At such the Immobiline consumption is drastically lowered, while maintaining the quality parameters of each gradient (see Figs. 3-6). Received May 13, 1991

5 References [l] Righetti, P. G., Isoelectric Focusing: Theoy, Methodology and Applications, Elsevier, Amsterdam 1983. [2] Righetti, P. G., Immobilized p H Gradients: T h e o y and Methodology, Elsevier, Amsterdam 1990. [3] Chiari, M., Ettori, C. and Righetti, P. G., J. Chromatogr. 1991, 559, 119-131. [4] Celentano, F. C., Gianazza, E. and Righetti, P. G., Electrophoresis 1991, 12,693-703. [5] Dossi, G., Celentano,F., Gianazza, E. and Righetti, P. G.,J. Biochem. Biophys. Methods 1983, 7, 123-142.

Electrophoresu 1991, 12, 1021-1027

IPG simulator: Nonlinear pH gradients

[6] Gianazza, E.,Dossi, G., Celentano, F. and Righetti, P. G., J. Biochem. Biophys. Methods 1983, 8, 109-133. [7] Celentano,F.,Gianazza,E.,Dossi,G. and Righetti,P.G., Chemometr. Intel. Lab. Systems 1987, 1, 349-358. [8] Gianazza, E., Celentano, F., Dossi, G., Bjellqvist, B. and Righetti, P. G., Electrophoresis 1984, 5, 88-97. [9] Gianazza, E., Astrua-Testori, S. and Righetti, P. G., Electrophoresis 1985, 6, 113-117. [lo] Gianazza, E., Giacon, P., Sahlin, B. and Righetti, P. G., Electrophoresis 1985, 6, 53-56. [Ill Righetti, P. G., Gianazza, E. and Celentano, F., J. Chromatogr. 1986, 336.9-14.

Pier Giorgio Righetti Carlo Tonani Chair of Biochemistry and Dept. Biomedical Sciences and Technologies, University of Milano

1021

[12] Mosher, R. A,, Bier, M. and Righetti, P. G., Electrophoresis 1986, 7, 59-66. [13] Gianazza, E., Celentano, F., Magenes, S., Ettori, C. and Righetti, P. G., Electrophoresis 1989, 10, 806-808. [14] Celentano, F. C., Tonani, C., Fazio, M., Gianazza, E. and Righetti, P. G., J. Biochem. Biophys. Methods 1988, 16, 109-128. [15] Righetti, P. G., Fazio, M., Tonani, C., Gianazza, E. and Celentano, F. C., J. Biochem. Biophys. Methods 1988,16, 129-140. [I61 Righetti, P. G., Gianazza, E. and Gelfi, C., in: Neuhoff, V. (Ed.), Electrophoresis '84, VCH, Weinheim 1984, pp. 29-48. [17] Bates, R. G., Determination of pH: Theory and Practice, Wiley, New York 1973.

Immobilized pH gradients (IPG) simulator an additional step in pH gradient engineering: 11. Nonlinear pH gradients While in the companion paper (Tonani, C. & Righetti, P. G., Electrophoresis 1991, 12,1011-1021) we gave the general outline of our new computer program, immobilized pH gradients (IPG) simulator, able to simulate and optimize linear pH gradients for isoelectric focusing in immobilized pH gradients, in the present report we extend the application of such a program to: (i) convex exponential gradients, (ii) logarithmic and (iii) polynomial gradients*. Such gradients are meant to give equal space to protein spots in complex protein mixtures (e.g.,cell lysates, biological fluids) and follow the statistical distribution of protein plvalues along the pH axis. They will prove of fundamental importance in two-dimensional maps, both because they optimize the spreading of spots in the two-dimensional plane and because of the excellent reproducibility of immobilized pH gradients. The following concave exponential recipes are given: pH 3-8, pH 3-9, pH 3-10, pH 3-11, pH4-7,pH4-8,pH4-9,pH4-10,pH4-11,pH5-8,pH5-9,and pH5-10,as well as the most extended pH 2.5-11 interval. Two interesting logarithmic gradients are described: pH 3-6 and pH 3-7 and one sigmoidal (derived with a polynomial of Shdegree): pH 3-11.

1 Introduction We described, in the companion paper [l],a new program able to simulate and optimize any wide (or narrow) recipe for isoelectric focusing (IEF) in immobilized pH gradients (IPG). This new computer program (the IPG simulator) is the outcome of 8 years of accumulated experience in modelling IPGs, perhaps the most powerful electrokinetic technique today available in protein separation. Our first program (called MGS, or monoprotic electrolyte gradient simulator) has produced essentially all the extended IPG recipes in use today (from 3 to 6 pH unit intervals, narrower Correspondence: Prof. P.G. Righetti, University of Milano, Via Celoria 2, Milano 20133, Italy

Abbreviations: f3, buffering power; CA, carrier ampholytes; CV, coefficient of variation; 2-D, two-dimensional; I, ionic strength; IEF, isoelectric focusing; IPG, immobilized pH gradients; SD, standard deviation; 1-12, quadratic moment around zero

0VCH Verlagsgesellschaft mbH, D-6940 Weinheim, 1991

gradients being quite easy to compute by ordinary means) [2].Over the years, we have become aware of the limitations of our program, especially with regard to the optimization algorithm, based on keeping the buffering power (p) along the pH scale constant. Since the best recipes are obtained by a peculiar sort of titration, by which the molarity of the buffering ions is varied in the two chambers of the gradient mixer, it transpired that the target function CV(p) utilized would not work properlyunder these conditions [3]; in fact, SD(pH) was proposed instead (see the companion article). Being aware of such limitations, we proposed, in the companion paper, a new optimization algorithm, based on minimizing the quadratic moment around zero of the residuals ( p z ) .It transpired that this target function would perform well in the pH 4-10 region (by reducing the deviation from linearity 2 to 4 times as compared to previously optimized recipes), and even better at extreme pH regions (here p2 was one order of magnitude better than CV(p) or SD (pH). *

Program available from Fluka Chemie ACi, tluchs, Switzerland 0173-0835/91/1212-1021 $3.50+.25/0