Effects of nonuniform column packing in analytical gel chromatography

ANALYTICAL

57, 578487

BIOCHEMISTRY

Effects

of Nonuniform Gel Ii.

Depnrtmerd

(1974)

Column Packing Chromatography Associating

in Analytical

Solutes

of Biochemistry, Clemson Univexity, Clemson, South Carolina .?9&31 AND

Department

Rcceiv4

of Biochemistry, Charlottesville, .June

7, 1973;

Liraivetd.y of T7irginia, F’irginia 22901 :ICW~I~~YI

July

17, 1973

Effeck of nonuniform column packing on boundary profiles for selfassociating systems have bren invest,igntc4 I,>- computer simulntion. lUigr:lSolute spwiescllangesalong tlw tion rate of each of the interconverting column as a result, of nonuniform packing, and the, difference irl wlocity of monomer and rl-mcI’ is not constant as thr> sample movc>s down the c~ohlmn. A grestcr amount of overall axial disprkon rcsul(s. as com~larrd to th(’ constant-rolumn case. Proc~edur*~s tlevelollcd in this stlltlp can be applietl to any c~s~~erimentally measurrtl column nonunilormit~.

Analytical gel clironintogra~~l~p is n useful technique for the study of interacting systems of macromolecules. For such systems the behavior of solute profiles provides n guide to the interaction parameters (stoichiomet’ry anal equilibrium coidant I as well :w the molccwlnr size parameters of t,lic interacting species ( 1-31. The shnlw anal migration l’ropcrties of solute I>rofilcs may also he influcnccvl l)y nonuniform f’acking of t#he columu, I,roducing Tarintions along the column axis in the cross-scctionsl nren aunilnt)lc to solute. Such nonuniformity is invariably I’rcscnt, in real c:w~s (41. Tlic~ ycvioue gaper (51 describes consequencesof nonuniform Ilacking in the chromatography of solutes comprising a single molcculnr species. For these systems the I)rofilcs of concentration wrsus distance hnw nearly the same sll,z~~Cas that cxl)ectcd for lionlmiform twckingr, when concentrations are expressed in terms of bulk solut,ion values. In this paper we report results of n study on the migration ’ To Copyright All rights

whom

inquiries

should

be addresard.

578 @ 1974 by Academic Press. Inr. of reproduction in any fol,m reserved

580

ZI~~MERMAN

Lion of gel chromatography

AND

ACKERS

is (I).

J’ = uTC!‘T - LpWy/d.r.

(1) J’ is the *olutc flux lwr unit crab,‘4 sectional area: 117’is the translational velocity along the ask of the column, L, is roffficicnt describing the axial dispersion, 5 is distance the llnrarneter, and C1’2’ is solute concentration in the total column franic of wfcrcnce.’ The velocity uj for any OI~(’ 11~0lec~11arslwcies j is Ilj = F/
(2) n-here C’j is the concentration of tlic jth specks in the total culunin frame of reference. The dispersion p:iramcter for each species, Li, is made up of three terms and may he n-rittcn as (81

n-here L,, is a coustant arising from the statistical summation of nonuniformities within the column, T)j is the free tliffusion coefficient, q is a gel particle-packing factor, (1 is the gel pnrticlv diarnetcr, and d is the column cross-section area. For a complctc discussion of thwc terms anrl bon- they arise see Refe. (7 and 8 1. The axial dispersion coefficient, t,.. for total solute is a gradient, average which may he written for 2 N-0 species (i1101101ti(‘r-/1-1t1cr 1 system ad

C-1) The parameter coe.fi

C’,, is related

to C’, t~llrough the coluw~z equilibG~m

ficien t Ii' C", ,, = A-'((",)".

The column equilibrium coefficient and the thcrmodynnmic constant, I< = C,/CC,‘lv, arc related 11y

(5)

equilibrium

concentrtltion C’T t,hat. is ohserved espcrimentxlly with t,hr scanning gel chromatogr,zph (4).

COLUMN

581

KONUNIFORMITT

If the parnttieter~ [,,, :mtl G1 i: are not, distance de~~cn&~ttt, l&l. (121 has a simple form (11. Han-ever, if [,, ant1 <, :~c functions of cliskmce, then R’ is also :t futtrtion of distance and Eq. (4) must be rmrittcn. The second term in the denominator bccontes

c = a[s’(c”,)“] = &-‘((lll)“-l a(“’ 1 ((” )” !E 1 ax . ax a.1 ax $ince Ii’ is also :t function

('7)

of distance,

(S) Ry cornbitting

ternis :itttl minor twirtmgetttcnt

or, at any point x xitliin

one tnay write

the colutnn.”

The present :mnlyAs diffw from previous ottcs (l-3) in this fortnuh~tion for the :rsi:iI dis~wsioll cocffiricnt :tlttl tltr cotmywttclin~ relation for t.he velocity tcrnt I(.,,. This c5plicit rcl:~tiottship for IJr n-as ttsctl it1 the simulxtions reported here.

Nutnericnl solubiotw to &I. I 1 ) wew ~wrfortncd on :ttt IBJI 36O/SO computer” wing the basic Ixogtxttt of (lox (91 as tttodified for gel chrotmtogrnphy (1 1. The computational tnetltotl, clcscribcd clscwhcrv (l-3, 9’) , itivolrcs nltcrn:~tc routtcls of diqwrdoti mtd trnttSl:xtiott bnscd on ;L finite

582

ZIMMERhfAN

Equilibrium

system 1 2 3 4 5

ACKERS

TABLIi 1 Constants for Monomer~Tetrnmer

Percentage, bl weight,, of monomer al. a loading concentration 0.1 mg/rnl 90 75 50 25 10

AND

Kquilibrium constant

8imulations

Equilibrium constant

Cml/mg)” 1.52 7.90 s.00 1 .92 0.00

x x x x x

10~ lo? 103 1W 106

M-3

1.87 9.70 9.83 2.36 1.11

x x x x x

10’4 10’4 10’6 10’7 10'9

approximation to Eq. (1). Initially the column is specified to be saturated so that at time t = 0, for z < 0, C2’ = 0 and for 5 > 0, CVs= C, g,.,z,. C, is the bulk loading concentration. At x = 0 there is an infinitely sharp boundary. Simulated dispersion and translation are then allowed to take place for the approprintc time and the resulting concentration profiles are recorded. Major modifications of the previous procedures (1-3 1 were required. Because of the distance dependence of K’, tables of Us,and L,, vs (I, could not be used. Therefore at erery concentration in every step the column concentration of monomer was cnlc*ulatetl using an interval dividing routine. The appropriate transport velociticts and dispersion parameters were then calculated directly. difference

Parameters of the C’oluttln and Solute S!yatem The computer simulwt~ionsare all based on n solute system in which a 17,000 -IN monomer of radius 18.9 ;i is in rapidly established equilibrium with its tetramcr of radius 30.0 -&. The fire equilibrium constants used are shown in Table 1. In all casesthe trailing boundary of a plateau experiment has been sirnulatctl, losing psramekrs pertaining to a typical Sephadex G-200 column (1,8). C’olumn parameters of 0.714 for the partition coefficient and 3.61 X IO-’ cm”/‘min for the dispersion coefficient were used for the monomer ; 0.474 and 4.86 X IO-’ ctn’/min for the partition coefficient. and dispersion cocfficicnt, respectively, for the tetramer. The parameter, (Y)had an initial value of 0.295 at s = 0 and was changed linearly at a rate of -O.O295/cm. The parameter, p, had an initial value of 0.670 and changed at’ a rate of O.O280/ctn. The total column was given unit cross-sectional area. The flow rate was 1.2 ml,/hr and the final time simulated n-as 192 min. Par:imeters ant1 proccdl!rcd u~d to calculate all of t’hesevalues are given in Refs. (1 and 2).

COLUMN

RESULTS

NONUNIFORMITY

AND

583

DISCUSSI0N

As shown in Figs. 1 and 2, the column packing has a pronounced effect on the quantitative nature of the reaction profile, although qualitative features (bimodsl charact,cr of derivat’ives, etc.1 are the wmcx. The pncking effect accelerates the movcrncnt of each solute species within the column but the extent of this acceleration d~~pc~r~ls on the size of the solute molecules. This results in nn incrcwc in the difference in velocit’y het#ween the monomw and t,hc Ijolymer itctrnmcr for these cxamplesl as the sample moves down the rollimn. A grcatcr amount of overall axial dispersion is observed. a5 compared with the constant column c:w (Fig. 2). This gre:~tc~r dispersion is not eliminated 1)~ reducing concentrations Pfy to bulk solution values Cr. The time clcvc~lopmcnt of one of the system9 (K = 9 X 10’; iml/‘mg 1’0 is shown in Fig. 3. The lnrgc effect, of nonuniformity on the position of the centroid for this one syst,cm is shown in Table 2. The cent,roid position, x, for uniform packing was calculated from the relationship s = Ft./& (6) where 2~ was taken as the value of &A at x = 0 for t,he

581

FIG.

ideal curve

ZIMMERMAiT

AND

ACKERS

2. Ideal column packin g some solutc~ and condilions as in Fig. 1 but column packing. Curvrs (A) are derivatives of concentration profiles (IS). is t.he same as Fig. 1 of Ref. (2).

3

5 'i,

7

with This

5

cm

FIG. 3. Time development of curl’c 5 is Fig. 1. Curves shown are for (1) 48 min, (2) 90 min, (3) 144 min, (4) 192 min. Loading concentration was 0.1 mg/ml. The equilibrium constant was 9 X 10’ (ml/mg)3. Curves (A) are derivatives of curves (B).

COLUMX

Comparison of Centroid

585

NONUKIFORMITT

TA13LR 2 Positions for Illliform

and N~~nlmiform Gel Packing Sonhform

packing

Time lrnirl)

s ((‘Ill )

Average velocit,j Ccm/min)

4x 96 144 192

1.5557 3.176” 4. s703 6.646%

0.03241 0.03309 0.03382 0.03462

nonuniform packing simulat’ion. The centroid position for nonuniform packing ww calculated from the relation

were (,~” refers to the ncight average [ at I(’ = 0 and x,, is any arbitrary position on the plateau. The datta in Fig. 3 can be t,ransformecl into the time independent coordinate system desrrihed previously ( l,lO,ll I by the use of the following equation

where X is the actual distance coordinate at t’imc t, S, is the distance coordinate pertaining to isolated monomer, and S, is tbe corresponding clistancc coordinate of the polymer (tetramerI at time t. The abovc transformation of Fig. 3 is S~OWII in Fig. 4. Here WCobserve an interestBing phcnonicnon, a hinge point. The finding that a hinge point, tsists is similar to that found previously for the ideal case of constant [ ( 1) except, t’hc coordinate of the hinge point is no longer equal to the position of the c>entroid. The boundaries imposed by the coordinate transformation recluire that, the position of the centroid for the polymer itetrnmer’j he located at # = 1 and the wntroitl for the monomer be at I) = 0. It woul11 secni tbnt, the position of the centroid for any intermediate case would fall hetwecn 0 and 1. For the case shown in curve 4 of Fig. 4 the ~nluc of I+ for t,he ccnt#roid at 192 mill is 0.862. Yet, for this part,icular case, the hinge point lies at a value slightly greater than I .O, inclicating the pronounced effect, of dispersion. It is obvious from these studif 1::that t’he struct,ure seen in solute profiles

586

ZIMMERMAN

ASD

ACKERS

FIG. 4. Time independent coordinak transformation of Fig. 3 into a time independent coordinate system. Coordinates for pure monomer and tctramrr arc given by abscissa values of 0 and 1. respectively. Curves l-4 are numbered in corrrspondence with Fig. 3.

of a t,railing boundary experiment may be substantially modified by variations in column packing. It is also obvious tha,t any attempt to fit a real experimental curve by simulation would give erroneous answers for t’he partition and dispersion coefficients if one did not include these effects. In this study we have developed the necessary simulation methods for realistic analysis of experimental systems. The function <(.T) need not be linear, but can be any arbitrary table of numbers determined experimentally by the equilibrium saturation technique described previously (4,6). ACKNOWLEDGMENT This work

has been supported

by USPHS

Grant

GM-14493

REFERENCES 1. ZIMMERMAN,

2. ZIMMERMAN. 4242. 3. ZIMMERMAN, 4. BRUMBAUGH:

5. WEISS, G.. 6. WARSHAW, 7. ACKERS, G.

Ii.. AND ACKERS. G. K. (1971) J. Bio/. f?hem. 246, 1078. J. Ii.. Cox. D. J.. AND ACECI~R~,G. K. (1971) J. Binl. Chem.

J.

J. K., AND APKERS, G. I(. (1971) J. Biol. Chem. 246, 7289. E. E., AND ACKERS, G. K. (1968) J. Biol. Chem. 243, 6315. ASD Acmns, G. K. (1974) Amzl. Biochenz. 57, 569. H. S., AND ACKERS. G. K. (1971) Anal. Biochem. 42, 405. K. (1970) Advau. Prot. Chem. 24, 343.

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AND

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(19%)

(+. Ii.,

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;