- Email: [email protected]
Gradient design to optimize rate zonal separations
ASALYTICAL
BIOCHEMISTRY
Gradient LYNN
Design
56, 370-382 (19731
to Optimize
CHIJRCHILL,2 Department
GARY of
BANKER,”
Psychobiology,
at Irvine, Received March
Rate
Irvine,
University
California
Zonal AND
Separations1
CARL of
W. COTMAN
California
92664
8, 1973; accepted June 21, 1973
An approach to the design of gradients which maximize resolution is developed by analyzing the sedimentation of particles in linear sucrose gradients. Our analysis establishes the fundamental principles of rate separations. These principles can assist in the successful design of preparative centrifugation procedures. Rate separations are always optimal in homogeneous media or very shallow gradients of low density. In homogeneous media, resolution of particles which differ only in sedimentation coefficients is determined by the ratio of their sediment,ation coefficients. Particles whose sedimentation properties oppose each other can, under certain conditions, not separate or barely separate unless conditions are carefully selected. Particle populations which differ more in density than in sedimentation coefficients clearly separate bett,er by rate than by isopycnic banding. Rate separations in gradients are considerably improved in a type of gradient where the viscosity decreased as the density increased.
The separation of particles by centrifugation depends on differences in density and sedimentation rate. Separations by density differences are achieved by isopycnic banding; separations by differences in sedimentation rates are achieved by differential or rate zonal centrifugation. Selection of media for optimal separation by isopycnic banding depends only on knowledge of the particle densities to be separated (1’1. Selection of media to optimize separations by rate differences depends on knowl-
‘This research was supported by a grant from the National Institutes of Health (NB 68597). ‘Lynn Churchill (Doyle) is the recipient of a National 1nstitut.e of Mental Health Predoctoral fellowship (MH 5169741). Current address: Department, of Pharmacology, University of Wisconsin Medical Center, Madison, Wis. 53766. This article is a portion of a dissertation submitted to the University of California at Irvine in partial fulfillment of the requirements for the degree of Doctor of Philosophy. “Gary Banker is the recipient of a National Institute of Mental Health Predoctoral fellowship (MH 50166-62). Current address: Department of Anatomy, Washington University School of Medicine, St. Louis, MO. 63110. 370 Copyright @ 1973 by Academic Press, Inc. All rights of reproduction in any form reserved.
GRADIENT
DESIGN
371
edge of t’he interactions among particle densities, sedimentation coefficients, and centrifugation media. Because the interactions among these variables are complex, it is not always clear how to select conditions to maximize resolution. If the sedimentation coefficients and densities are known, it is possible to compute the sedimentation rate of a particle in any medium of known viscosity and density as well as in gradients (2). These sedimentation properties have now been measured for a wide range of ljarticles including subcellular and cellular particles (3,4) as well as macromolecules (21. This information can be used to analyze the separation of particles in different media. In this way we determined the relationship between sedimentation properties and centrifugation media from which the conditions for best resolution can be predicted. Under some circumstances the equations normally used to describe particle sedimentation are not applicable (5,6). The causes of deviations from expected sedimentation have received extensive consideration. In practice, conditions leading to “anomalous” sedimentation can usually be avoided so that the sedimentation rate can be accurately computed. In preparing a fractionation strategy, it is important to understand in advance how variations in a procedure arc likely to affect particle separations. Although the sedimentation rate of a particle or a particle population can be determined theoretically, t,his approach has not been widely utilized in preparing subcellular fractionation procedures. This paper describes how the separation of particle populations can be computed as an aid in subcellular fractionation. In addition, the effects of varying gradient design for different combinations of particles have been systematically analyzed. Based on these results, generalizations are described for selecting optimal conditions for separation. From these theoretical principles, an investigator can gain the same understanding of the principles of subcellular fractionation that often comes only after extensive experience. Application of these principles should minimize the experimentation needed to achieve optimal separations. Preliminary reports of these findings have been previously presented (7,8). METHODS
The sedimentation of a particle can be completely described by an equation that relates ti’f (O = angular velocity in rad/sec, t = time in secl to the particle’s sedimentation coefficient corrected for variation in the viscosity and density of the medium. Therefore, the separat,ion between two particles sedimenting under identical conditions can be determined at any instant, in any medium or gradient. We have defined the separat,ion between two particles as the percent of the t,ot’al gradient path between the two particles at any given ant value. To
372
CHURCHILL,
BANKER,
AND
COTMAN
compute the separation between two particles, the Gt value required to sediment each particle along equal increments of the rotor path was computed. Then the position of the slower particle was calculated by linear interpolation, given the 2t value required to sediment the faster particle to a particular position. Resolution was then displayed as a function of the position of the faster particle, as in Fig. 1. The 2t value required to sediment a particle to a position, Ri, in the rotor is given by the following equation (2) :
I
In Ri
W2t= llS2O,w[(Pp - P20,w)/~20,w1
In R.
rim 6’~
-
d In R,
Pm)
Dl
where szO,,,, = sedimentation coefficient of the particle in water at 20°C; pP = particle density in g/cc; pzO,Wand pm = density of water at 20°C and the medium at 4”C, respectively; 7z0,W and rl,,, = viscosity in centipoises of water at 20°C and medium, respectively; R = radial distance from the center of rotation in centimeters ; R, = original position of the particle ; and Ri = the ith posit,ion of the particle. The integral in
FIG. 1. Separation of two particles that differ in sedimentation are equal in density. Resolution is shown as a function of the faster particle. Resolution in homogeneous media (10% sucrose constant sucrose concentration) is compared with resolution in gradients that increase in slope from an initial sucrose concentration exemplified particles have s~o,~ values of 9.4 and 5.6 X 109 S and g/cc. Thus the particles band at 38% sucrose (arrow). Computation as percent of gradient path between particles was accomplished by the method of Bishop (7) and Halsall and Schumaker (4) for the as described in Methods.
coefficients but position of the or any other linear sucrose of 10%. The densities of 1.17 of resolution modifications of PDP computer
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373
Eq. 1 was computed using trapezoidal approximation (2). These computations were performed on an Olivetti Programma 101 (8) or a PDP 12 computer (Focal 12 program). The program on the PDP 12 required about 5 min and on the Olivett,i Programma 101, about 20 min. Values for constants are typical for runs in the BXIV zonal rotor. The initial particle position, R 0, was 3.637 cm; the termination of the typical sedimentation path, Rf, was 6.426 cm. Values of Ri were chosen at equal increments between R, and Rf. For gradients, the weight percent sucrose was calculated for each Ri, given the initial and final sucrose concentration at R, and Rf, respectively. The corresponding density and viscosity values were computed from a program by Bishop (9) modified for the PDP 12. The accuracy of the trapezoidal approximation to the integral and of the linear interpolation of the slower particle’s position increases as the term Ri-Ri-1 decreases. The value of the term Ri-Riml used for theoretical calculations was 0.093 cm, 1/3Oth of the total sedimentation path. A decrease in Ri-Ri-, to 0.031 cm did not significantly alter the values obtained. For homogeneous media, where exact values for resolution can be obtained (see Appendix), the computer calculations were correct to within 1%. RESULTS
Separation of particles differing in szo,=.,To analyze the relationship between resolution and centrifugation media, we first considered particles that differ in sedimentation coefficients alone. This restriction enables analysis of rate separations as a function of media properties without the additional variable of particle density differences. Based on the results of this first simple case, the analysis was extended to include particles that differ in both s~,,,~~ and density. Figure 1 shows how two such particles separate in a variety of different media and gradients. Resolution by differences in sedimentation rates is great,est in homogeneous media. Resolution in gradients with shallow slopes approaches that in homogeneousmedia. In gradient’s with steep slopes, resolution is poor. The reason why a homogeneous medium provides better resolution than gradients can be understood from an analysis of the relative sedimentation rates. In homogeneous media, particle int,eraction with medium [q”,/ ( pp - p,,) in Eq. 11 is constant throughout the sedimentation path. The sediment,ation rates of the particles increase with increasing radial distance due to increases in centrifugal force. Since the leading particle is at a greater radial distance than the slower particle, its
374
CHURCHILL,
BANKER,
AND
COTMAN
sedimentation rate increases more. Therefore the leading particle accelerates away from the slower particle and resolution is good. In linear sucrose gradients, the medium viscosity and density increase as the particles sediment. The decelerating forces due to increased viscosity and density offset particle accelerat,ion due to increased centrifugal force. Since the leading particle enters the more concentrated medium first, its sedimentation rate decreases more t,han that of the slower particle. As a result, the slower particle tends to catch up to the faster one, and resolution decreases. The steeper the gradient slope, the greater the deceleration due to particle interactions with media. Therefore, gradients with steep slopes prevent adequate resolution. Even in shallow gradients the density of the medium can interact with particle density to decrease resolution. This decrease in resolution occurs when the range of gradient densities is close to t’he isopycnic banding densities of the particles. The reason for this decrease is that whenever pp is close to pm even small increases in pm cause large increases in ym/(pl, - p,) . The large increases in vrnJ(pP - pm) result in particle deceleration and decreased resolution. Therefore, media densities should be selected as far away from particle densities as possible when the two particles have similar densities and are to be separated by s20,w differences. Resolution of any pair of particles. In selecting a medium for a particular set of particles it is important to consider their sedimentation properties in relationship to the media. The exact relationships among particle sedimentation properties, separation media, and resolution are difficult to establish because of the number of variables involved. To reduce the number of variables we have analyzed the relationship between particle sedimentat.ion properties and resolution in homogeneous media. In homogeneous media, the relationship between the particle sedimentation rat.es and resolution can be solved algebraically (Appendix). Resolution is an exponential function of the ratio of sedimentation coefficients, S,/S?, and a ratio of differences between particle densities
and media densities, [ (pl - pm)/(p, - pm)I X 1(pz - P~~.~)J(~~ - pzo.,.,) 1. For convenience we refer to this latter term as the “ratio of density differences.” In Fig. 2, resolution is illustrated as a function of the ratio of sedimentation coefficients and the ratio of density differences. Resolution depends on the product of these two ratios (see Appendix). When t.he product of the ratio of sedimentation coefficients and the ratio of density differences is much different than 1.0, resolution is large. When the product of the ratios is close to 1.0, resolution is poor. If the ratio of sedimentation coefficients and density differences are both greater than
tiRADIENT
37.5
DESIGN
RATIO OF SEDIMENTATION COEFFICIENTS 3.0 1.5 IO
FIG. 2. Maximum resolution of two particles in homogeneous media. Resolution was computed as a function of the ratio of density differences (z-axis) and the ratio of sedimentation coefficients (contour lines) according to Eq. 4 (Appendix). Resolution of particles that differ in s.qO,Walone is illustrated by the darkrned y-axis. Resolution for particles that differ in density alone is shown by the darkened contour line in the upper half of the graph. Above this line (stippled area), particle sedimentation properties work together to achieve resolution since both ratios are greater than one. For particles in the clear area, sedimentation properties work against each other since the ratio of sedimentation coefficients is less than one. The ratio of density differences represents the ratio of differences hetween partirle and medium density: [(PI - PdlbZ
- Pm)1 x Lb2 - P20,W)Ibl - P20,w)l.
1.0 (stippled area), the particle sedimentation properties work together to improve resolution. However if one of the particle sedimentation
properties
counters
the other
(clear
area),
resolution
can he quite
poor. If densit,y differences override s20,wdifferences, then the particle
376
CHURCHILL,
BANKER,
AND
COTMAN
with the greater density moves faster (above the z-axis) and resolution is mainly by density differences. If the differences in sedimentation coefficient are larger than the differences in density, then the particle with the greater s20,Wsediments faster (below the s-axis) and separation is mainly by .saa,,,,differences. An important practical question is how to select a homogeneous medium that will optimally resolve two particles with known density and sm,w values. To answer this question, it is helpful to reexpress ISOPYCklC SANDING
20
IO
ISOPYCNIC 'C
%
SUCROSE
(W/W)
BANDING ,_.^ ^. ,518-h
(5’C)
FOR PARTICLES
FIG. 3. Resolution of two sucrose concentration. Particle The particle with a density sedimentation coefficients are resents a ratio of sedimentation in Fig. 2.
WITH DENSITIES I.17 GM/CC
_..^. XL,
1.14 AND
particles in homogeneous media as a function of densities were chosen as 1.17 (pJ and 1.14 g/cc (~2). of 1.14 g/cc bands at 31.8% sucrose. The ratios of indicated by the contour lines. The dashed line repcoefficients of l/1.7 as in Fig. 4. Other details as
GRADIER‘T
DESIGN
377
the data in Fig. 2 as a function of sucrose concentration. This requires that we hold part,icle densities constant since the ratio of density differences is a funct,ion of both particle density and the density of the medium. The results for a specific set of particle densities are shown in Fig. 3. As previously, resolution of particles whose sedimentation properties work together (stippled area) is excellent. When particle sedimentation properties oppose each other (clear area), separation can be either predominantly by density (above the axis) or by .szO,,, differences (below the axis). For a particular set of particles with a given ratio of sedimentation coefficients, separation by .s~~,,,~differences is optimal in low density media and separation by density differences is optimal in media near isopycnic banding density. Between these two extremes there is a medium density in which the part.icles cannot be resolved. The results we have presented so far can be applied to designing gradients for separation by s20,w differences. In general the best rate separations can be achieved in shallow gradients of low density media. For particles whose sedimentation properties oppose each other. the selection of the appropriate density range is especially important. In particular, in some media So,,,, differences can cancel out density differences and prevent separation. This principle has been shown in Fig. 3 for sedimentation in homogeneous media. Figure 4 shows how it applies to density gradients. For the particles illustrated in Fig. 4, the ratio of sedimentation coefficients equals l/1.7 (shown by a dashed line in Fig. 3). Initially when the particles are in low density media the particle with the larger s~~.,~ sediments faster. This point is illustrated by comparing the position of the particles at low w?t values in Fig. 3. As the particles enter the higher density media (25-30s sucrose), the particle with the larger density takes the lead. At all points in such a gradient, resolut’ion is poor. Resolzltion of particle po&ations. For separation of two particles differing in both szn,,,. and density some resolut,ion is always obtainable by isopycnic banding since the density differences will always be resolved. In actuality the more typical problem involves separating two particle populations that overlap in both s?,,,,,, and density. When the overlap in density is small, separation by isopycnic banding can still be used. The question is when the overlap in s?,,.,\-is less than in density will rate separations resolve the populations better than isopycnic banding. To answer this question we have selected two normally-distributed poplilations of particles that overlap 33% in density and 8% in s?~,,~ (Figs. 5A,B). How much would these populations overlap in a medium that
CHURCHILL,
35
40
45
BANKER,
50 ROTOR
RADIUS
AND
55
COTMAN
6.0
65
(CM)
FIG. 4. The sedimentation of two particles with density differences and SZO,~ differences that oppose each other. The position of each particle in a lD45% sucrose gradient is shown as a function of the total applied force, w’t. From l&29% sucrose, the particle with the larger sedimentation coefficient sediments faster. At 29% sucrose, the particles pass each other and the particle with the larger density sediments ahead of the other. Finally, the particles band at their isopycnic banding densities (31.8% and 37.7% sucrose).
separates mainly by s20,wdifferences, such as 10% sucrose? By using Fig. 2 and values of p and s?~,,”for selected part,icles in the populations, it is possible to compute this overlap. When this computation is made, the overlap is only 8%, considerably bet,ter than could be achieved by isopycnic banding. Knowing that, reasonably good separations by rate are possible, we can design a gradient to separat’e these populations. Since a homogeneous medium is not practical for rate zonal centrifugation, we considered a gradient that will resolve mainly by s~~.,~differences and also provide capacity and stability. We chose a shallow convex gradient from l&18% sucrose. In this gradient the overlap in the two populations is 13% (Fig. 5C). Thus rate sedimentation can give a het.ter separation than isopycnic banding. Furthermore the resolution calculated for homogeneous medium is a reasonably good approximation to that found in shallow gradients of similar density.
GRADIENT
DENSITY
(GM/CC)
103 x szo w (S)
ROTOR
FIG. 5. Rate xonal separation
379
DESIGN
RADIUS
(CM1
of two particle populations.
The particle populations
are representedashaving normal distributionsof ~~0,~ and density. Selected particles within each population were used to compute resolution in the convex gradient. The particle with the mean s20,Whad the mean density and particles with an SLWW above or below the mean had densitiesthat deviated a correspondingamount from the mean density. The means and standard deviations for ~~0,~ and density values were chosen so that populations overlapped in density more than in GO,%,. (A) The distribution of density values for each of the populations. The region of overlap (stippled), includes 33% of each population. Particles in this region cannot be resolved by isopycnic banding. (B) The distribution of sedimentation coefficients for each population. The overlap in S~.W of the two populations (strippled) is 8%. CC) The resolution of these particles in a shallow sucrose gradient. The overlapping region is only 13% of each population, considerably better than can be accomplished by isopycnic banding. DISCUSSION
By analyzing the resolution of particles sedimenting in linear sucrose gradients, we have developed some basic principles to facilitate the selection of appropriate centrifugation conditions that provide for optimal resolution. Some of the proposed principles of gradient. design have been previously derived by an empirical approach. Brakke (10) in his pioneering work with sucrose gradients observed that shallow linear gradients resolved plant viruses more effectively than steep gradients. Pretlow and
380
CHURCHILL,
BANKER,
AND
COTMAN
Boone (4) found t#hat shallow gradients with density ranges far less than particle densities were optimal for separating mammalian cells by rate. In macromolecular separations where particle densities are similar, shallow gradients with a density range less than the density of the particles are utilized (11,12). These empirical results combined with our theoretical study indicate that these principles of gradient design are of general application. Gradient design to maximize resolution depends on knowledge of sedimentation coefficients. There are a number of methods for computing sedimentation coefficients. Values of s m,w can be det.ermined from the sedimentation of particles by rate zonal centrifugation (2,8,11, and Churchill and Cotman, in press) or differential centrifugation (13) at different w’t values. Sedimentation coefficients can also be determined for particles distributed throughout the sedimentation path (2,14). Special gradients have been designed so that particles sediment at a constant rate proportional to their s20,w values (12). If direct measurements of sz’O,w values are not convenient, then sZo,, values can be approximated from quantitative electron microscopic analysis (15). Even without precise values for sedimentation properties, a semiquantitative approximation should enable a reasonable prediction of the best conditions for particle separation. To simplify the analysis of gradient design we have focused on rate zonal centrifugation in linear sucrose gradients. In some applications consideration of other gradient media and nonlinear gradients are advantageous. In sucrose gradients viscosity increases dramat’ically with increases in density. Increases in either viscosity or density decrease resolution. One way to improve resolution is to select media with low viscosity at high concentrat,ions, such as CsCI,. Another approach is to design a gradient of decreasing viscosity and increasing density. Such a gradient could be obtained by combining a shallowly-increasing sucrose gradient with a steeply-decreasing gradient prepared from a highly viscous substance of low density, such as methylcellulose. In such a gradient, the leading particle will accelerate away from the trailing particle, and resolution will be better than in homogeneous media. For example, a linear l&20% sucrose gradient combined with a linear 3.5-0.4s gradient of methylcellulose (av MW 26,000) provides 74% resolution for the particles in Fig. 1. Compared with 48% resolution in homogeneous sucrose and 37% resolut’ion in a linear 1@20% sucrose gradient,, this type of gradient offers quite an improvement,. When gradients are shallow so t.hat resolut,ion is good, they may not provide adequate capacity and stability. Also a sedimenting zone of particles will diffuse more in a shallow gradient (16). One solution to
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381
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the capacity problem is differential centrifugation and recycling (17). Alternately, zonal centrifugation in convex gradients provides increased capacity in the initial regions of the gradient where it is most needed (18). Figure 5 illustrates that resolution in a shallow convex gradient can approximate optimal resolution for t’wo particle populations. Thus shallow convex gradient.s provide a compromise that allows both resolution and adequate capacity. This analysis of particle separations by differences in sedimentation rates has provided some basic principles for selecting cent.rifugation media to optimize resolution. A method has been presented for computing resolution of particles with known sedimentation properties in any medium of known viscosity and density. This method along with the principles derived should aid in select,ing appropriate conditions for rate separations. APPENDIX:
RESOLUTION HOMOGENOUS
OF
TWO MEDIA
PARTICLES
The relation between applied force and particle homogeneous media can he calculated from Eq. 1.
u2t = [l/s~~.,l X [dbp
IN
sedimentation
in
- P,,~JIX I& - P?o,~.)/~~Po,~I X (In R; - In RJ PI
The separation of two particles in homogeneous media can then be obtained from the positions of each particle at any given applied force. Then (In RI - In R,)/(ln
R2 - In R,) = WS21
- pm)1
X [(PI - P,)/(Pz
x Lb2- P2O,w)/(Pl -
P20,w)l
= k'. [31
In homogeneous media, maximum resolution occurs when the faster part’icle has reached the end of the centrifugation path. Then if K > 1, one may solve for R,, R, = R, exp ((In R, - R,)/K). Therefore, resolution, the distance between the particles as percent of the gradient path, may be expressed as a. function of the position of the first particle:
RI - R, exp((ln RI - In R,)/K)/(R, For the computations presented in this paper R, = Rf = R = 3.637 cm. If K < 1, one may solve Eq. 3 for R, and Resolution = Rz - R, exp((ln Rz - In R,)/K)/(Rf Resolution
=
- R,).
[41
6.426/cm and by symmetry
- R,).
ACKNOWLEDGMENTS The authors acknowledge the assistance of Mr. Richard Roemer in designing programs for the PDP 12 computer. We also express our gratitude to Dr. Richard F. Thompson, who permitted us to use the PDP 12. We thank Mrs. Pat Lemestre for secretarial aid.
[Sl
382
CHURCHILL,
BANKER,
AND
COTMAN
REFERENCES 1. ANDERSON, N. G. (1966) Nut. Cancer Inst. Monogr. 21, 9. 2. HALSALL, H. B., AND SCHUMAKER, V. N. (1969) Anal. Biochem. 30, 368. 3. BEAUFAY, H., BENDALL, D. S., BAUDHUIN, P., WATTIAUX, R., AND DE DUVE, C. (1959) Biochem. J. ‘73, 628. 4. PRETLOW, T. G., AND BOONE, C. W. (1969) Exp. Mol. Pathol. 11, 139. 5. SCHUMAKER, V. N. (1967) Advan. Biol. Med. Phys. 11, 246. 6. DOYLE, L. C., AND COTMAN, C. W. (1972) Anal. Biochem. 49, 29. 7. COTMAN, C. W., DOYLE, L. C., AND BANKER, G. (1971) Fed. Proc. Fed. Amer. Sot. Exp. Biol. 30, 1181. 8. COTMAN, C. W. (1972) in Methods in Neurochemistry (R. Rodnight, and N. Marks, eds.), p. 45, Plenum Press, New York. 9. BISHOP, B. S. (1966) Nat. Cancer Inst. Monogr. 21, 175. 10. BRAKKE, M. K. (1966) Arch. Biochem. Biophys. 45, 275. 11. MARTIN, R. G., AND AMES, B. N. (1961) J. BioZ. Chem. 236, 1372. 12. NOLL, H. (1967) Nature 215, 360. 13. COTMAN, C., BROWN, D. H., HARRELL, B. W., AND ANDERSON, N. G. (1970) Arch. B&hem. Biophys. 136, 436. 14. SCHACHMAN, H. K. (1959) in TJltracentrifugation in Biochemistry, Academic Press, New York. 15. BAUDHUIN, P., AND BERTHET, J. (1967) J. Cell. Biol. 35, 631. 16. SCHUMAEER, V. N. (1966) Separ. Sci. 1, 409. 17. SCHUMAKER, V., AND REES, A. (1969) Anal. Biochem. 31, 279. lg. BERMAN, A. S. (1966) Nat. Cancer Inst. Monogr. 21, 41.
BIOCHEMISTRY
Gradient LYNN
Design
56, 370-382 (19731
to Optimize
CHIJRCHILL,2 Department
GARY of
BANKER,”
Psychobiology,
at Irvine, Received March
Rate
Irvine,
University
California
Zonal AND
Separations1
CARL of
W. COTMAN
California
92664
8, 1973; accepted June 21, 1973
An approach to the design of gradients which maximize resolution is developed by analyzing the sedimentation of particles in linear sucrose gradients. Our analysis establishes the fundamental principles of rate separations. These principles can assist in the successful design of preparative centrifugation procedures. Rate separations are always optimal in homogeneous media or very shallow gradients of low density. In homogeneous media, resolution of particles which differ only in sedimentation coefficients is determined by the ratio of their sediment,ation coefficients. Particles whose sedimentation properties oppose each other can, under certain conditions, not separate or barely separate unless conditions are carefully selected. Particle populations which differ more in density than in sedimentation coefficients clearly separate bett,er by rate than by isopycnic banding. Rate separations in gradients are considerably improved in a type of gradient where the viscosity decreased as the density increased.
The separation of particles by centrifugation depends on differences in density and sedimentation rate. Separations by density differences are achieved by isopycnic banding; separations by differences in sedimentation rates are achieved by differential or rate zonal centrifugation. Selection of media for optimal separation by isopycnic banding depends only on knowledge of the particle densities to be separated (1’1. Selection of media to optimize separations by rate differences depends on knowl-
‘This research was supported by a grant from the National Institutes of Health (NB 68597). ‘Lynn Churchill (Doyle) is the recipient of a National 1nstitut.e of Mental Health Predoctoral fellowship (MH 5169741). Current address: Department, of Pharmacology, University of Wisconsin Medical Center, Madison, Wis. 53766. This article is a portion of a dissertation submitted to the University of California at Irvine in partial fulfillment of the requirements for the degree of Doctor of Philosophy. “Gary Banker is the recipient of a National Institute of Mental Health Predoctoral fellowship (MH 50166-62). Current address: Department of Anatomy, Washington University School of Medicine, St. Louis, MO. 63110. 370 Copyright @ 1973 by Academic Press, Inc. All rights of reproduction in any form reserved.
GRADIENT
DESIGN
371
edge of t’he interactions among particle densities, sedimentation coefficients, and centrifugation media. Because the interactions among these variables are complex, it is not always clear how to select conditions to maximize resolution. If the sedimentation coefficients and densities are known, it is possible to compute the sedimentation rate of a particle in any medium of known viscosity and density as well as in gradients (2). These sedimentation properties have now been measured for a wide range of ljarticles including subcellular and cellular particles (3,4) as well as macromolecules (21. This information can be used to analyze the separation of particles in different media. In this way we determined the relationship between sedimentation properties and centrifugation media from which the conditions for best resolution can be predicted. Under some circumstances the equations normally used to describe particle sedimentation are not applicable (5,6). The causes of deviations from expected sedimentation have received extensive consideration. In practice, conditions leading to “anomalous” sedimentation can usually be avoided so that the sedimentation rate can be accurately computed. In preparing a fractionation strategy, it is important to understand in advance how variations in a procedure arc likely to affect particle separations. Although the sedimentation rate of a particle or a particle population can be determined theoretically, t,his approach has not been widely utilized in preparing subcellular fractionation procedures. This paper describes how the separation of particle populations can be computed as an aid in subcellular fractionation. In addition, the effects of varying gradient design for different combinations of particles have been systematically analyzed. Based on these results, generalizations are described for selecting optimal conditions for separation. From these theoretical principles, an investigator can gain the same understanding of the principles of subcellular fractionation that often comes only after extensive experience. Application of these principles should minimize the experimentation needed to achieve optimal separations. Preliminary reports of these findings have been previously presented (7,8). METHODS
The sedimentation of a particle can be completely described by an equation that relates ti’f (O = angular velocity in rad/sec, t = time in secl to the particle’s sedimentation coefficient corrected for variation in the viscosity and density of the medium. Therefore, the separat,ion between two particles sedimenting under identical conditions can be determined at any instant, in any medium or gradient. We have defined the separat,ion between two particles as the percent of the t,ot’al gradient path between the two particles at any given ant value. To
372
CHURCHILL,
BANKER,
AND
COTMAN
compute the separation between two particles, the Gt value required to sediment each particle along equal increments of the rotor path was computed. Then the position of the slower particle was calculated by linear interpolation, given the 2t value required to sediment the faster particle to a particular position. Resolution was then displayed as a function of the position of the faster particle, as in Fig. 1. The 2t value required to sediment a particle to a position, Ri, in the rotor is given by the following equation (2) :
I
In Ri
W2t= llS2O,w[(Pp - P20,w)/~20,w1
In R.
rim 6’~
-
d In R,
Pm)
Dl
where szO,,,, = sedimentation coefficient of the particle in water at 20°C; pP = particle density in g/cc; pzO,Wand pm = density of water at 20°C and the medium at 4”C, respectively; 7z0,W and rl,,, = viscosity in centipoises of water at 20°C and medium, respectively; R = radial distance from the center of rotation in centimeters ; R, = original position of the particle ; and Ri = the ith posit,ion of the particle. The integral in
FIG. 1. Separation of two particles that differ in sedimentation are equal in density. Resolution is shown as a function of the faster particle. Resolution in homogeneous media (10% sucrose constant sucrose concentration) is compared with resolution in gradients that increase in slope from an initial sucrose concentration exemplified particles have s~o,~ values of 9.4 and 5.6 X 109 S and g/cc. Thus the particles band at 38% sucrose (arrow). Computation as percent of gradient path between particles was accomplished by the method of Bishop (7) and Halsall and Schumaker (4) for the as described in Methods.
coefficients but position of the or any other linear sucrose of 10%. The densities of 1.17 of resolution modifications of PDP computer
GRADIENT
DESIGN
373
Eq. 1 was computed using trapezoidal approximation (2). These computations were performed on an Olivetti Programma 101 (8) or a PDP 12 computer (Focal 12 program). The program on the PDP 12 required about 5 min and on the Olivett,i Programma 101, about 20 min. Values for constants are typical for runs in the BXIV zonal rotor. The initial particle position, R 0, was 3.637 cm; the termination of the typical sedimentation path, Rf, was 6.426 cm. Values of Ri were chosen at equal increments between R, and Rf. For gradients, the weight percent sucrose was calculated for each Ri, given the initial and final sucrose concentration at R, and Rf, respectively. The corresponding density and viscosity values were computed from a program by Bishop (9) modified for the PDP 12. The accuracy of the trapezoidal approximation to the integral and of the linear interpolation of the slower particle’s position increases as the term Ri-Ri-1 decreases. The value of the term Ri-Riml used for theoretical calculations was 0.093 cm, 1/3Oth of the total sedimentation path. A decrease in Ri-Ri-, to 0.031 cm did not significantly alter the values obtained. For homogeneous media, where exact values for resolution can be obtained (see Appendix), the computer calculations were correct to within 1%. RESULTS
Separation of particles differing in szo,=.,To analyze the relationship between resolution and centrifugation media, we first considered particles that differ in sedimentation coefficients alone. This restriction enables analysis of rate separations as a function of media properties without the additional variable of particle density differences. Based on the results of this first simple case, the analysis was extended to include particles that differ in both s~,,,~~ and density. Figure 1 shows how two such particles separate in a variety of different media and gradients. Resolution by differences in sedimentation rates is great,est in homogeneous media. Resolution in gradients with shallow slopes approaches that in homogeneousmedia. In gradient’s with steep slopes, resolution is poor. The reason why a homogeneous medium provides better resolution than gradients can be understood from an analysis of the relative sedimentation rates. In homogeneous media, particle int,eraction with medium [q”,/ ( pp - p,,) in Eq. 11 is constant throughout the sedimentation path. The sediment,ation rates of the particles increase with increasing radial distance due to increases in centrifugal force. Since the leading particle is at a greater radial distance than the slower particle, its
374
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sedimentation rate increases more. Therefore the leading particle accelerates away from the slower particle and resolution is good. In linear sucrose gradients, the medium viscosity and density increase as the particles sediment. The decelerating forces due to increased viscosity and density offset particle accelerat,ion due to increased centrifugal force. Since the leading particle enters the more concentrated medium first, its sedimentation rate decreases more t,han that of the slower particle. As a result, the slower particle tends to catch up to the faster one, and resolution decreases. The steeper the gradient slope, the greater the deceleration due to particle interactions with media. Therefore, gradients with steep slopes prevent adequate resolution. Even in shallow gradients the density of the medium can interact with particle density to decrease resolution. This decrease in resolution occurs when the range of gradient densities is close to t’he isopycnic banding densities of the particles. The reason for this decrease is that whenever pp is close to pm even small increases in pm cause large increases in ym/(pl, - p,) . The large increases in vrnJ(pP - pm) result in particle deceleration and decreased resolution. Therefore, media densities should be selected as far away from particle densities as possible when the two particles have similar densities and are to be separated by s20,w differences. Resolution of any pair of particles. In selecting a medium for a particular set of particles it is important to consider their sedimentation properties in relationship to the media. The exact relationships among particle sedimentation properties, separation media, and resolution are difficult to establish because of the number of variables involved. To reduce the number of variables we have analyzed the relationship between particle sedimentat.ion properties and resolution in homogeneous media. In homogeneous media, the relationship between the particle sedimentation rat.es and resolution can be solved algebraically (Appendix). Resolution is an exponential function of the ratio of sedimentation coefficients, S,/S?, and a ratio of differences between particle densities
and media densities, [ (pl - pm)/(p, - pm)I X 1(pz - P~~.~)J(~~ - pzo.,.,) 1. For convenience we refer to this latter term as the “ratio of density differences.” In Fig. 2, resolution is illustrated as a function of the ratio of sedimentation coefficients and the ratio of density differences. Resolution depends on the product of these two ratios (see Appendix). When t.he product of the ratio of sedimentation coefficients and the ratio of density differences is much different than 1.0, resolution is large. When the product of the ratios is close to 1.0, resolution is poor. If the ratio of sedimentation coefficients and density differences are both greater than
tiRADIENT
37.5
DESIGN
RATIO OF SEDIMENTATION COEFFICIENTS 3.0 1.5 IO
FIG. 2. Maximum resolution of two particles in homogeneous media. Resolution was computed as a function of the ratio of density differences (z-axis) and the ratio of sedimentation coefficients (contour lines) according to Eq. 4 (Appendix). Resolution of particles that differ in s.qO,Walone is illustrated by the darkrned y-axis. Resolution for particles that differ in density alone is shown by the darkened contour line in the upper half of the graph. Above this line (stippled area), particle sedimentation properties work together to achieve resolution since both ratios are greater than one. For particles in the clear area, sedimentation properties work against each other since the ratio of sedimentation coefficients is less than one. The ratio of density differences represents the ratio of differences hetween partirle and medium density: [(PI - PdlbZ
- Pm)1 x Lb2 - P20,W)Ibl - P20,w)l.
1.0 (stippled area), the particle sedimentation properties work together to improve resolution. However if one of the particle sedimentation
properties
counters
the other
(clear
area),
resolution
can he quite
poor. If densit,y differences override s20,wdifferences, then the particle
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with the greater density moves faster (above the z-axis) and resolution is mainly by density differences. If the differences in sedimentation coefficient are larger than the differences in density, then the particle with the greater s20,Wsediments faster (below the s-axis) and separation is mainly by .saa,,,,differences. An important practical question is how to select a homogeneous medium that will optimally resolve two particles with known density and sm,w values. To answer this question, it is helpful to reexpress ISOPYCklC SANDING
20
IO
ISOPYCNIC 'C
%
SUCROSE
(W/W)
BANDING ,_.^ ^. ,518-h
(5’C)
FOR PARTICLES
FIG. 3. Resolution of two sucrose concentration. Particle The particle with a density sedimentation coefficients are resents a ratio of sedimentation in Fig. 2.
WITH DENSITIES I.17 GM/CC
_..^. XL,
1.14 AND
particles in homogeneous media as a function of densities were chosen as 1.17 (pJ and 1.14 g/cc (~2). of 1.14 g/cc bands at 31.8% sucrose. The ratios of indicated by the contour lines. The dashed line repcoefficients of l/1.7 as in Fig. 4. Other details as
GRADIER‘T
DESIGN
377
the data in Fig. 2 as a function of sucrose concentration. This requires that we hold part,icle densities constant since the ratio of density differences is a funct,ion of both particle density and the density of the medium. The results for a specific set of particle densities are shown in Fig. 3. As previously, resolution of particles whose sedimentation properties work together (stippled area) is excellent. When particle sedimentation properties oppose each other (clear area), separation can be either predominantly by density (above the axis) or by .szO,,, differences (below the axis). For a particular set of particles with a given ratio of sedimentation coefficients, separation by .s~~,,,~differences is optimal in low density media and separation by density differences is optimal in media near isopycnic banding density. Between these two extremes there is a medium density in which the part.icles cannot be resolved. The results we have presented so far can be applied to designing gradients for separation by s20,w differences. In general the best rate separations can be achieved in shallow gradients of low density media. For particles whose sedimentation properties oppose each other. the selection of the appropriate density range is especially important. In particular, in some media So,,,, differences can cancel out density differences and prevent separation. This principle has been shown in Fig. 3 for sedimentation in homogeneous media. Figure 4 shows how it applies to density gradients. For the particles illustrated in Fig. 4, the ratio of sedimentation coefficients equals l/1.7 (shown by a dashed line in Fig. 3). Initially when the particles are in low density media the particle with the larger s~~.,~ sediments faster. This point is illustrated by comparing the position of the particles at low w?t values in Fig. 3. As the particles enter the higher density media (25-30s sucrose), the particle with the larger density takes the lead. At all points in such a gradient, resolut’ion is poor. Resolzltion of particle po&ations. For separation of two particles differing in both szn,,,. and density some resolut,ion is always obtainable by isopycnic banding since the density differences will always be resolved. In actuality the more typical problem involves separating two particle populations that overlap in both s?,,,,,, and density. When the overlap in density is small, separation by isopycnic banding can still be used. The question is when the overlap in s?,,.,\-is less than in density will rate separations resolve the populations better than isopycnic banding. To answer this question we have selected two normally-distributed poplilations of particles that overlap 33% in density and 8% in s?~,,~ (Figs. 5A,B). How much would these populations overlap in a medium that
CHURCHILL,
35
40
45
BANKER,
50 ROTOR
RADIUS
AND
55
COTMAN
6.0
65
(CM)
FIG. 4. The sedimentation of two particles with density differences and SZO,~ differences that oppose each other. The position of each particle in a lD45% sucrose gradient is shown as a function of the total applied force, w’t. From l&29% sucrose, the particle with the larger sedimentation coefficient sediments faster. At 29% sucrose, the particles pass each other and the particle with the larger density sediments ahead of the other. Finally, the particles band at their isopycnic banding densities (31.8% and 37.7% sucrose).
separates mainly by s20,wdifferences, such as 10% sucrose? By using Fig. 2 and values of p and s?~,,”for selected part,icles in the populations, it is possible to compute this overlap. When this computation is made, the overlap is only 8%, considerably bet,ter than could be achieved by isopycnic banding. Knowing that, reasonably good separations by rate are possible, we can design a gradient to separat’e these populations. Since a homogeneous medium is not practical for rate zonal centrifugation, we considered a gradient that will resolve mainly by s~~.,~differences and also provide capacity and stability. We chose a shallow convex gradient from l&18% sucrose. In this gradient the overlap in the two populations is 13% (Fig. 5C). Thus rate sedimentation can give a het.ter separation than isopycnic banding. Furthermore the resolution calculated for homogeneous medium is a reasonably good approximation to that found in shallow gradients of similar density.
GRADIENT
DENSITY
(GM/CC)
103 x szo w (S)
ROTOR
FIG. 5. Rate xonal separation
379
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RADIUS
(CM1
of two particle populations.
The particle populations
are representedashaving normal distributionsof ~~0,~ and density. Selected particles within each population were used to compute resolution in the convex gradient. The particle with the mean s20,Whad the mean density and particles with an SLWW above or below the mean had densitiesthat deviated a correspondingamount from the mean density. The means and standard deviations for ~~0,~ and density values were chosen so that populations overlapped in density more than in GO,%,. (A) The distribution of density values for each of the populations. The region of overlap (stippled), includes 33% of each population. Particles in this region cannot be resolved by isopycnic banding. (B) The distribution of sedimentation coefficients for each population. The overlap in S~.W of the two populations (strippled) is 8%. CC) The resolution of these particles in a shallow sucrose gradient. The overlapping region is only 13% of each population, considerably better than can be accomplished by isopycnic banding. DISCUSSION
By analyzing the resolution of particles sedimenting in linear sucrose gradients, we have developed some basic principles to facilitate the selection of appropriate centrifugation conditions that provide for optimal resolution. Some of the proposed principles of gradient. design have been previously derived by an empirical approach. Brakke (10) in his pioneering work with sucrose gradients observed that shallow linear gradients resolved plant viruses more effectively than steep gradients. Pretlow and
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Boone (4) found t#hat shallow gradients with density ranges far less than particle densities were optimal for separating mammalian cells by rate. In macromolecular separations where particle densities are similar, shallow gradients with a density range less than the density of the particles are utilized (11,12). These empirical results combined with our theoretical study indicate that these principles of gradient design are of general application. Gradient design to maximize resolution depends on knowledge of sedimentation coefficients. There are a number of methods for computing sedimentation coefficients. Values of s m,w can be det.ermined from the sedimentation of particles by rate zonal centrifugation (2,8,11, and Churchill and Cotman, in press) or differential centrifugation (13) at different w’t values. Sedimentation coefficients can also be determined for particles distributed throughout the sedimentation path (2,14). Special gradients have been designed so that particles sediment at a constant rate proportional to their s20,w values (12). If direct measurements of sz’O,w values are not convenient, then sZo,, values can be approximated from quantitative electron microscopic analysis (15). Even without precise values for sedimentation properties, a semiquantitative approximation should enable a reasonable prediction of the best conditions for particle separation. To simplify the analysis of gradient design we have focused on rate zonal centrifugation in linear sucrose gradients. In some applications consideration of other gradient media and nonlinear gradients are advantageous. In sucrose gradients viscosity increases dramat’ically with increases in density. Increases in either viscosity or density decrease resolution. One way to improve resolution is to select media with low viscosity at high concentrat,ions, such as CsCI,. Another approach is to design a gradient of decreasing viscosity and increasing density. Such a gradient could be obtained by combining a shallowly-increasing sucrose gradient with a steeply-decreasing gradient prepared from a highly viscous substance of low density, such as methylcellulose. In such a gradient, the leading particle will accelerate away from the trailing particle, and resolution will be better than in homogeneous media. For example, a linear l&20% sucrose gradient combined with a linear 3.5-0.4s gradient of methylcellulose (av MW 26,000) provides 74% resolution for the particles in Fig. 1. Compared with 48% resolution in homogeneous sucrose and 37% resolut’ion in a linear 1@20% sucrose gradient,, this type of gradient offers quite an improvement,. When gradients are shallow so t.hat resolut,ion is good, they may not provide adequate capacity and stability. Also a sedimenting zone of particles will diffuse more in a shallow gradient (16). One solution to
GRADIENT
381
DESIGN
the capacity problem is differential centrifugation and recycling (17). Alternately, zonal centrifugation in convex gradients provides increased capacity in the initial regions of the gradient where it is most needed (18). Figure 5 illustrates that resolution in a shallow convex gradient can approximate optimal resolution for t’wo particle populations. Thus shallow convex gradient.s provide a compromise that allows both resolution and adequate capacity. This analysis of particle separations by differences in sedimentation rates has provided some basic principles for selecting cent.rifugation media to optimize resolution. A method has been presented for computing resolution of particles with known sedimentation properties in any medium of known viscosity and density. This method along with the principles derived should aid in select,ing appropriate conditions for rate separations. APPENDIX:
RESOLUTION HOMOGENOUS
OF
TWO MEDIA
PARTICLES
The relation between applied force and particle homogeneous media can he calculated from Eq. 1.
u2t = [l/s~~.,l X [dbp
IN
sedimentation
in
- P,,~JIX I& - P?o,~.)/~~Po,~I X (In R; - In RJ PI
The separation of two particles in homogeneous media can then be obtained from the positions of each particle at any given applied force. Then (In RI - In R,)/(ln
R2 - In R,) = WS21
- pm)1
X [(PI - P,)/(Pz
x Lb2- P2O,w)/(Pl -
P20,w)l
= k'. [31
In homogeneous media, maximum resolution occurs when the faster part’icle has reached the end of the centrifugation path. Then if K > 1, one may solve for R,, R, = R, exp ((In R, - R,)/K). Therefore, resolution, the distance between the particles as percent of the gradient path, may be expressed as a. function of the position of the first particle:
RI - R, exp((ln RI - In R,)/K)/(R, For the computations presented in this paper R, = Rf = R = 3.637 cm. If K < 1, one may solve Eq. 3 for R, and Resolution = Rz - R, exp((ln Rz - In R,)/K)/(Rf Resolution
=
- R,).
[41
6.426/cm and by symmetry
- R,).
ACKNOWLEDGMENTS The authors acknowledge the assistance of Mr. Richard Roemer in designing programs for the PDP 12 computer. We also express our gratitude to Dr. Richard F. Thompson, who permitted us to use the PDP 12. We thank Mrs. Pat Lemestre for secretarial aid.
[Sl
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REFERENCES 1. ANDERSON, N. G. (1966) Nut. Cancer Inst. Monogr. 21, 9. 2. HALSALL, H. B., AND SCHUMAKER, V. N. (1969) Anal. Biochem. 30, 368. 3. BEAUFAY, H., BENDALL, D. S., BAUDHUIN, P., WATTIAUX, R., AND DE DUVE, C. (1959) Biochem. J. ‘73, 628. 4. PRETLOW, T. G., AND BOONE, C. W. (1969) Exp. Mol. Pathol. 11, 139. 5. SCHUMAKER, V. N. (1967) Advan. Biol. Med. Phys. 11, 246. 6. DOYLE, L. C., AND COTMAN, C. W. (1972) Anal. Biochem. 49, 29. 7. COTMAN, C. W., DOYLE, L. C., AND BANKER, G. (1971) Fed. Proc. Fed. Amer. Sot. Exp. Biol. 30, 1181. 8. COTMAN, C. W. (1972) in Methods in Neurochemistry (R. Rodnight, and N. Marks, eds.), p. 45, Plenum Press, New York. 9. BISHOP, B. S. (1966) Nat. Cancer Inst. Monogr. 21, 175. 10. BRAKKE, M. K. (1966) Arch. Biochem. Biophys. 45, 275. 11. MARTIN, R. G., AND AMES, B. N. (1961) J. BioZ. Chem. 236, 1372. 12. NOLL, H. (1967) Nature 215, 360. 13. COTMAN, C., BROWN, D. H., HARRELL, B. W., AND ANDERSON, N. G. (1970) Arch. B&hem. Biophys. 136, 436. 14. SCHACHMAN, H. K. (1959) in TJltracentrifugation in Biochemistry, Academic Press, New York. 15. BAUDHUIN, P., AND BERTHET, J. (1967) J. Cell. Biol. 35, 631. 16. SCHUMAEER, V. N. (1966) Separ. Sci. 1, 409. 17. SCHUMAKER, V., AND REES, A. (1969) Anal. Biochem. 31, 279. lg. BERMAN, A. S. (1966) Nat. Cancer Inst. Monogr. 21, 41.