Sept., 1956
1211 OPERATING AND COMPARATING PROCEDURES IN ANALYTICAL ULTRACENTRIFUGATION
OPERATING AND CO.MPARATING PROCEDURES FACILITATING SCHLIEREN PATTERN ANALYSIS I N ANALYTICAL ULTRACENTRIFUGATION BY RODESTRAUTMAN The Rockefeller Institute for Medical Research, New York, New York Received March 18, 1956
The simultaneous introduction of several alterations in ultracentrifuge operation and in ineasuriiig the pliotograpliic plate are described which enable an over-all improvement and simpiification of the analysis. The alterations are: (a) centrifuging at a predetermined temperature; ( b ) using a double sector centerpiece; (c) using a combination phaseplate-wire schlieren diaphragm; (d) photographing a t predetermined integral values (in minutes) of the true sedimentation time wit,hout measui’ing or requiring reproducibility of acceleration time; (e) equipping a two-dimensional micrometer comparator with a 5 s projector and revolution counter reading directly to 0.01 mm. on the plate; ( f ) replacing the sclilieren pattern with a tabulation of a significant number of its two dimensional coordinates; (g) selecting the measured points of the schlieren pattern so that they occur at equal intervals of the cube of the radius; and ( h ) using a printing calculator t o record these coordinates. A special tabulation of squares and cubes of the radius is described for use with a comparator. A summary is given of the formulas for: (a) the concentration at any level; ( b ) the molecular weight determined in sedimentation velocity by the Archibald principle; (c) the true boundary position; ( d ) the sedimentation rate; ( e ) the initial concentrations in paucidisperse systems; and ( f ) the distribution of sedimentation rates in polydisperse systems.
The theory and equipment for analytical ultracentrifugation’-+ recently have been developed sufficiently t o warrant complex analysis of the entire refractive index gradient patterns obtained from solutions of biochemical interest. I n particular, it is now possible to determine: (a) the molecular weight over a broad range of weight averages without obtaining sedimentation equilibrium for every level in the cell; (b) the sedimentation rate of skew boundaries; (c) the relative composition with respect to the paucidisperse components in concentrated solutions; and (d) the distribution of Sedimentation rates in the polydisperse components even with significant diffusion ‘in the boundary region. It is the purpose of this paper to describe operating and cornparating equipment, procedures and formulas found useful in performing the above analyses. Especial attention is given to the selection of ordinates of the schlieren pattern spaced equally on a radius-cubed scale instead of uniformly on the radius scale, or a t random. The reason for this choice is that an increment of 7-3 corresponds t o an increment of r z dr since d(r3) = 3r2 dr. The integral f r2(bn/br) dr, where bn/br is the index of refraction gradient, can then be evaluated as a summation of ordinates each chosen a t the center of successive equal increments on this radius-cubed scalee4Only J r 2 ( b n / d r ) dr appears in the deterinination of the initial concentrations in paucidisperse system^,^ or of the molecular weight before loss of the plateau solution.6 Both integrals Jr?(bn/br) dr and J ( b n / b r ) dr are needed t’o obtain the true boundary position6 or the concentration a t any intermediate level in the approach to sedimentation equilibrium5J and spacing of ordinates uniformly on a linear scale or a radius-cubed scale will be equally convenient. (1) J. W. Williams, J . PoZymeT Sci.. 12, 351 (1954). (2) Technical Manual for Spinco Model E Ultracentrifuge, Spinco Division, Beckman Instruments, Inc., Belmont, California. (3) E. G. Pickels, in “Methods of Medical Research,” Vol. 5, A. C. Corcoran, ed., Year Book Publishers, 1953. (4) R. T r a u t m a n a n d V. N. Schumaker. J . Chem. Phys., 2 2 , 551 (1954). (5) S. M. Klainer a n d G. Kegeles, THIEI JOUBNAL, S9, 952 (1955). ( 8 ) R. Goldberg, ibid., S T , 194 (1953). (7) W. J. Archibald, ibid., SI, 1204 (1947),
In the determination of the concentration a t m y level in a velocity ultracentrifuge experiment, only J(bn/br) dr appears, in which case the radius-cubed scale is inconvenient. But this cnlculation is rarely needed, since concentrations corrected to zero time are generally of more interest,. The radius-cubed scale is also useful in calculating the apparent distribution function* from the concentration gradient since each ordinate must he multiplied by r3. Ultracentrifugation Equipment and Procedures The sedimentation analysis is simplified if experiments in a given series are all made at the same temperature, if the true sedimentation time is accurately and conveniently known, and if the base line is simultaneously on the photographic plate with the schlieren pattern from the solution. The following are modifications of the Rpinco Model E ultracentrifuge and operating procedure.2 (a) Whenever practicable the experiment is performed at 20.0”. To do this, the rotor is precooled in a refrigerator to a temperature below 20” and then brought to the desired value, as measured with the thermocouple in contact with the suspended rotor, with the aid of n 250 watt infrared lamp before closing the vacuum chamber. Such a lamp, about 10 cm. away from the rotor, heats the rotor about 20”/ hour. The adiabatic cooling of the rotor on acceleration must be c ~ n s i d e r e d . ~ The refrigeration for the vacuum chamber is set empirically such that the rotor temperature a t the completion of the experiment is within 1 0 . 1 O of the initial temperature. ( b ) Correction for the sedimentation tlrnt occurs during acceleration is made by maintaining the drive current constant during this period and starting the automatic camera when the speed is two-thirds of its final value.8 This procedure is simplified by the use of n mechanical stop10 which prevents the “exposure time adjustment dial” from rotating until the desired speed is reached. A microswitch under the mechanical stop also starts a running time meter1*reading in 0.01 minute, permanently installed between the vacuum gage and the viewing window and wired in parallel with the synchronous motor. The dial is held a t a position corresponding to half way between the shutter microswitches, so that the true sedimentation time of any exposure taken automatically is accurately an (8) R. L. Baldwin. ibid., 6 8 , 1081 (1954). (9) D. F. Waugh and D. A. Yphantis, Rev. S c i . I n s t r . , 23, 609 (1952). (10) Identical with the timing latch used in t h e Gofman lipoprotein procedure” to s t a r t the camera, in t h n t case, when acceleration isoom. pleted. (11) 0. de Lalla a n d J. W.G o f m a n , in “Methods of Biochemical Analyeis,” D. Gliok, ed.. Interscience Publishers, New York. N. Y., 1954. (12) R. W. Craetner Company, Inc., Centerhrook, Connecticut.
1212
RODESTRAUTMAN
VoI. 60
a b C Fig. I.-Modified schlieren diaphragm mount: (a) Mount as seen from the direction of light travel. Vertical centering of pattern on plate is accomplished by thumb screw a t lower right. Diagonal wire is on border of phaseplate. Mask can be positioned in oversized friction guides. This mask is used only in special cases of two cells used simultaneously. (b) Scliliereii pattern with mask in place. The same protein, pepsin a t 8.4 mg./ml. concentration, is in both cells. The lower pattern is displaced downwards due to a 1’ quartz prism as the upper window of the double sector centerpiece. The right hand reference is from the reference in the AX-D rotor. The position and width of the horizontal band across the pattern depend upon the vertical position and alignment, respectively, of the mask; speed 59,780 r.p.m., t = 52 min., 0 = 60’. ( c ) Schlieren pattern a t t = 128 min. and 0 = 45’. integral multiple of 2 minutes. The running time meter aids in recording the time of each exposure and in visual interpretation of the pattern during the experiment. ( c ) A Wolter phaseplatel3 modified by addition of 0.003” diameter wire along the edge of the MgFt coating on half of a polished plate glass“ is used as a schlieren diaphragm. (d) The double sector centerpiece of Beams and Dixon16 and hlilchl6 is used with solution in one side and solvent in the other. The loss in contrast due to the superposition of two schlieren patterns is minimized with the phaseplate and is more than compensated for by the gain in accurate, convenient registration of the baseline. For sedimentation, a wedge quartz* is used for the upper quartz window of the cell, oriented to deflect the light toward the center of rotation, thus displacing the pattern downward on the photographic plate relative to the baseline in the image of the reference holes .I7 To further enable greater utilization of the vertical space on the photographic plate, even when only one cell is used, the schlieren diaphragm mount hm been modified as shown in Fig. l a . The two screws allowing cross motion to the optical track formerly at the lower right, have been replaced by a single screw and knob extending outside the dust cover, so that the pattern may be centered a t will during the experiment. In using two cells, when the pattern from the plain and the wedge cells do not overlap, the original contrast for each patJtern may be restored by insertion of the “checkboard” mask shown in place in Fig. la. The entire group of deflectcd and undeflected rays from the wedge cell is displaced to the right on this mount (for a cell giving downward displacement 011 the screen) and interrupts the schlieren diaphragm edge as a group ahove the intersections of the rays from the plain cell. The upper part of the light source image contributing to the background from the plain cell is thus masked out, as is the lower part of the light source image from the wedge cell. ThiR optional procedure is for experiments wit,h two cells, not for two sectors in one cell. An example is shown in Fig. l b in which two double sector cells are used. Note the change in contrast in different parts of the pattern arising from overlap of various combinations of t.he light from the 4 sector openings. Idater in the experiment, Fig. IC, the maximum gradient in the lower pattern is less than the separation of the patterns brought about by the wedge quartz and the quality of the opticnl registration is the same as if each cell were used alone. (13) H. Wolter. Ann. Physik. 7, 182 (19601. (14) R. Trautnian and V. W. Burns, Biochirn. Biophya. A d a , 14,26 (1954). (15) J. W. Beams and H. M . Dixon, 111, Reo. Sci. Inafr.. 94, 228 (1953). (16) L. J. Milch. Lob.Inoest., 2,441 (1953). (17) This procedure is identical t o that of Gofman.11 One chooses a plain cell or one displacing the pattern upward or downward on the acreen depending upon whether flotation or sedimentation patterns are t o be viewed.
Cornparation Equipment and Procedures Two-dimensional Micrometer Comparator.-A photograph of the comparator used is shown in Fig. 2, rather than a schematic drawing giving the details of construction. It is intended to illustrate some features that can be added to existing comparators, not to suggest duplication in entirety. The image of a ortion of the schlieren pattern is projected onto a vertical wfhe cardboard at the top, back by means of a camera lens and front surface mirror. This 5X image can be shielded with an oscilloscope viewing hood (not shown) so that i t can be clearly seen in a room of normal illumination without eyestrain or loss of contrast. The quality of this optical system is not critical since only those pencils of light are used which reach the fiducial mark, chosen here to be a small circle. Each axis has a 1 mm. pitch screw. The axis of abscissas x, running left and right, and corresponding to the radial axis r in the cell, has been equipped with a revolution counter eared up 1 to 10 so that it reads directly in 0.01 mm. &e cross axis of ordinates y is also calibrated in 0.01 mm. on the handwheel. Because of the curved nature of the centrifuge baseline on a schlieren pattern, it is in general necessary to make the subtraction of the ordinate of the baseline ysOlvfrom the ordinate of the solution pattern ysOla. This difference will be denoted as Ay. A printing calculator18 has been found to be extremely well suited for the direct transcription and arithmetic operations of the coikdinates. Comparation Procedure.-The key simplifying rocedure in using the comparator is to set the abscissa scare so that its origin, z = 0 , occurs at the location of the center of rotation on the plate. Thus, the plate is positioned left or right so that the abscissa counter (or scale or handwheel) reads M,vinnerwhen the inner, and/or Motouterwhen the outer reference edge is on the comparator fiducial mark, where i l f ~is the camera magnification. A tabulation can be made of these settings as a function of speed for the various ultracentrifuges and counter balances used. It is possible to set the plate to within ~k0.005cm. (on the rotor) of its true position. Rather than using the actual values of x*,i t will be convenient to index the radius-cubed scale by a t most three digit integers proportional to 2 3 . Thus define for the comparator a fixed scale from which
z = (lOS/S,)*
(1)
x f dz = ( x , * / 3 ~d)Z (2) Z ran be called the “cubed-scale index number.” The Z ecale reference xr is a fixed number chosen to be just radially beyond the bottom of the cell when a plate from the ultracentrifuge with the largest MOis aligned on the cornparator. For M O less than 2.20, the value of xr = 160.00 111111. is convenient and is chosen here. It would be ideal to have (18) Olivetti Corporation of America, New
York. New York.
OPERATING AND COMPARATING PROCEDURES I N ANALYTICAL ULTRACENTRIFUGATION 1213
Sept., 1956
an additional counter on the comparator of Fig. 2, giving (x/16)3 directly. Since this is not available, the radiuscubed scale is represented by a tabulation, a n abbreviation of which is given in columns 1 and 4 of Table I. Whatever the choice of zr the tabulation must cover a factor of 2 in Z since the cube of the ratio of the radii of the bottom and the top of a standard Spinco cell is 2.0. The choice of 2 = 1000 near the bottom permits the z position of 500 intermediate points to be calculated, without interpolation, from standard tables of squares, cubes and roots. Three such intermediate points are given as 686, 687 and 688 in Table I.
TABLE I IOOZ - 2
Z'/'/IO
Z
(IOt/zr)'
500 20 40 60 80 600 20
40 60 80 680 687 688 700 20 40 60 80 800 20 40 60
80 900 20 40 60 80 1000
zr = 160.00 n i m . log(z/100) z(rnm.1
'8
2/2r
(zr/z)S
0,793701 ,804145 ,814325 ,824257 .833955 ,843433 ,852702 ,861774 ,870659 ,879360 ,881516 ,882373 .882801 .887904 ,896281 ,904504 .912581 ,920516 .928318 .935990 .943539 .950969 .958284 ,965489 .972589 ,979586 .986485 ,993288 1 .oooO0o
1 ,5874 1.5404 1.5080 1,4719 1 ,4379 1 ,4057 1.3753 1 ,3465 1.3192 1,2932 1 ,2850 1 ,2844 1.2831 1.2684 1 ,2448 1.2223 1.2008 1,1801 1.1604 1.1415 1.1233 1.1058 1 ,0890 1 ,0728 1 ,0572 1.0421 1 ,0276 1.0136 1 . 0000
120.99 128.06 130.29 131.88 133,43 134.95 136.43 137.88 139.31 140.70 141.11 141.18 141.25 142,07 143.41 144.72 146.01 147.28 148.53 149.76 150.97 152.16 153.33 154.48 155.61 156.73 157.84 158.93 160.00
0.10378 10945 1492 201s 2520 3017 31\12 395 1 4397 4829 -1.956 ,140i7 ,14998 .I5249 .I5056 ,16053 .16439 .16815 .I7182 ,17539 ,17888 .18229 .I8561 ,18887 ,19205 ,19516 ,19821 .20120
,20412
In making a reading of the ordinate A y a t a particular Z , say 600, the z axis handwheel is turned, moving the aligned plate until the counter reads the corresponding x value,
TABLE I1 Z
A u (mm.)
600 20
0.53 2.49 3.98 9.13 13.16
40 60
80
Z
700 20 40 60
Au (mm.)
11.94 7.41 3.49 1.63 0.70 0.36
80 800 ZAyj = 54.82 A Z tan OZAyj = 1096 W = 8 . 4 4 mg./ml. Z ( ~ , / z j ) Ayj ~ = 70.43 A Z tan O Z ( ~ , / z j ) ~Ayj = 1409 A = 7 . 1 2 mg./ml. ( ~ . / a =) ~1.2847 R = 141.16 mm. rmax. ordinate = 141.06 mm. Comparator readings and calculations for the lower pattern of Fig. IC.
F.F,
X 10-3(mg./ml.)/mm 9.79 X lO3(S min.) xa = 1 2 9 . 5 6 m .
= 5.05
F.
=
ultracentrifuge plate is shown on the bakelite holder which can be aligned by means of a small thumb screw just in front of the second frame from the left. The holder travels on two pairs of drill rods placed a t right angles with a three-point suspension on each pair. The handwheel at lower right moves the plate left and right and is equipped with a counter reading directly in 0.01 mm. Rapid scanning is done by sliding the plate in its holder. The handwheel at lower left moves the plate front and back and is calibrated in 0.01 mm. The screw in the center left just under the end of the plate releases the plate holder for rapid scanning front and back. Not shown is a n oscilloscope viewing hood which fits into the top of the black box housing the projector. namely, 134.95 mm. This value of Z and/or z can be indexed on the calculator tape using the non-add operation. The yaxis handwheel is then turned until the solution schlieren pattern is on the fiducial mark. The reading is entered as a plus theri, correspondingly, the reading when the solvent schlieren pattern is moved to the fiducial mark is entered as a minus, and the difference, A y , is taken. If the spacing A Z = 20 were chosen, then the counter would be advanced to 136.43 mm. corresponding to Z = 620, and the ordinates a t this location measured. The readings taken on the lower pattern of Fig. IC are given in Table 11. Since ordinates are measured only a t tabulated x values, which are actual distances from the center of rotation, the accessory relations of ra, rt and In r involved in the theory can also be tabulated. These are given in Table I in the more useful form of Z = (10 z/z,)J, column 1; (zr/z)z, column 3; and log z, column 5. The first 3 columns are universal, and the fifth differs only by a constant, for any choice of xI. The fourth column is calculated from column 2 merely by multiplying bv the chosen value of xr for the comparator at hand. All values in Table I are also independent of the magnification Mo. This special tabulation is a convenient complement to the two dimensional comparator. The necessity for cubes, squares and logarithms is strictly a geometrical consequence of the sector shaped cell in a centrifugal field and does not depend upon the position of the meniscus, bottom of the cell, or whether flotation or sedimentation is taking place.
Notation.-The two-dimensional coordinates actually measured on the plate at any particular time t will be denoted by x and y. Their respective counterparts in the rotor are the radius r and the difference in the refractive index gradient between solution and solvent &n/br. The difference
RODESTRAUTMAN
1214
in refractive index An nsotn- nsolvwill be taken as a measure of the concentration. Both x and r are zero a t the center of rotation and are not considered functions of the time. The radial distance measured on the x scale of an identifiable moving point will be denoted by Z on the plate and T in the cell. Here T and Z are functions of the time. An asterisk (*) will be used t o denote quantities calculated as though diffusion and concentration dependence were negligible. The superscript zero (O) when applied t o an s rate indicates infinite dilution of the sedimenting solute, but when applied to a concentration means the initial value in the cell. Let Mo be the magnification of the schlieren camera lens; M , the magnification of the cylindrical lens; a’, the optical path of the cell; b’, the optical lever arm; 0, the angle the schlieren diaphragm makes with the light source; w , the angular velocity of rotation; s, the sedimentation coefficient (‘(s rate”) in svedbergs S sec.); t, the true centrifugation time in minutes; in a; conventional cell, r,, the meniscus position for sedimentation or the bottom of the cell for flotation and Tb, the bottom of the cell for sedimentation and the meniscus for flotation; and, in a synthetic boundTo, the meniscus, and r,, the initial position ary of the synthetic boundary. Three apparatus constants will be given special symbols F, F, F.
= ~,/(3000Mo),
s =
60 u2t
_ F- ,
(4)
F, log - = - l o g xa 3t
z
z -Za
(5)
Calculable from 5
Here s* is the combination of the coordinate x with the time t of the exposure t o make an “s rate scale” (equation 5 ) , while 8* is the time average of the instantaneous sedimentation coefficient s of a particular moving point (equation 8). Calculable from y ban _ = br
tan 8 a,b,nl,
(ysoin
-
3
mV) = F , tan
8 AY (9)
Calculable from Zand 9
-= A ( Z , , Z,)
(10)
(19) E. G . Pickets, W. F. Harrington and H. IC. Scliaclriiian, Proc. N u l l . Acnd. Sci., U.S., 88,943 (1952). (20) T. Svedberg and K. 0. Pedersen, “ T h e Ultracentrifuge.” Clarendon Press, Oxford, 1940.
..
.
Ly
Ay dZ
w (Z1,ZZ)
(11)
The notations A(Z1, 2,)and W(Z,, 2,)have been introduced for brevity in listing the formulas in the following sections. A corresponds t o the area of the schlieren pattern on the linear x scale, and W to the ‘(area” on the x 3 or “warped scale.”4 If Z z - Z1 is divided into m equal intervals AZ, and the ordinate Ayj is measured a t the center of each of these intervals on the cubed-scale, then A and W can be evaluated as m
A ( Z , , 2,)
=
F,F, tan 8 AZ
(X,/X,)~ Ayj
(12)
j=l
m
bv(z1,Z Z ) = FxFy(x,/x.)2tan 9 A Z
Ayj
(13)
j=l
W is a simple summation, whereas A is the accumulated sum of the products obtained by multiplying each ordinate by the appropriate (x,/T)~from column 3 of Table I. If this factor varies only in the last 3 places over the range from Zl to Zz, then it may be more convenient t o subtract a number B from each value and use the following form for calculation m
m
(3)
r = x / M o = 300F,Z’/r 111 r / r a
F,F, tan 9 ( X , / X . ) ~
=
m
l/(a’b’ M e ) , = l O I 3 In 1 0 / ( 6 0 ~ * )
3
Using these notations the following functions of r and dn/dr can be calculated directly from the coordinates measured on the comparator.1 , 2 , 3 , 2 0 Calculable from x or Z -
Vol. 60
.
..,
The integrals A and W are independent of the particular value of the limits when they are in plateau regions where Ay = 0. Typically A 2 and the location Z 3 of the first ordinate to be measured are chosen using tabulated values of Table I. Then 2, = 23 - A Z / 2 and 2, = [z3 ( m - 1) A21 AZ/2. The summation from 2, to 2, may be appropriately divided into two or more parts a t different spacings. This is indicated when the continuation of a very close spacing, chosen to give about 10 points on a narrow peak, would be unnecessarily time consuming in reading a region containing a broad pattern. If the last ordinate reading were taken a t Zj for a series with spacing AZ,, then the next reading should be taken a t Zj+, = Zj A21/2 AZ2/2, and then a t Zi+? = Zj+l AZ,, . . . , for the new series a t a spacing of AZ,. Velocity Sedimentation Analysis The equations of velocity ultracentrifugation to be considered assume the existence of an allcomponent-plateau, a region of limited radial extent containing all the initial components but with no concentration gradients. The subscript p refers to any point selected in this plateau. It is no longer necessary to further assume that there are no concentration gradients a t the meniscus nor that sedimentatioii coefficients are constant with concentration or time during an experiment. Complete derivations, interpretations or limitations are iiot, given, instead the appropriate formulas are collected and converted t o the notation of the two dimensional comparator. The equations are applicable for flotation as well as sedimentation.
+
+
+
+
+
Sept., 1956
O P E R A T I N G A N D C O M P A R A T I N G P R O C E D U R E S IN
The Concentration a t Any Level in the Cell.Klainer and Kegeles5 indicate, i n effect, that the concentration a t any level T in the cell can be expressed in terms of the initial concentration and the Concentration gradient pattern for r, 5 r 5 rpas An = An0
-
lr:(:)z$ + frl dr
= Ano -
and for r p 5 r
+ A(%, 2 )
(14)
1215
of s / D from the experiment, or series of experiments a t several speeds, so that the weight average molecular weight M , of_ the solute can be co-mputed from7rZ0M , (1 - Vp) = R T s / D , where V is the partial specific volume of the solute, p the solution density, T the absolute temperature, and R the gas constant. Substitute equation 14, evaluated a t 2, into equation 18 and equation 15, evaluated a t Zb into equation 19, using also equations 3 and 9
I Tb as
An = Ano = Ano
W(Z,, 2,)
dr
ANALYTICAL ULTRACENTRIFUGATION
+
+
L:"(2)'T lrrb % dr
(Ta/rb)'
W(zp,zb)
-
- A(Z,Zb)
dr
(15)
where Ano is the initial concentration. I n calculating the concentration a t the starting level Z a and the bottom of the cell zb, the A term in equations 14 and 15, respectively, is zero. At sufficiently high speeds the concentration gradient a t r , drops to zero as the sedimentation overcomes the back diffusion. I n this case the concentration a t any level 2 is
+
These are of the form p h(s/D)p (s/D)Ano and a plot of p vs. p will yield s / D as slope and p o = Ano as intercept on the p axis. The type of plot obtained is indicated in Fig. 3 where the various
and equation 14 then yields the initial concentration4
The left-hand side of equation 17 is also given by the index of refraction difference, nsoln - nsotv, measured on a differential refractometer.21 If the synthetic boundary cell19 is used in the ultracentrifuge as a differential refractometer, the boundary will diffuse back toward the meniscus ro from the starting position r,. To include all the gradients the integration should start a t ro, and Ano = W ( Z 0 , 2,). If the displacement of the boundary is negligible due to sedimentation, then Ano is also given5 by A (Z0,2,). Molecular Weight Analysis.-Archibald7 proposed to utilize the approach to sedimentation equilibrium in a sector cell in order to obtain the weight average molecular weight. Klainer and I
