Weisberg, Sanford

1 online resource (PDF, 20 pages)Weisberg, Sanford. (1973). A New Method for the Statistical Analysis of Dual Labeled Isotope Polypeptides Separated by Gel Electrophoresis. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/199194

University of Minnesota Digital Conservancy

--A New Method for the Statis-tical Analysis of Dual Labeled Isotope Polypeptides Separated by Gel Electrophoresis Sanford Weisberg* Technical Report No. 202 March 1973 *The author is at the Department of Applied Statistics, University of Minnesota, St. Paul, Minnesota 55101. This work was supported in part by N.S.F. Grant GS-32327x to the Statistics Department, Harvard Univer-sity. The data used come from research in The Biological Laboratories, Harvard University, under the direction of R.P. Levine. The work is supported by grants to R.P. Levine from the N.S.F. (GB-29203) and the Maria Moors Cabot Foundation for Botanical Research at Harvard University. The author would also like to thank William G. Burton and David Hoaglin. - I. SUMMARY Analysis of cellular polypeptides following separation by electro-phoresis on polyacrylamide gels (7) often takes advantage of radioactive isotopes that are introduced into the polypeptides during cell growtho Comparisons of polypeptides between differing organisms, for example, wild type versus mutant (8), or studies of the rate of turnover of poly-peptides (.1), conveniently involve the use of two radioactive labels (such as 3H and 14c). The gels on which the polypeptides have been separated are cut into 1 mm slices and the radioactivity associated with each slice is detennined and expressed as 3H or 14c counts per minute (cpm). This paper described a simple method of detecting dif-ferences in the polypeptides in the two differently labelled organismso -1----II. MOTIVATION After electrophoresis, lei: 14c and 39 denote the cpm in each · ge·l! -slice, after correction for background and crossover (4). First consider an experiment in which both 3H and 14c are used to label polypeptides from identical organisms. We shall call this a control experiment. In con-trol experiments, we expect the ratio of 3H to 14c cpm. in each slice to be a ~onstant, which we will call r. The parameter r is essentially a function of the amount of radioactive material placed on the gelo Thus, for control experiments, we expect the following equation to hold in every slice (1) = r. In experiments in which 38 and 14c label polypeptides from different .. organisms (such as a mutant and a wild type respectively), the mathema-tical relationship (1) will fail to hold in some slices, indicating those polypeptides by which the two organisms differ. Even in a control experiment, we will not observe exactly r for the -ratio in each slice. Neither the measurement process nor the experi-mental method are without error, so that the .value of r in each slic-e will vary. Furthennore, the counts for each polypeptide will be distri-buted over several slices; thus, even if there were .!!.Q. experimental error, we would still find v~riability in the ratio 3H/14c. In principle, we could take equation (1), estimate the value of r, and assusr the fit of the model at each slice. This approach is very difficult because ratios of the form 3ff/l4c are very unstable, ranging from O to oo • Furthermore, the variability of the ratio is highly -2---3-dependent upon the size of each of the counts. Finally, no standard, widely accepted methods exist for handling these ratios.· A more useful approach can proceed as follows: We multiply both sides of equation (1) by 14c to get the equation (2) 3H = r 14c We immediately recognize the following fact: In a control experiment, in each slic~, the 3H cpm should be a constant multiple of the 14c cpm. Equivalently, !f ~ .!l!!J! .!2 graph ~ 14c cpm ~ the ! axis against the 3H cpm .2!l the .X axis, ~ would expect ~ graph !2 be !. straight ~ ~ slope equal !2, :£·· As an example, in one control experiment where both 3H and 14c label the same organism, this graph appears as Figixre 1. (Figure 1 about here] Notice several important charaoteristics of Figure 1. First, the graph of the points is basically a straight line, as predicted by e qua-tion (2 ). If we were to fit a line to the points by eye, about as many points fall above the line as below it. Finally, and most importantly, as the counts get larger, the variability between points also gets i. larger. ... -If a monotonic transformation of the data to a different scale is_ performed, it may be possible to make the variability of the data nearly independent of magnitude. For example, suppose the counts at each slice follow a Poisson distribution; that is, for 14c in each slice, p(14c = k) = e-m mk/k! k = o, 1, ••• where m = the expected 14c count for that slice. (A similar law will hold for 3H). This is the usual model for counted data. In this case .... ----4-we will also have that the standard deviation of 14c is equal to (in. Thus the variability of a count will increase as the square root of its size. It can be easily shown [Scheff~, (6)] that, if Y follows a Poisson distribution with mean m (and therefore standard deviation 'frn'), then the variable '(Y is distributed approximately normally with mean ffli" and stan-dard deviation about ~o 1he important point for the purpose of this analysis is that, if the Poisson model is true, then the graph of~ vs {14c will be a straight line with the added benefit that the variability a\\;ay from the line will remain constant as the size of the count increaseso This graph for the same control experiment is shown as Figure 2 • [Figure 2 about here] To summarize, in a control experiment, we expect equation (2) to hold for each slice number. To make variability independent of size of count, we take the square root of equation (2) and, letting a= fr, we get (3) ~3H = afl4c All f-urther analysis will use equation (3). -.. la -ial III. ANALYSIS In equation ( 3) we recognize a standard linear regression problem, · ;which is to fit a straight line of the fonn (4) Y = ax + b to a set of pairs of points (X,Y). However, the problem at hand differs from the standard problem in several ways. Usual regression problems '-' are primarily interested in estimation of the parameters a and b. In I.al --.. -our problem, a is simply an artifact of the experiment. Although it must be estimated, we consider it to be merely a nuisance parameter :since it is a function of the amount of radioactive material on the gel. The value of the parameter bin (4) is known to be zero since when 14c =Owe would also have 3H = o. However, we shall estimate bas if it were not zero because we are interested in the regression line in the region where the data points actually occur, and not in the fit of the line in the region of zero. The principal interest in the analysis is to examine deviations of 1a1 the observed (transformed) counts from the estimated line. If, for a set of ccintiguous slices, the deviations are "large" and of the same -i..l sign, we will be able to say that the model fails to fit in that region. Thus, in experiments where 14c and 3H label polypeptides from different organisms, regions of slices where large positive deviations occur will .i denote excesses of 3H (and hence excesses of a specific polypeptide in the 3H-labeJed organism) and large negative deviations will indicate .. ... lal ... .deficiencies of 3H (and hence deficiencies of specific polypeptid~ in the 311-labeled organism). -5-IV. ESTIMATION OF PARAMETERS The standard method for estimating the parameters a and bis hy least squares. Denoting Y = J 3H , X = J 14c , our estimates of a and b a·re ( 5) 8 = I: (X-X) (Y-Y) ): (X-X) 2 "' -b = Y - ax where the sunnnation is over all slices, and X and Y denote the arithmetic mean of the X's and Y's respectively. This method of estimation is probably not the "best" under the circumstances since heavy weight is given to the extreme observations that may occur in experiments where 3H and 14c label different organisms. Much current statistical literature, such as (2) and (.3), is concerned with techniques that are "robust" even when extreme observations occur. In the present problem, probably little additional information is to be gained by using a more resistant method of estimation ...i compared to the loss due to increase in complexity. We shall therefore estimate the regression line as (6) = .,.. with a and b defined. by (5). For each slice, we can create a deviation score, Q, such that D = (observed value of/¾) minus ("fit" value of J3H) or, for each slice, (7) o = J 3n - (a J 14c + b) The remainder of the analysis uses the D's. We make the following observations: a. ~ D = 0, where the sum is over all slices. This is an artifact of using least squares to estimate a and b. - 6 -b. The variability in the D's is independent or nearly independent of both slice ntunber and magnitude of the count in that slice, as is shown in Figure 2. This means that a "large" value of D in slice 52 has exactly the same meaning as a large value of D in slice ntunber 93 or any other slice. c. An estimate of variability of the D's can be obtained as s =J~ D2/n(n-2) , where the sunnnation is over all slices. The statistics is usually called the standard error of regression. In control experiments, s should be a reliable estimate of the real standard deviation of the D's. 3 In an experiment where H and 14c label different organisms, the computation of swill put too much weight on those values of D that are large (i.e. those places where the model (3) fails) and will therefore overestimate variability. d. In a control experiment, each Dis equally likely to be positive . Th. . 1 . h . . h 3- d 14c or negative. is imp ies tat, in experiments were a an label different organisms, long runs of D's of the same sign might indicate a difference in polypeptides between the two organisms. e. Ignoring the fact that the D's are probably correlated, we would hope that the D's from a control experiment could be regarded as a sample of size n (= number of slices) from an approximately normal distribution, mean zero, standard deviation estimated bys. Thus we would expect to see no more than about 5% of the D's greater than 2s or less than -2s, and only perhaps 1% beyond 2.5s. These are only to serve as approximate rules, because of violations in the necessary assumptions. 3 14 In experiments where Hand C label different organisms, we will simply have too many large D's (i.e. more than 1% larger than 2.Ss) and D's that are too large (i.e. some D's greater than 4s or so). - 7 -V. EXAMPLE: CONTROL EXPERIMENT Consider a control experiment ih which 3H and 14c label the same organism. For the data shown in Figure 2, the least squares fit of the regression line is (8) J3H = 1.510 {14c + o. 758 with standard error s = 1.417. We next compute the· D's. For example, in slice nwnber 59, the 3H count was 302 count~ per minute while the 14c count was 112. Thus, for slice 59, D = {SJi - {1.570~l4c 1- o. 758) = ~ - (1.5701112 + o. 758) = 17.370 - 17.347 = ·0.033 which is a close fit in this slice. Possibly the most informative single graph is shown in Figure 3. It gives the slice number on the X-axis and the deviation for that slice on the Y-axis. We note that the deviations o~cillate apparently randomly 1a1 · between positive and negative values, as we would expect. The only possible exception to this would be in the region from about 26 to 32, where the deviations are all large and positive; indeed, the four largest positive D's occur at slices 28, 29, 30, and 31. -To get a stronger feel for the size of these deviations, Figure 4 gives a normal probability plot of the D's. If the spread of the D's were as expected (more formally, if the D's formed a sample of size n from a normal distribution) then the graph in Figure 4 should be approx-imately a straight lineo As we see the four points at the high end of the curve a.r~ too far to the right to be thought of as lying on the line. ·- -a -j i : u u : I u I J ' I w u u l ! w I u u ; I w I ; LJ u Thus, we are tentatively led to conclude that slices 28-31 are unusual, and leave interpretation to the biologist. Beyond this single observation, the graphs yield the expected picture of a control experiment. [Figure 3 about here] [Figure 4 about here] - 9 -VI. EXAMPLE: A COMPARISON OF THE CHLOROPLAST MEMBRANE POLYPEPTIDES OF A NORMAL AND A MUTANT ORGANISM A wild-type strain of Chlamydomonas reinhardi labeled with 14c is compared to a mutant strain called F-54 labeled with 3H. The wild-type can perform chloroplast membrane-bound functions involved in ATP synthesis. The mutant strain has lost one of these functions and consequently it cannot synthesize ATP (5). It is of interest therefore to compare the composition of the membranes of the wild-type and the mutant organisms. Figure 5 gives the data resulting after electrophoresis. The counts shown here have been corrected for background and crossover. JT '14 The fit regression equation of H onJ C is A= 2.557 J 14c - 2.293 , '-' with standard errors= 3.335. As previously stated, s probably over--estimates the variability because too much weight is given to large deviations. [Figure 5 about here] [Figure 6 about here] Figure 6 is a graph of deviations against slice number (note that the scale of Figure 6 is not the same as Figure 3). Figure 6 does not seem to oscillate between positive and negative values as easily as does Figure 3. More importantly, there are several regions in which large deviations occur. In slices 52 to 55, the deviations are very large and negative (the deviation in slice 53 is about -14.4, or about -4s), indicating a deficiency of 3u and hence a deficiency of specific polypeptide (s)-in the mu.tant relative to the wild-type. Another less obvious region that will need careful consideration by the biologist is around slices 87 to 96. [Figure 7 about here] - 10 -... .. Finally, Figure 7 gives a normal probability plot of the deviations. Unlike Figure 4, the plot is more S-shaped than straight, with several _. points at both ends of the graph well away from lying on a line. This .... ~ .. _, -----... ail .. _, -.... .. is the characteristic shape of normal plots when the deviations do not resemble a sample from a normal distribution. This will be the case in experiments that show differences between the 3H-labeled organism and the 14c-labeled organism . - 11 --... .. 1.·II. REFERENCES 1. Arias, I. M., D. Doyle and R. T. Schimke [1971]. Studies on the synthesis and degradation of proteins of the endoplasmic reticulum of rat liver. J. Biol Chem. 14!!, 3303. 2. Forsythe, A. B. [1972]. Robust estimation of straight line regression coefficients by minimizing p-th power deviations. Technometrics 1~- 159. 3. Hoerl, A. E. and R. W. Kennard [1970]. estimation for nonorthogonal problems. Ridge regression: biased Technometrics ~2, 55. 4. Kobayashi, Y. and D. V. Maudsley, in Bransome, E. D. Jr. (ed.) [1970]. !h~ Current ~~~tus ~f Liquid Sci_ntillation C9un~ing. New York: Grune and Stratton. 5. Sato, V. L., R. P. Levine and J. Neumann [1971]. Photosynthetic phosphorlyation in Chlamydomonas ~einh~rdi: effects of a mutation altering an ATP synthesizing enzyme. ~iochim. ~ip~. t:_~IA J53, 437. 6, Scheff~, H. [1959]. The Analysis of Variance, New York: Wiley • 7. Shapiro, A. L., E. Vinuela and J. V. Maizel Jr. [1967]. Molecular weight estimation of polypetpide chains by electrophoresis in SDS-polyaclamide gels. Biochem. Biophys. Res. Connnun. 28, 815-20. 8. Shapiro, B. M., Antonio G. Siccardi, Y.~ Hirota and J. Fran~ois [1970]. On the process of cellular division in Escherichia col4. -J. ~01. _!!~_i. _52, 75. -12-.... ... .., -... ..., _, ---.. ---.. o. --... -1582 + * * * * + * 1 * * l * / 1 + * * 1 * * + 1/ 1 * * * * H3C?M + * * 1 * I * + * * 1 * 1 1 21 1 * 1 31 1 + 1 4 21 * 1 3 1 * 321 * 1 23 * 2 41 + 64 * 32 * 32 * 4 + •••• + •••• + •••• + •••• + •••• + •••• + •••• + •••• + •••• + •••• + o. C14CPM Figure 1. Graph of 14c cpm (X-axis) versus control experiment. The fitted line is the numbers indicate the number of observations e.g. a 4 indicates 4 points. 642.0 3H cpm (Y-axis) for one least squares line. The graphed at a point; -lat ... '---.... -39.77 + * * * 1 * 1 1 + 1 * * * * + * * * 1 * 1 +. * * 2 * 2 132 1 * 1 3 33 _SQRTH3 + 1 34 2 * 13 1 1 * 1 311 * 31 * 1 1 + 111 321 * 2 22 ·* 1 * 1 * • * * * * + * * o. + •••• + •••• + •••• + •••• + ••••••••• + •••• + •••• + •••• + •••• + o. SQRTC14 Figure 2. Graph of j 14c cpm (X-axis) versus J 3H cpm (Y-axis) for the same data as Figure 1. The fitted line is the least squares line • 25.34 [ [ I I ( 0 .::, .!"~j., .:::, I l I ( I ( I MEMBR9NE PR~TEIN ANALYSIS C~NTROL 1 ' ( ( I ( C.i ~n ::1' ------+-1-+---------------------------,,-----cnr~ z C) -)->,:-~ l .• .... CJ C"J ' ::r+-------.-----,-----r---~-----,r------r-------y---lu. c-c ~(!.(!(' oU. 00 :.30. 0~ 120. c,c SL IC[ -NO. Figure 3. Graph of slice number (X-axis) versus deviation from regression in that slice (Y-axis) for a control experiment. The horizontal lines are drawn for convenience at+ 2s. ( I I -... -.... . ~igure 4. Normal plot of the deviations for a control experiment. The -1 i . points graphed are (D(i)'t ( n+l )), wher~1D(i) is the i-th smallest devia~ion, n is the number of slices, and I is the inverse of the Standard Gaussian distribution function. The graph should approximate a straight line. . (' ... (;,j co . 0 > Li.Jo Da . _JCJ er :::E a: E)C.:, z~ 0 I ~:, L':' c:, :::1' ('J I J C'J MfMBR9Nf PROTfIN ANALYSIS C0~TROL 1 X f I X X X }( X X J X X X X '-;' - ---·---,- -·-,--·----6. tiO --!i. (!C -2.l'O ~: • [! (! ~ . t) C D E v If; T l Llf ~ S 4.C2 G ~:G ,--··-·-.~. ~'L! ._ .... 1-i 1-i --la:I -llal ._ -----... -..., .... Figure 5. Plot of 14c cpm (scale at left) and 3H cpm (scale at right--these counts indicated by x's) for all slices used in the analysis (counts have been corrected for crossover and background). F 54 X WT H~TMEMPReJ LSC=82 8/13/72 0 . 0 . 0 (D -0 0 . 0 :::1' ..... 0 0 . 0 (\J -0 0 . 0 ... 0 o-_.. X 0 .o LO .co CL . u 0 ;:rO c-4C) ' (D u 0 0 . 0 :::14 0 0 . 0 (\I 0 0 ·-4-------.-----,---------------...---------,-----_...c °o.oo 20.00 ll0.00 60. 00 80. 00 100.00 120.00 SLICE NeJ. 0 0 0 c:o 0 0 0 r--0 0 . 0 (0 0 Ocv oD LI) ,.--I ): o>< I:>\ r::r L . OQ_ 0 (~u \fl (r) I __c 0 ,o I I (\) 0 0 . 0 ( ( ( l [ 0 ::J' . ::J' -0 (0 . 0) 0 a:> . :I' U") zo 0~ ..... 0 t- I cc ..... > WO c::,a:> :I' I 0 (0 . en I 0 :I' . :l' ( [ ( ( ( I ( [ [ ( ( F Sij X WT H~TMEMPR~ LSC=B2 8/13/72 _-+-_____________ ...__ _____ -r--------y--------,,..--------~--10.00 20.00 110.00 60.00 80.00 100.00 120.00 ll!0.00 SLICE NCJ. Figure 6. Slice ntnnber· versus deviation from regression. i ( ( SNrJ I 11:H /\30 .... os·21 £CB L8.h ZO. l £8·2- ss·s- 11S ·o l - Oh.hl- 1 I\) . ,r:= ,... X (X) X X I X -X . X CX) X 0) X I -. I\) ,r:= I 0 ~z NG :D .--~ ~ .. •:,-:. :0 ,r 0 ~~ • CJ grr, ~ < . 0 .... . 0) N .... ..... . x~ I\) ,r:= X X X X X ..... . X ex, X 0) X i-, X N . ,r:= (X) i-, cL/£1/8 28=:JSl 08dW3WlrJH lM X nS d !all 

A New Method for the Statistical Analysis of Dual Labeled Isotope Polypeptides Separated by Gel Electrophoresis

https://conservancy.umn.edu/bitstream/handle/11299/199194/Technical%20Report%20202%20A%20New%20Method%20for%20the%20Statistical%20Analysis%20of%20Dual%20Labeled%20Sanford%20Weisberg.pdf?sequence=1&isAllowed=y

A New Method for the Statistical Analysis of Dual Labeled Isotope Polypeptides Separated by Gel Electrophoresis

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