influence. The latter occurs because most of the elements for which effective compounds have not been found are also activators. When these elements are incorporated into a n effective absorber, nearly all of the excitation energy is trapped by the predominant activator. At a dilution factor of 100, the predominant activator ion would constitute up to 1 of the sample, leaving little if any energy to be emitted by activators at ppm or sub-ppm concentration levels.
such as C, N, and the permanent gases, can often be separated from trace impurities without difficulty, leaving the impurities t o be incorporated into more suitable host compounds. Preliminary studies with graphite indicate that such an approach can be successfully developed. Continuing diligent searches for suitable host compounds should extend the list of elements which will support analytically useful rare earth fluorescence. ACKNOWLEDGMENT
SUMMARY
The four groups of hosts described above include a n impressive number of compounds for which activated fluorescence of trace rare earth constituents of potential analytical value have been verified. In this study, compounds of the 49 elements enclosed by boxes in Figure 7 were found to support activated fluorescence. In addition to these, several other elements,
The authors express their sincere thanks to Takeshi Taniguchi and Byron Strom who prepared several of the compounds and obtained the respective spectra.
RECEIVED for review March 19,1970. Accepted June29,1970. Work performed in the Ames Laboratory of the U. S. Energy Commission.
Computer Calculation of Escape Curves of Non-Ideal Solutes in Thin Film Dialysis Kent K. Stewart,’ Lyman C. Craig, and Robley C . Williams, Jr.2 The Rockefeller Unioersity, New York, N . Y . 10021 Computer simulation has been used to study the behavior of non-ideal solutes in analytical thin film dialysis. The system of interconvertible dialyzing monomers was found to give either a straight line or an upward curving escape plot. Heterogeneity and widely increasing sampling time intervals also can potentially result in escape curves with upward curvature. Concentration dependent aggregation is characterized by an escape curve with a downward curvature. Potentials and limitations of thin film analytical dialysis are discussed.
EXTENSIVE STUDIES (1-7) from this laboratory have demonstrated that thin film dialysis can be performed jn such a way that it is highly sensitive to small differences in the diffusional size of a wide variety of biochemical molecules in solution. Data have been presented (3)to indicate the limit of differences detectable by the method as being in the range of 2-3% of Stokes radii. This estimate was based on careful studies with Schardinger dextrins as models since they are rigid solutes whose size and shape have been well documented by X-ray diffraction studies. A selectivity of this order was achieved 1 Present address, The Human Nutrition Research Division, Agricultural Research Service, United States Department of Agriculture, Beltsville, Md. 20705 Present address, Department of Biology, Yale University, New Haven, Conn.
(1) L. C. Craig, T. P. King, and A. Stracher, J. Amer. Chem. SOC., 79, 3729 (1957). (2) L. C. Craig and A. Ansevin, Biochemistry, 2, 1268 (1963). (3) L. C. Craig and A. 0. Pulley, ibid., 1, 89 (1962). (4) L. C. Craig, E. J. Harfenist, and A. C. Paladini, ibid., 3, 764
(1964).
(5) L. C. Craig, J. P. Fisher, and T. P. King, ibid., 4, 311 (1965). (6) L. C. Craig and E. J. Harfenist, “Peptides,” Proceedings of the
Sixth European Symposium, Athens, Sept. 1963, Pergamon Press, New York, 1965, p 373. (7) L. C. Craig, Science, 144, 1093 (1964). 1252
only when the pore size of the membrane was adjusted to a critical range for the particular molecular sizes to be studied. Methods for adjusting the pore size to the desired range in order to accommodate molecular sizes varying from those of amino acids to proteins of over 100,000 in molecular weight have been described (8). The basis of the selectivity seems to be due largely to the probability of a molecule of the solute finding its way into pores which are only slightly larger than the effective diameter of the solute. The effective diffusional diameters of the solutes are apparently determined by the longest cross section of the solute because of tumbling. It is known from nuclear magnetic resonance data (9) that the solutes of sizes of interest here are tumbling at a rate of approximately lo7 to 1011 times per second in aqueous solution. Thus analytical dialysis rates will be influenced by any condition which will alter the long cross section of the solute. Two such influences are the alteration of conformation and firm solvation. For many different reasons it is becoming increasingly important to learn as much as possible about the shape or conformation of large biochemical molecules in solution and their interaction or association with other molecules, particularly in dilute solution. Many of the techniques available for such study require concentrations above those of the greatest interest. Analytical thin film dialysis ( I O , 11)has been shown to be useful for studying binding and association phenomena (8) L. C. Craig and William Konigsberg, J . Phys. Chem., 65,
166 (1961). (9) A. Carrington and A. D. McLachlan, “Introduction to Magnetic Resonance,” Harper and Row, New York, 1967, p 187. (10) L. C. Craig, H. C. Chen, and E. J. Harfenist, in “Progress in Separation and Purification,” Vol. 2, The0 Gerritsen, Ed., Interscience, New York, N. Y., 1969, p 219. (11) L. C. Craig, H. C. Chen, and W. I. Taylor, J. MacromoL Sei.Chem., A 3, 133 (1969).
ANALYTICAL CHEMISTRY, VOL. 42, NO. 11, SEPTEMBER 1970
inside A
inside
B
inside
B
A
membrane
membrane
k,
'
B (n-mer of A )
membrane \
Figure 1. Diagram of diffusing solutes. Concentration of solute A inside the membrane is A and the concentration outside the membrane is A ' . Similarly the inside concentration of solute B is B, and the concentration outside is B'. The first order diffusion rates of solutes A and B are a and (?, respectively A. Diffusion of two ideal solutes through a membrane. The dialysis of each solute is first order B. Diffusion of two forms of one solute. The dialysis rates of the two forms are different although first order. The overall escape pattern is complicated by the A to B interconversion C. Diffusion of an aggregating solute. Solute B (an aggregate of A) does not dialyze, but can be converted to A
a t concentrations of lo--*to 10-4M. These are lower concentrations than those of most binding studies, and we believe that molecular interactions could be studied with this technique at even greater dilutions with radioactive tracers. The technique is kinetic in nature and the data are presented as solute escape curves. The shapes and the slopes of these escape curves are the principal pieces of information used t o characterize the solutes under study. Often it is obvious whether or not the particular preparation of solute behaves ideally as a single pure solute or as a mixture of molecular sizes. The use of dialysis as a probe of heterogeneity has also been discussed by other workers (12, 13). Hoch and Miller ( 1 2 ) have used a computer for analyzing the dialysis data of mixtures. Our laboratory has been interested also in the dialysis behavior of a wide variety of pure solutes (7, 14) and it has been found that many solutes can be characterized by their dialysis behavior in difTerent solvent environments in a very informative way. However, frequently self association
and/or interconversion occurs and these phenomena lead to more complex escape patterns (5,10,14-16). In trying to attain a better basis for the interpretation of thin film dialysis escape patterns, we have explored the use of a Control Data Corporation 160 G computer to calculate simulated escape patterns of ideal and non-ideal solutes. This work has included a study of the behavior of aggregating systems, interconvertible, dialyzing monomers, and heterogeneous systems. EXPERIMENTAL All solvents were distilled before use and all other chemicals were of reagent grade. The "C-urea was obtained from New England Nuclear. All dialyses of the I4C-urea were done at 40 "C in 0.1M potassium chloride, p H 5.33 with a heavily acetylated membrane, The standard method of this lab(14) L. C. Craig, in "Methods in Enzymology," Vol. XI, C. H. W. Hirs, Ed., Academic Press, New York, 1967, p 870. (15) G. Guidotti and L. C. Craig, Proc. Narl. Acad. Sei.,50, 46 (1963). (16) M. A. Ruttenberg, T. P. King, and L. C. Craig, Biochemistry, 5, 2857 (1966).
(12) H. Hoch and P. 0. Miller, ANAL.CHEM., 38, 658 (1966). (13) S. Siggia, J. G. Hanna, and N. M. Serecha, ibid., 35, 365 (1963); 36, 638 (1964).
+
Figure 2. Flow diagram of the Fortran computer program used to calculate solute dialysis escape curves
Calculate:
Amount Amount Amount Amount Amount Amount Amount Amount
A dialyzed B dialyzed A ' dialyzed B' dialyzed A converted to B B converted to A A ' converted to B' B' converted to A '
ANALYTICAL CHEMISTRY, VOL. 42, NO. 11, SEPTEMBER 1970
1253
\\ B
20
,
I
,
40
60
80
C
,
IO0
I
,
120
20
40
60
80
IO0
I20
Figure 3. A . Calculated curves of three ideal solutes. Curve I is the escape curve of a solute with an (Y equal to 0.10, in Curve I1 CY equals 0.05, in Curve I11 a equals 0.01 B. Calculated escape curve for an equal molar mixture of solutes I and I11 C. Calculated escape curve for an equal molar mixture of solutes I, 11, and I11
tube is directly proportional to the inside concentration (C,) of the solute (Equation 1).
'"Am
c
C 0,
V
L 0,
a
I1
0
I
I
20
I
I
1
40
1
60
I
I
I
80
(1)
-dCo/dt = kCo
(2)
The back diffusion is directly proportional to the outside concentration (C,) of the solute (Equation 2). These two equations allow us to calculate the solute concentration inside and outside the membrane at any time. Experimentally in thin film dialysis, the back diffusion problem has been minimized by using a tenfold larger outside volume than inside volume, and by periodically replacing the outside solution with fresh solvent. This treatment results in a close approximation to a linear plot of log per crnt remaining vs. time as previously reported (1). Mathematically the back diffusion problem can be handled either by Vink's method (18) or by doing a series of calculations using a At such that Equations 1 and 2 may be treated independently and recomputing Cr and Co after each calculation. We have used the latter method with the CDC 160 G computer for the calculations. Three types of systems have been examined: ideal mixtures (Figure 1A ) , interconvertible dialyzing monomers (Figure l B ) , and aggregating systems (Figure 1C). Escape curves for each system have been calculated. The flow sheet for the computer program is shown in Figure 2.
Time (min)
Figure 4. Calculated escape curves of interconvertible dialyzable monomers where A and B are in rapid equilibrium. The dialysis rate of A is two-thirds that of B. Each line is labeled with the equilibrium constant of the A to B conversion
-dCr/dt = kCi
RESULTS
As Craig et al. (I) and others (12) have shown, the rate of dialysis of ideal solutes is a standard first order kinetic process. Thus the rate of forward diffusion of a solute inside a dialysis
The behavior of ideal solutes (Figure 1A ) was calculated to check the programming. The expected escape curves were generated and are shown in Figure 3. We next calculated a variety of the escape curves of the more interesting non-ideal solutes. The escape curves of two interconvertible dialyzable monomers (Figure 1B) were examined in three different situations: kl and kz much larger than, approximately equal to, and, finally, much less than a and 8. When ki and k z were much larger than a and p , a family of straight lines was generated (Figure 4). The slope of the line was a function of the equilibrium constant for the A to B conversion. The same result
(17) G.A. Bray, Ana/. Biochem., 1,279 (1960).
(18) H. Vink, Ark. Kemi, 19, 531 (1962).
oratory (8, 14) was used for dialysis with a few minor modifications. The I4C-urea concentrations were determined by liquid scintillation counting using Bray's solution (17). THEORY AND CALCULATIONS
1254
ANALYTICAL CHEMISTRY, VOL. 42, NO. 11, SEPTEMBER 1970
Table I. Dialysis of Interconvertible Dialyzable Monomers Where CY and 0 Are of the Same Order of Magnitude as kland kz Apparent Keq. ki kz a P straight line? 0.02 0.20 0.20 Yes 0.1 0.02 0.055 0.0110 0.0550 Yes 1.o 0.055 0.055 yeso 0.11 0 036 5.0 0.55 0.055 yeso 0.011 0.036 5.0 0.055 0.055 yesa 0.036 0.0011 5.0 0,0055 0.055 no (curve up) 0.0110 0.0550 0.2 0.0110 I
0
The slope of the escape curve was the same in each of the plots.
11 0
I
I
I
I
I
I
I
IO
20
30
40
50
60
70
Time (min) Figure 5. Calculated escape curves of interconvertible dialyzable monomers when A and B are rapidly dialyzing and slowly undergoing interconversion. Solute B dialyzes ten times faster than solute A. In curve A , the equilibrium constant is equal to 0.1 and in curve B, the equilibrium constant is equal to 10.0
was also observed for most of the situations where kl and k2 were approximately equal to a and p as is shown in Table I. Finally, if kl and k2were much smaller than cy and 0,then the upward curvature seen in Figure 5 was observed. The system in which a dialyzable monomer is in equilibrium with a nondialyzable aggregate (Figure IC)has a characteristic downward curvature in its escape curve as is shown in Figure 6. Examination of Figure 6 and a number of other escape curves of aggregating systems has shown that the downward curvature is not always obvious in the escape curve of an aggregating system, but is observable when the appropriate combination of total concentration, equilibrium constant, and concentration range covered is present (5, 16). An assumption in the experimental analytical dialysis is that the effect of back diffusion may be ignored if the outside volume is ten times greater than the inside volume and the outside solutions are replaced with sufficient frequency. Since
0 .I
I
100
I
I
I
200
300
400
500
600
Time ( m i n ) M o n o m e r ; Monomer-Dimer ( x I 0 - 2 Figure 6. Calculated escape curves of an ideal monomer and a concentration dependent aggregating system whose monomer has the same rate constant as the ideal monomer. The dimer does not dialyze Initial concentration was 0.1M; the equilibrium constant is 5 X 10'; kl is 0.055 min-l and the monomer-dimer equilibrium is assumed to be instantaneous. Under these conditions there is 99.9% dimer at the beginning and 99.9% monomer when 99.9% of the solute has dialyzed out. The time scale for the aggregating system escape curve has been multiplied by to permit comparison of the escape curves
the computer program was available, the consequences of this assumption were examined. The calculations suggested that a systematic increase in the sampling time could result in a significant upward curvature of the escape curve (Figure 7) when back diffusion is allowed to become significant. Back diffusion becomes significant when the sampling time interval approaches 60% of the half escape time of the solute. We were also able to demonstrate this effect experimentally. Figure 8 shows the escape curves observed for I4C-urea when the sampling time interval was fixed at 30 minutes or started at 2 minutes and doubled after each sampling. The 30-minUte interval is approximately equal to a half escape time.
ANALYTICAL CHEMISTRY, VOL. 42, NO. 11, SEPTEMBER 1970
1255
' 70O
O
'OOL 70
k
50
30
.-cC .-
E
L
c
C W V
IO
L
a"
I
0
20
40
60
I
80
Time (min) Figure 7. Effect of sampling time intervals on escape curves Calculated escape curves with a fixed sampling time interval not greater than 60 of a half escape time V--0 b. Sampling time interval which is increased by a fixed amount at each sampling, 0-0,to above 60z of a half escape time c. Sampling time interval doubled at each sampling to well beyond 60 of a half escape time 0 -0 . Inside volume is 0.1 that of the outside a.
z
DISCUSSION
The results of the computer calculations give us further insight into a n understanding of the range of usefulness of thin film analytical dialysis. If the sampling time interval is held to a time not greater than half the half escape time, we can make the following generalizations: If the escape curve curves upward, then the solute is probably either heterogeneous or is undergoing a slow equilibrium between two or more dialyzable forms. If the escape curve curves downward, the solute is probably undergoing concentration dependent disaggregation. If the escape curve is linear, then the sample is probably homogeneous with respect to size. A linear escape curve cannot, however, by itself always be used to determine whether the solute is monodisperse and ideal, aggregated over the concentration studied, or participating in a monomer-monomer equilibrium where both monomers are dialyzable. It is useful to examine the three types of escape curve in more detail. Three causes for the upward curvature of the dialysis escape curves have been indicated : heterogeneity, slow equilibrium between monomers differing significantly in diffusional size, and a n improper choice of the sampling time interval so that back diffusion becomes significant, i.e., more than half of a half escape time. The latter ordinarily becomes significant only when two or more half escape times are covered in the same plot. It results from both the back diffusion and the changing sampling time interval when the sampling time is too large. 1256
I
60
I
120
I
180
I
I
240
300
Time (min) Figure 8. Effect of sampling time interval on the escape curves of 14C-urea Experimental curves with fixed sampling time intervals of thirty minutes A-A and A-A. Experimental curves with the sampling time interval doubled after each sampling and 0-0,0 - 0 . The half escape time approximates 30 min
The slow monomer-monomer interconversion can be rather easily distinguished from heterogeneous solutes. Experimentally all that is required is that the rapidly dialyzing material and the slowly dialyzing material be recovered separately, held to allow equilibrium to take place, and then a new escape curve determined for each sample. If the system is heterogeneous, then the fast dialyzing sample will still be fast and the slowly dialyzing sample will still be slow, but if a slow equilibrium is occurring then the rapidly and the slowly dialyzing samples should have similar if not identical escape curves. Aggregating systems are the only systems examined to our knowledge which show downward curvature. Thus we reaffirm our belief (5, 16) that the presence of a downward curvature is a reliable indication of aggregation. The absence of a downward curve in the escape plot is not evidence for the absence of an aggregating system. Since the curvature is a function of the equilibrium constant, the diffusion rate and the time span of the experiment, there are a number of situations which could yield linear escape curves. Still, if the minimum molecular weight of the solute is known, dialysis studies of model systems are made, and the solute escape curves are determined for starting concentrations over a ten- to a hundredfold range, one should be able to obtain a considerable amount of information on the aggregation behavior of the system.
ANALYTICAL CHEMISTRY, VOL. 42, NO. 11, SEPTEMBE:R 1970
The linearityof the escapecurves of the monomer-monomer systems has the practical result that these systems will often be overlooked experimentally. However, if other data indicate that there is a system of this type, the linearity of the escape curves can be quite useful. The equilibrium system of monomer-monomer interconversion of Figure 1B can easily be shown to have a first order escape curve for the solute described by Equation 3. -dC dt
-=[
a+KeqP
l+Ke,]
(3)
The diffusion rates of solutes A and B are a and 0,respectively, C is the total inside solute concentration and KBpis the equilibrium constant of the A to B conversion. Since the escape curves are straight lines even when a and /3 are in the same range as kl and kz, all these systems can be treated by the approximation that A and B are in rapid equilibrium. That is, any time a monomer-monomer system gives a straight escape curve, the system may be treated as an equilibrium system.
Thus if a and /3 can be determined, the equilibrium constant can be calculated. Likewise, if the equilibrium constant can be determined, then a and /3 can be calculated in terms of each other. If neither a, 0,nor the equilibrium constant can be determined, then the method is more limited. Although we may be able to get a series of linear escape curves under a series of conditions, it becomes quite difficult, if not impossible, to determine whether we have only one species for each concentration or two states with fixed diffusion rates but a different equilibrium constant for each condition, or whether we have multistates each with its own diffusion constant and equilibrium constant. ACKNOWLEDGMENT
The skillful technical assistance of Mrs. Helen Kac is gratefully acknowledged. RECEIVED for review May 4, 1970. Accepted July 2, 1970. Investigation supported in part by a grant from the National Institutes of Health No. AM 02493.
I
NOTES
Thin Film Dialysis Studies with Highly Acetylated Cellophane Membranes Kent K. Stewart’ and Lyman C. Craig The Rockefeller University, New Y o r k , N . Y . 10021
INPREVIOUS REPORTS from this laboratory (1-5) we have shown that standard Cellophane dialysis casing can be modified easily so that it has the range of pore sizes optimum for studying the dialysis of organic compounds of any specific molecular size from a molecular weight of 100 to that of 135,000. Pore sizes can be altered in a useful way by mechanical stretching while the membrane is wet (I). They can be made much larger by treatment with zinc chloride or made much smaller by a simple acetylation procedure. In connection with certain projected investigations dealing with the binding of inorganic solutes to proteins and nucleotides, it seemed of interest to explore further the behavior of very highly acetylated cellophane casing. It has been demonstrated that Visking cellophane tubing carries very little if any 1 Present address, The Human Nutrition Research Division, Agricultural Research Service, United States Department of Agriculture, Beltsville, Md. 20705
(1) L. C. Craig and William Konigsberg, J. Phys. Chem., 65, 166 (1961). (2) L. C. Craig and A. 0. Pulley, Biochemistry, 1,89 (1962). ( 3 ) L. C. Craig and A. Ansevin, ibid., 2, 1268 (1963). (4) G. Guidotti and L. C. Craig, Proc. Natl. Acad. Sci. U.S., 50, 46 (1963). ( 5 ) L. C. Craig, William Konigsberg, A. Stracher, and T. P. King, in “Symposium on Protein Structure,” A Neuberger, Ed., IUPAC Symposium, July 1957, John Wiley and Sons, Inc., New York, N. Y., 1958, p 104.
fixed charge (3) and that diffusional size can be the major parameter determining whether or not dialysis will occur, It therefore seemed of interest to explore if this would necessarily be true for highly acetylated membranes showing high selectivity for inorganic solutes. Although the charge on the molecule with the larger organic solutes with ionizable groups seemed to have little or no direct effect on dialyzability, a different behavior might be expected in the case of the inorganic solutes with high charge density. Selective, low porosity membranes have been prepared and the diffusion of a number of solutes through these membranes has been measured. In this paper we present the results of our studies on the effect of molecular size and change of temperature on the diffusion rates of some low molecular weight solutes through these membranes. EXPERIMENTAL.
Materials. All solvents were distilled before use, and all chemicals were reagent grade. The isotopically labeled compounds were purchased from New England Nuclear. The dialysis membranes were purchased from the Food Products Division of the Union Carbide Corporation. Membranes. The dialysis casing (Visking seamless cellulose No. 23) was stretched almost to breaking with the stretching trough described earlier (1) and then acetylated. We have used 25% acetic anhydride and 75% dry pyridine a t temperatures of 80 to 90 O C for periods of 1 to 4 hours for the acetylation. The membranes appeared to melt against the
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