Dietmar Tietz

Andreas Chrambach

Section on Macromolecular Analysis

Laboratory of Theoretical and Physical Biology

National Institute of Child Health and Human Development

National Institutes of Health

Bethesda, Maryland, USA

- Abbreviations
- Summary
- Introduction
- Hardware and software requirements
- Design of program ElphoFit
- Computation of nonlinear Ferguson plots
- Operation of program ElphoFit
- Applications
- Discussion
- Acknowledgments
- References
- Note added in proof
- Fig.1
- Fig.2
- Fig.3
- Fig.4
- Fig.5
- Fig.6
- Table 1
- Table 2
- Table 3

**Abbreviations:**

**BF**, Fitted parameter of Eq.5; **bp**, base pairs; **Ferguson plot:** plot of log10µ vs. %T; **KR**, retardation coefficient, negative value of the slope of the linear Ferguson plot; **l'**, specific fiber length (cm/g dry gel matrix material); **mobility**, migration velocity (cm/s) divided by field strength (V/cm); **MF, MP**, fitted parameters of Eqs.5 and 3; **µ**, absolute mobility (cm^2 V^-1 s^-1); **µo**, µ at %T = 0; **PAR1, PAR2**, fitted parameters of Eq.3; **R**, particle radius (nm); **r**, fiber radius (nm); **%T**, concentration of the sieving medium in % (g/100 mL); **TBE**, 89 mM Tris, 89 mM boric acid, 2.5 mM Na2-EDTA; **VF**, specific fiber volume (mL/g dry gel matrix material)

**Summary**

A desktop computer program evaluating physical properties of DNA and bacteriophages is presented. The analysis is based on data obtained from capillary and submarine-type agarose electrophoresis. Native molecular/particle properties and properties of the gel (or polymer) medium can be derived from electrophoresis at several gel concentrations. This is done conveniently by a computerized evaluation of the semi-logarithmic plot of mobility vs. gel concentration, designated the Ferguson plot. In application to most proteins, this plot is linear and computer programs exist to evaluate it. However, nonlinear Ferguson plots have assumed great importance in view of the fact that the plots are concave for DNA. Similarly, convex plots are important since they prevail in the electrophoresis of large particles in agarose. The computer program reported here is the first to (i) address concave Ferguson plots and (ii) allow for the evaluation of both cases using a desktop computer. Program ElphoFit version 2.0, a Macintosh application, is available upon request.

**1 Introduction**

The semi-logarithmic plot of mobility vs. gel concentration (Ferguson plot) can be used to determine particle size and free mobility (related to surface net charge density) as well as parameters which are descriptive of the gel fiber, i.e., fiber radius, volume and length. User-friendly computer programs for the analysis of linear Ferguson plots based on the extended Ogston model (1,2,3) were developed by D. Rodbard two decades ago for application on IBM mainframe computers [the "PAGE-PACK", Fig.8 of(4)]. More recently, a software package for application on the IBM or Macintosh desktop computers has been published [program ElphoFit by D. Tietz (5)].

However, Ferguson plots are not necessarily linear. Those of DNA fragments in crosslinked polyacrylamide or in agarose gels are concave (6) or, when composite gels are used, sigmoidal (7). Analysis of such curves has been previously carried out either by limitation to the initial linear concentration range of the plot (8,9,10) or by empirical curve fitting (11,12,13). Purely empirical approaches, however, are unable to evaluate the plots with regard to molecular and fiber properties.

Similarly, Ferguson plots of spherical particles (proteins, viruses) can be convex both in polyacrylamide (14,15) and in agarose gels (16,17). Convex plots have been analyzed as linear ones by limitation to the initial linear range (16), by semi-empirical curve fitting linked to the extended Ogston model (18) and direct application of the Ogston model (19), using mainframe computers. One of the underlying rationales for modeling nonlinear Ferguson plots (19) was to replace the retardation coefficient, KR(R), a function of particle radius, in the Ferguson plot equation by a retardation coefficient which is also a function of gel concentration, KR(R,T) (Eq.1). Specifically for agarose gels, the working hypothesis was raised that the radius of the agarose supercoiled fiber diminishes and the total fiber length becomes larger with increasing gel concentration to approach the dimensions of the single-stranded agarose double-helix. This hypothesis is supported by independent physical evidence (20).

The principle of a variable retardation coefficient is also applied in the present study both to the previously analyzed convex plots and to the concave one. It is the aim of this study (i) to make the modeling of nonlinear Ferguson plots based on a physical model available to a wider audience by developing scientific software for desktop computers and (ii) to develop a model for the evaluation of concave Ferguson plots derived from electrophoresis of DNA fragments.

**2 Hardware and software requirements**

A Macintosh desktop computer (Apple Computer, Cupertino CA 95014) with an application memory size of 900 K is required. A 13" RGB high resolution color monitor and a LaserWriter IINT printer (both from Apple Computer) were used but are not obligatory.

Software is an extended version of program ElphoFit (5). Only the elements of the extension to concave and convex Ferguson plots are reported below.

The compiled and linked program ElphoFit version 2.0 is a 276 K application. The general structure of the program [Figs.3 and 4 of (5)] has been modified as follows: Program elements designated "Curve Fitting" have been extended to concave and convex Ferguson plots. Other program elements have been adjusted accordingly. The program element designated "Graphics" has been changed so as to display the graphic output in color and increased size. This change was necessitated by the need to distinguish between experimental data points and predicted curves of several particles.

**4 Computation of ****Ferguson plots**

**4.1 Concave Ferguson plots**

If the Ferguson plot is concave, KR(R) becomes a decreasing function of gel concentration T.

**[equ.1]** logµ(R,T) = logµo - KR(R,T)*T

The functions relating KR and T are given by Eqs.2 and 3.

**[equ.2]** KR(R,T) = 0.01*VF*(1 + PR(R,T)/r)^2

**[equ.3]** PR(R,T) = R * {(1+(T/MP)^[PAR1*EXP(-PAR2*R)]}^-1

PR(R,T), a function relating R and T (Eq.3), describes a sigmoidal decrease of effective particle radius with gel concentration (see Section 7.2), a relationship which has been previously suggested (3). This type of sigmoidal equation is a modification of a function previously applied to the relation between molecular weight and mobility of SDS-proteins [program SIGMOID, Table V of (21)].

Parameters of Eqs.1 to 3 are defined as follows: µ = particle mobility (cm^2 V^1 s^1); µo = µ extrapolated to T = 0; KR is the retardation coefficient, the slope of the line connecting a point on the Ferguson plot with µo (19); VF is the total fiber volume (mL/g of dry polymer fiber) (18); R is the particle radius (nm) and r the fiber radius (nm) (2); MP has the dimension of T and determines the point of inflection of the sigmoidal relationship (19); PAR1 and PAR2 determine the slope around the point of inflection (19). The exponent of T/MP is related to particle radius R. This allows one by using only three parameters (MP, PAR1, PAR2) to simultaneously model the nonlinear decrease of all effective particle radii (10 values in Section 6.1) as a function of increasing polymer concentration. The assumption of shrinking effective particle radii (see Section 7.2) is in accordance with the observed increase of the contour length of DNA with increasing agarose concentration [Table 2 of (8)].

**4.2 Convex Ferguson plots**

The modeling of the convex Ferguson is based on the hypothesis that the effective fiber radius, r, decreases with increasing concentration of polymer fibers (18,19,22). This hypothesis is embedded in Eqs.4 and 5

**[equ.4]** KR(R,T) = 0.01*VF*(1 + R/Fr(T))^2

**[equ.5]** Fr(T) = r * [(1+(T/MF)^BF]^-1

Fr(T), a function relating r and T (Eq.5), describes a sigmoidal decrease of fiber radius with increasing T. MF (with the dimension of T) determines the point of inflection, and BF the slope around that point.

**5 Operation of program ****ElphoFit**

5.1 Data files: The input of experimental data, consisting of particle radii, measured mobilities and gel concentrations, is entered in the format of Fig.5B of (5). Further instruction is provided by program ElphoFit [File Handler (option 2) of the Main Menu, Demo Data Files of the Sub Menu (item 2), see Fig.3 of (5)]

5.2 Fitting linear Ferguson plots: The procedure starts with the simplest approach which is fitting of linear plots. The user-interactive operation of the program follows the sequence given in Section 4.1, (ii) to (viii), of (5).

5.3 Assumption of a common free mobility: This option may be selected, if the results of Section 5.2 suggest that the Ferguson plots under analysis share a common free mobility intercept, or if one wishes to assume a common µo value on the basis of literature data [e.g. (23)]. Under this assumption, it is possible to repeat the evaluation of linear Ferguson plots (Section 5.2) or to proceed to nonlinear fitting (Section 5.4).

5.4 Nonlinear Ferguson plots: The curve fitting algorithm uses Eqs. 1 to 5 rather than polynomial functions (11,12). Parameters representing physical properties of the particle and the polymer fiber are fitted instead of coefficients which convey no physical meaning. Rather than considering each curve separately all of the plots are evaluated at the same time using a simultaneous curve fitting strategy [Fig.5 of (2)]. Its principle is that all the plots share fiber-specific parameters (radius, volume), whereas particle-specific parameters (radius, free mobility) are unique for each plot. The procedure requires the following input: (i) An interactive response to the question whether to consider the particle radius, the fiber radius or both as functions of gel (polymer) concentration. (ii) Initial estimates of parameters and setting of constraints (upper values) may be automatic or provided by the user. Parameter values are always constrained to be larger than zero. The automatic choice of initial parameter estimates and constraints is based on the linear fit. It assumes that computed parameters derived from fitting of linear Ferguson plots to nonlinear curves may not be accurate, but that values are at least of the right order of magnitude. If not, the automatic choice may produce a misfit. It is important to perform curve fitting with different initial estimates, since several solutions may exist when nonlinear Ferguson plots are evaluated. The curve fitting algorithm may arrive at a local, but not a global minimum of sum of squares. It is also essential to check whether the results may depend on only a few data points (perhaps outliers). For that purpose, a data file may be prepared which, e.g., contains only every other data point. Setting of constraints is recommended in order to prevent a shift of parameter values into unrealistic ranges at the beginning of the curve fitting procedure. (iii) No matter how (ii) was decided, the computer program offers the possibility of interfering while curve fitting is in progress: Pressing key L or l removes the upper limit of the constraints; pressing key S or s terminates the curve fit after conclusion of the current iteration. (iv) If a common µo intercept has not been chosen previously, the fit of nonlinear Ferguson plots may be repeated using this option.

**Fig.1:** Concave Ferguson plots of DNA fragments in agarose
solutions, 40 oC, TBE buffer, using capillary electrophoresis:
Data of (24). Agarose: SeaPlaque GTG (FMC). phiX174 RF
DNA/Hae III fragments (GIBCO-BRL); component numbers are defined
by DNA length (bp) and geometric mean radius (nm) as follows: 1)
72, 3.14; 2) 118, 3.7; 3) 194, 4.37; 4) 234, 4.65; 5) 271 and
281, 4.92; 6) 310, 5.11; 7) 603, 6.38; 8) 872, 7.21; 9) 1078,
7.74; 10) 1353, 8.35. The concave plots were modeled by Eqs.1 to
3, using the assumption that the effective radii of DNA fragments
diminish as the gel concentration increases (see Fig.2). Fiber
parameters are considered constant. Dependency values indicate
the quality of the model. Values usually range from 0 (no
dependency, parameter is of no importance) to 1 (absolute
dependency, variation of a parameter value can be compensated for
by adjusting (an)other parameter(s)). Dependency values >=
0.9999 indicate unreliable results. Descriptive output of
ElphoFit version 2.0 in the format provided by the computer is
shown.

**6 Applications**

**6.1 Concave ****Ferguson plots of DNA fragments**

Mobility values of DNA fragments ranging in length from 72 to 1,353 bp obtained by capillary electrophoresis in agarose solution (SeaPlaque GTG, FMC, Rockland ME) above its gelling temperature (40 oC) were used (24). Agarose concentrations ranged from 0.3 to 2.6% in TBE buffer (24). For application to Eqs.1 to 3, base pair (bp) values were converted to geometric mean radii, R, according to Eq.10 of (8); the program ElphoFit performs that calculation [Particle Radius etc (option 5), Main Menu]. The concave Ferguson plots of the 10 DNA fragments resulting from that input are shown in Fig.1 together with the numerical output describing the results of curve fitting. The assumption of no common free mobility intercept is in agreement with the observation of two peaks for preparation phiX174 during capillary electrophoresis in buffer in the absence of agarose (Dr. P. Bocek, personal communication). Effective mean pore radii and particle radii calculated by program ElphoFit are plotted as a function of agarose concentration in Fig.2. The figure shows diminishing effective radii of DNA fragments as the concentration of the sieving medium increases. The decrease is sharper for larger fragments than for smaller ones. Effective pore radii of the sieving matrix decrease as the gel concentration increases and approach the medium range of molecular radii of the particles under investigation.

**Fig.2:** Comparison of effective pore and particle radii
which are derived from the concave Ferguson plots of Fig.1. Data
were computed by program ElphoFit version 2.0 using Eqs.1 to 3;
no common free mobility intercept is assumed. The dotted lines
refer to the DNA fragments 1 (bottom) to 10 (top) specified in
the legend of Fig.1. The order in which they are depicted is
necessarily the opposite of that in Fig.1, since DNA size and
migration rate are inversely related. It should be noted that
the sizes of pores in a fiber network is not constant (comparable
to a grid), but follows a Poisson distribution. Therefore,
fragments 4 to 10 are able to migrate and to access a
considerable portion of the spacings [Fig.17 of (3)], although
their effective radius exceeds the mean pore radius at higher gel
concentrations.

When a common free mobility of all DNA species is assumed [in accordance with (23)], the concave curve fit of the data is nearly as good as that derived without the assumption - with a determination coefficient of 0.997 instead of one of 0.998, where values usually range from 0 (very poor) to 1.0 (perfect fit). Correspondingly, the fiber parameters derived from the fit are similar ( click here to see Fig.3). Results indicate an agarose fiber with dimensions more closely related to a single stranded double helix (radius of 1 to 2 nm, total volume of about 1 mL/g) than a supercoiled fiber [radius of 20 to 30 nm, volume of 5 to 8 mL/g based on log10(mobility)] (16,19). The obtained fiber radii are comparable with agarose fiber radii of 2.8 to 4.4 nm for the uncoated fiber obtained from freeze-fracture electron microscopy using a similar buffer system (25).

Program ElphoFit can also be applied to determine the size of unknown DNA fragments. For this purpose, component 2 with a geometric mean radius of 3.7 nm (118 bp) and component 5 with a geometric mean radius of 4.92 nm (271 and 281 bp) were treated as unknowns. Using the nonlinear curve fitting algorithm, the sizes and free mobilities of the fragments were determined as shown in Table 1 (click here to see this table). Confidence ellipses for component 2 are shown in Fig.4 (click here to see the figure).

**Fig.5:** Convex Ferguson plots derived from electrophoresis
of bacteriophages in agarose gels. A one-dimensional
submarine-type electrophoresis apparatus (37) and phosphate
buffer, 0.03 M ionic strength, pH 7.4, 25 oC, was used (16). Data
are those provided for (19) by Dr. P. Serwer (University of
Texas Health Science Center, San Antonio, TX). Virus species and
particle radii (nm) determined by low-angle X-ray scattering are
given in parentheses: 1) R17 (13.3), 2) glufix T7CI (26.1), 3)
9-P22 (31.4), 4) T3 (30.1), and 5) T5 Hd (41.9). The convex
plots were modeled by Eqs.1, 4 and 5, assuming variable gel fiber
parameters (click here to see Table 2) and constant particle sizes. Sample
output of program ElphoFit version 2.0 is shown. Dependency is
explained in the legend of Fig.1.

**6.2 Convex Ferguson plot of bacteriophages**

Agarose gel electrophoretic mobility values of five spherical bacteriophages of Dr. P. Serwer (University of Texas Health Science Center, San Antonio, TX), previously applied to a Ferguson plot analysis on mainframe computer (19), were used to construct and evaluate convex Ferguson plots by program ElphoFit version 2.0. Graphic and numerical computer output is shown in Fig.5. Convex Ferguson plots of bacteriophages can be modeled by considering the fiber radius as a decreasing function of gel concentration. Since the fiber volume is constant, the total fiber length has to increase necessarily (click here to see Table 2). The estimated fiber dimensions indicate a supercoiled agarose fiber (16,19). Fiber dimensions vary from the ones reported in Section 6.1, since (i) agarose gels instead of agarose solutions and (ii) phosphate buffer instead of TBE buffer have been used.

The mathematical approach is different from the previously reported one (19), since the apparent fiber dimensions are not considered as a function of the size of the sieved particle. The simplification is possible, because the particles investigated in this study have sizes within one order of magnitude. This is in contrast to the model of (19) which also considered the sieving of the much smaller proteins.

The model used for analyzing the phages in Fig.5 is also applicable to the characterization of unknown viruses. For that purpose, phages 3 and 4 were treated as unknowns, while the other three particles were considered as size standards. Results are shown in Table 3 (click here to see the table). The radii determined by X-ray scattering and the geometric mean radii predicted by analysis of convex Ferguson plots (Eqs.1, 4 and 5) are within 1.7 and 3.4 % of variation for phages T3 and 9-P22, respectively. It should be noted that the conventional size determination of phages 3 and 4 using only a single gel concentration would lead to erroneous results: Since the smaller phage T3 has a much larger free mobility and, therefore, migrates considerably faster than phage 9-P22, their sizes would be estimated in reverse order. Treatment with sodium dodecyl sulfate to achieve a similar surface net charge density is inapplicable, because it would cause a degradation of the viruses into their subunits.

**7 Discussion**

**7.1 Advances made in this study**

The usefulness of conducting electrophoresis at several gel (4,26,8,27) or soluble polymer (28,29,24,30) concentrations lies in the information with regard to particle and effective fiber properties which a semi-logarithmic plot of mobility vs. gel/polymer concentration (Ferguson plot) yields. In application to concave Ferguson plots derived from electrophoresis of DNA fragments, considering a sufficiently wide concentration range (6), this information had been unavailable prior to this study. The convex Ferguson plots of spherical particles in agarose gel electrophoresis (16) were interpretable with regard to particle and fiber properties only by means of mainframe computers (18,19,22). This report presents the analysis of both concave (Fig.1) and convex Ferguson plots (Fig.5) by means of a desktop computer. DNA fragments are characterized according to their size and free mobility (related to surface net charge density). The software, called ElphoFit version 2.0, is designed for Macintosh and is available upon request on a floppy disk (3.5", double density, to be provided by the requestor). The program, by contrast with the previous one for linear Ferguson plots (3), is not as yet available for IBM PC and compatibles.

**Fig.6:** Explanation of the phenomenon of DNA band inversion
in terms of Ferguson plots (top panel). MOB3 to MOB7 represent
simulated mobility data of DNA fragments, having an effective
radius of an equivalent sphere, RAD3 to RAD7, as shown in the
bottom panel. IRR indicates a fragment with anomalous migration,
i.e., it migrates disproportionately faster at higher gel
concentrations. In the depicted example, this anomaly is caused
by a disproportionately steep decline of effective particle
radius (RAD6 IRR, bottom panel) with polymer concentration
compared to the other species. This is possibly due to a
conformational difference. Alternatively, band inversion could
also be due to a disproportionately shallow decline of effective
particle radius (not shown). Eqs.1 to 3 were employed for
simulating the data assuming a continuous electric field and the
following realistic parameters: fiber radius r = 4 nm, fiber
volume VF = 4 mL/g, free mobility µo = 40 x 10^-5
cm^2 V^-1 s^-1, effective particle radii (RAD3 to RAD7), R, at
0% polymer concentration are 3 to 7 nm, PAR1 = 10 and PAR2 = 0.2
(MOB3, MOB4, MOB5, MOB7) or PAR1 = 3 and PAR2 = 0.1 (MOB6 IRR).

**7.2 Assumptions and properties of the applied model**

The data evaluation is based on the following assumptions: (i) the sieving medium can be approximated by a fiber network, (ii) inert fibers are randomly and uniformly distributed, (iii) the particles do not aggregate, decompose or change their surface net charge density, (iv) the quotient of measured and free mobility can be related to the available fractional volume in the polymer matrix. A more detailed discussion has been given in Section 6.4 of (5). In contrast to the original extension of the Ogston model (2), the model of the present study does make a provision for a variation of fiber and particle dimensions as a function of gel concentration. In the case of the convex Ferguson plot obtained from agarose electrophoresis of bacteriophages (19), the present model assumes an increase of total fiber length and a diminishing fiber radius with increasing gel concentration (click here to see Table 2). In the case of the concave Ferguson plots derived from capillary agarose electrophoresis of DNA fragments, it is assumed that the radius of the equivalent sphere of the DNA fragment diminishes with increasing concentration of the sieving medium (Eq.3). The equivalent sphere (1) of nonspherical particles is defined as one having access to the same percentage of spacings (available fractional volume) for a particular set of experimental conditions as the nonspherical particle. A possible explanation for the reduction in particle radius could be the independently observed stretching of the DNA molecule in inverse relation to pore size of the gel [Table 2 of (8)]. Such a stretched molecule is capable of reptation and would have access to a larger part of the available fractional volume, just like a spherical particle having a smaller radius.

A potential application of the approach presented in this study lies in the elucidation of band inversion (31) which is the faster migration of larger DNA fragments than some smaller ones in a continuous electric field at higher gel concentrations. Assuming the data points of Fig.6, the crossing concave Ferguson plot would explain band inversion. The underlying concept is that of a random matrix structure (fiber network) and a non-random particle shape (equivalent sphere). Band inversion has also been modeled by using the biased reptation model (32) which assumes a non-random matrix structure (tubes within the polymer) and a random particle shape ["chain made of N freely-jointed and rigid primitive segments of length a" (32)]. One of the two kinds of geometric assumptions has to be made, since mathematical modeling of both random polymer and particle structures is impossible due to infinite degrees of freedom.

The advantages over empirical curve fitting of using a mathematical approach based on Eqs.1 to 5 in combination with a simultaneous curve fitting strategy are: (i) Physical parameters related to the sieved particle and the sieving matrix can be determined (e.g., Figs.2 and 3, Tables 1 to 3); (ii) predictions beyond the range of experimental data are more accurate; (iii) the number of fitted parameters is considerably smaller than that of a polynomial approach: modeling the 10 nonlinear curves of Fig.1 (no assumption of common free mobility intercept) requires a total of 15 parameters whereas a polynomial approach would depend on 30 (2 nd order) or 40 (3 rd order) adjustable parameters. The advantage of a polynomial approach is that almost any curve shape can be fit whereas a model based on physical parameters is more difficult to develop, since it requires a hypothesis with regard to what causes nonlinearity and how it can be modeled.

**7.3 Biomedical significance and futuristic outlook**

The analysis of concave and convex Ferguson plots by means of desktop computer based on a model which is related to physical parameters descriptive of the polymer matrix and the sieved particle represents progress. However, it does not extend yet to more complex shapes such as to sigmoidal or multiple-sigmoidal (ondulating) ones (7,17). Characterizing more complex plots is important, if one considers that the shape of the Ferguson plot may reflect the conformation of the particle. In support of this hypothesis it has been shown, e.g., that rodshaped virus particles generate concave Ferguson plots (33) as does DNA at elevated polymer concentrations (6), i.e., at concentrations capable of stretching the molecule [Table 2 of (8)]. Similarly, ondulating Ferguson plots of plant virus particles have been interpreted in terms of their molecular malleability (17). Evidently, the possibility to gauge conformation of DNA and its complexes with proteins based on the shapes of Ferguson plots or the equivalent Ferguson curves on transverse pore gradient gels (12) merits the further development of computerized Ogston-model-based Ferguson plot analysis to apply to plots and curves of any shape. Modeling of complex curve shapes will require the introduction of more fitted parameters at the risk that these parameters may not be adequately defined. Program ElphoFit circumvents this problem by allowing the user to control the quality of the applied model (see dependency values in Fig.1). Empirical Ferguson plot analysis is already at this time capable of defining in terms of a polynomial function any shape of Ferguson plot (11) or Ferguson curve (12), but without being able to derive from that function information regarding the properties of the particle or polymer fiber under study.

One of the disadvantages of conventional Ferguson plot analysis has been that this technique requires the determination of mobility values at several gel/polymer concentrations which is laborious and time consuming. The Ferguson curve technique (12) using transverse pore gradient gels reduces the required efforts, but at the price of uncertainty in the local gel concentration. The application of capillary electrophoresis to Ferguson plot analysis (29) shortens its duration significantly, i.e., the determination of mobilities for the DNA fragments reported in the present study only requires 5 to 15 minutes per data point. Developments are under way which promise an even greater efficiency (34): Dovichi et al. are designing a parallel capillary system supporting the simultaneous electrophoresis in 32 capillaries. Assuming that the tube properties of the 32 are very similar, capillaries filled with sieving media of different concentration could generate data for several Ferguson plots in about 20 minutes inclusive of their evaluation by program ElphoFit. The detection level can be very low, i.e., 10^-20 mol in the case of fluorescein-labeled DNA fragments (35). If 8 values per plot are appropriate, four samples each containing a number of DNA fragments or other particle species may be evaluated at the same time. The analysis would provide the number of components, their size and free mobility (surface net charge density) and the shapes of Ferguson plots characteristic of conformation.

**Acknowledgments**

Many thanks are due to Drs. Petr Bocek (Czechoslovak Academy of Sciences, Institute of Analytical Chemistry, Brno, CSFR) and Barry L. Karger (Barnett Institute, Northeastern University, Boston MA) for their helpful discussions.

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43) Ruchel, R., Brager, M.D., Anal. Biochem. 1975, 68, 415-428.

44) Zimm, B.H., in: Sarma, R.H., Sarma, M.H. (Eds.), Structure & Methods, Vol. 1: Human Genom Initiative & DNA Recombination, Adenine Press, Schenectady NY 1990, pp. 15-17. ISBN 0-940030-29-2

**Note added in proof**

The concept of an equivalent sphere for nonspherical particles has been discussed recently in (38). The authors of this paper conclude that there is no simple relationship between particle size, shape and the radius of an equivalent sphere and the latter concept, therefore, is dismissed. However, a simple relationship has never been postulated. Ogston (1) postulated an equivalent sphere which has the same probability of "no fiber contact" as the nonspherical particle. The results of the present study also indicate that the size of an equivalent sphere is a nonlinear function of particle size, conformation and gel concentration (Figs.2 and 6). But this complexity is of no significance for determining the size of DNA, since geometric mean radii extrapolated to 0 %T (absence of a gel) are considered. Under these conditions most of the DNA species assume a near spherical configuration at an ionic strength comparable to that of electrophoresis [(39), Rampino and Chrambach, J. Chromatogr. submitted].

Another question of interest is, whether the radius of gyration instead of the geometric mean radius should be applied, as has been postulated recently (40). This question has been previously addressed and resolved in favor of geometric mean radii by (41,42), since geometric rather than hydrodynamic properties appear involved. Radii of gyration are 1 to 2 orders of magnitude larger than geometric mean radii, depending on the number of DNA base pairs and the ionic strength of monovalent ions. If radii of gyration (persistence length of 146.6 nm for 5 mM sodium) would have been used in this study, an unrealistic large radius of 195 nm (results not shown) is computed for the agarose fiber, a dimension which would actually be visible in the light microscope. The apparent agreement between the radius of the "holes" in polyacrylamide gels (43) and pore radii determined on the basis of DNA sizes specified in terms of radii of gyration (40) seems irrelevant to the degree that the channels connecting the "holes" appear to be limiting the passage of particles ["lakes and straits" model (44)].