Dietmar Tietz Section on Macromolecular Analysis
Andreas Chrambach
Laboratory of Theoretical and Physical Biology
National Institute of Child Health and Human Development
National Institutes of Health
Bethesda, Maryland, USA
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.
3 Design of program ElphoFit
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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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)].
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