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RESEARCH PAPER
Modeling and experimental characterization of peak tailing
in DNA gel electrophoresis
Yi Sun ÆYien Chian Kwok ÆNam-Trung Nguyen
Received: 19 August 2006 / Accepted: 21 October 2006
Springer-Verlag 2006
Abstract Capillary electrophoresis (CE) is an effi-
cient separation method in analytical chemistry. It
exploits the difference in electrophoretic migration
velocities between charged molecular species in aque-
ous or diluted polymer solution when an external
electric field is applied to achieve separation. Despite
the standard assumption that electrophoretic data ob-
tained from pulse-loaded molecular species should
have Gaussian peak shapes, experimentally observed
peaks are frequently distorted or highly asymmetric.
Interaction of charged species with the wall of the
capillary is the primary source for serious band
broadening and peak tailing. This paper reports a
mathematical model for the peak profiles in capillary
electrophoresis, taking adsorption on capillary wall
into account. The model is based on the advection–
diffusion equation, Langmuir second order kinetic
equation and appropriate boundary conditions. It is
applied to simulate the gel electrophoretic separation
of the 11 fragment FX174-Hae III double stranded
DNA ladders in a polymeric microchip. By using the
migration velocities and diffusivities from the mea-
surement, and properly selecting two fitting parame-
ters, namely adsorption and desorption coefficients, the
simulated peak shapes show remarkable similarity
with the experimental electrophoretic results. The
effect of adsorption and desorption coefficients are also
investigated and the result shows that adsorption of
analytes from the main analyte zone and desorption of
these analytes appear to be the reasons of peak tailing,
with the latter being the major cause.
Keywords Peak tailing DNA Gel electrophoresis
Microchip
1 Introduction
Capillary electrophoresis (CE) is known as a fast
separation method for inorganic ions, nucleic acids
and protein analysis. It is an effective tool in the
analysis of polymerase chain reaction (PCR) products
(Zhou 2005), DNA sequences (Shi 2003), and geno-
typing (Sun and Kwok 2006). CE separation usually
exploits the difference in electrophoretic migration
velocities between charged molecules in aqueous or
low-viscosity polymer solutions when an external
electric field is applied to achieve separation. The
migration of the analytes through a detector cell is
recorded versus time as an electropherogram. The
basic shape of a peak in an electropherogram is usu-
ally assumed to be ‘‘symmetrical’’ and can be
approximated by a Gaussian equation. In practice, due
to analyte-wall interaction, the peak shapes may not
always be symmetrical. Skewed or highly asymmetric
peaks with prolonged tails do occur frequently in the
laboratory (Schwinefus 1998). In recent years, micro-
chip electrophoresis has attracted attention as the next
advancement in capillary electrophoresis. When using
conventional CE instrumentation, most routine ana-
lysts would regard the peak tailing as malfunctions or
Y. Sun Y. C. Kwok
National Institute of Education, 1 Nanyang Walk,
Singapore 637616, Singapore
N.-T. Nguyen (&)
School of Mechanical and Aerospace Engineering,
Nanyang Technological University, 50 Nanyang Avenue,
Singapore 639798, Singapore
e-mail: mntnguyen@ntu.edu.sg
123
Microfluid Nanofluid
DOI 10.1007/s10404-006-0126-3
artifacts of the instrumentation, and ignore it as long
as it does not affect the quantification. However, for
microchip CE, as the separation length is much shorter
than the conventional CE machine, peak tailing usu-
ally leads to large peak dispersion and severe reduc-
tion in resolution.
A few previous research works investigated the peak
shapes and took analyte-wall interaction into account.
A thorough analysis of wall adsorption in capillary
zone electrophoresis (CZE) was done by Schure and
Lenhoff (Schure 1993). They were able to assess the
zone broadening or even peak shapes using the Lapi-
dus–Amundson kinetic model. Analytical results on
peak profiles in CZE in the presence of wall adsorption
have been presented by Gas et al. (Stedry 1995; Gas
1995). The model was based on physico-chemical
relations such as mass balance equations and adsorp-
tion rate equations. Recently, Ghosal et al. (Shariff
2004; Ghosal 2003) presented numerical simulations on
peak tailing in CZE. Numerical solutions of the cou-
pled electro-hydrodynamic equations for fluid flow and
the advection–diffusion equation for analyte concen-
tration were illustrated in the limit of thin Debye lay-
ers. The modification of the zeta-potential due to
analyte adsorption and the hydrodynamic flow field
were considered. Besides models for CZE, an alter-
native two-state model for gel electrophoresis was
suggested. This model, originally formulated by Grid-
ding and Eyring (1955) in a simplified form, was gen-
eralized later to include more complicated non-
Markovian effects (Weiss 1985). In this model one
imagines that a single protein molecule in the gel can
be in one of two phases, mobile or entangled. In the
mobile phase the protein is assumed to diffuse as a
Brownian particle; in the entangled phase it remains
stationary until it eventually disentangles from the gel
to resume motion.
So far the theoretical discussion of the tailing phe-
nomenon has been focused on CZE or gel electro-
phoresis for the application of proteins, little effort has
been put into the modeling of peak tailing frequently
observed in DNA gel electrophoretic separation.
Moreover, although several equations have been sug-
gested to simulate skewed peak shapes, none of them
have been claimed to be successful in matching with
the experimental peaks. In addition, in most models
only a single species is simulated. This is not the case in
real practice where multispecies need to be separated
during one CE process. In this paper, a mathematical
model is used to simulate the gel electrophoretic sep-
aration of 11-fragment double-stranded DNA ladders
(uX174-Hae III dsDNA digest) in the presence of wall
interactions. The evolution of analyte concentration in
a cylindrical microchannel is formulated and solved
numerically. The experiment of electrophoretic sepa-
ration of DNA ladders is carried out. It is found that
the numerical solution is well fitted to the experimental
result. The effect of adsorption and desorption coeffi-
cients are also studied and the result shows that peak
asymmetry can be largely attributed to the desorption
of the analyte behind the main analyte zone.
2 Mathematical model
2.1 Description of the model
We consider a microchannel with a simple cross-sec-
tional geometry for sample injection and separation as
shown in Fig. 1. The microchannel defined by points A
and D was the separation microchannel and that de-
fined by points B and C was the injection microchan-
nel. At the end of each microchannel, there were
reservoirs for sample, buffer or waste. These reservoirs
also provided access of the electrodes for high-voltage
input. Although the later mathematical analysis is non-
dimensional and general, following parameters are ta-
ken for the comparison with experimental data. The
microchannel has a hydraulic radius of 0.055 mm. The
lengths of separation and injection microchannels were
80 and 20 mm, respectively. Both injection channel and
separation channel are filled with diluted polymer
55
70
10
10
A
B
Detection
point
C
D
Unit: mm
Fig. 1 Schematic of the micro CE chip. The short and long
channels are injection and separation channels, respectively. The
buffer, sample, sample waste and buffer waste reservoirs are
indicated as A,B,C,D, respectively
Microfluid Nanofluid
123
solution. The sample is 11-fragment FX174-Hae III
dsDNA ladders, ranging from 72 to 1,353 bp. For the
simulation, only separation process is considered, i.e., it
is supposed the sample has been transferred electr-
okinetically into the cross junction. The DNA migrates
under the influence of an external electric field acting
in the axial direction of the separation channel. DNA
molecules can be adsorbed to and desorbed from the
inner wall of the microchannel. Two forces affect the
analyte in the solution: the concentration gradient of
the analyte which causes a diffusional flux, and the
electrical potential gradient which leads to a migration
flux of the analyte. There are two movements inside
the microchannel, the electro-osmotic flow (EOF)
velocity of the polymer solution and the electropho-
retic velocity of the DNA molecules. In this paper, the
analyte velocity refers to the apparent velocity of DNA
molecules which include both the EOF and electro-
phoresis. Normally in polymer devices, the electro-
phoretic velocities of negatively charged compounds
will be greater than EOF generated by the surface
charges, causing the net migration from the cathodic to
the anodic reservoir. For example, the zeta potential of
PMMA substrate is –20 mV to –30 mV (Brian 2004),
dynamic viscosity of 3.5% hydroxypropylcellulose
(HPC) solution was measured to be 0.880 N s/m
2
by
an automated Ubbelohde-type capillary viscometer
(Schott Gerate, hofheim, Gremany) at 25C. Thus, the
electroosmotic mobility of HPC solution in the laser-
ablated PMMA substrate was in the order of 10
–4
mm
2
/
V s, while the electrophoretic mobility for the 118-bp
fragment was measured to be 4 ·10
–2
mm
2
/V s (Sun
et al. 2006). Although the zeta potential is altered due
to the adsorption of charged DNA molecule during
electrophoretic separation, we assume that DNA
molecules move at constant speed as electrophoretic
flow always dominates over EOF. DNA migration in
sieving matrix is much more complicated than in
free solution, and there are at least three different
regimes for motilities and diffusion coefficients of
DNA molecules: the Ogston regime where the size of
the molecule is smaller than the mean pore size; the
reptation regime where the molecule remains in a
random coil conformation but is larger than the mean
pore size; the reptation with orientation regime where
large molecules are oriented in the field direction and
no separation is possible. So far no theory-based gen-
eral equation can apply to all known regimes (Mercier
2006), therefore, to reduce complexity, our model is
focused on the concentration profiles of DNA frag-
ments, and velocities and diffusion coefficients of
DNA molecules are taken from experiment. Other
possible effects contributing, in practice, to sample
dispersion, e.g., thermal effects or Taylor dispersion,
will not be considered here.
For convenience, we assume a circular cross-section
for the microchannel, however, the qualitative or
quantitative effects are equally valid for channels of
any cross-sectional shape. The variables of interest
are the concentration of the analyte in the fluid,
c(mol m
–3
), and the concentration of the analyte
adsorbed to the surface of the wall, s(mol m
–2
), the
analyte velocity u, the radical distance from the cen-
terline r, distance from the inlet x, and the time t. For
non-dimensional analysis, all the above variables are
normalized as follow:
r¼r=rc;x¼x=rc;t¼tumin=rc;u¼u=umin ;c
¼c=cmax;and=;s¼s=ðcmax rcÞ;
where r
c
is the capillary radius, u
min
is the velocity of
the slowest DNA fragment (the 1,353 bp fragment)
and c
max
is the maximum value of concentration at the
initial time t*= 0.
As the radial distribution of the concentration in the
microchannel is not uniform due to the analyte inter-
action with the wall during migration: the concentra-
tion of the analyte in solution is thus a function of three
variables: the spatial x*andr* coordinates and the
time t*, i.e., c
i
*
=c
i
*
(x
*
,r
*
,t
*
). Since there are 11 species
in the sample, c
i
*
is used to represent the dimensionless
concentration of the ith component. The evolution of
concentration of any particular species is described by
the advection-diffusion equation:
@c
iðx;r;tÞ
@tþu
i
@c
iðx;r;tÞ
@x¼1
Pe r2c
iðx;r;tÞ
ð1Þ
where Pe = u
min
r
c
/Di is the Peclect number with the
diffusion coefficient D
i
of the i–th component.
The actual surface concentration, s*
i
, of the ad-
sorbed analyte depends on time t* and the x* coordi-
nate, i.e., s*
i
=s*
i
(x*, t*). The mass flux density in the
direction perpendicular to the wall is equal to the time
derivative of the surface concentration, ¶s
*i
/¶t
*
.
According to Langmuir second-order kinetic law, it can
be written as
@s
i
@tðx;tÞ¼k
ac
ws
max s
iðx;tÞ
k
ds
iðx;tÞð2Þ
where c
*w
,k
*a
and k
*d
are the dimensionless concen-
tration at the wall, the dimensionless Langmuir rate
constants of adsorption and desorption, respectively,
and s
max
is the maximum value of the concentration of
Microfluid Nanofluid
123
adsorbed analyte which corresponds to the total
monolayer coverage of the surface. The dimensionless
parameters can be readily related to the corresponding
dimensional parameters by k
*a
=(c
max
r
c
/u
min
)k
a
and
k
*d
=(c
max
r
c
/u
min
)k
d
, where k
a
and k
d
are the
dimensional counterparts.
As the electric field acts tangentially to the channel
wall, the radial flux transporting the analyte to the wall
is only the diffusional radial flux, which can be ex-
pressed as 1
Pe @c=@r:At the capillary wall, the ra-
dial diffusional flux equals to the mass flux of the
adsorbed analyte as described by Eq. 2. Consequently,
the boundary condition expressing the mass balance at
the channel wall can be formulated as:
@s
i
@tðx;tÞ¼1
Pe
@c
i
@rðx;1;tÞð3Þ
For boundary conditions in the x-direction, the
concentration is supposed to be zero at the two ends,
i.e., c
*i
(0,r
*
,t
*
) = 0 and c
*i
(L
*
,r
*
,t
*
)= 0.Lis the
length of the separation channel and L
*
=L/r
c
.
Initial conditions describing the distribution of the
analyte in the microchannel at time t* = 0 can be given
in the form
c
iðx;r;0Þ¼c
i0ðxÞð4Þ
sðx;0Þ¼s
max
c
i0ðxÞ
c
i0ðxÞþk
d=k
að5Þ
These initial conditions indicate that there is no ra-
dial flux at time t* = 0 when the migration starts.
2.2 Concentration distribution
The equations presented above with appropriate
boundary and initial conditions should provide com-
plete information about the evolution of the concen-
trations of both the analyte in solution and the
adsorbed analyte on the microchannel wall. However,
direct numerical integration is inefficient due to the
smallness of the microchannel radius in comparison to
its length (L*~10
3
–10
5
). The system could be reduced
to partial differential equations with only one spatial
variable x* using Ghosal’s asymptotic theory (Ghosal
2003). The detailed derivation is included in the
Appendix section. The asymptotic theory is valid only
if axial variations in all dependent variables occur on a
characteristic length scale that is very much larger than
r
c
and all temporal variations occur on a characteristic
time scale that is very much larger than the diffusion
time t>>r
c
2
/D, where Dis the diffusivity of the
analyte. Discussion from Ghosal (Ghosal 2003) showed
that assumption of slow variations is valid everywhere
in the channel except for a relatively short region
(~few millimeter) near the inlet section.
In this paper, the overbar will indicate an average
over the cross-section, i.e, for any variable H(x
*
,r
*
,t
*
)
Hðx;tÞ¼2Z1
0
rHðr;x;tÞdrð6Þ
The cross-sectional averaged analyte concentration
cican be expressed by the one-dimensional partial
differential equation:
@ci
@tþu
i
@ci
@x¼1
Pe
@2ci
@x22@s
i
@tð7Þ
Equation 7 shows that ciobeys the advection–dif-
fusion equation with a source term to account for
analyte-wall interaction effects.
The distribution of concentration in the micro-
channel is expressed by:
c
iðx;r;tÞ¼ciðx;tÞþPe
4ð12r2Þ@s
i
@tð8Þ
The value of c
*i
at the wall is obtained by setting
r* = 1, thus
c
w¼ciðx;tÞPe
4
@s
i
@tð9Þ
Substituting Eq. 9 into Eq. 2, the interaction of the
analyte with the wall is depicted by
@s
i
@tðx;tÞ¼k
aciðx;tÞPe
4
@s
i
@tðx;tÞ
s
max s
iðx;tÞ
k
ds
iðx;tÞð10Þ
Thus, only two coupled partial differential equations
Eq. (9) and Eq. (10) need to be solved numerically.
Assuming that fluorescence intensity is proportional to
analyte concentration, the overall fluorescence inten-
sity function of the 11 fragments in the DNA ladder
can be estimated as:
cðx;tÞ¼aX
11
i¼1
ciðx;tÞð11Þ
where ais an arbitrary factor with the unit of the
intensity.
A finite volume approach is adopted for the numer-
ical solution. For simplicity, all spatial derivatives are
Microfluid Nanofluid
123
approximated by second-order finite differences. The
differential equations are solved using the Crank-
Nicholson method with time increments chosen to fulfill
the stable condition. The simulations were performed
using Matlab 7.0 (The MathWorks Inc., MA, USA).
3 Experimental section
3.1 Microchip fabrication and system setup
Microchips were fabricated by using laser ablation
technique in poly (methyl methacrylate) (PMMA). The
microfluidic pattern was designed using CorelDraw 10
(Corel Co., Canada). The pattern was then sent to a
laser scriber for direct micromachining on PMMA
substrate. The commercial CO
2
laser scriber (Universal
M-300 Laser Platform, Universal Laser Systems Inc.,
AZ, USA) was used to engrave the PMMA substrate.
Access holes were also drilled by CO
2
laser to allow
fluid access to the microchannels. To form the micro-
fluidic device, the trench was sealed by the thermal
bonding with pressure of 20 kPa at 165C for 30 min.
The microchannels were 0.15 mm wide and 0.087 mm
deep. The lengths of separation and injection micro-
channels were 80 and 20 mm, respectively.
Microchip electrophoresis with laser induced fluo-
rescence (LIF) detection was performed on an inverted
confocal microscope (TCS SP2, Leica Microsystems
AG, Germany). The output radiation (488 nm) from
an Argon ion laser first passed through a 488 nm band
pass filter. The laser beam was then reflected by a
500 nm acousto optical beam splitter (AOBS) and into
a25·long field objective (0.4 N.A). The laser beam
was focused onto the separation channel 55 mm from
the intersection point, resulting in an effective sepa-
ration length of 55 mm. Subsequently, fluorescence
was collected by the same objective and transmitted
back through the AOBS. The emission beam was then
focused through a pinhole of 340 lm and passed
through a programmable 525 nm spectrophotometer
prism (SP) barrier filter. The emission beam was finally
detected by a highly sensitive side-window photomul-
tiplier tube (PMT) with a gain of 800 V/V. Confocal
imaging of the detection volume of the microchannel
was accomplished by beam scanning with a proprietary
K scanner. Data was collected and processed by Leica
confocal software (LCS) program.
3.2 Microchip electrophoretic separation
The separation running buffer consists of 3.5%
Hydroxypropylcellulose (HPC, 100000 MW, Sigma,
MO, USA) in 80 mM MES/40 mM tris (hydroxy-
methylamino) methane (TRIS) (Sigma, MO, USA).
Intercalating dye YOPRO-1 (Invitrogen, USA) was
added to a concentration of 10 lM. The DNA sizing
ladder, FX174-it Hae III dsDNA digest (Invitrogen,
USA) was dissolved in autoclaved DI water to a final
concentration of 5 lg/ml. The DNA electrophoretic
separation was run with reverse polarity (injection end
negative polarity; detection end positive polarity). The
voltages for the four reservoirs were generated by a
commercial 4-channel kilovolt power supply (MCP 468,
CE resources Pvt Ltd., Singapore). Platinum wires
(Goodfellow Corporation, England) were inserted to
reservoirs as electrodes. For DNA loading, an electric
field of 350 V/cm was applied between the sample
reservoir ‘B’ and the sample waste reservoir ‘C’ for 30 s,
while both the buffer reservoir ‘A’ and the buffer waste
reservoir ‘D’ were allowed to float. Following sample
injection, electric field was then switched to the sepa-
ration channel and subsequent separation occurred. An
electric field strength of 300 V/cm was applied across
the buffer and the buffer waste reservoirs.
4 Results and discussions
We first performed electrophoretic separation of
FX174-Hae III dsDNA digest. Migration velocities
and diffusion coefficients for the 11 fragments were
calculated from the electropherogram. These values
were in turn used in the simulation model. We then
compare experimental and simulated peaks by evalu-
ating two easily measured parameters that characterize
electrophoretic peaks. These are the full width at half-
maximum (FWHM) of the peak, and the degree of
peak asymmetry.
4.1 Microchip electrophoretic separation of DNA
fragments
The microfabricated PMMA chip was used for elec-
trophoretic separation of DNA fragments. Figure 2a
shows the electropherogram of FX174-Hae III dsDNA
digest separated under an electric field strength of
300 V/cm. The result was repeated three times, and the
relative standard deviations were less than 0.8 and 5%
for migration time and half peak width, respectively.
As seen from the electropherogram, marked tailing can
be observed, and 271/281 fragments cannot be baseline
resolved. The results indicate that DNA samples were
severely adsorbed to the PMMA channel wall during
the experiment. This is due to the rugged channel
Microfluid Nanofluid
123
surface fabricated by direct CO
2
laser ablation. The
machined trench walls were estimated to have a sur-
face roughness of 5–10 lm (Malek 2006). The peak
tailing phenomena caused by DNA adsorption signifi-
cantly reduces the resolution of DNA separation.
Migration velocities and diffusion coefficients of the
11 fragments are needed for simulation. Migration
velocities can be easily calculated by the formula: u=
L/t, where Lis the separation length and tis the
migration time which can be measured directly from
the electropherogram. Molecular diffusion coefficients
of DNA fragments in polymer solutions were calcu-
lated from the total peak variance in the measured
FWHM. Electrophoretic separation of the fluores-
cently labeled DNA ladders was performed at three
different separation lengths (L= 50, 55 and 60 mm).
FWHM was measured for each run. Slow moving
fragments move across the detector slower than faster
moving fragments and consequently, lead to ‘broader’
peaks. To compensate for this effect, we converted the
‘‘time width’’ of the peaks to actual peak width by
multiplying with the velocity of the migrating frag-
ment. The total peak variance in this case is given by
r2¼r2
static þ2DðL=uÞð12Þ
where r
static
is the static (or time-independent) con-
tribution to band broadening and uis the velocity of
the migrating species. Plotting the total variance (r
2
)
as a function of retention time (L/u) results in a
straight line with a slope equal to twice the dispersion
coefficient. With migration velocities and diffusion
coefficients, Peclet numbers could be calculated
according to the equation Pe = u
min
r
c
/D. Migration
velocities, diffusion coefficients and Pe numbers of the
11 fragments are listed in Table 1.
4.2 Numerical simulations
4.2.1 Concentration of 11 DNA fragments
We will now investigate the influence of analyte-wall
interaction on peak widths and shapes by numerically
solving the equations presented in Sect. 2‘‘Mathe-
matical Model’’. The separation microchannel is sup-
posed to be filled with the diluted polymer solution and
a hypothetical sample. The sample is a mixture of the
11 species of DNA fragments. The initial concentration
profile for each species is Gaussian shape and can be
expressed as:
ciðx;0Þ¼c
ið0Þexp ðxx0Þ2
2r2
0
"# ð13Þ
where the initial variance r
0
= 10, and initial position
x
0
= 182. This physically corresponds to the case in
which the analyte is located at the intersection that is
Fluorescent intensity (normalized)
Time (sec)
Measurement
Simulation
Fluorescent intensity (arbitrary)
1353
1078
872
603
310
281
271
224
194
118
72
1353
1078
872
603
310
281
271
224
194
118
72
Time (s ec)
(a)
(b)
Fig. 2 Electropherograms of
the 11 DNA fragments. aCE
experiment of 5lg/ml FX174-
Hae III dsDNA digest in
80 mM MES/40 mM TRIS
buffer with 3.5% HPC. The
separation electric field
strength was 300 V/cm. b
Simulated intensity signal in t-
domain based on Eq. 11. The
figure shows the result of the
temporal evolution of
concentration peaks at a
hypothetical detector placed
at a distance x
d
= 1,182 from
the inlet
Microfluid Nanofluid
123
10 mm from the inlet of the separation channel (10/
0.055 mm = 182). c
i
(0) represents the normalized
maximum value of concentration for each fragment at
initial time, and the values are shown in Table 2. They
are determined according to the original composition
of the DNA sample. The other input parameters for
numerical simulations are given in Table 3.K*
a
,k*
d
and s*
max
are closely related to the material and the
fabrication process and here the values were chosen to
fit the experiment.
The results of the simulations are functions of cðx;tÞ
which can be presented either in the t-domain or in the
x-domain. In this paper, the presentation of the func-
tion cðx;tÞis in the t-domain, which is consistent with
the real experiments. The depiction in the t-domain
presents peak shapes during passage through the
detector. Figure 2b shows the temporal distribution of
cat a hypothetical detector placed at 65 mm from the
inlet. The figure shows the result for x*
d
= 1,182. The
graph refers to the multiple peaks from the multi-
component sample in t-domain. In the signals the
horizontal axis represents time of arrival at a fixed
detector location so that the sharp edge appears first,
and then the gradually decaying tail. The long tails are
caused by the cumulative effect of the analytes
adsorption and desorption long after the main peaks
have passed. It can be seen that as the number of base
pairs increases, the peak width increases and the peak
shape becomes noticeably asymmetric.
The simulated peak shapes have a striking similarity
with experimentally observed peak shapes from the
microchip separation of the DNA ladder. In making
the comparison, FWHM and the asymmetrical factor
for each DNA fragment are evaluated for both
experimental and simulated peaks. Asymmetrical fac-
tors were calculated according to the equation
D¼wþw
wþþw
;ð14Þ
where w
+
is the time from the peak maximum to the
trailing edge at 50% maximum and w
–
is the time from
the leading edge at 50% maximum to the peak maxi-
mum. An asymmetrical factor of zero indicates a
symmetric band, a positive asymmetry indicates tailing
and a negative asymmetry indicates fronting. Com-
parisons of experimental and simulated FWHM and
asymmetrical factors are presented in Table 4. The
simulation for concentration is seen to be in good
agreement with the experimental result. The effect of
gel matrix on DNA is reflected through the Pe number.
As the length of DNA fragment increases, Pe number
becomes bigger, and the peak shape tends to be
broader and more asymmetrical.
4.2.2 Effect of k*
a
and k*
d
The Langmuir second order kinetics is assumed for the
wall interaction. To study the effect of k
a
and k
d
,we
simulated three cases of analyte–wall interactions for
the fragment 603 bp with values of the parameters k*
a
,
k*
d
and s*
max
as shown in Table 5. In case I, there is no
interaction at the channel wall, while in case II, only
adsorption of analyte is considered and in Case III,
both adsorption and desorption processes are taken
into account. In Fig. 3, the concentration profile at the
detector position is shown for all the three cases. For
ease of comparison, the profiles are normalized so that
the maximum value is unity in all the cases. Obviously,
Case I gives symmetric and the narrowest peak. This
shows that if there is no analyte–wall interaction, the
distribution of the analyte is supposed to be Gaussian.
In the other two cases, the peaks are broadened to
various degrees, implying the interaction at the channel
Table 1 Measured migration velocities, diffusion coefficients
and Pe numbers of the 11 FX174-it Hae III dsDNA fragments.
DNA fragment (bp) Migration
velocity (mm/s)
Diffusion
coefficient
(10
–6
mm
2
/s)
Pe
72 0.395 3.5 26.4
118 0.354 3.25 28.43
194 0.311 2.9 31.86
224 0.292 2.75 33.6
271 0.274 2.55 36.24
281 0.269 2.5 36.96
310 0.259 2.4 38.5
603 0.194 2.08 44.42
872 0.178 1.9 48.63
1078 0.172 1.84 50.21
1353 0.168 1.7 54.44
Table 2 Normalized initial concentrations for the 11 DNA fragments.
DNA fragment(bp) 72 118 194 224 271 281 310 603 872 1078 1353
c*
i0
0.2 0.25 0.5 0.5 0.5 0.5 0.5 1 1 1 1
Table 3 Values of input parameters for simulation.
Parameter L*k*
a
k*
d
s*
max
Value 1455 0.5 0.12 0.2
Microfluid Nanofluid
123
wall will cause band broadening and distortion of the
Gaussian shape. The peak obtained in Case III is much
wider and more asymmetric than the peak in Case II,
showing that the desorption of analytes behind the
moving main plug appears to affect peak asymmetry
more significantly than the absorption of analytes.
5 Conclusions
In this paper, a mathematical model was developed to
describe the evolution of analyte concentration in gel
electrophoretic separation of 11-fragment FX174-Hae
III dsDNA ladders while wall interactions were taken
into consideration. The model was formulated based
on advection-diffusion equation and Langmuir second
order kinetic equation. By using the migration veloci-
ties and diffusivities from the measurement and
choosing proper values for adsorption and desorption
coefficients, the numerical solution fits very well to the
electrophoretic data obtained from the experiment
carried out in a PMMA microchip. The study of
adsorption and desorption coefficients showed that
adsorption and desorption of the analyte behind the
main analyte zone could lead to peak tailing, and the
latter was identified to be the primary cause. The
mathematical model provides an efficient way to
investigate the effect of wall interaction on peak
profiles. Once adsorption, desorption coefficients and
maximum adsorbed concentration are identified for a
microchannel, the model could be used to estimate
peak shapes as well as resolution of analytes for gel
electrophoretic separation, which will save a lot of
effort compared with experimental trial-and-error
method. Further investigations are being pursued and
other possible causes of peak deformation such as
change of zeta potential due to analyte adsorption will
be considered to further improve the model.
6 Appendix
An asymptotic theory has been exploited to derive
equations for cross-sectional averaged concentration
ci:The theory is valid provided axial variations have
characteristic length scales that are much larger than
the capillary radius and temporal variations have a
characteristic time scale much larger than the charac-
teristic diffusion time over a capillary radius. It is as-
sumed that the dependent variables are functions of r
and the slow variables T=bt*andX=bx*, where b
is a small parameter. By rewriting Eqs. 1 and 2 in terms
of the variables r,X,T,c, and u, and the final solution
could be found using asymptotic series in powers of b,
in the form /=/
0
+b/
1
+b
2
/
2
+ ....
Equations 1 and 2 can be rewritten in terms of the
slow and fast variables as:
Table 4 Comparisons of experimental and simulated FWHM
and asymmetrical factors for the 11 FX174-Hae III dsDNA
fragments.
DNA
fragment (bp)
FWHM (s) Asymmetrical factor
Experiment Simulation Experiment Simulation
72 1.38 1.41 0.33 0.35
118 1.49 1.52 0.4 0.43
194 1.61 1.59 0.45 0.47
224 1.75 1.79 0.45 0.46
271 1.84 1.81 0.5 0.51
281 1.99 2.04 0.5 0.49
310 2.18 2.24 0.56 0.54
603 2.49 2.38 0.6 0.59
872 3.98 3.87 0.67 0.65
1078 5.67 6.04 0.75 0.71
1353 7.95 8.16 0.875 0.82
Table 5 Three cases for the analyte-wall interaction.
Case k*
a
k*
d
s*
max
I 0 0 0.2
II 0.5 0 0.2
III 0.5 0.12 0.2
850 900 950 1000 105 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
k* =0, k* =0, s*=0.2
admax
k* =0.5, k* =0, s*=0.2
admax
k* =0.5, k* =0.12, s*=0.2
admax
Concentration c
Dimension less time t*
Fig. 3 The concentration profiles of the 603 bp fragment at the
detector position (x
d
= 1,182) for the three wall interaction cases
Microfluid Nanofluid
123
1
r
@
@rr@c
i
@r
¼bPe @c
i
@Tþu@c
i
@X
b2@2c
i
@X2ðA:1Þ
1
Pe
@c
i
@r
r¼1¼b@s
i
@TðA:2Þ
At the lowest order, Eqs. A.3 and A.4 are obtained
for c*
i0
.
1
r
@
@rr@c
i0
@r
¼0ðA:3Þ
1
Pe
@c
i0
@r
r¼1¼0ðA:4Þ
The solution is c*
i0
=c*
i0
(X,T), which is indepen-
dent of r. At the next order, equations for c
i1
can be
written as
1
r
@
@rr@c
i1
@r
¼Pe @c
i0
@Tþu@c
i0
@X
ðA:5Þ
1
Pe
@c
i1
@r
r¼1¼@s
i0
@TðA:6Þ
Averaging both sides of (A.5) over the cross-section
of the capillary using the boundary condition (A.6), the
left-hand side becomes 2Pe @si0
@T:Thus, Eqs. A.5–A.6
can be solved provided the following solvability con-
dition is satisfied:
@c
i0
@Tþu@c
i0
@X¼2@s
i0
@TðA:7Þ
Eq. A.7 is the evolution equation for c
i0
. The solu-
tion to Eqs. (A.5) and (A.6) is obtained by straight-
forward integration of (A.5) with respect to r*,
c
i1ðX;r;TÞ¼c
i1ðX;0;TÞPe
2r2@s
i0
@TðA:8Þ
Averaging both sides of this equation over the cross-
section of the capillary, the concentration on the cen-
terline of the capillary, c
*i1
(X,0,T) can be expressed
in terms of ci1ðX;TÞ;
c
i1ðX;0;TÞ¼ci1ðX;TÞþPe
4
@s
i0
@TðA:9Þ
so that Eq. A.8 may also be written as
c
i1ðX;r;TÞ¼ci1ðX;TÞþPe
4ð12r2Þ@s
i0
@TðA:10Þ
To obtain an evolution equation for c*
i1
, the second
order b
2
terms need to be considered in Eqs. A.1 and
A.2, which gives
1
r
@
@rr@c
i2
@r
¼Pe @c
i1
@Tþu@c
i1
@X
@2c
i0
@X2ðA:11Þ
1
Pe
@c
i2
@r
r¼1¼@s
i1
@TðA:12Þ
Averaging both sides of Eq. (A.11) over the cross-
section and using the boundary condition Eq. (A.12)
the solvability condition is obtained as
@ci1
@Tþu@ci1
@X¼2@s
i1
@Tþ1
Pe
@2ci0
@X2ðA:13Þ
As c*
i0
is a function of Xand tonly, so that @c
i0
@r¼0
and c
i0¼ci0:
Next we combine the evolution equations for ciat
zeroth and first order to obtain a single equation for ci:
This is achieved by multiplying Eq. A.13 by band adding
the result to Eq. A.7. The higher-order terms may be
added or dropped without affecting the asymptotic
validity of the resulting equation. Thus, we obtain
@ci
@tþu@ci
@x¼1
Pe
@2ci
@x22@s
i
@tðA:14Þ
The distribution of concentration can be expressed
as:
c
i¼ciðx;tÞþPe
4ð12r2Þ@s
t
@tðA:15Þ
The value of concentration at the wall is obtained by
setting r* = 1, thus
c
w¼ciðx;tÞPe
4
@s
i
@tðA:16Þ
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