Anal. Chem. 2002, 74, 4456-4463
Equilibrium Gradient Methods with Nonlinear Field Intensity Gradient: A Theoretical Approach H. Dennis Tolley,*,† Qinggang Wang,‡ David A. LeFebre, and Milton L. Lee‡
Departments of Statistics, Chemistry and Biochemistry, Brigham Young University, Provo, Utah, 84602-5700, and Biohmics, Camino, California, 95709
Equilibrium gradient methods belong to a family of separation techniques in which analytes are forced to unique equilibrium points by a force gradient and a counter force along the separation pathway. The basic theory for equilibrium gradient methods where the force gradient is induced by a field gradient is developed in this paper. The results indicate that peak capacity can be dynamically improved by using a nonlinear field-intensity gradient in which the first section is steep, and the following section is shallow. Using electromobility focusing (EMF) as an example, a separation model is established. EMF is an equilibrium gradient method that uses an electric field intensity gradient to induce a force gradient on charged analytes, such as proteins, and a constant hydrodynamic flow as an opposing force. Equations relating operating parameters with separation performance are given. Although simulation results show that a peak capacity of over 10 000 is theoretically possible using a single channel in a separation time just under 2 months, if 100 parallel separation units are utilized in an array format under the same operating conditions, the same peak capacity can be obtained in just over 12 h. Although a number of powerful techniques, such as chromatography, electrophoresis, and field flow fractionation, are available for the separation of mixtures of chemicals and small particles, there is a growing need for new techniques that can provide even greater resolving power. For example, with the recent completion of the rough draft of the human genome, researchers are turning their attention to the proteome. The enormity of the task of characterizing the complete proteome can be appreciated by the fact that although there are ∼34 000 genes in humans, it is estimated that there are between 500 000 and 1 000 000 proteins.1 There is no analytical technique available today that can come close to separating even 10 000 proteins in a relatively short time, let alone 500 000 or more. Among various developing new separation techniques designed to improve resolving power, equilibrium gradient methods are quite promising. * To whom correspondence should be addressed. Fax: 801-422-0635. E-mail:
[email protected]. † Department of Statistics. ‡ Departments Chemistry and Biochemistry. (1) Binz, P. A.; Muller, M.; Walther, D.; Bienvenut, W. V.; Gras, R.; Hoogland, C.; Bouchet, G.; Gasteriger, E.; Fabbretti, R.; Gay, S.; Palagi, P.; Wilkins, M. R.; Rouge, V.; Tonella, L.; Paesano, S.; Rossellat, G.; Karmime, A.; Bairoch, A.; Sanchez, J. C.; Appel, R. D.; Hochstrasser, D. F. Anal. Chem. 1999, 71, 4981-4988.
4456 Analytical Chemistry, Vol. 74, No. 17, September 1, 2002
An equilibrium gradient method as defined by Giddings and Dahlgren2 is “... a method in which a gradient or combination of gradients causes each species to seek an equilibrium position along the separation path.” In an equilibrium gradient method, the net force on each analyte species changes monotonically along the separation path (e.g., column or channel) and, more importantly, changes direction at some point in the channel. Thus, wherever the analyte species is placed in the separation path, the net force will draw it to its unique equilibrium point where the force is 0. Unlike the common linear separation techniques in which the standard deviation of the bandwidth continually increases as the separation continues, the standard deviation of the bandwidth in equilibrium gradient methods remains constant after equilibrium has been reached. Since the net force on the analyte species goes to 0 only at the equilibrium point, any deviation from this point caused by diffusion, etc., gives rise to a restoring force that tends to keep the analyte species focused in a narrow region around the equilibrium point. The key objective of an equilibrium gradient method is to form a net force gradient on analyte species along the separation channel. From this general characteristic, we can classify equilibrium gradient methods into two categories. In one category (constant field), a constant external field is applied along the separation path. A gradient in some property, such as pH or density, also exists along the path. The constant external field combined with the property gradient exerts a force gradient on the analyte species. This category covers most of the known equilibrium gradient methods, such as density gradient sedimentation3-5 and isoelectric focusing.6-10 Both of these methods were established in the mid 1900s, and they have played important roles in the separation of biological samples. In density gradient sedimentation, a density gradient is established along a tube subjected to a centrifugal force. The analyte species moves in the tube under the action of the centrifugal field and stops at the (2) Giddings, J. C.; Dahlgren, K. Sep. Sci. 1971, 6, 345-356. (3) Meselson, M.; Stahl, F. W.; Vinograd, J. Proc. Natl. Acad. Sci. U.S.A. 1957, 43, 581-588. (4) Meselson, M.; Stahl, F. W. Proc. Natl. Acad. Sci. U.S.A. 1958, 44, 671682. (5) Schildkraut, C. L.; Marmur, J.; Doty, P. Proc. Natl. Acad. Sci. U.S.A. 1959, 45, 430-443. (6) Kolin, A., J. Chem. Phys. 1954, 22, 1628-1629. (7) Kolin, A. In Electrofocusing and Isotachophoresis; Radola, B. J., Graesslin D., Eds.; de Gruyter: Berlin 1977, 3-33. (8) Svensson, H. Acta Chem. Scand. 1961, 15, 325-341. (9) Svensson, H. Acta Chem. Scand. 1962, 16, 456-466. (10) Svensson H. Arch. Biochem. Biophys. Suppl. 1 1962, 132-140. 10.1021/ac020027w CCC: $22.00
© 2002 American Chemical Society Published on Web 07/31/2002
position where its density is equal to the density of the surrounding medium. In isoelectric focusing, a pH gradient is established along the separation channel. The analyte species moves along the separation channel under the action of an electric field, and stops at the pH value where it becomes neutral (isoelectric point). Since bulk flow is usually not present in the separation channel, methods in this category have the advantage that no diffusion except molecular diffusion occurs, and the technique should produce very sharp bands. However, since the gradient results from some physical or chemical property, it is not easy to change the shape of the gradient once it is formed. In the other category (field gradient), the force gradient results directly from a gradient in the external field or combination of fields. As opposed to the constant field category, the shape of the gradient is easy to change, which makes it possible to easily change the equilibrium points of analyte species. As we show below, improved resolving power becomes possible on the basis of this characteristic. However, since a field gradient is not always easy to form in the first place, methods in this category have not developed as quickly. In the 1990s, excited by the original idea of O’Farrell,11 Ivory and co-workers introduced a new equilibrium gradient method for charged molecules, which they called “field gradient focusing”.12-16 In this method, an electric field intensity gradient was formed along the separation channel. The electrophoretic velocities of charged analyte species changed proportionally with the electric field intensity gradient and were opposed by a hydrodynamic flow from a pump. Thus, the analyte species would move to and be focused at positions along the channel where their electrophoretic migration velocities balanced the pump flow. Following the idea behind the name of isoelectric focusing, in which the analytes are focused according to their isoelectric points, we will refer to this technique as “electromobility focusing” or EMF, which means that the analytes are focused according to their electrophoretic mobilities. Although many of the advantages of equilibrium gradient methods have been known for over 50 years, their potentials in separation have not been fully explored. Giddings proved that equilibrium gradient methods in the constant field category have a similar resolving power when compared to their kinetic counterparts.2 However, in this paper, we show that equilibrium gradient methods in the field gradient category could be used to produce a much higher peak capacity by applying a nonlinear field intensity gradient. Using EMF as an example, a separation model was established with a piecewise electric field intensity profile. Separation performance, that is, peak capacity and separation time, was simulated based on the model, and the simulation results showed that high peak capacity could be achieved in a relatively short time by using a parallel separation array. BASIC THEORY OF EQUILIBRIUM GRADIENT METHODS WITH FIELD GRADIENT Throughout this paper, we assume only low concentrations of the analytes, and we assume that uncoupled (linear) flux equations (11) O’Farrell, P. H. Science 1985, 227, 1586-1589. (12) Koegler, W. S.; Ivory, C. F. Biotechnol. Prog. 1996, 12, 822-836. (13) Koegler, W. S.; Ivory, C. F. J. Chromatogr. A. 1996, 229, 229-236. (14) Greenlee, R. D.; Ivory, C. F. Biotechnol. Prog. 1998, 14, 300-309. (15) Huang, Z.; Ivory, C. F. Anal. Chem. 1999, 71, 1628-1632. (16) Ivory, C. F. Sep. Sci. Technol. 2000, 53 (11), 1777-1793.
are adequate to represent transport of the analytes. We also assume that transport and separation of analyte occurs in only one dimension. Taking the axis of transport as the x coordinate, we define the following: J ) flux density of the analyte c(x) ) concentration of the analyte at point x u ) velocity of the bulk flow P(x) ) intensity of the external field at point x q(x) ) negative value of the gradient of the external field intensity and can be expressed as
q(x) ) -
∂P(x) ∂x
(1)
m ) velocity of the analyte induced by the external field with unit intensity DT ) dispersion coefficient that represents the sum of all contributions to effective diffusion W(x) ) translational velocity of the analyte at point x and can be expressed as
W(x) ) mP(x) + u
(2)
Actually, the velocity of bulk flow, u, induced by either a pump or an electroosmotic flow can also be treated as a velocity induced by an external field where the value of m is the same for all analytes. If there is more than one external field applied to the analyte in the channel, a similar expression can be obtained by summing all of the velocities induced by these external fields. The basic theory of equilibrium gradient methods with a field gradient can be derived from the general transport equation17
J ) W(x)c(x) - DT
∂c(x) ∂x
(3)
In equilibrium gradient methods, a steady-state band will be formed for each analyte when the analyte is focused. For such a steady-state band, we have
J)0
(4a)
W(x0) ) 0
(4b)
and
where x0 is the focusing position of the analyte. Substituting eq 4a into eq 3 and rearranging terms, we obtain
W(x) ∂c(x) - c(x) )0 ∂x DT
(5)
Under the assumptions described in the beginning of this section, W(x)/DT can be treated as independent of c(x), and the general solution for eq 5 is18 (17) Giddings, J. C. Sep. Sci. Technol. 1979, 14 (10), 871-882. (18) Braun, M. Differential Equations and their Applications; Springer Verlag: Heidelberg, 1983, pp 11-20.
Analytical Chemistry, Vol. 74, No. 17, September 1, 2002
4457
(∫ ( ) ) x
c(x) ) c0 exp
x0
W(x) dx DT
(6)
where c0 is the concentration of the analyte at the focusing position, x0. To obtain a clearer picture of the focused band, we describe W(x) as a Taylor expansion about the position, x ) x0, where the analyte is focused.
W(x) ) W(x0) +
( ) ∂W(x) ∂x
(x - x0) +
x)x0
( ) 2
1 ∂ W(x) 2 ∂x2
(x - x0)2 + ‚‚‚ (7) x)x0
If W(x) is linear in x, the second and higher order terms are 0. Alternatively, if the focused band is sufficiently narrow, the second and higher orders are negligible and may be ignored.2 If these two conditions are not satisfied, as illustrated below, a conservative estimation can be obtained based on eq 7. Substituting eq 4b into eq 7, we obtain
W(x) =
(
)
∂W(x) ∂x
(x - x0) ) - mq(x0)(x - x0)
x)x0
(8)
In this case, we assume that m does not change with x, which is true for equilibrium gradient methods with a field gradient, such as EMF. On the other hand, m does change with x in equilibrium gradient methods with a constant field, such as isoelectric focusing. Substituting eq 8 into eq 6 after integration, we obtain
c(x) ) c0 exp
(
)
- mq(x0)(x - x0)2 2DT
(9)
The concentration at the focusing position, c0, is determined by the total amount of the analyte
M)
∫
x
+∞
-∞
Ac(x)dx ) Ac0
2πDT
(10)
mq(x0)
where M is the total amount of the analyte, and A is the crosssectional area of the channel. The integration is carried out over the domain -∞ < x < +∞ instead of the actual channel domain, since this simplification has a negligible impact on accuracy. Substituting eq 10 into eq 9, we obtain
M
c(x) ) A
x
2πDT
exp
(
)
- mq(x0)(x - x0)2 2DT
(11)
mq(x0)
Therefore, the focused band is a Gaussian distribution with a standard deviation of
σ) 4458
x
DT
mq(x0)
(12)
Analytical Chemistry, Vol. 74, No. 17, September 1, 2002
As can be seen from eq 11, a positive value of mq(x0) is required to obtain a focused band, which means that restoring forces will be exerted on both sides of the band. An approximate expression for resolution can also be obtained by combining eqs 8 and 12
Rs )
∆x ) 4σ
|W(x′0)|
xmq(x0)DT
(13)
where x′0 is the focusing position of the second analyte. It can be found from eq 12 that the width of the band is inversely proportional to the square root of the field intensity gradient, which means a steeper gradient will produce a sharper peak and, thus, higher peak capacity. However, from eq 13, we know that the resolution is also inversely proportional to the square root of the field intensity gradient, which means that a shallower gradient will give higher resolution. It may appear that resolution and peak capacity cannot be improved simultaneously. However, if we assume a nonlinear field intensity gradient in which the first part of this gradient is rather steep, the analytes can be tightly ordered in this segment according to eq 12. If the following part of the gradient is rather shallow, the analytes can be separated with high resolution as they move into this segment according to eq 13. In this manner, both peak capacity and resolution can be improved simultaneously. However, this is valid only if the sample components can be moved from the first segment to the second after focusing in the first. Even though a nonlinear gradient has been used in equilibrium gradient methods with constant field to improve resolution, the above idea cannot be applied, since the focusing positions cannot be easily changed in these methods. On the other hand, as mentioned earlier, focusing positions can be easily changed in equilibrium gradient methods with field gradient by changing the shape of the field gradient; therefore, dynamic improvement of peak capacity can be realized. ELECTROMOBILITY FOCUSING WITH A STEPWISE LINEAR ELECTRIC FIELD GRADIENT Here, we use EMF as an example to illustrate the idea described above. Although Ivory and co-workers have demonstrated that resolution in EMF can be improved by controlling the electric field intensity,15 dynamic improvement of peak capacity by using a nonlinear electric field intensity gradient has never been reported. Since EMF can focus only analytes with the same sign charge, either positive or negative, at one time, we consider only negatively charged analytes in this section. A treatment for positively charged analytes can be derived easily in a similar manner. For negatively charged analytes, a monotone nondecreasing electric field intensity profile is required to focus these analytes in EMF according to eq 12. Meanwhile, to demonstrate dynamic improvement of peak capacity, the gradient of the electric field intensity should be steep at the beginning and shallow at the end of the channel, which means that the electric field intensity should be concave. Although such a concave, monotone nondecreasing electric field intensity profile can have different shapes, to simplify the model, we will start with one of the most simple cases, which is a piecewise linear profile. However, as illustrated later in this paper, we show that the results obtained from this
simple case will be valid for any concave, monotone nondecreasing electric field intensity profile. Channel Design. We consider the simple case of a piecewise electric field defined as follows:
E(x) ) a1 - b1x
for 0 e x < l1
(14a)
E(x) ) a2 - b2x
for l1 < x e l1 + l2
(14b)
E(x) ) 0
for x < 0 or x > l1 + l2
(14c)
The channel starts at x ) 0, and the direction of the electric field is the direction of the x axis. The bulk flow has the same direction as the electric field, and the sample (in this case, containing negatively charged analytes) is introduced into the channel at x ) 0 by the bulk flow. Since the analytes are negatively charged, the electrophoretic migration of the analytes is toward the beginning of the channel. The first part of the channel has a steep electric field intensity gradient defined by eq 14a with a length of l1, and the second part of the channel has a shallow electric field intensity gradient defined by eq 14b with a length of l2. Here we define b1 and b2 as the negative values of the gradients of the electric field intensity in the first and second parts of the channel, respectively, where |b1| . |b2|. To focus the negatively charged analytes, the electric field intensity must increase monotonically with distance from the start of the channel. Therefore, the electric field intensity gradient must be positive in both parts of the channel, which means that b1 and b2 are both negative. The total length of the channel is L ) l1 + l2, and detection is achieved at the end of the channel, that is, at x ) l1 + l2. Such an electric field can be established in several ways, such as by using a channel with changing cross-sectional area.12,13 We assume
E(x) ) 0
at x ) 0
(16)
We also assume E(x) is continuous, implying that the two linear segments join at x ) l1, and therefore,
- b1l1 ) a2 - b2l1
(20)
Vf )
V C+1
(21a)
Vs )
CV C+1
(21b)
we obtain
and
where V ) Vf + Vs is the total voltage drop over the channel. As should be pointed out, C is only determined by the structure of the channel and remains the same once the channel is built, although V may be altered. Combining eqs 18 and 21a, we obtain
b1 ) -
V 2 C+1l2
(22)
1
Combining eqs 16, 17, 19 and 21b, we obtain
b2 ) -
(
)
(23)
)
(24)
2l2 V 2 C2 C+1l l1 2
and
a2 )
(
2V 1 2 l1C + C + 1 l1 l2 l 2 2
(15)
Therefore,
a1 ) 0
Vs Vf
C)
(17)
Thus, the parameters that define the electric field intensity gradient (b1, b2, a1, and a2) can be obtained from the parameters of the channel structure (l1, l2, and C) and the total voltage drop, V. During the separation, the parameters of the channel structure will be kept constant, but the total voltage drop will decrease with time, which makes the electric field intensity gradient also change with time in the same manner. Separation Procedure. According to eqs 2 and 4b, the focusing position of the analyte j in an EMF method satisfies
W(x0j) ) u + µjE(x0j) ) 0
(25)
The voltage drop over the first part of the channel is
Vf )
∫
l1
0
b1l12 (- b1x)dx ) 2
(18)
x0j )
The voltage drop over the second part of the channel is
Vs )
By defining
∫
l1+l2
l1
b2l22 - b2l1l2 (a2 - b2x)dx ) a2l2 2
where µj is the electrophoretic mobility of the analyte j, and x0j is the focusing position of the analyte j. Substituting eqs 14a and 14b into eq 25, we obtain
(19)
u + µjai µjbi
for i ) 1 or 2
(26)
Since E(x) is monotonically increasing along the channel, there can be either only one solution for x0j between 0 and l1 + l2, or no solution. No solution means that analytes with weak electrophoretic mobilities move out of the channel with the bulk flow. Analytical Chemistry, Vol. 74, No. 17, September 1, 2002
4459
According to eq 12, the focused band of the analyte j in EMF is usually a Gaussian distribution with a standard deviation of
σ)
x
DT biµj
for i ) 1 or 2
l1 + l2 - 4σ ) (27)
Because of the existence of bulk flow from a pump, the dispersion coefficient, DT, can be very large in EMF. To reduce dispersion, the channel can be packed with some particles. Dispersion effects in packed channels have not been well-studied; however, a very simple approximation was given by Ivory and co-workers14 as
DT ) 2udp
second resolved analyte will satisfy
u + a2µ2 b2µ2
(34)
Substituting eqs 23, 24, and 27 into eq 34, and solving for µ2, we obtain
µ1
µ2 ) 1-4
(28)
x (
) Kµ1
)
DT Cl1 - 2l2 ul2 Cl1 - l2
(35)
where where dp is the diameter of the particles. To set up a strategy for separation, we specify a range of electrophoretic mobilities for analytes that will be separated in the channel. Let µ1 represent the least negative value and µm represent the most negative value; then, for any analyte j that can be focused inside the channel, we have
µm e µj e µ1 < 0
(29)
In performing a separation, the sample is first introduced into the channel by the bulk flow. The bulk flow velocity, u, is set to be constant throughout the separation, and the initial total voltage drop, V1, is set to let the focusing position of the first analyte (µ1) be exactly at the end of the channel, that is, at x ) l1 + l2. According to eq 26, we have
l1 + l2 )
u + µ 1a 2 µ1b2
(30)
Solving for µ1, we obtain
µ1 ) -
u a2 - b2(l1 + l2)
(31)
(
)
u 2µ1 Cl1 - l2 C + 1 l1l2
(
)
)
(36)
From eq 36, we find that K is always >1. Substituting eq 28 into eq 36, we obtain
1
K) 1-4
x (
)
2dp Cl1 - 2l2 l2 Cl1 - l2
(37)
In this case, K is determined by the channel structure only and has nothing to do with the operating parameters, such as the linear velocity of the bulk flow. Thus, in the initial conditions, the first analyte (µ1) is focused at the end of the channel, and the second analyte (µ2) with unit resolution from it has a mobility of µ2 ) Kµ1. In the second step, the total voltage drop is decreased from V1 to V2 to move the first analyte (µ1) out of the channel and the second analyte (µ2) to the end of the channel. From eq 33, it is easy to see that V2 must satisfy
u 2µ2 Cl1 - l2 C + 1 l1l2
(
)
(38)
Since µ2 ) Kµ1, we obtain
(33)
To obtain unit separation, the resolution between the first analyte (µ1) and the second analyte (µ2) should be 1, which corresponds to a distance of 4σ between the focusing positions of the two analytes. Therefore, the electrophoretic mobility of the 4460
x (
DT Cl1 - 2l2 ul2 Cl1 - l2
(32)
Thus, to set the focusing position of the first analyte (µ1) at the end of the channel, the initial total voltage drop, V1 must be set to
V1 ) -
1-4
V2 ) -
Substituting eqs 23 and 24 into eq 31, we obtain
u µ1 ) 2V1 Cl1 - l2 C + 1 l1l2
1
K)
Analytical Chemistry, Vol. 74, No. 17, September 1, 2002
V2 ) -
u ) V1/K 2µ1 Cl1 - l2 K C + 1 l1l2
(
)
(39)
Thus, decreasing the total voltage drop to V1/K moves the second analyte (µ2) to the end of the channel. Continuing to decrease the total voltage drop systematically to V1/K2, V1/K3, etc., moves each resolved analyte to the end of the channel. In general, the sequence of resolvable analytes has electrophoretic mobilities that satisfy the difference equation
µj ) Kj-1µ1
(40)
To resolve these analytes, the total voltage drop should be decreased in the manner of
Vj ) V1/Kj-1
Table 1. Simulated Peak Capacities and Minimum Separation Times in EMF with Bilinear Gradienta peak capacity
(41) voltage (kV)/ linear velocity (cm/s)
Thus, the peak capacity between µ1 and µm, n, satisfies
µm ) K
n-1
(42)
µ1
5/0.0091 10/0.0182 20/0.0364 30/0.0546
- b2 (V/cm2)/time (h) 5/26 1/128 0.1/1278 634 896 1267 1551
1416 2003 2833 3470
4480 6336 8961 10 975
a Calculated for channel packed with 3-µm particles, l ) 9 cm, and 1 l2 ) 1 cm.
or
n)1+
ln(|µm|) - ln(|µ1|) lnK
(43)
From eqs 36 and 37, K is mainly determined by the channel structure; therefore, ideally, the peak capacity can be very high, whereas practically, it is determined by the accuracy of the channel construction and how precisely the operating parameters, such as V and u, can be controlled. It should be mentioned that as a result of the nonuniform distribution of electrophoretic mobilities of the analytes, the real number of peaks that can be separated will be less than the number given by eq 43, which is the same as in all other separation techniques. Separation Time. The separation time is the sum of the times required for each separation step. The accurate time can be calculated by integrating the translational velocity, W(x), and the lower bound time for each step can be obtained by assuming that each analyte moves at the same velocity of the bulk flow, u. According to our unpublished experimental results, this lower bound is a good approximation in most cases. Therefore, the time required for the first step, t1, which is the time required to move the first analyte (µ1) from the beginning of the channel to the end of the channel, can be expressed as
t1 )
l1 + l2 u
(44)
For all other steps, tc, which is the time required to move the analyte from its penultimate position to the end of the channel, can be expressed as
tc )
4σ 4 ) u u
x ( 2dpl2
)
Cl1 - l2 Cl1 - 2l2
(45)
Therefore, the lower bound of the total separation time, t, can be expressed as
simulation is now described. Consider a simulated channel with an electric field intensity gradient constructed with two linear segments (l1 ) 9 cm, and l2 ) 1 cm) as described above. The channel is packed with 3-µm gel filtration particles to reduce dispersion. Considering analytes with electrophoretic mobilities between -10-5 and -10-3 cm2/Vs, the peak capacity was determined by incrementally decreasing the voltage to drive analytes to the end of the channel, as described above. Table 1 gives the resulting peak capacities for a packed channel for various voltage drops, V1, and electric field intensity gradients in the second part of the channel, b2. The reason we use b2 instead of C is because b2 can give a much clearer idea of how shallow the electric field intensity gradient is. Also given is the lower bound of the time for separation (in hours). Note that the time is constant across various voltage drops for specific values of b2. The linear velocities of the bulk flow, which are constant across values of b2 for specific voltage drops, are also listed in the table. The entries in Table 1 are interesting in that the peak capacity can be very high. For example, a total voltage drop of 30 kV with b2 ) - 0.1 V/cm2 can produce a peak capacity of over 10 000 in a little less than 2 months (1278 h). On the other hand, a total voltage drop of 5 kV with b2 ) - 5 V/cm2 can result in a peak capacity of 634 in 26 h. Separation in Parallel. The long separation times for singlechannel EMF limit the potential of EMF in separating large numbers of proteins. However, the EMF channel can be specifically designed to focus analytes within only a specific mobility range. Supposing Emin e E(x) e Emax, then from eq 25, we obtain
-
u u eµeEmin Emax
(47)
Performance Simulation. Dynamic improvement of peak capacity in EMF with a nonlinear electric field intensity gradient has been previously demonstrated from both simulation and experimental results.19,20 However, to give a clearer idea of how the peak capacity can be significantly improved, a more detailed
From eq 47, only analytes with mobilities in this range can be focused in the EMF channel. All analytes with more negative electrophoretic mobilities will remain at or near the injection end, and all with electrophoretic mobilities less negative will be pushed out of the detection end without being focused. This characteristic allows focusing analytes with different electrophoretic mobilities in parallel channels. From eq 45, we find that the number of peaks within a specific range of electrophoretic mobilities is based on the difference in the natural logarithms of the bounding mobilities. Considering analytes with electrophoretic mobilities ranging
(19) Wang, Q.; Tolley, H. D.; Lefebre, D. A.; Lee, M. L. Anal. Bioanal. Chem. 2002, 373, 125-135.
(20) Wang, Q.; Lin, S.-L.; Warnick, K. F.; Tolley, H. D.; Lee, M. L. J. Chromatogr. A., accepted.
t ) t1 + (n - 1)tc
(46)
Analytical Chemistry, Vol. 74, No. 17, September 1, 2002
4461
Table 2. Simulated Peak Capacities and Minimum Separation Times Using Parallel Channels in EMF with Bilinear Gradienta peak capacity/time (h) - b2 (V/cm2) voltage (kV)
5
1
0.1
5 10 20 30
20 Parallel Units 640/1.6 1420/6.6 900/1.5 2020/6.6 1280/1.3 2840/6.4 1560/1.3 3480/6.4
4480/63.9 6340/63.9 8980/64.0 10 980/63.9
5 10 20 30
100 Parallel Units 700/0.5 1500/1.6 900/0.4 2100/1.5 1300/0.3 2900/1.3 1600/0.3 3500/1.3
4500/12.8 6400/12.9 9000/12.8 11 000/12.8
Calculated for channel packed with 3-µm particles, l1 ) 9 cm, and l2 ) 1 cm.
works well either when W(x) is linear or when bands are sufficiently narrow, which is valid in most cases. However, the dynamic focusing strategy described above consists of a series of steps in which each analyte is sequentially moved to a region of the channel where the electric field intensity gradient (b2) is very shallow, resulting in a wide peak. Although the electric field intensity profile illustrated above is a piecewise linear function, other nonlinear profiles can also be used to dynamically improve peak capacity, as long as they are concave and monotone nondecreasing. To include more general case, a conservative estimation is given below. The analysis above based on a piecewise linear profile of electric field intensity will still hold if the value of b2 is substituted with b2*, formulated as follows. Let E*(x) be a line defined as
E*(x) ) -b(x - x0)
(50)
a
where x0 is the focusing position of the analyte. Since E(x) is monotone, nondecreasing and concave, b can be chosen to satisfy between -10-5 and -10-3 cm2/Vs, the range of values is from 3 ln(10) ) - 6.9078 to - 5 ln(10) ) - 11.5129, or 4.6052 units. Dividing this range by N, for N channels operating in parallel, each channel has a range of 4.6052/N in logarithm units. If we build an array of N channels, the structural parameters of each channel can be specifically designed to focus analytes within only a specific mobility range each. Therefore, each channel will cover a nonoverlapping electrophoretic mobility range individually and collectively cover the whole range of electrophoretic mobilities of interest. In this case, the peak capacity, n′, can be expressed as
(
n′ ) N 1 +
4.6052 N ln K
)
(48)
and the separation time, t′, is
4.6052 t′ ) t1 + tc N ln K
(49)
Table 2 gives the total peak capacities and separation times for channel arrays operating in a parallel format. Compared with the results in Table 1, for the same operating conditions, that is, the same initial voltage and linear velocity, the peak capacities are almost the same for both single channel and parallel channel array. However, the separation times are dramatically different. For example, a voltage drop of 30 kV with a b2 equal to - 0.1 V/cm2 can produce peak capacities of over 10 000 in both single channel and parallel channel arrays. However, the separation time for a channel array with 20 parallel units is ∼64 h, and that for a channel array with 100 parallel units is a little over 12 h. In comparison, a single channel requires 1278 h to achieve the same separation. ADJUSTMENTS FOR LINEAR APPROXIMATION The general solution for the concentration profile of an analyte, c(x), is given by eq 6. However, the derivations following eq 6 are all based on the linearized form of the translational velocity, W(x), which is obtained by ignoring the second and higher orders of the Taylor expansion of W(x) given in eq 7. Such approximation 4462
Analytical Chemistry, Vol. 74, No. 17, September 1, 2002
|E*(x)| < |E(x)|, for |x - x0| < 3
x
DT bµ
(51)
Let b2* be the largest value of such b. Using this value of b2*, all results above will be conservative. The actual peak capacity will be larger than the calculated value using b2*. Similarly, the required voltage change will be smaller than calculated. However, if the sequential pattern of separation is followed as described, only using b2* in place of b2, the realized peak capacity will be the same as described, although the peak shapes will not necessarily be Gaussian. CONCLUSIONS Equilibrium gradient methods belong to a family of separation techniques in which analytes are forced to their unique equilibrium points by a force gradient. Although certain equilibrium gradient methods have been practiced for 50 years, their potentials in achieving high peak capacity have not been fully explored. In this paper, the basic theory for equilibrium gradient methods with a field gradient has been developed. Equilibrium gradient methods with a field gradient have several advantages. First, since the restoring force produced by the field gradient focuses the analyte species at their equilibrium points, peaks do not broaden with time. The separation parameters can be designed to overcome much of the band broadening from thermal diffusion and laminar flow. Second, the peak capacity is limited only by the quality of the separation system. In theory, with slight slopes, large voltage drops, and small particles in the channel, nearly any is attainable. Practically, however, one cannot currently use ultrahigh voltages and high pressures in the separation channels. The current theory indicates, however, that as the technical capabilities increase, equilibrium gradient methods with field gradients hold even greater promise. Third, equilibrium gradient methods with field gradients can be operated in parallel. The parameters characterizing a channel delineate the rate at which proteins can be separated. Since these can be determined in advance, setting up a parallel array is straightfor-
ward. In this paper, we have illustrated the savings in time attainable using a parallel array. EMF uses an electric field intensity gradient to induce a force gradient on charged analytes, such as proteins, and a constant hydrodynamic flow as an opposing force. A separation model for EMF with a simple piecewise electric field intensity gradient was established in this study. Charged analytes are tightly packed in order of electrophoretic mobility in the first section of the channel, where the gradient is steep, and then sequentially resolved as they move to the second section of the channel, where the gradient is shallow. The process can be easily controlled by carefully controlling the operating parameters, such as voltage drop along the channel. Equations relating operating parameters with separation performance, that is, peak capacity and separation
time, have been derived. Simulation results show that high peak capacity (over 10 000) is theoretically possible, but this application is not practical as a result of the long separation time (1278 h). However, by operating in a parallel configuration, the separation time can be dramatically reduced while still maintaining high peak capacity. Although the dynamic improvement of peak capacity was only illustrated with EMF in this paper, it is clear that other field gradient techniques are also possible as long as the field gradient involved can be easily and continuously changed. Received for review January 15, 2002. Accepted June 13, 2002. AC020027W
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