Biotechnol. Prog. 1993, 9,273-284
273
Effect of Intraparticle Convection on the Chromatography of Biomacromoleculest Douglas D. Frey,* Eberhard Schweinheim, and Csaba Horviith Department of Chemical Engineering, Yale University, New Haven, Connecticut 06520
The effect of intraparticle convection in chromatographic columns packed with gigaporous > on the band spreading of unretained particles (i.e., where dpore/dparticle biomacromolecules is investigated both experimentally and theoretically. A model is developed for the analysis of mass transfer in spherical particles of bidisperse pore structure when both convection and diffusion take place in the larger pores but only diffusion occurs in the smaller pores. The predictions of the model were experimentally verified. I t is demonstrated that gigaporous particles have advantages over conventional < for applications that do not require high porous particles (i-e.,where dpore/dparticle resolving power, to bring about fast separation. This is because columns packed with gigaporous particles can be operated a t high flow velocities without significant loss of efficiency due to the enhancement of mass transfer by intraparticle convection. The results of the model are used to examine t.he effectiveness of gigaporous column packings for rapid analytical chromatography and for the concentration and recovery of a dilute solute in a saturation-regeneration cycle utilizing frontal chromatography.
1. Introduction A number of investigators have addressed the influence of the configuration of the adsorbent on the performance of chromatographic processes. In the separation of biological substances by liquid chromatography, pellicular particles made from fluid-impervious microspheres (Horvhthand Lipsky, 1969;Kalghatgi and Horvlth, 1987,1992), gigaporous particles where dporeldpartic~e > (Afeyan et al., 1990a,b;Lloyd and Warner, 1990),continuous polymer phases ( H j e r t h et al., 1989),and fibers (Hegedus, 1988) have all been considered as means to improve the operating characteristics. An important feature of gigaporous particles is the presence of intraparticle convection, the effect of which on the mass transfer of slowly diffusing substances and consequently on the efficiency in biopolymer chromatography has only recently been investigated and exploited. Effects of this sort were first recognized by Kreveld and Van Den Hoed (1978) and examined in detail by Afeyan et al. (1990a,b), Liapis and McCoy (1992), and Rodrigues et al. (1991a,b). Related work in the field of reaction engineeringis discussed by Lu et al. (1992),Nir and Pismen (1977),Princeet al. (1991),RodriguesandFerreira (1988), and Rodrigues et al. (1982). According to these studies, columns with gigaporous packings have significant performance advantages over columns with conventional porous packings due to the high mass-transfer rates which are achieved when the former type of column is operated under conditions where intraparticle convection is operative. High-efficiency chromatography columns of this type are likely to have numerous applications in the biotechnology industry, both for analytical work and for the process-scale concentration and purification of proteins. Several investigators (Afeyanet al., 1990a,b;Nugent
* Author to whom correspondence should be addressed.
Presentedin part at the 14thInternationalSymposium in Column Liquid Chromatography (HPLC ’go), Boston, May, 1990,and at the NATO Advance Studies Institute Symposium on Chromatographic and Membrane Process Biotechnology, Azores, September, 1990. +
and Olson, 1990)have discussed the potential advantages of gigaporous column packings in such applications. The major goals of this study are to develop a model for predicting the performance of chromatography columns for the case where intraparticle convection is present, to test the model experimentally with unretained eluites, and to make predictions about the potential advantages of gigaporous column packings in chromatographic practice. Two types of commerciallyavailable 8-pmgigaporous particles are investigated: cross-linked polystyrene particles and alumina particles with a poly(butadiene)coating having 4000 and 5000 A as the larger pore diameters, respectively. The model developed in this work is based on a simple description of the governing transport processes and can be applied to particles having a bidisperse pore structure, where both convection and diffusion occur in the larger pores but only diffusion occurs in the smaller pores. Our approach for the prediction of transport rates in these systems entails the derivation of an apparent diffusion coefficient to account for the combined effect of diffusion and convection in a bidisperse pore network. The theoretical relations developed in this study are used to investigate the band spreading in columns packed with gigaporous particles for potential use in rapid, highresolution analytical chromatography of biomacromolecules. In addition, the usefulness of such columns for the process-scalerecoveryand concentration of a protein using frontal chromatography is examined.
2. Plate Height Relations for Linear Chromatography First we consider the plate height relations in order to describe the performance of analytical chromatography and to compare theoretical results with experimental data. This section briefly reviews these relations for a column containing particles of uniform pore structure with diffusion as the sole transport mechanism inside the particle. Linear chromatography is carried out at eluite concentrations sufficiently low so that Henry’s law applies. The Henry’s law equilibrium constant, Keq,is given by the
8756-7938/93/3009-0273$04.00/0 0 1993 American Chemical Society and American Institute of Chemical Engineers
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ratio of the concentration of adsorbed eluite (i.e., the amount of adsorbate per unit volume of solid adsorbent) to the eluite concentration in the mobile phase at equilibrium. If t R is the experimentally measured retention time of an eluite and t o is the retention time of an unadsorbed eluite which explores the entire mobile-phase space in the column, then the dimensionless retention factor, k’, is evaluated as
Since at equilibrium the concentration of the eluite in the pore fluid is assumed to be the same as that in the interstitial fluid, the phase ratio 4 is given by
where t is the intraparticle and a the interstitial porosity. The efficiency of the column is measured by the plate height, H, which is made dimensionless upon dividing by the particle diameter to obtain the reduced plate height, h. For uniformly porous column packings where intraparticle convection is absent, the reduced plate height is expressed by the following relationship (Giddings, 1965; Horvdth and Lin, 1976, 1978; KuEera, 1965): = hdisp + hext + hint + hkin
(3)
The magnitudes of the individual plate height increments on the right side of eq 3 can be evaluated as discussed below. The combined contribution to the plate height from deviations from plug flow in the column, from extracolumn effects, and from axial diffusion is denoted by hdisp and can, in its simplest form, be expressed as
hdisp= A
+ BiPe
(4)
More precise relations for hdispwhich account for coupling between the mechanisms just mentioned and radial molecular diffusion are discussed by Giddings (1965).The parameters A and B in eq 4 are characteristic of the packing, and Pe is the particle Peclet number given by ud,JD,, where D, is the diffusivity of the eluite in the bulk mobile phase, u is the interstitial fluid velocity, and d, is the particle diameter. The Peclet number is frequently termed the reduced velocity in chromatographic literature. Typical values of A and B in a well-packed column are 1.5 and 1.6, respectively (Katz et al., 1983). For such columns at moderate to high Peclet numbers, hdispis generally a minor contribution to the reduced plate height. The effect of the extraparticle (external) mass-transfer resistances, hex,,can be evaluated from the relationship hex,
=
(1- CY)(“~’)’CU’’~ pelt . j 3.27[c~+ (1- ~r)K,,’3~
(5)
Equation 5 was derived by combining the expression for the plate height contribution from external mass transfer with the mass-transfer correlation for a bed of particles developed by Wilson and Geankoplis (1966). The distribution parameter Keq’is the amount of eluite inside the particle per unit particle volume divided by the equilibrium concentration of the eluite in the external
a
b
FLOW
SUBSIDIARY TICLE
/
GIGAPORE
PARTICLE SURFACE
PARTICLE CENTER
Figure 1. Schematic illustration of the structure of a gigaporous particle (a) and of the two geometrical parameters for a gigapore (b), given by the angle forfied between the pore axis in the direction of flow and the tangent to the surface ( w ) and the distance of the gigapore from the center of the particle ( E ) .
fluid and is given by
+
Keq’ = t ( 1 - €)Keq (6) For moderate to high Peclet numbers, hext is also a relatively small contribution to the reduced plate height. In chromatographic practice, the contribution from intraparticle mass transfer, hint, has the greatest influence on the reduced plate height. Consequently, a major goal of the design of chromatography systems is to reduce the magnitude of this term. Intraparticle diffusion gives rise to the following contribution to the reduced plate height (Giddings, 1965; Horvdth and Lin, 1976, 1978; KuEera, 1965): ‘internal
-
Oa(1- C Y ) ( K , ~ ’ ) ~
- 30cX[a + (1- a)Keq’l”Pe
(7)
where the tortuosity 0 corrects the diffusion coefficient for geometrical effects in the porous media, including the presence of dead-end pores, and the factor X corrects for both frictional drag and steric exclusion effects caused by the presence of the pore wall. Slow adsorption kinetics can also contribute a plate height increment given by 30,
hki, = -Pe 8Kdp2 where K is the desorption rate constant (Giddings, 1965). This term is mentioned here only for completeness and will not be discussed further. The preceding relations have been developed to describe diffusion phenomena contributing to the efficiency of traditional chromatographic columns. Equation 7, however, does not apply to the case where two scales of particle porosity exist or where convection takes place in the gigapores. The goal of the present work is to derive a modified form of that equation which accounts for these effects.
3. Transport Processes in Porous Media The idealized structure of a typical gigaporous particle is illustrated in Figure 1,which shows a set of subsidiary particles defining the interconnected network of the gigapores. In the case illustrated, the subsidiary particles have a uniform network of intermediate-sized pores. Within the subsidiary particles a still smaller pore structure
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may exist, but this third scale of porosity will be ignored in this study. In the following development a single prime superscript (with the exception of k’ defined above) will refer to a property of either the network of gigapores or the particle itself. On the other hand, double prime superscripts will denote a property of either the pore network inside the subsidiary particles or the subsidiary particle itself. Thus, c’ is the volume fraction of gigapores in the particle, whereas &’isthe volume fraction of pores in the subsidiary particles. Similarly, the tortuosity and the fluid velocity are 8’ and u‘ in the gigapores and 8” and u” in the pores in the subsidiary particles, respectively. For all of the cases considered in this study, the fluid velocity in the pores of the subsidiary particles is assumed to be zero, Le., u” = 0. An effective diffusivity in the gigapores and in the pores of the subsidiary particles can be defined, respectively, as
and the distribution parameter Keql is given by
Keql = e’
DJ”
D/=,I,
Although the tortuosities 6’ and 8“ in eqs 9 and 10 can in principle be calculated if the pore structure is known, in most cases these parameters must be treated as empirical constants and determined experimentally. In contrast, the hindrance parameters A’ and A”, which are functions of the relative size of the eluite molecule and the pore, can usually be predicted with reasonable accuracy (Brenner and Gaydos, 1977). The eluite concentration at a particular point in the gigapore network will be denoted by the symbol C’. Similarly, q’ is the combined amount of adsorbed and unadsorbed eluite per unit volume a t a particular point in the particle, Le., it is the volume average of the eluite concentration at a particular point. Concentration variables averaged over the entire particle are denoted by an overbar, and when they are averaged over an entire single longitudinal pore (see Figure l ) , an overbar with a subscript gp is used. Note that variables with an overbar are functions of time only, whereas variables with an overbar and subscript gp are functions of both the time and the spatial variable 5. This last variable denotes the shortest distance between two longitudinal pores, one of which passes through the center of the particle as shown in Figure 1.
Since the total amount of eluite in the particle is the sum of the amount in the pore fluid and the amount adsorbed, it follows that
4’ = e’p + (1- 6’)“’
(11)
q‘can also be determined by averaging qgpl over the volume of the particle as follows: 24JOd’ ‘ q , , ’ [ d m dE(12) (d,,’I3 For bimodal pore structures, the phase ratio is given by Q’
=
(1- t’)(l - t ” ) ( l - a ) 4 = k‘lKeq= a (1- a ) [ &+ (1- e’)”’]
+
(13)
(14)
For a column packed with spherical particles, the Darcy’s law specific permeability (Bird et al., 1960) is given by the Blake-Kozeny equation as follows
B
E
uapLlAP =
(d,’)’a3
l 8 0 ( l - a)’ where u is the interstitial fluid velocity, p is the liquid viscosity, and LIAP is the reciprocal of the axial pressure gradient in the column. It is assumed that the fluid velocity in the gigaporescan also be estimated by the Blake-Kozeny equation, so that a relation similar to eq 15 applies to the gigapores when the subsidiary particles are spherical. The permeability of the gigaporous particles, B’, can then be expressed by
B’
and
+ (1- €’)[e’’ + (1- d’)Keq]
u‘c’ud-’lAP’ = ‘ P
(d,”)
’(
6’)
(16) . , 180(1- E’)’ where d,’/AP’ is the reciprocal of the pressure gradient which exists across an individual particle. When averaged over all of the particles in the bed, this quantity can be taken to be LIAP. For arbitrarily shaped subsidiary particles, an approximate relation for u’ results from replacing d,,” in eq 16 by 6 / M , where M is the external surface area per unit volume of the subsidiary particle (Bear, 1972). Accordingly, the permeability of gigaporoils particles having cylindrical subsidiary particles is approximated by eq 16, with the numerical factor 180 replaced by 120. Thus, if a and t’ are fixed, the intraparticle fluid velocity is approximately one-third higher with cylindrical than with spherical subsidiary particles. From eqs 15 and 16 it follows that, when both the particles and the subsidiary particles are spherical, the ratio of the intraparticle and interstitial fluid velocities is given by 3
The average pore size in a particle can be estimated as the diameter of a cylinder for which the surface area to volume ratio is the same as the surface area to void volume ratio of the particle (Bear, 1972). Therefore, for spherical subsidiary particles, dpore’= [2/3(t’/(l - &))Id(’, and for e’ = 0.4 it follows that dporel= d,”/2, i.e., the diameter of the gigapores is one-half that of the subsidiary particles.
4. Mass Transfer in Gigaporous Particles 4.1. Diffusion in Bimodal Porous Media. We consider the case where diffusion is the sole transport mechanism inside the particle and the subsidiary particles are nonporous. Under these conditions the concentration gradient of the eluite in the gigapores will be zero a t the particle center and largest a t the particle surface. Near equilibrium when the driving force for mass transfer is small, it is reasonable to assume that the eluite concentration in the gigapores is a parabolic function of radial distance from the center of the particle to its exterior surface. The eluite concentration profile can be written, therefore, as (Yang, 1987; Do and Rice, 1986)
C’(t,r) = a,(t) + a,(t)r’ (18) Combination of eq 18with a material balance for the eluite and the requirement that the concentration gradient is
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276
zero at the particle center, we obtain that
where C,’ is the concentration a t the outer surface of the particle. Equation 19 is identical to the linear driving force approximation proposed by Glueckauf (1955). With porous subsidiary particles, the diffusional flux of eluite at the outer surface of the particle can be expressed as
Equation 20 assumes that the eluite enters the particle only through the mouths of the gigapores. The factor f in eq 20 is less than unity and corrects for diffusional resistances in the subsidiary particles. According to Haynes and Sarma (1973),f can be evaluated as follows:
+ K,,)’ + (1- t”)t’(l + Keq)]
(1- t’)(t”)’(l
i=[l+(
[e’
J5.(
(”)I-’
De t f’ f
(21)
f
The preceding relations strictly apply only to the case of linear adsorption equilibrium. Biomacromolecules, however, often exhibit nonlinear adsorption equilibrium even at low concentrations. For this case the relations presented here can be applied in an approximate manner if K,, is evaluated as the slope of the isotherm a t the average concentration in the particle. In addition, matrix generalization methods, such as those described by Wong and Frey (1989), can be used to extend this treatment to competitive adsorption behavior. 4.2. Convective Intraparticle Mass Transfer. This section addresses the case when convection is the sole mechanism for mass transfer inside gigaporous particles, i.e., diffusion is neglected and no radial concentration gradients are present in the gigapores. Our approach for modeling this case is to develop a linear driving force approximation, analogous to that described in the previous section, for the case where the eluite concentration in the interstitial (i.e., extraparticle) fluid varies linearly with time. Although this method can be applied to particles of arbitrary geometry, including deformable particles which tend to have a flattening spherical shape when under compression ( H j e r t h et al., 1991),only relations for slab and spherical geometries will be presented here. For convenience, we first examine convective mass transfer in a single gigapore which traverses the particle in the axial direction from the upstream to the downstream side (see Figure 1). Since the eluite concentration at the upstream pore mouth is the same as that in the interstitial fluid, C,, and since the net mass flow is the difference between mass flows at the upstream and downstream pore mouths, the mass flow of eluite into the particle per unit area of particle surface is given by
where Cd,gplis the concentration at the downstream mouth of a particular pore and w is the angle between the tangent to the particle surface and the axis of the gigapore (Figure 1). Recall that the subscript gp refers to a variable pertinent to any single longitudinal gigapore, for which w is assumed to be the same a t both the upstream and downstream pore mouths.
When the interstitial fluid concentration varies linearly with time and when there is no mass transfer between individual gigapores, the eluite concentration in a particular gigapore is a linear function of distance between the upstream and downstream pore mouths. This condition together with eq 22 yields the relation
where qu’ is the value of q’ at the upstream pore mouth. Equation 23 with w = nJ2 is an exact description of mass transfer under the conditions described for a particle having a slab geometry. However, for spherical geometry each gigapore corresponds to a different value of w , and the average concentration of the eluite inside a longitudinal pore, qgpl, will be a function of the distance between that pore and the particle center which, as mentioned previously, is denoted by the symbol 6 in Figure 1. In particular, since the concentration of the eluite at the upstream pore mouth is the same for all of the gigapores, and since the condition that the fluid velocity is uniform in the particle implies that the derivative of the eluite concentration with respect to distance inside a gigapore is the same for all of the gigapores, it follows that qgplis a linear function of the length of the gigapore. Furthermore, as the pore length approaches zero, qgp’approaches the concentration at the upstream end of the pore (i.e., qu’). This implies that the average concentration in a longitudinal gigapore can be written as
When qgpl(t) is averaged over the volume of the particle by using eq 12, we obtain (25) If eq 25 is solved for ijgP,po’ and substituted into eq 24, the result can be written as 4(q’
- q,’)dd,’’
- 45’
+ Q”’ (26) 3dp‘ The flux associated with a gigapore can be expressed solely in terms of 5, qu’, and 4’ by substituting eq 26 into eq 23. This result can then be integrated over the surface of the particle and divided by the total surface area of the particle to yield
Qgpl =
(27) Equation 27 forms the basis for determining mass-transfer rates when convection inside the particle is present. 4.3. Combined Effects of Intraparticle Diffusion and Convection. If the convection and diffusion mechanisms for mass transfer are independent and additive, comparison of eqs 20 and 27 shows that the relative importance of convection and diffusion when the subsidiary particles are nonporous (i.e., f = 1)is measured by JconvlJdiff = 2pe’/45 (28) where Pe’ is a pore Peclet number based on the particle diameter, the intraparticle convective velocity, and the effective diffusion coefficient of the eluite in the gigapores, i.e., Per = u‘dp’/De‘. By combining eqs 20 and 28 we obtain an apparent effective diffusion coefficient which incorporates the effects of both diffusion and convection in the
Biotechnol. Prog., 1993, Vol. 9,No. 3 10
277 /
(
i
Afeyan
’
”
’
1
,’
/
100 \
4 -
a”
3 0
,,,A
without intraparticle convection (u’=O), ,,’
1
,?/
j
h
3-
%Sphere 1
1
2
10
20
200
0
100 200
400
600
800
1000
Pe
Pe’
Figure 2. Dependence of the normalized apparent intraparticular diffusivity on the pore Peclet number (Pe’ = u’d,//D,‘) according to Afeyan et al. (1990), Rodrigues and Ferreira (1988), and this work for both slab and spherical geometries. gigapores of a spherical particle as Dapp= f(1+ =)Dl 2Pe‘ where the factor f , which accounts for the diffusional resistances in the subsidiary particles, can be determined by substituting (1+ 2Pe’/45)D,‘ for D,’ in eq 21. For the case of slab geometry, eq 20 applies but with the particle diameter replaced by the slab width and with the numerical factor 10 in that equation replaced by 6. The procedure just described can then be repeated to yield an apparent diffusivity of the form Dapp= f(1+ Per 6ID:
(30)
where Pe’ = u‘dslab/Dmand dslabis the slab width. It is of interest to compare the relations developed here with approximations for DaPqemploying alternative assumptions since such a comparisonillustratesthe influence of model assumptions on mass-transfer rate predictions. In particular, Rodrigues and Ferreira (1988) developed a method for estimating Dappin a porous slab by comparing transfer functions which relate the surface concentration to the average concentration in the slab for the cases where only diffusion is present and where diffusion and convection are both present. This comparison led to the result
--]
1 1 -1D,‘ (31) tanh (Pe’12) (Pe’/2) As Pe’ approaches infinity, eq 31 yields values of Dappthat are equivalent to those from eq 30 for the case of nonporous subsidiary particles (i.e., f = 1). For finite values of Pe‘, eq 31 yields values that are somewhat lower than those from eq 30. Recently, Rodrigues (1991) used an analogy with the relation between the Thiele modulus and the catalytic effectivenessfactor to propose that eq 31 can be empirically modified to apply to spherical geometry by replacing the width of the slab in that equation by d,’/3. Another approach for the case of spherical geometry was taken by Afeyan et al. (1990b),who estimated Dappon the basis of order-of-magnitude arguments involving characteristic times for diffusion and convection. Their result, which does not account for diffusion in the subsidiary particles, has the same form as eq 29 but with the much larger numerical coefficient of 112 instead of 2/45. Figure 2
Figure 3. Illustration of the reduced plate height for a retained eluite (k’ = 2) as a function of the particle Peclet number for various types of porous column packings with 0’ = 4,cy = e’ = 0.4, and A‘ = 1. Nonporous subsidiaryparticles (f = 1)were assumed where applicable. For gigaporous particles, the increasing level of intraparticle convection is expressed by the increasing values of the subsidiary particle diameter to particle diameter, d,”/d,’. The results for pellicular packings are also shown for comparison. compares the various methods for predicting Dappfor both slab and spherical geometries in the range of 0 < Pe’ < 200, which is believed to apply to gigaporous particles in chromatographic practice. In the remainder of this work, eq 29 will be used to predict mass-transfer rates when intraparticle convection is present, since the assumptions underlying this relation appear to be reasonable and since this relation agrees closely with eq 31 modified to apply to spherical geometry.
5. Plate Height Relations for Gigaporous Stationary Phases Equation 7 gives the plate height increment due to intraparticle mass-transfer resistances when convection inside the particles is absent and the subsidiary particles are nonporous. In order to include the effect of these two phenomena on hint, we substitute IIDappfor the combination of terms W/(D,X’) in eq 7 to yield 6”a(l- a)(Keq’)* D,‘
hint
=
-Pe 30~’X’[a+ (1- (~)K,qll~Dapp
(32)
It is of interest to note that eq 32 with Dapp evaluated using eq 29 yields a flow rate dependency for the plate height similar to that resulting from the coupling theory of Giddings (1965), which accounts for the effect of molecular diffusion on axial dispersion. Although the other plate height increments may also be affected by intraparticular convection, we assume that the effect is sufficiently slight such that hdispand hextare adequately represented by eqs 4 and 5, respectively, for columns packed with gigaporous particles. Figure 3 illustrates the relation between the reduced plate height and the Peclet number calculated using eqs 4,5, and 32 for a retained eluite (k’= 2) and for nonporous subsidiary particles cf = 1). The results were obtained using typical parameter values and are shown for five cases as follows: (1)conventional porous particles with both u’/u and d//d,,’ approaching zero, such that there is essentially no intraparticle convection; (2) gigaporous particles having a moderately large pore size, i.e., d,”/d,’ (3) gigaporous particles of tke and u’lu = =3X
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278
d,"/d,'
= 7.7 x 400
15
1
A
300
h 200
100
0
0
50
100
150
Pe Figure 4. Reduced plate height as a function of the particle Peclet number for a ratio of subsidiary particle diameter to particle diameter of 7.7 X 10-3. Theoretical lines were calculated with A = 2.5,B = 1.6, 0 = 8,a = e = 0.4, X = 0.88, k' = 0, and u' = 0. A: Experimental data obtained with tryptophan on PLRP(300 A) under conditions of no retention.
type investigated in this study, i.e., d,"ld,' = 10-1and U'IU = (4) gigaporous particles of the type described by Carpenter et al. (1986) where the pore size is a substantial fraction of the particle size, Le., d,"/d,' = 2.5 X lo-' and u'/u = 5 X and ( 5 ) pellicular particles for which the contribution hint to the reduced plate height is assumed to be negligible. As shown in Figure 3, for a particular gigaporous packing, intraparticle convection yields a significant reduction in the reduced plate height only if the Peclet number is above a limit determined by the ratio d,"/d,' for the packing under consideration. In particular, if a = 6' it follows from eqs 17 and 29 that the plate height contribution from intraparticle mass transfer is reduced by one-half when Pe = (45/2)(d,'/d,")*.
6. Experimental Section 6.1. Instruments. The liquid chromatograph was assembled from a Model 1004 precision metering pump (Beckman, Altex Division, San Ramon, CA),a Model 7413 sampling valve (Rheodyne, Cotati, CA) with a 1-pL loop, and a Model LC 85B variable-wavelengthdetector (PerkinElmer, Norwalk, CT) with a 1.4-pL flow cell. In all experiments the wavelength and the time constant of the detector were set a t 214 nm and 20 ms, respectively. The system dead volume between the injection valve and the detector was minimized by using short 0.12 mm i.d. connecting tubes. All chromatographic runs were performed at room temperature between 23 and 25 "C. The chromatograms were obtained with a Model B41 stripchart recorder (Kipp and Zonen, Delft, The Netherlands). The column-packing device was assembled from a No. DSHF-302 air-driven pump (Haskel, Burbank, CA), a Craftsman 4HP 20-gal air compressor, and a 100-mL No. 316 stainless steel reservoir rated at 25 000 psi. 6.2. Materials. Chemicals. Acetonitrile (ACN), methanol, trifluoroacetic acid (TFA), uracil, and fructose were purchased from J. T. Baker (Phillipsburgh, NJ). Water was purified by a Barnstaedt Nanopure unit. Tryptophan, bovine carbonic anhydrase, chicken egg lysozyme, and a-amylase were obtained from Sigma (St. Louis, MO). Stathnary Phases. Macroreticular copolymers of polystyrene-divinylbenzene (PLRP) having 300,1000, and
0
0
500
1000
1500
2000
Pe
Figure 5. Reduced plate height as a function of the particle Peclet number for a ratio of subsidiary particle diameter to Theoretical lines were calculated particle diameter of 7.7 X withA = 2.5, B = 1.6,0= 8, a = e = 0.4, k' = 0,u' = 0, and with different values of X as indicated. Experimental data were carbonic anhydrase obtained on PLRP(300A) withlysozyme (o), (+), and amylase (v)under conditions of no retention.
4000 A mean pore diameters were kindly donated by Polymer Laboratories (Shropshire, UK). The nominal diameter of the spherical particles was 8pm. A Unisphere PDB 4.6 X 250 mm column was a gift from Biotage (Charlottesville, VA). It was packed with 8-pm alumina particles having an open (gigaporous) structure and a surface coated with a thin poly(butadiene) layer. Columns. No. 316 stainless steel columns of 100 X 4.6 mm were packed with a 10% (w/v) methanol slurry of the PLRP particles at 3500 psi by using methanol as the packing fluid. Before the column was removed from the reservoir, it was perfused with water for 20 min. Such conditioning of the column with water at the packing pressure was found to improve the reproducibility of the packing procedure with the styrenic material. Unisphere particles were obtained by carefully unpacking the commercial column. They were slurried in methanol (10% w/v) and repacked in the column of the above dimensions at 6000 psi by using methanol as the packing fluid. 6.3. Methods. All experiments were carried out by isocratic elution with acetonitrile containing 30 % (v/v) water and 0.1 % (v/v) TFA at room temperature. Tryptophan, carbonic anhydrase, lysozyme, and a-amylase were eluted in the mobile-phase volume, i.e., their retention volume was the same as that of uracil and fructose used as unretained tracers. The substances were dissolved in the mobile phase, and the concentrations of the samples were in the range from 1to 3 pg of protein per pL. The chromatograms were evaluated by measuring the retention time a t the apex, as well as the area and the half-height width of the peaks. The peaks were generally Gaussian except at the highest flow rates where they were slightly skewed in shape.
7. Comparison of Theory and Experiments 7.1. Conventional Macroporous Adsorbent. Ex-
perimental data obtained with PLRP(300 A) using tryptophan, lysozyme,carbonic anhydrase, and amylase having molecular diameters of 6,41,50.4, and 60 A,respectively, in the native form (Tanford, 1961) are shown in Figures 4 and 5. The eluent composition was 70% ACN and 0.1 % TFA (v/v). Under these conditions all eluites were effectively unretained and eluted in the same volume.
Blotechnol. hog., 1993, Vol. 9, No. 3
279 10
I
r yptup
101I
Lysozyme
n e
8
Brenner and Gay1
with intr apar t iclc convection
6
h
1
4
A
Amylase o
0 ‘ 0
0.1
\
2
0
0.2
20
0
40
data with the proteins indicated to the theoretical predictions by Brenner and Gaydos (1977) for different molecular to pore diameter ratios.
The van Deemter curves shown in Figures 4 and 5 were calculated using eqs 3-6 with the assumption that intraparticle convection was absent. A value of 8 = 8.0 was used for the tortuosity in the calculations because with this value the magnitude of A which gave the best fit for tryptophan (A = 0.88) conformed to the predictions of Brenner and Gaydos (1977). Data for tryptophan is the least affected by hindered diffusion among the eluites because of its small molecular size and therefore should yield the most accurate value for 8. For A and B the respective values of 2.5 and 1.6 were used to give the best overall fit to the experimental data. The value of A used is somewhat higher than the value of 1.5 suggested by Katz et al. (1983), which reflects the fact that the gigaporousparticles used here were packed less efficiently than the silica particles used in their study. The tortuosity of 8.0 used for the calculations in Figures 4 and 5 is in general agreement with measured tortuosities for consolidated porous media having a moderate to small pore size when a significant number of dead-end pores are present. For example, Yang (1987) has reviewed diffusion data for several types of silica and alumina particles and found tortuosities to be in the range of 2-6. In addition, if the empirical relations used by Katz et al. (1983) to account for intraparticle diffusion in silica are compared to eq 5, they correspond to a tortuosity of 6.3. In Figure 6 the dependence of the diffusionalhindrance factor, A, on the ratio of the molecular diameter of the eluite to the pore diameter (deluite/dpore) is illustrated for tryptophan and the three proteins when 0 = 8.0. In the calculations we have assumed that the diameters of the eluite molecules are unaffected by the denaturing conditions employed. Although the molecular diameter of the denatured protein is expected to be larger than that of the native form, Figure 6 indicates that the use of the molecular dimension of the native protein still leads to reasonable estimates of the degree to which hindered diffusion is present even under the conditions employed in this study. 7.2. Gigaporous Adsorbents. Small Eluite Molecule. Figures 7-9 illustrate experimental and theoretical results on the flow rate dependence of the plate height for tryptophan in columns packed with the gigaporous adsorbents PLRP(1000 &, PLRP(4000 &, and Unisphere. In calculating the theoretical curves depicted in Figures 7-9, hint was determined using eq 32, A’ and A“ were
60
100
80
Pe
detuite/dpore
Figure 6. Comparison of A values calculated from experimental
2.5 x lo-’
d,”/d,’
Figure 7. Reduced plate height as a function of the particle Peclet number for a ratio of subsidiary particle diameter to particle diameter of 2.5 X Theoretical lines correspond to A = 2.5, B = 1.6,8’ = 5.1, 8“ = 8, a = e’ = e” = 0.4, A‘ = 0.96, A” = 0.88,and k’ = 0; the solid line corresponds to the presence, and the dashed line to the absence, of intraparticle convection. Experimental data were obtained with tryptophan on PLRP(1000 A) under conditions of no retention. 10 d,”/d,’
= lo-’ without
8 -
h
,*
intraparticle convection
20 /0
20
40
60
80
100
Pe
Figure 8. Reduced plate height as a function of the particle Peclet number for a ratio of subsidiary particle diameter to particle diameter of 1 X 10-l. Theoretical lines correspond to A = 3.4, B = 1.6, 8’ = 3.4, 8” = 8, (Y = e’ = e“ = 0.4, A’ = 0.99, A’’ = 0.88,and k’ = 0; the solid line applies to the presence, and the dashed line to the absence, of intraparticle convection. Experimental data were obtained with tryptophan on PLRP(4000 A) under conditions of no retention.
calculated according to Brenner and Gaydos (1977), f was obtained from eq 21, and the parameter B was taken as 1.6. Since PLRP(4000 A) and PLRP(1000 A) are structurally similar to PLRP(3OO A), except that the last adsorbent has no gigapores, we used for the subsidiary particles of the two styrenic adsorbents as well as for the Unisphere adsorbent the value of 8” = 8 as discussed in the preceding section. From the relations in section 4, and using d / / d < values of lP1,1.2 X 1P2, and 2.5 X 10-3, the respective u’/u ratios were estimated as 1.0 X for PLRP(4000 A), 2.0 X 10-2 for Unisphere, and 6.25 X 10-4 for PLRP(1000 A). The subsidiary particles in the two PLRP adsorbents were assumed to be spherical, while those in Unisphere were assumed to be cylindrical. The values of A used were 2.5 for PLRP(1000A), 3.4 for PLRP(4000 A), and 3.7 for Unisphere, which indicates that the last two adsorbents were packed into columns less efficiently than the first one. The tortuosity of the gigapores,
280
Biotechnol. Prog., 1993, Vol. 9, No. 3 200
10 d,"/d,'
= 1.2 x
d,"/d,'
lo-'
= 2.5 x lo-*
without 150
6 -
h 100
h
intraparticle convection
4 -
I
I 50
I
0
J
I 20 60 80
0
100
40
Pe
Figure 9. Reduced plate height as a function of the particle
Peclet number for a ratio of subsidiary particle diameter to particle diameter of 1.2 X lo-'. Theoretical lines correspond to A = 3.1, B = 1.6, 8' = 3.1, 8" = 8, CY = e' = t" = 0.4, A' = 0.99, and A" = 0.88; the solid line applies to the presence, and the dashed line to the absence, of intraparticle convection. Experimental data were obtained with tryptophan on Unisphere under conditions of no retention. 0', was chosen to yield the best agreement between
experimental and theoretical results, and the values were 5.1, 3.4, and 3.1 for PLRP(1000 A), PLRP(4000 A), and Unisphere, respectively. The decreasing tortuosity with increasing pore size is often observed for porous media and probably reflects a decreasing number of dead-end pores as the pores become larger. Proteins. The results discussed above offer all of the physical properties needed to predict the efficiency of columns packed with gigaporous adsorbents over a wide range of Peclet numbers and to assess the effect of intraparticle convection. In Figures 10-12, experimental and theoretical van Deemter plots of three proteins (lysozyme,carbonic anhydrase, and amylase) are depicted for each of the gigaporous adsorbents. All of the parameters were evaluated as before. For comparison, Figures 10-12 also show van Deemter curves calculated for lysozyme in the hypothetical case where no convection takes place in the gigapores, i.e., u' = 0 and Dapp= De'. I t is seen from Figures 10-12 that there is a good agreement between the experimental data and the theoretical predictions for each adsorbent. The differences in the theoretical results for the three proteins are due to the corresponding differences in the amount of hindered diffusion present. These differences, however, are rather small in Figures 10-12 since diffusion in the subsidiary particles is relatively fast compared to the mass-transfer rate in the gigapores for all of the cases investigated (i.e., f was always greater than 0.9) and since the diameter of the gigapore is much larger than the molecular size of the eluite. The particular results obtained in Figure 11 for PLRP(4000 A) are similar to those reported by Lloyd and Warner (1990) for this adsorbent when myoglobin was used as an eluite under conditions of no retention. The relatively small plate heights shown for the PLRP(4000 A) and Unisphere packings in Figures 11 and 12 are partially due to the fact that the eluites were not retained under the conditions of the experiments. For retained eluites the plate heights are expected to be significantly larger at the same value of the Peclet number, as illustrated in Figure 3 for tz' = 2.
d,"/d,'
= lo-'
,,>Lysozyme
II
i
11
(u'=O) Amvlase
30
h
20
10 /
8. Prospects for Gigaporous Column Packings 8.1. Analytical Chromatography of Biomacromolecules. It has been shown in the preceding sections that intraparticle mass-transfer resistances in a gigaporous particle are commensurate with that of a standard porous particle which is smaller in diameter by the ratio (D,'/Dapp.)112.For example, in the case where the ratio D,'/Dappis 25, the diameter of the gigaporous particle can be 5 times larger than that of a conventionalporous particle in order to obtain the same efficiency. In turn, the column
.
281
Biotechnol. Prog., 1993, Vol. 9, No. 3
h
L
j>
r,
,
0
0
Lysozyme (u'=O)
I
,
,
,
500
,
,
,
di'idp';,
1000
1 2 , x, IO-: 1500
,
1
0'
"
0
2000
10
"
20
containing the gigaporous packing will have 25 times the permeability of the column with the conventional packing. We now investigate the feasibility of exploiting this effect when designing chromatographic systems for rapid analysis of protein mixtures with high resolution. To facilitate comparisons, it is assumed that the value of k' is the same for all of the columns under consideration. For a fixed pressure drop and number of plates, the analysis time is shortest when the column is operated at the optimal flow velocity, i.e., at the minimum plate height (Guiochon, 1980). Figure 13 shows the relation between the reduced plate height and the Peclet number near the plate height minimum for columns packed with conventional porous, gigaporous, and pellicular particles. For the gigaporous particles the ratio of d,"/d,' was taken to be lo-' to reflect the properties of the material used in the experiments. The results are depicted in Figure 13 for the same conditions as in Figure 3. As seen, the minimum plate heights and optimum velocities are essentially identical for gigaporous and conventional porous particles. This implies that the minimum analysis time and the corresponding particle size are also nearly identical for the two cases. Consequently, there is little advantage in using gigaporous particles in the neighborhood of the optimal flow velocity. However, since D, is very small for macromolecules, the minimum separation time for their mixtures at a given pressure drop and plate number would be obtained with particles too small (d; = 0.1 pm) to be of practical use a t the present level of technology (Antia and Horvlth, 1988). Thus, comparisons of various columns on the basis of the minimum separation time is of limited use in the case of biomacromolecular separations. There is another way of looking a t the problem, however, by comparing the separation time as a function of the plate number with columns which are packed with particles having dimensions that correspond to the current practice of liquid chromatography and by examining the effect of intraparticle convection. The results of such an investigation are illustrated in Figure 14. Calculations were made
30
"
40
" 50
Pe
Pe
Figure 12. Reduced plate height as a function of the particle Peclet number for a ratio of subsidiary particle diameter to particle diameter of 1.2 X 10 I . Theoretical lines correspond to A = 3.7, B = 1.6, 0' = 3.1, 0" = 8, LY = er = e" = 0.4, k' = 0, and A' = 0.934, 0.920, and 0.906 as well as A'' = 0.51, 0.42, and 0.35 for lysozyme, carbonic anhydrase, and amylase, respectively. The solid lines apply to the presence of intraparticle convection. The dashed line was calculated for lysozyme assuming the absence of intraparticle convection. The theoretical line for carbonic anhydrase is not shown; it lies between those for lysozyme and amylase. Experimental data were obtained on Unisphere with lysozyme (o),carbonic anhydrase ( 0 ) ,and amylase (v)under conditions of no retention.
"
Figure 13. Dimensionless van Deemter plots a t low particle Peclet numbers for conventional porous packings (u' = 0), gigaporous packings (d,"/d,' = lo-'), and pellicular packings. Conditions are the same as in Figure 3. 2000
I
'
'
'
I
1000
100
200
A
1000
2000
N Figure 14. Separation time versus plate number calculated for 5-cm-long columns packed with 8-,um particles of various pore sizes as indicated. Conditions correspond to D, = cm2s-l, del,,,, = 40 A, f = 1, 0' = 4, k' = 3.3, and u'/u =
for three 5-cm-long columns packed with 8-pmparticles having 4000-, 1000-,and 300-Apore diameters, assuming f = 1, k' = 3.3,O' = 4, and D, = For the adsorbent with 4000-A pores, U'IU was taken to be lo-*, but no intraparticle convection was considered for the other two cases. It is seen from Figure 14 that the column packed with particles having 4000-A pores yields significantly shorter analysis times than the column packed with particles having 1000-8,pores, only when the number of theoretical plates required to bring about the separation is moderate, i.e., less than 200 in this particular case. This is because the value of the Peclet number needed becomes so small with decreasing plate height that the mass-transfer advantages of gigaporous adsorbents vanish (cf. Figures 3 and 13). Figure 14 also indicates that, when the required number of theoretical plates is large, it is important to have pores sufficiently large to provide for unhindered diffusion of the eluite. In particular, for the eluite of 40-A molecular size used in the calculations, X increases from 0.5 to 0.85 when going from 300- to 1000-Apores. However, as seen in Figure 14, a further increase in pore size to 4000 Aresults in only a small additional reduction in separation time, unless a moderate number of theoretical plates
Biotechnol. Prog., 1993, Vol. 9, NO. 3
282
suffices for the separation. This also implies that the advantages of gigaporous packings diminish considerably when mixtures with many components have to be separated, and thus a large number of theoretical plates is required for complete resolution. As discussed by several workers (Synder, 1980; Frey, 1990), the column efficiency under isocratic conditions determines the efficiency in gradient elution. Therefore, the conclusion reached in this analysis on the basis of an isocratic elution model applies as well to the comparison of gigaporous and conventional porous particles under gradient elution conditions. In particular, if k’ differs significantly among the eluates over the range of eluent composition used in a gradient protocol, a moderate number of theoretical plates may be adequate to produce the desired separation. Gigaporouspackings can be useful if employed under such conditions to yield rapid separations with gradient elution. In Figure 14 we also consider the behavior of a 5-cm column containing 8-gm pellicular particles for which the reduced plate height is taken as the sum hdisp + hext accordingto eqs4 and 5. Comparison of the plots in Figure 14suggests that a column packed with pellicular adsorbent particles yields substantially shorter analysis times when high resolving power, Le., a large number of theoretical plates, is needed than columns packed with conventional or gigaporousparticles. The advantages of micropellicular sorbent particles for rapid high-resolution chromatography of biological macromolecules have been demonstrated previously (Kalghatgi and Horvlth, 1987; Kalghatgi et al., 1992). Moreover, micropellicular sorbents are expected to be stable at elevated temperatures, such that they offer the possibility of performing liquid chromatography at high temperatures where the decrease in eluent viscosity and the increase in eluite diffusivity yield additional advantages in terms of column efficiency (Antia and Horvlth, 1988). Thus, the use of micropellicular particles would appear to be the preferred route for achieving rapid, high-resolution, analytical separations of biomacromolecules. On the other hand, the relatively low loading capacity of columns packed with pellicular stationary phases circumscribes their employment in preparativescale separation. In such applications, particularly when the requirements for high column efficiency are extenuated, gigaporous column packings may have advantages due to the high speed of separation and relatively high column loading capacity. An example of such an application is given in the next section. 8.2. Frontal Chromatography, As shown previously, gigaporous particles can be expected to be useful in applications which benefit from high flow velocities for rapid separation but do not require a large number of theoretical plates. One situation where this condition nearly always applies is frontal chromatography with packed columns for the concentration and isolation of proteins when the sorbent exhibits high selectivity for the product to be recovered. In this process the feed is introduced into the column, and the “light-end” components pass through without retention whereas the product is “captured” by the stationary phase. The feed process is halted when breakthrough of the product front occurs and the product begins to appear in the effluent. Thereafter, the mobile-phase composition is changed to desorb nearly all of the product and recover it in an effluent fraction much smaller than the original feed volume. By appropriate selection of conditions for product recovery, the “heavy-end” feed components remain bound to the stationary-phase surface and are removed in the ensuing
’‘ -9
0’4
0.2
0
1
,
, 300
i\,
,
,
r
0
2000
4000
6000
,
1 8000
Pe
Figure 15. Illustration of the dependence of the adsorbent utilization with 98% product recovery on the particle Peclet number with the pore size as the parameter. Conditions correspond to 20-fim particles in a 20-cm column, a Langmuir adsorption isotherm with a = 5 and b = 4 X lo? cmj g-I,and an eluite feed concentration of 10 g cm- I . Other conditions are the same as in Figure 14.
regeneration step. Separation processes with biospecific sorbents such as “affinity chromatography”usually employ similar schemes, with the product being the only retained feed component. The effectiveness of the adsorbent in frontal chromatography is largely determined by the dynamic sorption capacity of the column. The “adsorbent utilization”, which is always less than unity, is the ratio of the dynamic saturation capacity and the saturation capacity measured under static conditions. The efficiency of the process is higher with columns having a higher sorbent utilization because of the concomitant higher production rate and lower regeneration costs. When a Langmuir isotherm with parameters a and b applies [i.e., q = aCm/(1+ bCm)l,and for sufficiently long columns where a constant pattern is achieved, the concentration profile in the column and the adsorbent utilization can be determined by relations described by Vermeulen et al. (1984). In order to compare adsorbent utilizations estimated for conventional porous and gigaporous column packings for a range of Peclet numbers, calculations were made assuming the same parameters used for the calculation of the results in Figure 14, as well as 20-cm columns packed with 20 hm diameter particles having 4000-, 1000-, and 300-Apores. A 98% product recovery was also assumed in the calculations since a constraint of this type conforms to typical industrial practice. The results are shown in Figure 15 by plotting the adsorbent utilization versus the Peclet number. It is seen that at sufficiently large Peclet numbers the columnpacked with particles having 1000-A pores is better in terms of adsorbent utilization than the column containing particles with 300-A pores. This is because with the 40-A protein considered in the calculations hindered diffusion is nearly absent in 1000-8, pores but is significant in 300-A pores. Figure 15 clearly demonstrates that the column packing with 4000-A pores has a substantial advantage over the others at high flow velocities. This advantage arises from the fact that such a column exhibits high adsorbent utilization under the constraint of 98 % product recovery even at very high Peclet numbers. This result suggests that gigaporous stationary phases may indeed have useful applications in various types of process chromatography of biological macromolecules, including affinity chroma-
Biotechnoi. hog., 1993, Vol. 9, No. 3
tography. This conclusion is qualitatively supported by the recent work of Liapis and McCoy (1992),who investigated affiiity chromatographywith gigaporousparticles. Quantitative agreement between their results and ours is not expected, however, because in their work the particles were assumed to have a slab geometry, the effects of hindered diffusion were ignored, and the constraint of a fixed recovery was not enforced, and because in our work the effects of slow sorption kinetics are not considered.
9. Conclusions A model leading to an apparent diffusion coefficient is developedfor predicting and analyzingmass-transfer rates in columns packed with gigaporous particles. Such adsorbents have a bidisperse pore structure in which convection and diffusion occur in the larger pores but only diffusion occurs in the smaller pores. The results are in agreement with experimental data measured on columns packed with various gigaporous adsorbents with proteins under conditions of no retention. The analysis strongly suggests that gigaporous particles have performance advantages over particles with standard-sized pores for applications which do not require high resolving power to accomplish fast separations. In particular, model calculations for analytical chromatography indicate that typical gigaporous adsorbents yield a significant reduction in separation time in comparison to other porous column packings when a low to moderate number of theoretical plates suffices for the separation. In contradistinction, rapid, high-resolution analytical chromatography of biopolymers can be most effectively achieved through the use of micropellicular particles. Columns packed with gigaporousparticles can be effective for the concentration and isolation of proteins at high levels of adsorbent utilization and high product recovery by frontal chromatography since such separations generally do not require high plate efficiencies, and the column can therefore be operated at high flow velocities with small loss of efficiency.
283
Pe'
pore Peclet number for intraparticle convection (U'dp'JDe') eluite concentration in particle, g cm-3 9 r radial distance in particle, cm t time, s elution time for an inert eluite, s to tR elution time, s U interstitial fluid velocity, cm 5-1 U' interparticle fluid velocity, cm 5-1 Greek Symbols interparticle void fraction intraparticle void fraction tortuosity correction to diffusivity to account for hindered diffusion and steric effects fluid viscosity, g cm-1 5-1 distance between a particular longitudinal gigapore and the particle center phase ratio angle between axis of gigapore and the particle surface Subscripts conv contribution from convective transport diff contribution from diffusional transport axial dispersion disp downstream end d external ext internal int kinetic kin m mobile phase longitudinal gigapore gP outer surface of particle 8 upstream end U
Notation a1, a2
a b
B C Cm DaPP De Dm dP deluite dpore
f h H J
k' Ke, Keql L M N
P Pe
functions of time in eq 18 Langmuir isotherm parameter Langmuir isotherm parameter, cm3 g-' permeability, cm2 concentration of eluite in pore fluid, g cm-3 concentration of eluite in interstitial mobile phase, mol cm-3 apparent diffusivity, cm2 s-1 effective diffusivity in pore fluid, cm2 s-1 diffusivity in bulk mobile phase, cm2 5-1 diameter of particle, cm molecular diameter of eluite, cm diameter of pore, cm correctionfactor for diffusion in subsidiary particle reduced plate height plate height, cm mass flux, g cm-2 s-1 equilibrium parameter defined by eq 1 equilibrium constant distribution parameter defined by eqs 6 and 14 column length, cm external surface to volume ratio for subsidiary particle, cm-l number of theoretical plates pressure, g cm-l particle Peclet number for interstitial convection (Udp'lDm)
Acknowledgment We thank Alirio Rodrigues of the University of Porto for many helpful comments on this work and for pointing out an algebraic error in our original development of eq 30. We also thank Frank P. Warner of Polymer Laboratories Ltd. for supplying the polystyrene particles used in this study. E.S.was a recipient of the Feodor Lynen Fellowship given by the Alexander von Humbolt Foundation. This work was supported by Grants No. CTS 9008746 and BCS 9014119 from the National Science Foundation and by Grant No. GM 20993from the National Institutes of Health.
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