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Influence of buoyancy-driven convection on protein separation by free-flow electrophoresis
Adv. Space Res. Vol. 12, No. 1, pp. (1)373—(l)383, 1992 Printed in Great Britain. All rights rewved.
0273-1177)92 $15.00 Copyright (~)1991 COSPAR
INFLUENCE OF BUOYANCY-DRIVEN CONVECTION ON PROTEIN SEPARATION BY FREE-FLOW ELECTROPHORESIS M. J. Clifton, N. Jouve and V. Sanchez Laboratoire de Genie Chimique et Electrochimie (U.R.A. CNRS 192), Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse cedex, France
ABSTRACT In free-flow electrophoresis, the stability and reproducibility of the flow field are prerequisites for a satisfactory separation. Due to the presence of the species to be separated and due to mass- and heat-transfer phenomena, non-uniformities in density inevitably appear, both within the carrier buffer and between the carrier buffer and the sample stream. On earth they give rise to buoyancy-driven convection which interferes with the separation. These effects have been quantified by numerical modelling. Agreement has been found between the numerical results and experimental observations. INTRODUCTION Electrophoresis is widely used as a technique for analysing solutions of biological molecules. In this process, only small quantities of matter (~1~.tg)are treated, generally in a buffer solution immobilized on a gel support. The use of a free-flow process allows greater quantities of matter to be separated (1 10 g/hour) so that the process becomes a preparative one /1/. In the free-flow version of zone electrophoresis, a buffer solution of uniform pH is made to flow slowly, in a vertical direction, through a thin rectangular chamber; this is the carrier buffer. On either side of this chamber are electrode compartments, which are used to apply a DC electric field across the width of the chamber. The protein mixture to be separated is injected into the flow atthe entrance to the chamber through a fine tube. The proteins are carried by the flow along the length of the chamber and at the same time migrate across the chamber in a direction perpendicular to the flow. The lateral distance through which each protein molecule migrates is proportional to its mobility, to the field strength and to its residence time in the cell. At the outlet of the chamber, the various proteins in the mixture have been separated into different streams according to their mobilities and can be collected in the series of tubes through which the buffer leaves the chamber. However, there are a number of secondary phenomena which can impair the separation achieved by free-flow electrophoresis and some of these effects are gravity-dependent: this is why microgravity has been considered as a suitable environment for this operation. The most important of these effects are the following ones. -
Non-uniform Residence Times and Eleciro-osmosis The flow in the separation chamber is laminar and fully-developed, so the velocity profile across the thickness of the chamber is parabolic (or approximately so). This means that the liquid near the centre of the chamber flows faster than the liquid near the walls. Thus the protein molecules near the centre will spend less time in the electric field and migrate less than the molecules near the wall. The result of this is that the initially circular cross-section of the protein stream at the injection point is deformed into a crescent shape. This spreading causes a loss of resolution. To reduce this effect, the protein must be kent away from the walls of the chamber: this can be done by using a small injection radius. but
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rise to fixed electric charges on their surfaces. The combined effect of this surface charge with the applied electric field is to cause an electro-osmotic flow along the wall in the direction of the applied field; this produces a flow velocity with a parabolic profile perpendicular to the direction of forced convection. The effect on a protein stream having no electrophoretic mobility would be to produce yet anothercrescent deformation in the cross-section of the protein stream. This second crescent deformation can either cancel out or worsen the crescent effect due to the non-uniform residence times. Buoyancy-driven Flow Differences in density capable of giving rise to natural convection can be found both within the carrier buffer and between the carrier buffer and the protein stream. The protein injected into the chamber has usually been dissolved in a buffer solution of the same type but not necessarily of the same concentration as the carrier buffer. So there is usually a difference in density between the injected solution and the carrier solution which is due both to the presence of the protein and to a difference in ion concentration. This means that on earth there can be a difference in velocity between the injected solution and the carrier buffer which is not negligible; it is a difference which can vary along the length of the channel as the buffer ions and the protein migrate differently. The trajectory followed by the protein under the influenceof gravity need not be straight and the separation predicted on the basis of a straight trajectory may not be attained. There are also non-uniformities in density within the carrier buffer. These arise because the ion transport of the buffer is disturbed by the presence of membranes between the electrode compartments and the separation chamber and also because of the heat dissipated in the carrier buffer by the passage of the electric current (Joule heating). The membranes serve to prevent the gas bubbles given off at the electrodes from entering the separation chamber and they also allow a high circulation rate to be maintained in the electrode compartments without it interfering with the carrier buffer flow. This means that the ion concentrationwill rise on one side of the membrane and fall on the other; this behaviour is known as concentration polarisation. If a layer of less conducting solution forms on the membrane, the solution in that region will heat up more than the rest of the carrier buffer. Thus the density differences within the carrier buffer are due to non-uniformities both in temperature and in ion concentration. They imply a flow distribution which can be quite different from the ideal flow pattern which one would have in the absence of gravitational effects and there are operating conditions for which the flow becomes unstable and flow reversal can occur. So operation on earth imposes certain limits on the operating conditions: use of very thin separation chambers, moderate field strengths, injection of fairly dilute protein solutions etc. One way of overcoming these limitations would be to operate under microgravity conditions. Elecirokinetic (Kohlrausch) Effects So far the migration of the protein has been discussed as though it were independent of the surrounding medium. In fact the application of a DC field to an electrolyte solution means that all the ions present will migrate. The migrations of the different ions in the solution are related by the necessity of maintaining local electrical neutrality as well as by the conditions of conservation and chemical equilibrium. This means that local ion concentrations in the vicinity of the protein stream can vary in often surprising ways /2 4/. -
Electrohvdmdvnamics The ionic composition of the injected stream will always necessarily be different from that of the carrier buffer, ifonly because of the presence of the protein. This means that in and around the protein stream, due to variations in conductivity, there will be a distortion of the elecric field and this will give rise to shear stresses in the solution. The shear stress tensor for the buffer already contains terms due to viscous stresses and must now be modified to include stresses due to the electric field: -‘-
.\__
Buoyancy Convection in Electrophoresis
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delta and E is the field strength /5/. This electrohydrodynamic phenomenon tends to flatten the protein stream out into a ribbon /6/. The result is a dispersal of the protein stream and a consequent loss in resolution. The only solution to this problem is to adjust the conductivity of the injected solution to make it equal to that of the carrier buffer, but this equality will not be preserved all along the length of the channel as the protein and the buffer ions migrate at different rates. NATURAL CONVECTION iN THE CARRIER BUFFER The presence of natural convection in the carrier buffer during free-flow electrophoresis has been observed experimentally in our laboratory, but the very narrow geometry of the separation chamber means that precise experimental measurements are almost impossible. So we have made use of numerical modelling to explore some of the implications of this phenomenon. A three-dimensional model of the flow under steady-state conditions has been developed /7/. In this model the coupled momentum (Navier-Stokes) and heat-transfer equations are solved by a finite-difference technique. The variations in ion concentration due to membrane polarisation are represented by a concentration profile across the width of the cell; this profile was obtainedfrom experimental measurements at the outlet of the cell. However concentration variations in the direction of flow and through the thickness of the chamber are less importantand have been neglected. These concentration profiles have a direct influenceon the density of the solution, but they also have an indirect effect via the Joule heating. The more electrically resistive (more dilute) zones of the solution heat up more than the less resistive ones. The resulting difference in temperature is the second source of density variations. Our calculations have shown that, contrary to a widely held belief, the direct effect of concentration on the density is of greater importance than the effect due to temperature variations, as long as the separation chamberhas adequate cooling. The influence of these concentration profiles on the carrier buffer flow has been summarised in Fig. 1. If the membranes can be chosen so as to be of opposite polarity, i.e. cation-exchange on the anode side, anion-exchange at the cathode, then it is possible to have the situation shown on the left-hand side of the figure. There is no ionic depletion of the solution within the chamber but the concentration increases near each membrane in a roughly symmetrical manner; here the depleted solutions are in the electrode compartments where their effect is minimized by the high flow rate of the electrode buffer. If the concentration varies sufficiently and the carrier buffer is flowing upwards (Fig. ib), then back flow can occur. A dimensionless correlation for predicting back flow has been derived and will be presented below. If the carrier buffer flows downwards, then the flow is accelerated near the membranes (Fig. ic); this is not as undesirable as back flow but can be a source of instability. For both flow directions there will be a modification of the effective residence time of the buffer in the chamber (i.e. the residence time experienced by the protein stream). The situation represented on the right-hand side of Fig. 1, arises if the membranes are both of the same polarity, e.g. both cation-exchange. Then the solution on one side of the chamber is enriched in ions while on the other side it is depleted. The presence of a layer of dilute solution on one membrane can cause an important increase in cell resistance with a consequent loss in effective field strength; this is not desirable. The lack of symmetry in the density field means that the flow will be accelerated near one membrane and slowed down or even reversed near the other and this occurs whether the flow direction is upwards or downwards (Figs. le or it). Since a uniform flow distribution is imposed at the inlet and outlet of the chamber, there will be lateral “compensating” movements near the inlet and outlet which will give the proteins a non-linear trajectory (Fig. 2). Such trajectories have indeed been observed experimentally in our laboratory. It might be thought that the presence of these compensating flows would modify the crescent deformation of the protein stream, but this is not the case. The lateral velocity component introduced by the compensating flow has a parabolic profile across the chamber thickness with a zero value at the wall. This means that the ratio between this lateral component and the axial velocity does not vary across the thickness, so that, in the
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the wall. When the protein migration compensates for this wall velocity, then the electro-osmotic flow no longer causes any deformation. The compensating flows can, however, modify the position at which a protein leaves the separation chamber. But for any important displacement to occur, the injection point would have to be situated at quite some distance beyond the line that marks the beginning of the electric field; in that case the protein stream would undergo only the lateral movement of the outlet zone. This is not the usual design of a free-flow electrophoresis chamber as it implies that a certain length of the electric field is of no use in separating the protein species. C
p.
p.
figla
)‘
figld
fig lb
y
fig le
figic
y
figif
Fig. 1. Effect of buffer concentration on flow pattern. a: polarization layer with membranes of opposite polarities; b: velocity profile with carrier buffer flowing upwards; c: carrier buffer flowing downwards; d: polarization layer with membranes of same polarity; e: carrier buffer flowing upwards; f: carrier buffer flowing downwards. CONDITIONS FOR BACK FLOW One of the important results of our study of a three-dimensional model for the flow pattern in electrophoresis was that over a large part of the length of the chamber (i.e. outside the inlet and outlet zones) the flow was well described by a simpler two-dimensional model /7/. From this model, it was possible to derive a dimensionless correlation which predicts the conditions under which back flow can occur. Flow reversal should be avoided in free-flowelectrophoresis because it introduces the possibility nf rP~mi,rinath~nrnu~inerre~mwith thp, rnrri~rhiiffetr if thp. nrntAin miornu~efnr e.nnisrh tnwnrds th~
Fig. 2 Non-Ihear trajectory of the protein stream due to Iareralcompensating movements
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~i~
.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
20.0
U
Fig. 3. Types of velocity profiles for different values of Gr/Re and 1: increased velocity in the polarization layer; 2: Poiseuille flow; 3: limiting condition for start of back flow; 4: polarization layer giving back flow. ~.
1,14
1,02
r
1~j1~
Buoyancy Convection in Electrophoresis
function of follows:
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where the dimensionless Grashof number Gr and Reynolds number Re are defined as
~,
Gr=~ Z3p~g 0
j.t
0
Re=
~.to
Here ~P = Pm P0 is the difference in density between the membrane surface and the bulk solution and c is the dimensionless thickness of the polarisation layer at the membrane. The two curves are for two extreme values of the aspect ratio YJZ, where Y is the chamber width and Z is the chamber half-thickness. The insets in this figure show the type of velocity profile that can be expected for a particular combination of Gr/Re with c. Flow reversal occurs when the Gr/Re value lies below the curve. —
VARIATIONS IN BUFFER RESIDENCE TIME All of the phenomena of natural convection described so far can cause modifications in the effective residence time of the carrier buffer in the chamber. However it should first be noted that in the absence of any density variations, the effective residence time of the carrier buffer (i.e. the residence time of the buffer in the part of the chamber where the protein passes) will vary with the aspect ratio YIZ of the chamber. The fluid velocity at the membrane surface is necessarily zero and this introduces end effects which are all the more important the narrower is the chamber. For very wide chambers, as YIZ oo, the flow tends towards an ideal Poiseuille flow between two infinite flat plates. In that case the flow velocity varies in only one dimension (no variations with x or y) and the mean residence time becomes —
T00X1Z where X is the chamber length. For finite values of YJZ, in the absence of buoyancy effects, the mean residence time in the central part of the cell ‘r~can be calculated from the two-dimensional Navier-Stokes equation for the axial velocity: 2u a2u
a
=Re— az2 ax where aP/ax is a constant. A correlation between t~ and the asymptotic residence time ‘r~yjcan be found as a function of the aspect ratio Y/Z. This correlation is shown in Fig. 4. The physical significance of this correlation is illustrated by Fig. 5. —+—
ay2
In the presence of buoyancy-driven convection, the two-dimensional Navier Stokes Equation becomes
a2u —
ay2
a2u az2
+—
~P =Re—
ax
Or ——
Re
where Or is the local Grashof number Gr(y). From the numerical solution of this equation, it is found that the mean residence time ‘t of the carrier buffer in the central zone of the chambercan be calculated with quite good precision from the following correlation:
F~
‘r
1
/~1_
•
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M. J. Clifton etaL
YIZ = 20 _______
_____________
YfL
=
Y/Z
->00
40
yp
.to~ to Fig. 5. Effect of YJZ ratio on residence time in fully developed flow. U
Fig. 6. Effect of various types of velocity profile on the effective residence time ‘r. 800
700
6Ocmx6cmxi.Smm
600
~xcmx3m~j
500 400
3Ocmx4cmxl.Smm
300 200
.
100• 0• -2.0
.
.
—1.5
.
—1.0
I
—0.5
.
0.0
I
0.5
I
1,0
1,5
2.0
Buoyancy Convection in Electrophoresis
(1)381
right-hand sides respectively. The left-hand side of this equation represents the fact that the additional flow in the central zone introduced by the buoyancy-driven flows near the membranes is spread out over the distance Y. This means that for chambers that are very wide in comparison with the thickness c of the polarisation layers, the extra flow has little effect. The physical relationship between the residence times ‘r and t~ for different velocity profiles is illustrated by Fig. 6. In Fig. 7, the mean effective residence time ‘r of the carrier buffer is plotted as a function of the average density difference between the bulk solution and the solution at the membrane surfaces: = +
Ap
p2)!2 p0 —
The three curves correspond to three different chamber geometries which have been used in our laboratory experiments. To sum up, the flow pattern obtained with a chamber 3 mm thick is most sensitive to natural convection but it offers the advantage of allowing the protein stream to be kept away from the walls, thus reducing spreading by crescent deformation. The chambers which are 1.5 mm thick give a more stable flow pattern but since the diameter of the protein stream can hardly be reduced below 1 mm for technical reasons, the spreading of the protein stream by the crescent effect is quite important. The solution to this dilemma could well be the use of a 3 mm thick chamber in a microgravity environment. The advantages of operating in microgravity will be confirmed by considering the other buoyancy-induced effect in electrophoresis. BUOYANCY EFFECTS IN THE PROTEIN STREAM The presence of the protein in the protein stream raises its density and this means that there will be a difference in density between the protein stream and the surrounding carrier buffer. Even ifthe protein is dissolved in a more dilute buffer to compensate for this effect, the protein will migrate differently from the other ions in the solution and will soon be found in a zone having a similar ion concentration to that of the carrier buffer. This means that in a down-flowing buffer the protein will descend more rapidly than the carrier solution, whereas in an up-flowing buffer it will rise more slowly (see Fig. 8). As a result, the residence time for the protein in the chamber will be either shorter or longer than the residence time of the buffer. In order to quantify this effect, we have once again considered the two-dimensional 2u a2u Navier-Stokes equation: Gr
a~
a
=Re——— ax Re where Or now quantifies the density difference between the protein stream and the carrier buffer. This equation was solved for cylindrical protein streams, with an abrupt change in density between the protein stream and the surrounding solution. The calculation was performed over a range of radii for the protein stream varying from 0 to 0.7 Z. For an infinitely small radius the buoyancy effect tends towards zero and the velocity u tends towards the parabolic Poiseuille profile, but as the protein stream increases in diameter the buoyancy effect becomes more and more important. Strongly negative values of Or (i.e. with an up-flowing buffer) can lead to a negative velocity at the centre of the stream; this has no strict physical meaning, of course, but it corresponds to the fact that when a sufficiently dense protein solution is injected into a slowly up-flowing buffer it is seen that the protein solution falls back and is not carried upwards by the buffer flow. —+—
ay2
az2
In Fig. 9 we have plotted the dimensionless velocity u at the centre of the protein stream as a function of q the dimensionless flow rate of the protein stream; q also corresponds to the rate of injection of the protein solution.
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M. J. Clifton etaL
Down
GriRe
flow
0
>
q.
~
,~
j~
,~
Fig. 8. Disturbance in the local velocity proffle due to the density diffei~nce between the sample and the carrier buffer.
~
,~
1.5
carrier buffer
carrierbuffer protein stream
Up flow
GriRe
__________
<
0 1.5
__________
10
9 8
Fig. 9.centre Dimensionless velocity atthe as a function of ofthe theprotein dimensionless streamu rate of injection of the protein solution q, for various values of Or/Re.
6
0
S 4 ~0
3 Gr/Re0
.1
~
0
I 0.0
0.2
0.4
I
0.6
0.8
q
I
1.0
1.2
I
1.4
1.6
Buoyancy Convection in Electrophoresis
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The value of ~ corresponds to the maximum injection rate that can be obtained in the absence of buoyancy effects. As q increases, the protein stream increases in diameter and the buoyancy effect becomes more marked. For vanishing q or zero Or/Re, the value u = 1.5 corresponds to the case of undisturbed Poiseuile flow, as it would be observed, for example, in zero gravity. In a down-flowing buffer (Or positive), the change in velocity can be quite important, whereas for up-flowing buffer (Or negative) the range of values of q that can be used is more and more limited as the Or/Re value increases in magnitude. In this case too, precise experimental observations are difficult to perform, but the approximate measurements of protein residence times made in our laboratory are in agreement with this correlation. It is not unusual to observe in the down-flow case that the protein residence time is half that of the buffer. As it is the protein residence time that determines the distance of migration, it can be appreciated that this buoyancy effect introduces a considerable uncertainty into the separations that can be obtained by electrophoresis. CONCLUSION It seems inevitable that in free-flow electrophoresis there should be density differences both within the carrier buffer and between theprotein stream and the carrier buffer. These differences, under the influence of earth’s gravity, can cause serious difficulties in the use of electrophoresis as a separative technique. This is why several projects are under way for performing electrophoresis separations in the microgravity environment available in spacecraft. We have seen that there are serious reasons for preferring the microgravity environment but it remains to be seen whether the resulting improvements in resolution of the process can justify the expense of such an operation. ACKNOWLEDGEMENT This work was performed with the aid of a contract with the Centre National d’Etudes Spatiales (CNES), the French space agency. REFERENCES 1. K. Hannig, Preparative electrophoresis, in: Electrophoresis: Theory, Methods and Applications, Volume II, ed. M. Bier, Academic Press, New York, 1967, pp. 423-471. 2. M. Bier, O.A. Palusinski, R.A. Mosher and D.A. Saville, Electrophoresis: mathematical modeling and computer simulation, Science, 219, # 4590, 1281-1286 (1983). 3. D.A. Saville, O.A. Palusinski, Theory of electrophoretic separations, Part I: formulation of a mathematical model, AIChE J., 32, # 2, 207-214 (1986). 4. O.A. Palusinski, A. Oraham, R.A. Mosher, M. Bier and D.A. Saville, Theory of electrophoretic separations, Part II: construction of a numerical simulation scheme and its applications, AJChE J., 32, # 2, 2 15-223 (1986). 5. L.D. Landau and E.M. Lifschitz, Electrodynamique des Mileux Continus, Editions Mir, Moscow,
1969. 6. P.H. Rhodes, R.S. Snyder and 0.0. Roberts, Electrohydrodynamic distortion of sample streams in continuous flow electrophoresis, J. Colloid Interface Sci., 129, # 1,78-90 (1989). 7. N. Jouve and M.J. Clifton, Three-dimensional modelling of the coupled flow field and heat transfer in free-flow electrophoresis, mt. J. Heat Mass Trans., submitted for publication (1990).
0273-1177)92 $15.00 Copyright (~)1991 COSPAR
INFLUENCE OF BUOYANCY-DRIVEN CONVECTION ON PROTEIN SEPARATION BY FREE-FLOW ELECTROPHORESIS M. J. Clifton, N. Jouve and V. Sanchez Laboratoire de Genie Chimique et Electrochimie (U.R.A. CNRS 192), Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse cedex, France
ABSTRACT In free-flow electrophoresis, the stability and reproducibility of the flow field are prerequisites for a satisfactory separation. Due to the presence of the species to be separated and due to mass- and heat-transfer phenomena, non-uniformities in density inevitably appear, both within the carrier buffer and between the carrier buffer and the sample stream. On earth they give rise to buoyancy-driven convection which interferes with the separation. These effects have been quantified by numerical modelling. Agreement has been found between the numerical results and experimental observations. INTRODUCTION Electrophoresis is widely used as a technique for analysing solutions of biological molecules. In this process, only small quantities of matter (~1~.tg)are treated, generally in a buffer solution immobilized on a gel support. The use of a free-flow process allows greater quantities of matter to be separated (1 10 g/hour) so that the process becomes a preparative one /1/. In the free-flow version of zone electrophoresis, a buffer solution of uniform pH is made to flow slowly, in a vertical direction, through a thin rectangular chamber; this is the carrier buffer. On either side of this chamber are electrode compartments, which are used to apply a DC electric field across the width of the chamber. The protein mixture to be separated is injected into the flow atthe entrance to the chamber through a fine tube. The proteins are carried by the flow along the length of the chamber and at the same time migrate across the chamber in a direction perpendicular to the flow. The lateral distance through which each protein molecule migrates is proportional to its mobility, to the field strength and to its residence time in the cell. At the outlet of the chamber, the various proteins in the mixture have been separated into different streams according to their mobilities and can be collected in the series of tubes through which the buffer leaves the chamber. However, there are a number of secondary phenomena which can impair the separation achieved by free-flow electrophoresis and some of these effects are gravity-dependent: this is why microgravity has been considered as a suitable environment for this operation. The most important of these effects are the following ones. -
Non-uniform Residence Times and Eleciro-osmosis The flow in the separation chamber is laminar and fully-developed, so the velocity profile across the thickness of the chamber is parabolic (or approximately so). This means that the liquid near the centre of the chamber flows faster than the liquid near the walls. Thus the protein molecules near the centre will spend less time in the electric field and migrate less than the molecules near the wall. The result of this is that the initially circular cross-section of the protein stream at the injection point is deformed into a crescent shape. This spreading causes a loss of resolution. To reduce this effect, the protein must be kent away from the walls of the chamber: this can be done by using a small injection radius. but
M. J. Clifton et aL
(1)374
rise to fixed electric charges on their surfaces. The combined effect of this surface charge with the applied electric field is to cause an electro-osmotic flow along the wall in the direction of the applied field; this produces a flow velocity with a parabolic profile perpendicular to the direction of forced convection. The effect on a protein stream having no electrophoretic mobility would be to produce yet anothercrescent deformation in the cross-section of the protein stream. This second crescent deformation can either cancel out or worsen the crescent effect due to the non-uniform residence times. Buoyancy-driven Flow Differences in density capable of giving rise to natural convection can be found both within the carrier buffer and between the carrier buffer and the protein stream. The protein injected into the chamber has usually been dissolved in a buffer solution of the same type but not necessarily of the same concentration as the carrier buffer. So there is usually a difference in density between the injected solution and the carrier solution which is due both to the presence of the protein and to a difference in ion concentration. This means that on earth there can be a difference in velocity between the injected solution and the carrier buffer which is not negligible; it is a difference which can vary along the length of the channel as the buffer ions and the protein migrate differently. The trajectory followed by the protein under the influenceof gravity need not be straight and the separation predicted on the basis of a straight trajectory may not be attained. There are also non-uniformities in density within the carrier buffer. These arise because the ion transport of the buffer is disturbed by the presence of membranes between the electrode compartments and the separation chamber and also because of the heat dissipated in the carrier buffer by the passage of the electric current (Joule heating). The membranes serve to prevent the gas bubbles given off at the electrodes from entering the separation chamber and they also allow a high circulation rate to be maintained in the electrode compartments without it interfering with the carrier buffer flow. This means that the ion concentrationwill rise on one side of the membrane and fall on the other; this behaviour is known as concentration polarisation. If a layer of less conducting solution forms on the membrane, the solution in that region will heat up more than the rest of the carrier buffer. Thus the density differences within the carrier buffer are due to non-uniformities both in temperature and in ion concentration. They imply a flow distribution which can be quite different from the ideal flow pattern which one would have in the absence of gravitational effects and there are operating conditions for which the flow becomes unstable and flow reversal can occur. So operation on earth imposes certain limits on the operating conditions: use of very thin separation chambers, moderate field strengths, injection of fairly dilute protein solutions etc. One way of overcoming these limitations would be to operate under microgravity conditions. Elecirokinetic (Kohlrausch) Effects So far the migration of the protein has been discussed as though it were independent of the surrounding medium. In fact the application of a DC field to an electrolyte solution means that all the ions present will migrate. The migrations of the different ions in the solution are related by the necessity of maintaining local electrical neutrality as well as by the conditions of conservation and chemical equilibrium. This means that local ion concentrations in the vicinity of the protein stream can vary in often surprising ways /2 4/. -
Electrohvdmdvnamics The ionic composition of the injected stream will always necessarily be different from that of the carrier buffer, ifonly because of the presence of the protein. This means that in and around the protein stream, due to variations in conductivity, there will be a distortion of the elecric field and this will give rise to shear stresses in the solution. The shear stress tensor for the buffer already contains terms due to viscous stresses and must now be modified to include stresses due to the electric field: -‘-
.\__
Buoyancy Convection in Electrophoresis
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delta and E is the field strength /5/. This electrohydrodynamic phenomenon tends to flatten the protein stream out into a ribbon /6/. The result is a dispersal of the protein stream and a consequent loss in resolution. The only solution to this problem is to adjust the conductivity of the injected solution to make it equal to that of the carrier buffer, but this equality will not be preserved all along the length of the channel as the protein and the buffer ions migrate at different rates. NATURAL CONVECTION iN THE CARRIER BUFFER The presence of natural convection in the carrier buffer during free-flow electrophoresis has been observed experimentally in our laboratory, but the very narrow geometry of the separation chamber means that precise experimental measurements are almost impossible. So we have made use of numerical modelling to explore some of the implications of this phenomenon. A three-dimensional model of the flow under steady-state conditions has been developed /7/. In this model the coupled momentum (Navier-Stokes) and heat-transfer equations are solved by a finite-difference technique. The variations in ion concentration due to membrane polarisation are represented by a concentration profile across the width of the cell; this profile was obtainedfrom experimental measurements at the outlet of the cell. However concentration variations in the direction of flow and through the thickness of the chamber are less importantand have been neglected. These concentration profiles have a direct influenceon the density of the solution, but they also have an indirect effect via the Joule heating. The more electrically resistive (more dilute) zones of the solution heat up more than the less resistive ones. The resulting difference in temperature is the second source of density variations. Our calculations have shown that, contrary to a widely held belief, the direct effect of concentration on the density is of greater importance than the effect due to temperature variations, as long as the separation chamberhas adequate cooling. The influence of these concentration profiles on the carrier buffer flow has been summarised in Fig. 1. If the membranes can be chosen so as to be of opposite polarity, i.e. cation-exchange on the anode side, anion-exchange at the cathode, then it is possible to have the situation shown on the left-hand side of the figure. There is no ionic depletion of the solution within the chamber but the concentration increases near each membrane in a roughly symmetrical manner; here the depleted solutions are in the electrode compartments where their effect is minimized by the high flow rate of the electrode buffer. If the concentration varies sufficiently and the carrier buffer is flowing upwards (Fig. ib), then back flow can occur. A dimensionless correlation for predicting back flow has been derived and will be presented below. If the carrier buffer flows downwards, then the flow is accelerated near the membranes (Fig. ic); this is not as undesirable as back flow but can be a source of instability. For both flow directions there will be a modification of the effective residence time of the buffer in the chamber (i.e. the residence time experienced by the protein stream). The situation represented on the right-hand side of Fig. 1, arises if the membranes are both of the same polarity, e.g. both cation-exchange. Then the solution on one side of the chamber is enriched in ions while on the other side it is depleted. The presence of a layer of dilute solution on one membrane can cause an important increase in cell resistance with a consequent loss in effective field strength; this is not desirable. The lack of symmetry in the density field means that the flow will be accelerated near one membrane and slowed down or even reversed near the other and this occurs whether the flow direction is upwards or downwards (Figs. le or it). Since a uniform flow distribution is imposed at the inlet and outlet of the chamber, there will be lateral “compensating” movements near the inlet and outlet which will give the proteins a non-linear trajectory (Fig. 2). Such trajectories have indeed been observed experimentally in our laboratory. It might be thought that the presence of these compensating flows would modify the crescent deformation of the protein stream, but this is not the case. The lateral velocity component introduced by the compensating flow has a parabolic profile across the chamber thickness with a zero value at the wall. This means that the ratio between this lateral component and the axial velocity does not vary across the thickness, so that, in the
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M. J. Clifton etaL
the wall. When the protein migration compensates for this wall velocity, then the electro-osmotic flow no longer causes any deformation. The compensating flows can, however, modify the position at which a protein leaves the separation chamber. But for any important displacement to occur, the injection point would have to be situated at quite some distance beyond the line that marks the beginning of the electric field; in that case the protein stream would undergo only the lateral movement of the outlet zone. This is not the usual design of a free-flow electrophoresis chamber as it implies that a certain length of the electric field is of no use in separating the protein species. C
p.
p.
figla
)‘
figld
fig lb
y
fig le
figic
y
figif
Fig. 1. Effect of buffer concentration on flow pattern. a: polarization layer with membranes of opposite polarities; b: velocity profile with carrier buffer flowing upwards; c: carrier buffer flowing downwards; d: polarization layer with membranes of same polarity; e: carrier buffer flowing upwards; f: carrier buffer flowing downwards. CONDITIONS FOR BACK FLOW One of the important results of our study of a three-dimensional model for the flow pattern in electrophoresis was that over a large part of the length of the chamber (i.e. outside the inlet and outlet zones) the flow was well described by a simpler two-dimensional model /7/. From this model, it was possible to derive a dimensionless correlation which predicts the conditions under which back flow can occur. Flow reversal should be avoided in free-flowelectrophoresis because it introduces the possibility nf rP~mi,rinath~nrnu~inerre~mwith thp, rnrri~rhiiffetr if thp. nrntAin miornu~efnr e.nnisrh tnwnrds th~
Fig. 2 Non-Ihear trajectory of the protein stream due to Iareralcompensating movements
M. J. Clifton etaL
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~i~
.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
20.0
U
Fig. 3. Types of velocity profiles for different values of Gr/Re and 1: increased velocity in the polarization layer; 2: Poiseuille flow; 3: limiting condition for start of back flow; 4: polarization layer giving back flow. ~.
1,14
1,02
r
1~j1~
Buoyancy Convection in Electrophoresis
function of follows:
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where the dimensionless Grashof number Gr and Reynolds number Re are defined as
~,
Gr=~ Z3p~g 0
j.t
0
Re=
~.to
Here ~P = Pm P0 is the difference in density between the membrane surface and the bulk solution and c is the dimensionless thickness of the polarisation layer at the membrane. The two curves are for two extreme values of the aspect ratio YJZ, where Y is the chamber width and Z is the chamber half-thickness. The insets in this figure show the type of velocity profile that can be expected for a particular combination of Gr/Re with c. Flow reversal occurs when the Gr/Re value lies below the curve. —
VARIATIONS IN BUFFER RESIDENCE TIME All of the phenomena of natural convection described so far can cause modifications in the effective residence time of the carrier buffer in the chamber. However it should first be noted that in the absence of any density variations, the effective residence time of the carrier buffer (i.e. the residence time of the buffer in the part of the chamber where the protein passes) will vary with the aspect ratio YIZ of the chamber. The fluid velocity at the membrane surface is necessarily zero and this introduces end effects which are all the more important the narrower is the chamber. For very wide chambers, as YIZ oo, the flow tends towards an ideal Poiseuille flow between two infinite flat plates. In that case the flow velocity varies in only one dimension (no variations with x or y) and the mean residence time becomes —
T00X1Z where X is the chamber length. For finite values of YJZ, in the absence of buoyancy effects, the mean residence time in the central part of the cell ‘r~can be calculated from the two-dimensional Navier-Stokes equation for the axial velocity: 2u a2u
a
=Re— az2 ax where aP/ax is a constant. A correlation between t~ and the asymptotic residence time ‘r~yjcan be found as a function of the aspect ratio Y/Z. This correlation is shown in Fig. 4. The physical significance of this correlation is illustrated by Fig. 5. —+—
ay2
In the presence of buoyancy-driven convection, the two-dimensional Navier Stokes Equation becomes
a2u —
ay2
a2u az2
+—
~P =Re—
ax
Or ——
Re
where Or is the local Grashof number Gr(y). From the numerical solution of this equation, it is found that the mean residence time ‘t of the carrier buffer in the central zone of the chambercan be calculated with quite good precision from the following correlation:
F~
‘r
1
/~1_
•
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M. J. Clifton etaL
YIZ = 20 _______
_____________
YfL
=
Y/Z
->00
40
yp
.to~ to Fig. 5. Effect of YJZ ratio on residence time in fully developed flow. U
Fig. 6. Effect of various types of velocity profile on the effective residence time ‘r. 800
700
6Ocmx6cmxi.Smm
600
~xcmx3m~j
500 400
3Ocmx4cmxl.Smm
300 200
.
100• 0• -2.0
.
.
—1.5
.
—1.0
I
—0.5
.
0.0
I
0.5
I
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2.0
Buoyancy Convection in Electrophoresis
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right-hand sides respectively. The left-hand side of this equation represents the fact that the additional flow in the central zone introduced by the buoyancy-driven flows near the membranes is spread out over the distance Y. This means that for chambers that are very wide in comparison with the thickness c of the polarisation layers, the extra flow has little effect. The physical relationship between the residence times ‘r and t~ for different velocity profiles is illustrated by Fig. 6. In Fig. 7, the mean effective residence time ‘r of the carrier buffer is plotted as a function of the average density difference between the bulk solution and the solution at the membrane surfaces: = +
Ap
p2)!2 p0 —
The three curves correspond to three different chamber geometries which have been used in our laboratory experiments. To sum up, the flow pattern obtained with a chamber 3 mm thick is most sensitive to natural convection but it offers the advantage of allowing the protein stream to be kept away from the walls, thus reducing spreading by crescent deformation. The chambers which are 1.5 mm thick give a more stable flow pattern but since the diameter of the protein stream can hardly be reduced below 1 mm for technical reasons, the spreading of the protein stream by the crescent effect is quite important. The solution to this dilemma could well be the use of a 3 mm thick chamber in a microgravity environment. The advantages of operating in microgravity will be confirmed by considering the other buoyancy-induced effect in electrophoresis. BUOYANCY EFFECTS IN THE PROTEIN STREAM The presence of the protein in the protein stream raises its density and this means that there will be a difference in density between the protein stream and the surrounding carrier buffer. Even ifthe protein is dissolved in a more dilute buffer to compensate for this effect, the protein will migrate differently from the other ions in the solution and will soon be found in a zone having a similar ion concentration to that of the carrier buffer. This means that in a down-flowing buffer the protein will descend more rapidly than the carrier solution, whereas in an up-flowing buffer it will rise more slowly (see Fig. 8). As a result, the residence time for the protein in the chamber will be either shorter or longer than the residence time of the buffer. In order to quantify this effect, we have once again considered the two-dimensional 2u a2u Navier-Stokes equation: Gr
a~
a
=Re——— ax Re where Or now quantifies the density difference between the protein stream and the carrier buffer. This equation was solved for cylindrical protein streams, with an abrupt change in density between the protein stream and the surrounding solution. The calculation was performed over a range of radii for the protein stream varying from 0 to 0.7 Z. For an infinitely small radius the buoyancy effect tends towards zero and the velocity u tends towards the parabolic Poiseuille profile, but as the protein stream increases in diameter the buoyancy effect becomes more and more important. Strongly negative values of Or (i.e. with an up-flowing buffer) can lead to a negative velocity at the centre of the stream; this has no strict physical meaning, of course, but it corresponds to the fact that when a sufficiently dense protein solution is injected into a slowly up-flowing buffer it is seen that the protein solution falls back and is not carried upwards by the buffer flow. —+—
ay2
az2
In Fig. 9 we have plotted the dimensionless velocity u at the centre of the protein stream as a function of q the dimensionless flow rate of the protein stream; q also corresponds to the rate of injection of the protein solution.
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M. J. Clifton etaL
Down
GriRe
flow
0
>
q.
~
,~
j~
,~
Fig. 8. Disturbance in the local velocity proffle due to the density diffei~nce between the sample and the carrier buffer.
~
,~
1.5
carrier buffer
carrierbuffer protein stream
Up flow
GriRe
__________
<
0 1.5
__________
10
9 8
Fig. 9.centre Dimensionless velocity atthe as a function of ofthe theprotein dimensionless streamu rate of injection of the protein solution q, for various values of Or/Re.
6
0
S 4 ~0
3 Gr/Re0
.1
~
0
I 0.0
0.2
0.4
I
0.6
0.8
q
I
1.0
1.2
I
1.4
1.6
Buoyancy Convection in Electrophoresis
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The value of ~ corresponds to the maximum injection rate that can be obtained in the absence of buoyancy effects. As q increases, the protein stream increases in diameter and the buoyancy effect becomes more marked. For vanishing q or zero Or/Re, the value u = 1.5 corresponds to the case of undisturbed Poiseuile flow, as it would be observed, for example, in zero gravity. In a down-flowing buffer (Or positive), the change in velocity can be quite important, whereas for up-flowing buffer (Or negative) the range of values of q that can be used is more and more limited as the Or/Re value increases in magnitude. In this case too, precise experimental observations are difficult to perform, but the approximate measurements of protein residence times made in our laboratory are in agreement with this correlation. It is not unusual to observe in the down-flow case that the protein residence time is half that of the buffer. As it is the protein residence time that determines the distance of migration, it can be appreciated that this buoyancy effect introduces a considerable uncertainty into the separations that can be obtained by electrophoresis. CONCLUSION It seems inevitable that in free-flow electrophoresis there should be density differences both within the carrier buffer and between theprotein stream and the carrier buffer. These differences, under the influence of earth’s gravity, can cause serious difficulties in the use of electrophoresis as a separative technique. This is why several projects are under way for performing electrophoresis separations in the microgravity environment available in spacecraft. We have seen that there are serious reasons for preferring the microgravity environment but it remains to be seen whether the resulting improvements in resolution of the process can justify the expense of such an operation. ACKNOWLEDGEMENT This work was performed with the aid of a contract with the Centre National d’Etudes Spatiales (CNES), the French space agency. REFERENCES 1. K. Hannig, Preparative electrophoresis, in: Electrophoresis: Theory, Methods and Applications, Volume II, ed. M. Bier, Academic Press, New York, 1967, pp. 423-471. 2. M. Bier, O.A. Palusinski, R.A. Mosher and D.A. Saville, Electrophoresis: mathematical modeling and computer simulation, Science, 219, # 4590, 1281-1286 (1983). 3. D.A. Saville, O.A. Palusinski, Theory of electrophoretic separations, Part I: formulation of a mathematical model, AIChE J., 32, # 2, 207-214 (1986). 4. O.A. Palusinski, A. Oraham, R.A. Mosher, M. Bier and D.A. Saville, Theory of electrophoretic separations, Part II: construction of a numerical simulation scheme and its applications, AJChE J., 32, # 2, 2 15-223 (1986). 5. L.D. Landau and E.M. Lifschitz, Electrodynamique des Mileux Continus, Editions Mir, Moscow,
1969. 6. P.H. Rhodes, R.S. Snyder and 0.0. Roberts, Electrohydrodynamic distortion of sample streams in continuous flow electrophoresis, J. Colloid Interface Sci., 129, # 1,78-90 (1989). 7. N. Jouve and M.J. Clifton, Three-dimensional modelling of the coupled flow field and heat transfer in free-flow electrophoresis, mt. J. Heat Mass Trans., submitted for publication (1990).