Acknowledgment
= density, gram dry solid/cc. wet bed = cycle time, sec. = dimensionless fluid phase concentration, C/C, +f = dimensionless fluid phase concentration in equilibrium with the solid, C,*/C, d8 = dimensionless solid phase concentration C,P,K/C, w = frequency, sec.-l
pe T
The authors thank R. W. Rolke, R. A. Gregory, T. J. Butts, and D. L. Guttormson for many valuable discussions, and J. E. Sabadell for assistance with experimental work. Nomenclature
A
constant in equilibrium function constant in equilibrium function B fluid concentration, ml. toluene/ml. fluid C Cf* fluid concentration in equilibrium with adsorbent, ml. toluene/ml. fluid C, = arbitrary fluid concentration, ml. toluene/ml. fluid C, = solid concentration, ml. toluene/gram dry solid D = constant in equilibrium function f ( t ) = dimensionless fluid velocity F = equilibrium function (Equation 13) h, = mass transfer coefficient i = position increment i j = time increment j j~ = “j-factor” N T = number of time increments N Z = number of position increments R = interphase mass transfer rate expression t = time, dimensionless T = temperature, O C . V = magnitude of f ( t ) for square wave velocitp z = distance, dimensionless = = = =
GREEK LETTERS a e 77
0 K
X
dimensionless velocity coefficient phase angle, radians dimensionless axial diffusivity a parametei in the distribution function-e.g., perature = reciprocal of column fluid fraction = dimensionless mass transfer coefficient
= = = =
tem-
+*
literature Cited
Acrivos, A,, Ind. Eng. Chem. 48, 703 (1956). Alexis, R. W,,Chenz. Eng. Progr. Symp. Ser. 63, N o . 74, 51 (1 OK?\ \ * Y ” ’ , .
Bird, R. B., Stewart, W. E., Lightfoot, E. N., “Transport Phenomena,” Riley, Sew York, 1960. Jenczewski, T. J., Meyers, -4. L., A.I.Ch.E. J . 14, 509 (1968). Lapidus, L., “Digital Computation for Chemical Engineers,]’ McGraw-Hill, New York, 1962. hlchndrew, 11. A., Ph.D. dissertation, Princeton University, (1967). Rice, A. R., Ph.D. dissertation, Princeton University, 1966. Robins, G. S., B.S.E. thesis, Princeton University, 1967. Rolke, R. IT., Ph.D. dissertation, Princeton University, 1967. Skarstrom, C. W., Ann. lV.Y. Acad. Sci. 72, 751 (1959). Sweed, N. H., Ph.D. dissertation, Princeton University, 1968. Wakao, S . , Matsumoto, H., Suzuki, K., Kawahara, A,, Kagaku Koguku 32, 169 (1968). Wilhelm, It. H., Rice, A. IT.,Bendelius, A. R., ISD. ESG.CHEX. FUNDAMESTAL~ 6, 141 (1966). Kilhelm. R. H.. Rice. A. IT.. Rolke. R. IT., Sweed. S . H.. IND. EKG.%HEM. ’FL7SDAMEXT.& 7, 387 (1968). Wilhelm, R. H., Sweed, N.H., Science 169, 522 (1968). RECEIVED for review December 27, 1968 ACCEPTED February 17, 1969 Studies supported by NSF Grant GK-1427X. Additional support to one of the authors by XSF graduate fellowships is acknowledged. Computations performed in the Princeton University Computer Center, aided by NSF Grant GP-579.
Editor’s Note: The research reported in this paper was carried out jointly by R. H. Wilhelm and S . H. Sweed and is reported more completely in Professor h e e d ’ s Ph.D. thesis writ,teii a t Princeton in 1968.
PARAMETRIC PUMPING AND THE LIVING CELL HElN L. BOOIJ Laboratory of Medical (‘hemistry, Ilniz’ersity of Leidm, Leden, The iVetherlnnds Wilhelm suggested that parametric pumping might serve as a candidate model for active biological transport. In scaling the model to cell dimensions one encounters several difficulties. It i s not easy to fit a process of this kind into the biological membrane, as in doing so the number of adsorbent sites must be drastically reduced. Perhaps this decrease of effective sites will make the process inefficient on the biological level. The next difficulty has to do with the coupling of the oscillatory fields. While oscillatory electrical fields of high frequency at the cell surface are easily conceivable (and perhaps measurable), oscillatory mass transfer fields of high frequency are less likely in the very few (if any) pores of the biological membrane. Beside the experimental approach indicated by Wilhelm, a computational approach, aimed at scaling the mathematical model to a low number of adsorbent sites, might serve as a basis for verification or rejection of the model.
13 1966 a n unusual group of scientists-biochemists, cellular biologists, electron microscopists, geneticists, engineers, mathematicians, physical chemists, and physiologists-met in Frascati, Italy, for a Symposium on Intracellular Transport, organized by the International Society for Cell Biology
(Warren, 1966). The study of energy transductions in the cell, the relationships among structures and functions, movements within cells, and the transport from cell to cell or from medium to cell requires a n increasingly sophisticated integration of the natural, physical, and mathematical sciences. A t VOL.
8
NO. 2 M A Y
1969
231
the symposium, scientists of widely diverging disciplines focused their discussions on the processes and mechanisms underlying transport in the living cell. Despite semantic difficulties it soon became apparent that the engineer concerned with the analysis and design of chemical processing plants has many interests in common with the cellular biologist who tries to unravel nature’s way of chemical processing. One of the highlights of this symposium was Wilhelm’s lecture on parametric pumping, a model for active transport. Since my own scientific work had been aimed a t the relationship between structure and function of biomembranes, I felt that this new approach to the century-old enigma of transport across the cell border should be explored as fully as possible. Rilhelm’s invitation to come to Princeton as a visiting professor provided the opportunity for a series of discussions on the supposed role of parametric pumping in the living cell. This visit was also the beginning of a profound friendship, which was broken off by Kilhelm’s untimely death a few days before we left Princeton to return to the Netherlands. Though we did not reach a definite conclusion, this report may provide some suggestions for scientists who wish to continue Rilhelm’s stimulating train of thought. The reader will detect a certain bias caused by my 30 years of experience with the problem of permeability of the living cell. This, however, should not discourage the scholars who want to pursue the study initiated by Wilhelm. Any new concept should encounter a lot of opposition in order to get the incentive to produce as much experimental evidence as possible. Parametric Pumping
Parametric pumping is a dynamic separation technique. A liquid mixture is separated by the coupled action of one oscillatory field upon another. I n principle the technique is applicable to any fluid mixture that adsorbs on a second phase, or more generally, to any interphase equilibrium function which is dependent on Some intensive thermodynamic variable. The extent of separation depends highly on the system and the details of operation. The separation of toluene and heptane may serve as an example (Wilhelm and Sweed, 1968). The equipment (Figure 1 ) consists of a jacketed glass column (100 by 1 cm.) packed with silica gel particles and an infusion-withdrawal pump with coupled pistons. A programmed-cycle timer periodically reverses the direction of the fluid stream and also the jacket temperature by connection to hot or cold sources. Both oscillatory actions have the same frequency and are in phase. At the beginning of a run the bottom syringe is filled with 30 ml. of a mixture containing 20% by volume of toluene in n-heptane. The interstitial fluid in the packed bed of adsorbent particles has the same constitution and approximately the same volume. Hot water is circulated through the jacket as the upflow starts. When 30 ml. has been displaced, the jacket is switched to the cold-water source and the pump reverses direction The separation factors (defined as the ratio of mixed average toluene concentrations by volume in a syringe, top compared to bottom) are surprisingly high. Wilhelm and Sweed separated toluene from heptane to the extent of a lo5to 1 concentration ratio of toluene (cycle time 30 minutes, number of cycles 51, temperature limits 70’ and 4’ C.). Obviously the separation depends on the difference in
141
I
r--*Y I
Figure 1. Arrangement for separation by thermal parametric pumping Redrawn with slight alterations from Wilhelm and Sweed ( 1 968)
adsorption of toluene and n-heptane onto the silica gel particles. -4s a first approximation one niight consider the actions involved in separation in the following way. As the solution flows upward, an upward flux of toluene is caused by the fact that the heating of the system displaces toluene from the adsorbed to the dissolved state. I n the downward stroke the system is cooled and toluene is adsorbed, decreasing the downward flux of this substance. .4 succession of these adsorption-desorption cycles tends to cause accumulation of toluene at one end of the column and depletion a t the other. For set temperature-cycling limits and adsorbent 131 opei ties the separation punip-ul> per cycle is greatest for the longest cycle period. The adsorbent column (Figure 1 ) is used as e, batch separator with total reflux. The general idea of paranietric pumping may also be applied in various other arrangenientse.g., in continuous-flow, open-system columns. I n the case discussed the “pump” is identified as the oscillatory thermal field; the flow displacements between fluid and solid phases simply keep the system in a state of disequilibrium. Wilhelm (1966) has stressed that instead of the oscillatory thermal field other fields might be used as long as there are coupling relations between the fields involved. These niight include oscillatory electrical, magnetic, chemical potential (such as p H ) , and similar fields. Finally, although Wilhelm’s experimental \vork i\as with columns of particles, there is nothing that requires the solid phase to be stationary. All that is required is that there be a relative displacement between solid and fluid phases. Active Transport
I
232
l&EC
FUNDAMENTALS
The contents of the living cell differ considerably froin its medium. The red blood cell, for example, contains iiiuch more potassium than sodium ion, whereas in the blood plasma the reverse prevails. Several organic niolecules may be taken u p against a concentration gradient. Two conflicting theories have tried to explain this state of affairs. According to one theory every substance can enter the cell freely. The
I
0 0
C
B
A
0
Figure 2.
I..
D 0 4 0
O-
bolisrn
Suggested mechanisms of permeability
A. Diffusion through hydrophobic core of a biomembrana B. Diffusion through selective pores C. Facilitated diffusion b y selective carriers D . Coupling of metabolism and transport
cell is in equilibrium with its surroundings, but certain substances are bound specifically to cell conslituents and the solubility in the cell water is less than in the medium. The other theory states that the cell is separated from the medium by a harrier or membrane. Thus a state of nonequilibrium is maintained. This barrier must show a pronounced selectivity and living cells must have a mechanism which allows for the selective uptake of certain substances. The balance of experimental evidence favors the latter theory (Booij, 1962; Stein, 1967). One of the most important features of the supposed membrane is a hydrophobic core (lipids, perhaps mixed with apolar parts of proteins). This hypothesis (Overton, 1895) is strengthened by the fact that the outer layers of the cell have a high electrical resistance (Loewenstem, 1966). The difficulty is, however, that no substance of vital importance t o the cell (ions, sugars, amino acids, etc.) will pass this hydrophobic barrier a t a n appreciable rate. Moreover, some substances are transported into the cell against a concentration gradient, a phenomenon which costs energy. I t has been extremely difficult to explain the permeability phenomena on a nioleculai level. A discussion of the schemes pictured in Figure 2 will show the trend of thought of many scholars. A siniple diffuqion through the hydrophobic layer ie ; the larger the solubility in “oil,” the means ~ o i ~ eelectivitl easier the passage. Ohviouhly this type of selectivity is not very helpful for the uptake of the hydrophilic substances needed by the cell (Figure 2 , A ) . The next step is the assu~nptionof selective poreq (Figure 2 , B ) . Here the probleni is that a large number of different pores should be present in order t~ account for the biological selectivity. For this reason the concepl of the inobile carrier (Figure 2 , C) has found more adherents. I t ia supposed that a membrane constituent (protein or lipid) binds a substance preferentially a t one side of the menibrane and then “rotates” to the other d e (facilitated diffusion). r e i t h e r of these models permits the acc~uinulationof a substance a t one side of the membrane. This would be p o s d ~ l eif, for example, by some metabolic reaction the carrier i, changed in such a way that it cannot bind the substance a t the inner side of the membrane (Figure 2, D). Then it enters the cell loaded and returns unloaded; the substance accumulates in the cell. -4number of variations on this theme are possible. Selective enzymes may bind the substance to the carrier: in other hypotheses these enzymes are themselves embedded in the membrane. These hypotheses are highly speculative. Until now the supposed carriers have not been isolated. TVilhelm (1966) emphasized that parametric pumping is a
form of active transport, a process b y which a mass flux against a n apparently adverse fluid-phase concentration gradient is developed in a localized structure at the expense of some form of energy. It proved possible to construct a mathematical model of the process. H e suggested that applicability of the model may be explored beyond the immediate macroscopic separation process for which the model was derived. I n a n endeavor to apply the principle to the partition of sodium and potassium ions between the cell and its medium he compared experimental findings of Hodgkin (1964) and Huxley (1964) on giant squid axon cells and responses typical of a system of pwametric pump equations. A series of metabolism-associated steps will supply the energy for Na+ and K+ separation, undoubtedly through the production of a n energy-rich phosphate. I n a n arbitrarily chosen example the chemical driving forces act through electrical potentials as intermediates to cause separation. An electrokinetic mechanoelectrical transducer analog constructed by Teorell (1966) shows the possibility of the intimate coupling of the chemical potential gradient, the electrical potential gradient, and the pressure gradient. I n Rilhelm’s model too it is suggested that a n intimate coupling exists between an oscillatory electrical field and flow alterations. I n turn the oscillatory electrical field may have a large influence on the adsorptive equilibria of the ions. Small differences in adsorption of the two ions involved may in the course of a number of cycles lead to the concentration of one ion species a t one system terminal and of the other ion at the other terminal. I n comparing the classic voltage clamp experiments on squid axon cells with the mathematical model Wilhelm noted a gross similarity in the time course between experiment and model. Other features (a limiting value of separation and a transient minimum in the curves relating time and separation) also show a qualitative similarity. This analogy between experiment and model justifies the design of experiments which might serve as a basis for verification, rejection, or modification of parametric pumping as a candidate model for active biological transport. TKOtypes of experiments have been suggested by Wilhelm : measurement of oscillations in the electrical potentials of living cells, superimposed on a mean value, and effect of metabolic inhibitors and other substances related to the energy source on rapid-acting electrical outputs of the cell. Unsolved Problems
When trying to scale the proposed model to the dimensions of the living cell we encounter several difficulties. The height of the bed of adsorbent particles must be brought down from 50 to 100 cni. in the macroscopic experiments to 50 to 100 A. in the biomembrane, a factor of lo8. As the system must operate b y an alternating flow of water along adsorbent sites, one of the possibilities would be a water-filled pore in the lipid core of the membrane. The lining of the pore might consist of protein bearing the adsorbent sites. I n view of the small dimensions, the number of adsorbent sites in the system will be highly restricted. This leads to the first question. How does the height of the adsorbent bed influence the effectiveness of the parametric pumping process?’ Intuitively one would suppose that the effectiveness of the process would decrease strongly when the system contains only 2 to 10 adsorbent sites in line, but this guess should be substantiated by computation. One might argue that the height of the bed per se is not VOL.
8
NO.
2
MAY
1 9 6 9
233
8
A
-7 AFigure 3.
I I Possible oscillations in pores
A. Oscillatory fluid displacements in a stationary pore
B. Oscillating protein in a water-filled pore
important, since dimensionless equations can be employed to analyze a parapumping column. The question is, however, whether the conditions for these equations are met in small channels, having very few adsorption sites. These channels are so narrow that cations can barely pass. Thus diffusion to and from the adsorption sites is highly restricted. I n comparison to Figure 1 a parametric pumping process in the living cell would be a n open system consisting of very small “adsorbent columns” and extremely large “receivers” at both ends of the system. It may be asked how large the ratio between the total number of adsorbent sites and the volume of the smallest receiver (the cell) must be in order to get a separation between the ions in a reasonable lapse of time. This is a very important question, because from biological experiments it has been deduced that the number of pores varies between nil and very few. Solomon (1960) computes from his experiments on erythrocytes that the pore area is far less than ly0 of the total surface. The same experiments may even be interpreted on the basis that there are no water-filled pores a t all. At first sight one would expect that the capacity of the parametric pumping process will be linearly dependent on the number of effective pores. From the biological point of view a computational approach to this question too is of fundamental importance. It is expected that the oscillatory fields connected with parametric pumping in the living cell will have a very high frequency (lo6 to lo8 c.P.s.). This high frequency also poses some interesting questions. For set conditions the separation puriip-up per cycle is greater for a long than for a short cycle. At very short cycles the diffusion of the ion to and from the adsorbent sites may become a limiting factor. Moreovei,, the oscillatory electrical field must be in p1ia.e with the mass transfer field. The di.placement of fluid along the adsorbent phase should also have thia high frequency, The pores in biomembranes (if any are present) will have a diameter only slightly larger than the diameter of small ions. I t is difficult to see how fluid aould be displaced in these pores a t the frequency expected (Figure 3, A ) . The way out of this difficulty may be found in Wilhelm’s suggestion that the fluid is stationary, but that a protein (or part of i t ) shows oscillatory displacements (Figure 3, B ) . This would also fit the concept (Vilhelni et al., 1968) that the field driving the active transport would be a pH field, directly coupled with metabolisni. This field would change the charge of the protein and thus influence the adsorption of the cations to be separated. I n that case, however, a change of p H in the medium would strongly affect active transport. Experimental evidence does not seem to confirm this view. Oscillations of pH about a mean value have been postulated. Since, for example, differences of adsorption of Na+ 234
I&EC
FUNDAMENTALS
and K+ will depend on variations in charge of certain ionized groups it must be expected that this mean will have a critical value. I t will be very difficult to get conclusive evidence in favor of the concept of parametric pumping from biological experiments. The nerve cell does not seem to be a suitable object for these studies. It is a highly specialized cell, showing a pronounced excitability. When excited, electrical potential lvaves travel along the surface. These potential waves are directly coupled lvith the displacement of S a + into the cell and K+ from the cell to the medium. These displacements are of a passive nature; they are not directly coupled with metabolism, and ATP formation is not necessary (Rossini et al., 1966). Metabolic inhibitors do not influence the electrical activity (Keynes, 1961). Even the cytoplasm of the axon is not essential for this phenomenon (lleves, 1966). The original situation, on the other hand, must be restored by an active exchange of Ka+ and K+. Thus it seems advisable to study the electrical phenomena supposedly correlated with parametric pumping in a cell not showing the electrical phenomena of nerve cells. The reasoning also shows that the similarity between the experiments on squid axons and the mathematical model may be fortuitous, because in the biological experiments passive elements play an important role. I n general one might say that the active exchange of Na+ and K+ is directly related to the presence of ATP or another energy-rich phosphate, but only indirectly to metabolism. This would mean that experiments with metabolic inhibitors will not give straightforward information about the possibility of parametric active transport. Finally a study of a possible separation of Na+ and K+ ions in a macroscopic parapumping system would be very interesting.
Conclusions
iictive transport is one of the most difficult problems of biology. The selectivity of the process is high. The cell distinguishes between S a + and K+ or between L- and D-isomers of amino acids and sugars. Contemporary biological studies are aimed a t the isolation of “carriers” which supposedly bind the substrate specifically and translocate the bound substrate from the outside of the cell to the interior or in the opposite direction. The specificity of these binding membrane constituents must be in the order of that of enzymes. The concept of parametric active transport does not require this great selectivity. This theoretical advantage is counteracted by the difficulties one encounters when trying to scale the experimental procedure to cell dimensions. At present it is not possible to judge whether parametric pumping occurs in living cells. Two lvays may be indicated to solve this dilemma. First, an effort should be made to scale the mathematical model to extremely small dimensions by computation. Special attention must be given to the number and spatial arrangement of the adsorbent sites and the frequency of the oscillatory fields. The results should be in harmony with known properties of biomembranes. Secondly, in keeping with Wilhelm’s suggestion, an effort should be made to measure electrical oscillations a t the surface of cells actively exchanging Kaf and Kf. Though this finding nould not be a definite proof, it would give a strong indication in favor of parametric active transport.
literature Cited
Booij, H. L., “Conference on Permeability,” p. 5, Tjeenk Willink, Zwolle, The Netherlands, 1962. Hodgkin, A. L., Science 146, 1148 (1964). Huxley, A . F., Science 146, 1154 (1964). Keynes, R . D., “Membrane Transport and Metabolism,” A. Kleinzeller and A. Kotyk, eds., p. 131, Academic Press, New York and London. 1961. Loeweristein, M‘.R.,’An&.X . Y . Acad. Sci. 137, 441 (1966). Meves, H., Ann. K.Y.Acad. Sci. 137, 807 (1966). Overton, E., Vjschr. Naturforsch. Ges. Zurich 40, 159 (1895). Rossini, I,., Cohen, H . P., Handelman, E., Lin, S., Terauolo, C. A,, Ann. X . Y . Acad. Scz. 137, 864 (1966). Solomon, A. K., Sei. Am. 203, 146 (1960). ~
~~~~
~
~
~
Stein, m7. D., “The Movement of Molecules across Cell Membranes,” Academic Press, ?Jew York and London, 1967. Teorell, T., Ann. X . Y . Acad. Sci. 137, 950 (1966). Warren, K. B., ed., “Intracellular Transport,” Symposia of the International Society for Cell Biology, Vol. 6, Academic Press, New York and London, 1966. Wilhelm, R. H., “Intracellular Transport,” K. B. Warren, ed., Symposia of the International Society for Cell Biology, Vol. 6, u. 199. Academic Press. New York and London. 1966. WIlhelmj R. H., Rice, A.’W., Rolke, R. W., Sweed, N . H., IND. ESG.CHEM. FUXDAMENTALS 7, 337, 1968 Wilhelm, R . H., Sweed, N. H., Science 169, 522 (1968). RECEIVED for review December 23, 1968 ACCEPTED February 7, 1969
RECUPERATIVE PARAMETRIC PUMPING Model Development and Experimental Evaluation ROGER W. ROLKE’ A N D RICHARD H. WILHELMZ Department of Chemical Engineering, Princeton Uniilersity, Princeton, N. J . 08640
A computational model, based on finite difference solution of partial differential equations, i s developed to describe thermal and solute concentration behavior within a recuperative parametric pumping column. Experimental data for the dilute NaCl solution-mixed bed ion exchange resin system are used to evaluate parameters in the model, which successfully simulates the time- and position-varying column profiles during an experimental NaCI-HzO separation run. To model accurately this system in which interphase solute transfer is strongly intraparticle diffusion-controlled and alternating in direction, intraparticle profiles must be taken into account.
PARAMETRICpuniping
is a new separation principle depending 011 dynamic coupling of a periodic relative displacement between two phases and a periodic interphase solute flux. This coupling gives rise t o a net axial solute flux and hence a sustained separation. An initial description and a more complete overview of parapumping have been presented (Wilhelm et nl., 1966, 1968). Methods of providing the relative axial displacement and inducing the interphase flux were discussed, as were various system characteristics. R’Iany of these characteristics were determined using computational models supplemented by experimental data such as those reported by Wilhelm and Sweed (1968), showing parapumping’s large separating power. A computational model is desirable for many reasons. First, it provides a conceptual framework which can help in understanding the coupling phenomena responsible for the parametric pumping separation. Secondly, it can yield detailed information which is difficult or impossible to measure experimentally-Le,, the time-varying solid phase composition. Thirdly, and most importantly, a computational model is able to simulate the physical system, and therefore it can yield information about the process with a minimum of timeconsuming experimentation. These features of the recuperative parapumping model have been used to investigate the effects of system variables and to determine the ultimate separation which could be Present address, Shell Development Co., Emeryville, Calif. 94608. Deceased.
achieved using a given systeiu. It has also helped to identify factors which limit the experimentally observed separation and has been used to investigate the possibility of eniploying reflux or additional columns in a cascade to enhance single column separation capacity (Rolke, 1967; Wilhelm et a!., 1968). This paper presents the development of a coniputational model for the thermal mode of recuperative parametric pumping. Initially, a mathematical description is formulated for the packed bed adsorptive bystein, followed by the development of a numerical method for solution of the resulting equations. Then experimental data for the dilute KaC1-ion exchange resin separating system are used, first to determine parameters in the computational model and then to evaluate the model’s predictive ability. Recuperative Parapumping Model
The recuperative parapumping model is based on the adsorptive, packed bed system shown in Figure 1. Periodic relative displacement of the phases is imposed by upflow and downflow of the interstitial fluid. The periodic interphase solute flux is brought about by cyclically varying the adsorbent temperature, affecting the solute equilibrium. Use of a two-phase system other than a packed bed-e.g., two immiscible liquid phases-or varying pressure or p H rather than temperature to affect the equilibrium will change the equations relating behavior of the two phases, but the basic parapumping principles discussed previously (Wilhelm et al., 1968) still apply. The recuperative system shown in Figure 1 VOL.
0
NO.
2
MAY
1969
235
