Quasi-linear gradients for capillary liquidchromatography with mass and tandem massspectrometry

Jie Zhou2, Felicia Rusnak2, Tim Colonius3 and Gary M. Hathaway1*1The Beckman Institute, California Institute of Technology, Pasadena, CA 91125, USA2Divisions of Biology, California Institute of Technology, Pasadena, CA 91125, USA3Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA

SPONSOR REFEREE: Professor Ken Standing, University of Manitoba, Winnipeg, Canada

Gradient elution, capillary liquid chromatography mass spectrometry was performed with linear, staticgradients constructed by laminar flowing ten, 1.5mL volume steps of decreasing organic concentration intotubing of small internal diameter. Sample loading, gradient formation, and sample elution wereaccomplished entirely by means of a commercially available micro-autosampler and single-syringe drivepump. The procedure was simple, fast, stable, and reproducible. Essentially linear gradients were producedwithout the use of additional valves, mixers, pumps or software. It took less than 10 minutes to form agradient and less than 30 minutes to construct the set of individual buffer vials. The gradients were shown tobe stable to storage. One hour after forming, peak retention times were reproduced to�0.5%. Long-termretention time reproducibility was found to vary by �2%. Chromatographic resolution was comparable orsuperior to that obtained by gradient elution with conventional dynamic mixing and split flow.

The procedure was adapted with a ‘peak parking’ method which extended the time for generatingpeptide fragmentation data up to 10 minutes per peptide with the triple quadruple mass spectrometer.Using this technique, collision data were collected at the 25 femtomole level on nine of ten trypticpeptides in a single run. Copyright# 2000 John Wiley & Sons, Ltd.

Received 3 January 2000; Accepted 5 January 2000

The electrospray quadrupole mass spectrometer’s ability toanalyze samples at the femtomole level makes it eminentlysuited for the study of subpicomole quantities of proteins1–3

and the complex peptide mixtures resulting from in situproteolysis of proteins separated by polyacrylamide gelelectrophoresis.4–7 Despite its ability to resolve mixtures,valid reasons exist for coupling a second dimension such asreverse phase, high performance liquid chromatography(LC/MS) to the electrospray source.8,9 Among these is theconcentrating effect achieved through the use of smalldiameter, reverse phase columns.10 Signal enhancement is adirect consequence of the fact that electrospray massspectrometry is concentration responsive.11 Since capillarycolumns require flowrates in the nanoliter/min range, suchlow flow gradients are usually achieved through flowsplitting. However, split flow is subject to complicationsfrom unpredictable fluctuations in column backpressures.Hence, direct gradient flow is the preferred, but difficult toachieve, method. Several approaches to direct elution havebeen proposed and carried out including storing gradientsoff-line as static gradients.12–14 The technique of formingstatic gradients was first proposed by Ishii and co-workers12

and applied to mass spectrometry by Daviset al.13 Thesimplicity of the method suffered somewhat from thenecessity of premixing the gradient prior to flowing it into

the loop. This required a means for pumping solvent A andB through a dynamic mixer to a switching valve, a second

Figure 1. Sketch of experimental setup for LC/MS with loop storedstep gradient. (P) single-syringe drive pump, (A) syringe needle of theautosampler, (S1) autosampler switching valve, (S2) switching valvefor peak parking, (T) ‘T’ splitter for peak parking, (R) restrictor line forpeak parking, (V) stainless steel Valco zero dead volume union, (C)C18 column packed in pulled sprayer capillary, and (N) nebulizer/sprayer body.

*Correspondence to: G. M. Hathaway, California Institute ofTechnology, The Beckman Institute, Mail Code 139-74, 400 S. WilsonAve., Pasadena, CA 91125, USA.Contract/grant sponsor: The Caltech Beckman Institute.

Copyright# 2000 John Wiley & Sons, Ltd.

RAPID COMMUNICATIONS IN MASS SPECTROMETRYRapid Commun. Mass Spectrom.14, 432–438 (2000)

valve for sample injection, and a high precision syringedrive pump.13

The work presented here describes a method for forminglinear gradients off-line from a series of steps of decreasing

organic concentration without mixers of any kind. Anautosampler with its single switching valve and a precisionpump was sufficient to perform sample loading, gradientconstruction and elution. The mechanism introduced bylaminar flow generated virtually linear gradients suitable forLC/MS and LC/MS/MS of peptides prepared by tryptic, in-gel digestion of proteins. Computer-generated models wereused to select the physical dimensions of the gradientstorage loop for optimal smoothness and linearity (seeappendix). We show that, in practice, a static gradientproduced by laminar flow of closely spaced steps givesresults comparable to those obtained with dynamicallymixed gradients.

METHODS

Gradient formation

The experimental setup is illustrated in Fig. 1. Theautosampler was used both to inject a small amount ofsample (usually 2mL) to a pre-equilibrated column, andthen to construct the gradient as follows. First, the 88 cmloop was filled with 60%B (22mL of 54% acetonitrile(ACN)). Then a program with three autosampler com-mands: (1) load (2) step and (3) aspirate was used to draw1.5mL steps from individual gradient vials. Generally, tenvials were used so that ACN concentration decreased by 5%

Table 1. Preparation of step gradient vials. Buffers A and B weremixed v/v in tapered bottom, glass autosampler vials inthe ratios shown. Total volume was 1.6 mL. 3� 20mL ofvial #1 were aspirated followed by 1� 1.5mL from vials2–17 and 6� 1.5mL of Buffer A

Vol. Ba Vol. Ab [B] [ACN]Vial No. (mL) (mL) (%) (%)

1 960 640 60 54.82 800 800 50 463 720 880 45 41.64 640 960 40 37.25 560 1040 35 32.86 480 1120 30 28.47 400 1400 25 24.08 320 1280 20 19.69 240 1360 15 15.2

10 160 1440 10 10.811 080 1520 05 06.412c 000 1600 00 02.0

a Buffer A: 2% ACN/0.1% acetic acid/0.02% TFA/98% H2O.b Buffer B: 90% ACN/0.1% acetic/0.02% TFA/10% H2O.c Vials 13–18 contained the same mixture.

Figure 2. Computer-modeled static gradients (A, left panels) compared with experimentally observed gradientsusing UV absorption at 265 nm (B, right panels). Loops were of i.d. 178 (top), 254 (middle) and 508mm (bottom).

Copyright# 2000 John Wiley & Sons, Ltd. Rapid Commun. Mass Spectrom.14, 432–438 (2000)

QUASI-LINEAR GRADIENTS FOR CAPILLARY LC/MS 433

per vial starting with 50% B (46% ACN). Finally, 10mL ofbuffer A were drawn to completely traverse the syringeneedle dead volume (10mL) and ensure the gradient waspositioned fully in the loop. During gradient formation,buffer A was directed to the column continuously to removeany nonbinding salts and other sample contaminants.Elution was commenced by activating the autosampler’sswitching valve which moved the low end of the gradient in-line with the column and pump. Gradient vials wereprepared by filling with a mixture of buffer A (2% ACN/0.1% acetic acid/0.02% trifluoroacetic acid (TFA)) andbuffer B (90% ACN/0.1% acetic acid/0.02% TFA) v/v asshown in Table 1.

Gradient stability and reproducibility

Gradient stability was measured by forming a gradient anddeveloping the column immediately, and comparing theresults with those obtained with a second run in which thegradient had been paused for 60 min after its formation.

To test long-term reproducibility, peak retention times forfive injections spaced over a 12-h period were measured18

and compared with those obtained with our commercialHPLC instrument.

Peak parking

Data collection for tandem mass spectrometry with thetriple quadrupole was extended by slowing the emergenceof a peak by modulating gradient flow in the followingmanner. A fast acting (<100 ms) switching valve connectedto a ‘T’ loop was inserted between the pump and theautosampler (electrical actuator Valco model EH and two-position valve Valco cat. no. EPC6CW Houston, TX, USA).One side of the ‘T’ was connected to a small column toprovide reduced backpressure. The other side was con-nected to the sample column flow path. Activating the valve

caused an almost instantaneous drop in pressure at thecolumn head from approximately 200 to 10 psi whichimmediately reduced the flow rate to 7–10 mL min. In thisway a peptide could be analyzed in LC/MS/MS for up to10 min in a manner analogous to nanospray.5,17

Mass spectrometry

Mass spectrometry was performed with a triple quadrupolemass spectrometer (API 365, PE-Sciex, Ontario, Canada)equipped with a modified Protana nanospray source(Odense, Denmark). To accommodate the outlet of thechromatography column, a conical adapter was machined tofit the Protana source, hold the union/column/sprayercombination, and provide a nebulizing gas. The barrel of aValco stainless steel union was turned on a lathe to fit theconical adapter. It held the column/sprayer and providedelectrical pickup for electrospray. The degree to which thesprayer needle extended beyond the 1 mm opening of theadapter was adjustable for optimal spraying. Details areshown in the schematic in Fig. 1. For LC/MS, capillarysprayer/column combinations, 0.1� 60 mm, were self-packed from the same batch of resin used in the 0.5 mmstainless steel columns obtained from Column Engineering.A small plug of hydrophilic membrane (Millipore Durapore0.65mm), was pushed to the 5–10mm opening of a laser-pulled (Sutter Labs, Novato, CA, USA), 360mm o.d. fusedsilica tubing. Resin was suspended in neat methanol in amicrocentrifuge tube housed in a high pressure bombconstructed in-house and similar to that described by Yates.4

Packing was initiated at 500 psi for a few minutes and thengradually increased to 1000 psi. The packed sprayer/columnwas connected via a Valco union to the switching valve ofthe autosampler by 50 cm of 30mm i.d. fused silica line.

Preparation of tryptic peptides

Tryptic peptides of horse apomyoglobin were prepared bydenaturing 5 nanomoles of the protein in freshly deionized 8M urea. The sample was diluted to a final concentration of 2M urea, 0.1 M ammonium bicarbonate buffer (Ambi), pH8.5. Trypsin (1%) was added and incubated overnight at30°C. Peptides were desalted using a Brownlee C18 guardcartridge, aliquoted, lyophilized, and stored dry atÿ15°C.Total peptides were quantitated from reconstituted aliquotsby amino acid analysis in triplicate.

In-gel tryptic digestion was performed in 25 mM Ambi,on 1.8 pmoles of bovine carbonic anhydrase quantitated bytriplicate amino acid analysis. The procedure of Hellmanetal.19 was modified as follows. Neutralized triscarboxyethyl-phosphine (2 mM in 50% ACN/25 mM Ambi) wassubstituted for dithiothreitol. In an effort to minimizeextractive losses, reduction was incorporated into the firstdye extraction step. The gel piece was rinsed once withAmbi, and alkylation with iodoacetamide was carried out byincorporating the reagent (25 mM) into the second extrac-tion step (in the dark). Digestion was initiated by adding0.25mg of trypsin to the dried gel piece in 5mL of Ambibuffer. A detailed description of this procedure may befound at our website.20

MATERIALS

The autosampler was purchased from LC Packings, SanFrancisco, CA, USA (FAMOS model 920). The sampling

Figure 3. (A) HPLC on 10 picomoles of apomyoglobin trypticpeptides. 0.5� 50 mm column, PE Biosystems Model 172, UVdetection at 200 nm and (B) TIC trace of 25 femtomoles of the samedigest injected onto a 0.1� 60 mm column with elution by staticgradient, 10 steps 2 to 46% ACN. Both columns were packed from thesame batch of 3mm Vydac C18.

Rapid Commun. Mass Spectrom.14, 432–438 (2000) Copyright# 2000 John Wiley & Sons, Ltd.

434 QUASI-LINEAR GRADIENTS FOR CAPILLARY LC/MS

capillary (syringe needle) of the autosampler was replacedwith one of 178mm i.d. Sample loops were constructed from0.178, 0.254, 0.406, and 0.508 mm i.d. polyether etherketone (PEEK) tubing. Limited volume (�2 mL), 12�32 mm glass autosampler vials were purchased fromAlltech, (cat. no. 72016).

A single-syringe drive pump model MV 10 with 2.0 mLsyringe capacity was obtained from Eldex Laboratories, Inc.

(Napa, CA, USA). To help reduce flowrate fluctuations, thesolvent compartment of the syringe drive pump wascontrolled at 36°C and insulated with 0.25 in. foam plastic.The case seams were sealed with tape and connecting PEEKand fused silica tubings were routed through 0.125 in. o.d.Teflon tubing for insulation. For all LC/MS experiments, thepump was operated in constant flow rate mode at 200 nL/min. A model 172 HPLC (Perkin-Elmer Biosystems’Division, Foster City, CA, USA) was modified for low flowby the use of an LC Packings ‘Acurate’ splitter and lowvolume (35 nL) flowcell. Small bore columns for the model172 were packed by Column Engineering (0.5� 50 mmVydac 3m bead size, 300mpore C18 resin, Ontario, CA,USA). Fused silica tubing was purchased from PolymicroTechnologies, Inc. (Phoenix, AZ, USA). Gel electrophoresiswas carried out with a Novex (San Diego, CA, USA) ‘XcellII’ gel apparatus using their Tris/glycine buffers and 12% 8�8 cm preformed gels. Chemically modified porcine trypsinwas obtained from Promega. ‘Omnisolve’ acetonitrile wasfrom EM Science. Trifluoroacetic acid andb-lactoglobulinwere obtained from Perkin-Elmer Biosystems’ Division(Foster City, CA, USA). Bovine carbonic anhydrase andequine apomyoglobin were obtained from Sigma-AldrichChemical Co. Deionized water (‘Milli-Q 18 megohm’) wasused throughout. All chemicals were reagent grade and usedwithout further purification.

RESULTS

Initial experimental results, obtained with various internaldiameter loops, compared well with computer-generatedmodels (Fig. 2). The experimental traces appeared smooth

Figure 4. Test of in-loop, static gradient stability in LC/MS mode with 25 femtomoles of apomyoglobin trypticpeptides injected. TIC shown before (upper trace) and after 60 min pause (lower trace). 10 steps, 15mL gradient, 2 to54% ACN in 75 min.

Table 2. Comparison of static and split-flow gradient elutionreproducibility. Static gradient experiments were car-ried out by LC/MS using a self-packed 0.1� 60 mm 3mVydac C18 column while the split-flow experiment wasconducted by UV absorbance and a commerciallypacked 0.5� 50 mm column of the same resin. Samplewas a tryptic digest of apomyoglobin. Tabulated datarepresent the average of five injections

Direct flow, in-loop gradient Split flow, dynamic mixed gradient

PeakStandard deviation

PeakStandard deviation

(min) (min) (%) (min) (min) (%)

16.10 0.51 3.2 19.62 0.24 1.217.70 0.50 2.8 23.42 0.14 0.5918.72 0.52 2.8 25.60 0.15 0.5722.88 0.49 2.1 28.63 0.14 0.4823.98 0.54 2.2 34.23 0.15 0.4424.62 0.55 2.2 37.30 0.15 0.4130.95 0.54 1.8 41.29 0.17 0.4232.48 0.58 1.8 51.75 0.15 0.3036.08 0.61 1.7 55.40 0.17 0.3148.15 0.45 0.9 66.16 0.21 0.3149.32 0.36 0.7

Averages 0.51 min 2.0% 0.17 min 0.50%

Copyright# 2000 John Wiley & Sons, Ltd. Rapid Commun. Mass Spectrom.14, 432–438 (2000)

QUASI-LINEAR GRADIENTS FOR CAPILLARY LC/MS 435

with little ‘step’ character and indicated full organicconcentration had been achieved. On the basis of theseexperiments we concluded that tubing with 178m i.d.provided the best results. The small amount of curvatureseen over the final 10% of the gradient (panel B1 of Fig. 2)was eliminated by including, as the final step, an 8%increase from 46 to 54% ACN. This was providedconveniently by forming the gradient with the loop initiallyfilled with 54% ACN (see Table 1). The final gradient gavea measured 96% linearity to>50% ACN. However, sincetheory predicts a distribution which lies between curvili-nearity and step-like character, we have used the term quasi-linear to describe static gradients formed in this manner.

Resolution was examined by comparing UV tracingsobtained by conventional HPLC and split flow with the totalion current chromatogram (TIC) produced with a staticgradient. As shown in the data of Fig. 3 for a mixture oftryptic peptides, comparable resolution was obtained withboth methods.

Gradient stability was investigated initially by examiningthe absorbance profiles of gradients in which 0.5, 1.0, and14 h delays were incorporated following gradient construc-tion. The results predicted there would be little effect ofdiffusion with delays in gradient application up to an hour(data not shown). To test the effect in experiment, a mixtureof peptides was chromatographed in two runs made intandem but with the second gradient application delayed by60 min after its formation. As shown by the data of Fig. 4,remarkably little change was observed in resolutionbetween the two TIC traces, and retention times werereproduced to 0.5%.

Long-term reproducibility was examined by recording

data for five duplicate runs with loop gradient elution andcomparing it with data obtained with our conventionalHPLC, and split flow with UV detection (Table 2). Themean deviation for peak time reproducibility for the staticgradient (TIC) was�2%. Our conventional HPLC withdynamic mixed flow splitting and UV detection showed areproducibility of 0.5% in this experiment.

Sensitivity in LC/MS was measured by injectingdecreasing amounts of the apomyoglobin digest, and asignal-to-noise (S/N) ratio of 15:1 was obtained with 20femtomoles of peptides injected (data not shown).

To illustrate the practicality of quasi-linear staticgradients in proteomic studies, tandem mass spectrometrywith collision activated dissociation (CAD) was performedon peptides obtained from an in-gel digest of 1.8 picomolesof bovine carbonic anhydrase. The data in Fig. 5 wereobtained by direct LC/MS/MS on 80 femtomoles of acarbonic anhydrase tryptic peptide. A complete y ion serieswas read, corresponding to the sequence VLDALDSIK.This sequence was matched to residues 158 to 166 of bovinecarbonic anhydrase as the top hit using NCBI BLAST.21

Although good fragmentation data was obtained by directLC/MS/MS at the<100 femtomole level, sensitivity andS/N ratio was limited by the time data could be accumu-lated. Even at the relatively low flowrate of 200 nL/min,peak widths of eluting peptides were of the order 10–20s.Hence a ‘peak parking’ method was adapted to gain moreanalysis time for signal acquisition. The data shown in themiddle panel of Fig. 6 were obtained by ‘parking’ nine often apomyoglobin peptides displayed in the TIC of the firstLC/MS run (upper panel). The peptide eluting withretention time 24.6 min in the upper panel (31.0 min in the

Figure 5. MS/MS with CAD performed on 80 femtomoles of an in-gel digest of bovine carbonic anhydrase. TICspectrum (upper trace). LC/MS/MS analysis on peak eluting at 44.86 min (lower trace). The sequence, VLDALDSIK,read from the y ion series corresponded to residues 158–166 of bovine carbonic anhydrase which was the number 1 hitfrom a BLAST search using only this sequence.21

Rapid Commun. Mass Spectrom.14, 432–438 (2000) Copyright# 2000 John Wiley & Sons, Ltd.

436 QUASI-LINEAR GRADIENTS FOR CAPILLARY LC/MS

middle panel) was selected and fragmented. Fragmentationdata are shown in the lower panel of the figure. Althoughneither a complete y or b ion series was obtained, fragmentions masses were used to unambiguously identify thepeptide as the sequence GHHEAELKPLAQSHATK (re-sidues 79–95 of myoglobin, calc. mass 1852.9) (UCSF MassSpectrometry Facility, Prospector MS-Tag22 and TheRockefeller University, PepFrag23).

DISCUSSION

The mechanism by which simple laminar flow produceslinear gradients from closely spaced organic steps is thoughtto be radial diffusion during gradient delivery. Gradientdelivery occurs at a flow rate sixty times slower than stepproduction. The lack of lateral diffusion (related to gradientstability) is a consequence of the radius-to-length ratio. Asingle step of the gradient for the 178mdiameter tubing usedin this work was contained in 6 cm of tubing with acalculated radius/length ratio of 677. As the gradientpotential for diffusion depends on dc/dx, it is predictablylarge in the radial dimension where the distance betweenlaminar phases is small.

A possible source of peak time variability is thetemperature dependence of flow. It was possible tothermostat the pump head and to insulate the connectinglines, but given the column configuration, i.e. as an integralpart of the spraying needle, it was not possible to thermostatthe column. This undoubtedly produced some variation inflow through fluctuations in room temperature. Frictionaleffects could be neglected as a source of heat as columnbackpressures seldom rose above 2 megapascals.24 Assum-ing peak retention reproducibility (�2%) was due solely and

directly to the change in the rate of flow, this wouldcorrespond to a variation of 4 nL/min. The presumption thatpeak elution times with static gradients should be directlyproportional to flowrate was confirmed by measuring peakretention times for six peptides at double the usual flowratebut with the same gradient volume. Average retention timesdeclined a factor of 2.00� 0.03 as expected, and gradientslope (as measured by UV absorbance) was observed to bedirectly proportional to flowrate (data not shown). It isimportant to note that the dependence of gradient steepnesson flowrate is an intrinsic property of loop stored gradientswhich we regard as a useful way to manipulate gradientslope providing flowrates can be modulated rapidly.

CONCLUSIONS

Construction of the set of the gradient buffer vials was fast,requiring less than 30 minutes, and less than 10 minuteswere required to form the static gradient which was stablefor at least one hour. The low volume used per gradient, pervial of 1.5mL, ensured hundreds of gradients from a singleset of buffer vials which were stable at room temperature forat least two weeks. Finally, the entire procedure of sampleloading, gradient production and elution is highly auto-matable and readily accomplished with the use ofcommercially available equipment.

APPENDIX

Gradient theory and computer modeling

In order to guide the choice of experimental setup (i.e.,tubing diameter, number of gradient steps, etc.), and guide

Figure 6. Demonstration of peak parking with 25 femtomoles of apomyoglobin tryptic peptides. LC/MS TICspectrum (upper trace). A second 25 femtomole injection with peak parking of nine of the ten peptides (middle trace).Tandem mass spectrum (lower trace) of the triply charged precursor ion (m/z619) from the peptide peak parked at31 min (arrows added as a visual aid). This peptide corresponded to the peak at 24.6 min in the first LC/MS run.

Copyright# 2000 John Wiley & Sons, Ltd. Rapid Commun. Mass Spectrom.14, 432–438 (2000)

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the interpretation of the experimental results, a simplemodel was constructed of the mixing process in the gradientloop. This model and its comparison with the experimentalresults are described here.

First, it is important to distinguish between the mixingprocesses that occur during the sequential drawing of themicro-liter steps of organic into the closed loop, and thosewhich occur during the subsequent flow of the concentrationgradient for elution. During elution, flow rates aresufficiently slow such that radial diffusion across any crosssection of the tube is essentially complete before any mixingoccurs due to axial transport by the nonuniform (varyingprincipally with radius) velocity in the tube.

During the filling of the loop, on the other hand, mixingoccurs by both processes and is governed by the advection-diffusion equation:

@c@t� u�r� @c

@x� 1

r@

@r@c@r

�1�

where c(x,r,t) is the instantaneous local concentration ofsolute,x is the axial distance along the tube,r is the radialdistance from the centerline,t is time, and g is thecoefficient of molecular diffusion. The steady velocitydistribution in the tube isu(r). Since the Reynolds numberof the flow was calculated to be near unity for theexperimental configurations described here, it is reasonableto assume that fully developed laminar flow prevails,whence, for a round tube, the Hagen-Poiseuille velocitydistribution is:

u�r� � u0�1ÿ �r=R�2�; �2�where R is the radius of the tube,u0 is the maximum(centerline) velocity, and is related to the overall volumeflow rate,Q, by:

Q� u0�R2

2: �3�

The mean concentration over any cross section is givenby:

Cm�x; t� � 2R2

Z R

oCrdr; �4�

For elution, where radial diffusion is fast compared toaxial transport, the concentration does not vary over thecross section (c(x,r,t) = Cm(x,t)), and transport of thegradient is governed by an advection equation with constantvelocity equal to the average velocity (flowrate divided bycross-sectional area):

@Cm

@t� Uavg

@Cm

@x� 0 �5�

Inthat case, the concentration gradient at any axial locationis simply transported downstream at the average flow speed.This means that the concentration gradient exposed to thecolumn during elution is governed completely by theprocess of filling the loop. Thus, our modeling effort hasbeen directed at modeling mixing during the sequentialdrawing of the micro-liter steps of organic into the closedloop. We solve Eqn. (1) directly using a 6th-order accuratefinite-difference method. The simulation code was validatedby comparing with the analytical solutions given byTaylor.15,16 In Fig. 1 we compare the evolution of the

simulated mean concentration with several experiments thathad varying tube diameters. It is seen that the model wellcaptures the trends of the actual measurements. For smallerdiameter tubes, where radial diffusion is more important, theshape of the curve exhibits remnants of the steps at earlytimes, but remains linear, increasing to the maximumconcentration at late times. For the larger diameter tubes,axial transport smooths the steps, but the rate of increase ofconcentration falls off at late times. Thus the shape of thegradient curve is thus determined by a competition betweenradial diffusion and axial transport by the radially varyingvelocity field. This competition is illustrated pictorially inFig. 2, which shows contours of concentration throughoutthe loop a short time after a series of ten, 5% gradient stepsare drawn into the tube (the initial condition is also shown).

Acknowledgements

Funding for this research was provided by funds from The CaltechBeckman Institute. The authors thank the City of Hope, Department ofImmunology, Protein Core Laboratory for providing the amino acidanalyses.

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