Proc. Natl. Acad. Sci. USAVol. 91, pp. 7330-7334, July 1994Biophysics

Kinetics of exchange processes in the adsorption of proteins onsolid surfaces

(homomolecular exchange)

V. BALLt, P. HUETZt, A. ELAISSARIt, J.-P. CAZENAVE§, J.-C. VOEGELt, AND P. SCHAAF¶II**tCentre de Recherches Odontologiques, Institut National de la Sant6 et de la Recherche Medicale, Contrat Jeune Formetion 92-04 1, Place de l'H6pital-67000Strasbourg, France; tUnitM mixte Centre National de la Recherche Scientifique-Biom6rieux, Ecole Normale Sup6rieure de Lyon 46, Aile d'Italie-69364 LyonCedex 07, France; Centre R6gional de Transfusion Sanguine, Institut National de la Sante et de la Recherche Medicale, Unit6 311 10, rue Spielmann-67085Strasbourg Cedex, France; lInstitut Charles Sadron, Centre National de la Recherche Scientifique-Universitt Louis Pasteur 6, rue Boussingault-67083Strassbourg Cedex, France; and IlEcole Europ6enne des Hautes Etudes des Industries Chimiques de Strasbourg (Unite de Recherche Associe6 405-CentreNational de la Recherche Scientifique) 1, rue Blaise Pascal, BP 296F-67008 Strasbourg, Cedex, France

Communicated by Howard Reiss, April 11, 1994 (received for review March 3, 1994)

ABSTRACT The homogeneous exchange process wherebyIgG molecules adsorbed onto latex particles are replaced byIgG molecules from the bulk solution was studied by means ofts radiolabeling. The exchange mechanism was Investigated onsurfaces saturated with either labeled or unlabeled proteins inthe presence of a solution of the opposite species in two sets ofindependent experiments. After rinsing of the surface by purebuffer followed by supplementary IgG adsorption, the ex-change process.followed a kinetic law of first order with respectto the IgG molecules from the bulk solution, and the apparentexchange rate constant was (2.3 ± 0.4) x 10-5 cm hr'1.

One of the main differences between adsorption processes of"small" molecules and of macromolecules on solid surfacesis that macromolecules can reach the surface by two differentmechanisms (1): by direct adsorption or by a progressiveexchange reaction. In the first case the molecules reach theadsorption surface directly on free available space. In con-trast, for the exchange mechanism the molecules reach thesurface by progressive removal of already-adsorbed macro-molecules. This occurs in particular when the surface issaturated with preadsorbed molecules. This second processhas never been observed in the adsorption of small moleculeson surfaces.The physical origin of this second adsorption process lies

in the fact that macromolecules interact with a surfacethrough several links. These links along the macromoleculechange with time, even if the molecule is in an equilibriumconfiguration. Interactions between macromolecule and sur-face must thus be considered as a dynamic and not a staticprocess. Moreover, macromolecules are not rigid bodies.Thus, if such a molecule enters into the vicinity of a surfacealready covered by adsorbed macromolecules, it can changeits conformation and diffuse into the adsorbed layer. Oncepart of the backbone of the diffusing molecule reaches theadsorption surface, it interacts with it. This interaction canthen be followed by sequential, segmental, reptation-likeadsorption of the molecule, in competition with the dynamicprocess of the formation and annihilation of links of previ-ously adsorbed macromolecules. Finally, the diffusing mol-ecules progressively replace those already adsorbed on thesurface, completing the exchange process.Such processes have been observed for synthetic polymers

(2) as well as for proteins (3, 4) by means of radioactivelabeling techniques. NevertHeless, and in spite of a greatnumber of investigations, little is known about adsorptionprocesses of macromolecules (5). Let us rapidly summarize

the main results that have been established. (i) The affinity ofproteins for a given surface usually increases with theirmolecular weight. High molecular weight proteins usuallyexchange preferentially with lower molecular weight ad-sorbed proteins (6, 7). (ii) The higher the hydrophobicity ofthe surface is, the larger is the adsorbed quantity (8, 9). (iii)These amounts are maximal for pH values close to theisoelectric point of the protein-coated surface (10, 11). (iv)For synthetic homopolymers of the same chemical nature,the affinity for a surface increases with the molecular weight(12). (v) It has been shown experimentally that for homopoly-mers of the same nature and the same molecular weight, theexchange process can be modeled by a first-order chemicalreaction with respect to both the bulk molecules and theadsorbed polymers (2). This result has been explained theo-retically by De Gennes (13). However, it has never beenextended to other situations-for example, a mixture of twodifferent molecular weight polymers of the same nature. Forproteins, on the other hand, the order of the exchangereaction has never been determined for any situation, despitethe importance of this mechanism in the adsorption ofproteins onto biosurfaces.

It is the goal of this article to present initial experimentalresults devoted to this problem. We will not only determinethe order of the exchange process but also give an evaluationof the exchange rate constant. The experiments were per-formed for a system where human IgG molecules in the bulkwere in contact with human IgG molecules adsorbed onpolystyrene latex particles. This type of protein has beenchosen mainly for its importance in biological and pharma-ceutical fields-for example, in immunological tests.

MATERIALS AND METHODSAdsorbents and Adsorbates. Adsorption was onto mono-

disperse polystyrene latex particles prepared under emulsi-fier-free conditions (14) using potassium persulfate as initia-tor. Latex particle size was measured by transmission elec-tron microscopy with Hitachi Ha 12A equipment. Fromsizing of about 100 particles at the Centre de MicroscopieAppliqude a la Biologie (University Claude Bernard, Lyon,France), a number-average diameter (D.) of 790 nm wascalculated with a polydispersity index of 1.001. The specificarea was deduced to be 7.3 m2 gel. The surface charge of theparticles originates from the initiator residues (-OSO3)anchored at the water/polymer interface. After the latexparticles were cleaned by ionic exchange in mixed-bed resins(Duolite), the amount ofcovalently bound surface groups wasdetermined by following the conductivity upon titration with

**To whom reprint requests should be addressed at: Institut CharlesSadron 6, rue Boussingault, 67083 Strasbourg, Cedex, France.

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0.02 M NaOH; the deduced surface charge density was -0.68.C.cm-2. The dry weight content of the particles in the initial

suspension (pH 7.9) was 9.1%. No flocculation occurredwhen the latex particles were suspended in the Tris/HClbuffer used throughout this study (50 mM Tris, pH 7.35/1mM NaCl/1 mM NaN3).The proteins employed were polyclonal IgG (weight-

average molecular weight -150,000) isolated from humanserum of about 1000 donors. These proteins were provided bythe Centre Regional de Transfusion Sanguine (Strasbourg,France). They were characterized by SDS/PAGE withCoomassie blue staining; purity was judged to be at least95%.The IgG molecules were radiolabeled according to the

method of McFarlane (15), in which iodine monochloride isthe iodinating agent. The amount of NaI was chosen so thatabout 1 out of 3,000 IgG molecules was labeled. All proteinsolutions were then stored in concentrated form [-20 x10-2% (wt/wt)] at -200C. Before use, they were quicklythawed and diluted to their final concentrations in Tris/HClbuffer. The radioactivities of the labeled solutions weredetermined by y counting (Minimaxi y, United Technologies,Packard). From comparison of the radioactivity with theabsorbance at 280 nm [E = 1.38 cm2 mg-1 (16)] as measuredby a spectrophotometer (Beckman, model 34), the specificactivity was determined.

Experimental Procedure. All the experiments were per-formed in 2-ml Eppendorf tubes. The initial suspension oflatex particles was diluted in filtered and deaerated Tris/HClbuffer, with a dilution factor of =15. The precise dilutionfactor for each individual experiment was determined pre-cisely by weighing. All the experiments were performed atroom temperature, 23 ± 10C.

Direct adsorption experiments. Prior to exchange experi-ments, the direct adsorption conditions must be determined.First, the absence of particle aggregation in the presence ofproteins was verified by optical microscopy. Further, toavoid direct adsorption during the exchange process, satu-rated surfaces should be used and thus plateau adsorptionconditions, in particular both characteristic adsorption timesand the bulk protein concentration necessary for saturation,had to be evaluated.

Adsorption kinetics were first assessed: 1 ml of dilutedlatex particles was mixed with 1 ml of a labeled IgG solutionof known concentration [2.3 x 10-2% (wt/wt)] in an Eppen-dorf tube. After a given adsorption time t, under gentlerotation of 8-10 rpm (Agitest 34050, Bioblock Scientific,Illkirch, France), the tube was centrifuged at 8000 x g for 12min and the mean activity was determined from three pre-cisely weighed 200-,ul samples of supernatant solution. Theactivity difference before and after the adsorption processallows the determination of the amount of adsorbed proteinas a function of time. Great care was taken to avoid thepresence of latex particles in the supernatant after centrifu-gation. Turbidimetric control (turbidimeter model 800, En-gineered Systems & Designs, Newark, DE) showed a loss of<0.02% of the particles during this step. All experimentswere duplicated. Fig. 1 shows a typical kinetic curve; equi-librium seems to be reached within <2 hr and the final bulkconcentration at equilibrium is (0.45 ± 0.05) x 10-2% (wt/wt).To determine the adsorption isotherm, the same kind of

experiments as just described were performed, but varyingthe bulk concentration of the labeled proteins (cG*) insteadof the adsorption time t, which was held constant at 3 hr.Again, each experiment was duplicated and Fig. 2 representsthe evolution of the quantity of adsorbed proteins as afunction of cw*. From this, one can conclude that theadsorption plateau of the isotherm at 0.54 Lg-cm-2 is reachedfor a bulk concentration of the order of 2 x 10-2% (wt/wt).

U

C)

0

%-.

0.4

0.8

0.2

0.1

0.0

..... ...I...I....... I...

0 0 @ 8

T1 2 ( 4 6 6

Time (hour.)

FIG. 1. Adsorption kinetics of labeled IgG, with an initial proteinconcentration (in the Eppendorf tube) cgcj* of (1.5 + 0.05) x 10-2%(wt/wt), performed at 23 ± 10C. Each experimental point wasduplicated. The final protein bulk concentration in the tube wasfound to be (0.45 ± 0.05) x 10-2% (wt/wt).

Moreover, the reproducibility of these experiments can beevaluated to be of the order of 5%.Exchange experiments. Two types of exchange experi-

ments were performed. (Set I) Kinetic experiments in whichthe surface of the particles was first saturated with unlabeledIgG molecules, followed by exchange with labeled ones. Thebulk concentration of labeled proteins (cfo*) was chosen sothat its relative variation was <15% over the whole experi-mental time. For each concentration, six Eppendorf tubescontaining latex particles coated with unlabeled proteinswere prepared. The amount of adsorbed labeled moleculeswas then determined after a different reaction time for eachtube (in the 1- to 5-hr time interval). This set of experimentswas complemented by additional measurements at a constantreaction time of 4 hr. (Set II) Exchange experiments at aconstant reaction time t with labeled IgG molecules adsorbedon the particle surface, but in contact with solutions ofunlabeled proteins at varying bulk concentrations cI .For all these experiments, IgG molecules were first ad-

sorbed onto the latex particles for 3 hr at an initial bulkconcentration of (3.5 ± 0.3) x 10-2% (wt/wt). This corre-sponds to adsorption plateau conditions. This step wasperformed as described in the "direct adsorption" section.After 3 hr of contact between the particles and the proteins,followed by centrifugation, the unadsorbed IgG moleculeswere removed, avoiding any particle loss. To do this, thelatex particles were separated from the solution by centrif-ugation, and a portion (1.2/2 ml) of the liquid was removed;great care was taken that all latex particles remained in thetubes. A turbidimetric control on the eliminated liquidshowed that the loss of particles during this step was <0.5%.The same buffer volume v was then added to the Eppendorftube and the latex suspension was redispersed. After further

.1-1N

N.-Co

0.6

0.6

0.4

0.9

0.2

0.1

0.00 2 9 4 5

C *IG ( 1 0 2% (Wf/W) )

FIG. 2. Adsorption isotherm of labeled IgG at (23 ± 1)0C after 3

hr of adsorption.

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centrifugation and redispersion, a control with an opticalmicroscope showed that the particle redispersion was satis-factory (no aggregates were present in the solution). Theremoval-dilution-redispersion-centrifugation step was re-peated four times (five times for the pure desorption exper-

iments, when cWg- 0) and led to a dilution ratio of 30-40(75

for the pure desorption experiments) for the protein solutionremaining from the direct adsorption step.

After the last removaWdilution-redispersion-centrifuga-tion step (on average 2 hr after the beginning of the first step),a volume v of protein solution was added to replace thebuffer. The protein concentration present in the bulk solutionwas then coo (or cWg*). The particles were then redispersedand this was taken as time t = 0 for the exchange reaction.This reaction was allowed to take place under a gentlerotation of 8-10 rpm. After a time t, the solution was again

centrifuged and the activity ofthe supernatant was measured.For the experiments of set I, the difference between the

activity of the supernatant after the exchange step and theinitial activity ofthe solution corresponds to the fixed amountof labeled proteins during the exchange step. In set II, theactivity of the supernatant was found to be very small. Caremust be taken to account for the activity due to any- labeledproteins remaining in the solution from the initial adsorptionstep. Hence, we proceeded as follows: after the directadsorption step with the labeled proteins, the adsorbedamountrF- was determined as described above. UnlabeledIgG molecules were added to the solution (after the fourremoval-dilution-redispersion-centrifugation steps) to allowfor the exchange. At the end of the exchange step and beforecentrifugation, the activity of the solution (composed by boththe latex particles and the exchanging protein solution) wasmeasured. The solution was then centrifuged and the activityof the supernatant determined. The difference between thesetwo quantities corresponds to the amount of labeled proteinsr* that remained on the surface of the particles, and thedifference between r-* (value of direct adsorption at theplateau) andr* represents the amount ArF* (where subscript"r99 means "'released") of proteins that were released fromthe surface during the exchange step. This method requiresthe determination of three activities, which leads to morescatter in these experimental results compared with thosefrom experimental set I.For all the exchange processes taking place in these

experiments, the relative variations of the bulk protein con-

centration and the surface concentration of adsorbed proteinswere small (<15%). Consequently, we assumed these con-

centrations to remain constant for the analysis of the kineticlaws.

RESULTS AND DISCUSSIONWe first performed experiment set I. Fig. 3 shows the typicalevolution of Al'F0*, the protein amount that adsorbs on thesurface of the particles during the exchange step, as a

function of the reaction time t. Two characteristic periodsappear on the graph: (i) a rapid process taking place duringapproximatively the first 2 hr (for the experimental conditionsunder study) followed by (ii) a slower and quasilinear in-crease with time (of slope Se, the symbol "e" being used for"exchange") ofthe amount oflabeled proteins that adsorb onthe surface. Independent experiments were made for 10different bulk concentrations. The values of Se were deter-mined from the kinetic curves by a linear least-squares fittingprocedure; the linear correlation coefficients were in therange 0.93-0.98. A linear dependence of Se with bulk con-

centration cWo* was observed (Fig. 4). During the exchangeprocess, the relative variations of cG* were <15%. More-over, the experiments were realized with IgG* bulk concen-

trations that were close to or within the adsorption plateau

U11

el

0.20

0.16

0.12

0.08

0 1 2 3 4 5 6

Time (hours)

FIG. 3. Examples of kinetic experiments according to set I.ArIp*, surface concentration of labeled IgG for various bulk con-centrations cwoG: v, (1.22 0.05) x 10-2%; o, (1.97 0.05) x10-2%; *, (3.60 0.10) x 10-2%. Straight lines correspond toleast-squares fits in the interval 2-5 hr.

domain. Due to the large amount of unlabeled adsorbedproteins, the surface concentration of the latter is assumed tobe constant. Consequently the linear dependence of Se withcG* indicates that the process under study can be modeledby a chemical reaction of order 1 for the IgG* molecules inthe bulk. Thus, the amount of exchanged proteins A1'Wc*varies as

d(Al'IgG*)/dt kef(rlgj,To)C -*, [1]

where f("4Tr ) is an unknown function of the amount ofadsorbed protein rIwG,TOt (considered as constant over theexperimental period between 2 and 5 hr), and is the kineticconstant for the exchange. The value of kCf(I'WG,Tot) can bededuced from Fig. 4, yielding (2.3 ± 0.4) x 10- cm-hr-1.To prove that the adsorption process described in the

experiments of set I is indeed due to an exchange mechanism(and not to further adsorption), it is necessary to demonstratethat-correlated to the adsorption process-proteins arereleased from the saturated surface of the particles. This wasthe aim of the experiments of set II lasting over an exchangeperiod of 4 hr, with unlabeled proteins in the bulk solution.The direct preadsorption was performed with labeled pro-teins. Fig. 5 represents the amount ZLr' of molecules releasedfrom the particles during the exchange mechanism as afunction of the bulk concentration cG measured at thebeginning of the experiment. The experimental data wereplottedagainst the initial and not the final bulk concentrationof proteins (as we have done above). Indeed, the finalconcentration could not be evaluated because the proteins

t.

vu

C)

_m

0.012

0.008

0.004

0.000

0 1 2 3 4

C~g X 10 2% (W/W)CIgG

FIG. 4. Slopes (Se) from Fig. 3 (and the seven experiments notpresented), plotted against final bulk concentration cwo*. Straightline corresponds to a least-squares fit with a linear correlationcoefficient of 0.95. A Student's test showed that the intercept withthe y axis was statistically indistinguishable from zero.

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0

0

0)10

a)

Nrdi

0.20

0.15

0 0.10

wo AII0.05

0.00

0 2 4 6 8

Bulk IgG concentration

( x102% (w/w) )

FIG. 5. O, Released amounts of labeled IgG* (Ar,.) from latexparticles (saturated with IgG*) versus cWo at a constant reaction timeof 4 hr. a, Pure desorption experiments (c o- 0). Solid line was

calculated from the kinetic law deduced from Fig. 4 with k1(FIso,Tot)= (2.3 + 0.4) x 10-5cm hr-1 and a y intercept of 0.06 pg cm2. m,Fixed amounts of labeled IgG* (Afto*) versus cftop at the samereaction time. Data include the values from the 10 kinetics measure-ment (set I) and from 11 additional measurements (see text).

were unlabeled. However, as in set I the relative variationsin c% should be <15%. A value of about 0.06 Ag.cm-2 wasmeasured for Arl* when the particles were brought in contactwith pure buffer (cG = 0) instead of an unlabeled proteinsolution. This value corresponds to almost pure desorption,since at the end of the experiment, the concentration oflabeled proteins in the buffer, which is due to pure desorp-tion, was <0.2 x 10-2% (wt/wt). The values for Ar,. increaseas co is increased. The large scatter in the experimentalvalues does not allow us to affirm conclusively a lineardependence of Al'r* on cW, but such a dependence iscompatible with the observed data.

If one compares the exchange processes performed in theexperimental sets I and II, the amount adsorbed (set I) andthe amount released (set II) for a given bulk concentrationshould be identical within the experimental precision. This isnot the case, as can be observed in Fig. 5 (difference between* and O). However, if one assumes a linear dependence ofArr* with cG, with slope equal to its value determined fromset I and passing through the value of Arr* at cW = 0

(quantity released by pure desorption), one recovers a goodagreement for set II (straight line in Fig. 5) with the experi-mental data of set I.To understand the origin of the differences between the

adsorbed and released amounts, given by experimental setsI and II, respectively, we reanalyze the data from the kineticexperiments (set I) by focusing our attention on the differencebetween the amount AI'r.* adsorbed during the exchangestep and the amount kef(rI G,Tot).cfG*t actually attributed tothe exchange mechanism. Fig. 6 shows the evolution ofArl'* - ke~f(rlgo,Tot)jcigG*.t as a function of cG*. It can firstbe noticed that these quantities are almost independent oftime and thus depend only on cw*. This again shows that atleast two mechanisms with different time scales are involvedduring the exchange step: the slow exchange process, char-acterized by a linear dependence with time and with bulkconcentration, and a more rapid process whose contributionis plotted in Fig. 6. The curve shows a plateau domain forprotein concentrations above (2-2.5) x 10-2o (wt/wt) as forthe direct adsorption isotherm (Fig. 2). The observed plateauvalue in Fig. 6 is close to 0.125 ,ugCcm-2.Due to its resemblance to the adsorption isotherm, we

suggest that this rapid process is due to additional adsorption.This adsorption occurs after the four removal-dilution-redispersion-centrifugation steps as soon as the surface

0

0.20

o _-% 0.15E- CQ2

_ Wo

,0 0.05

tio0.00

0 1 2 3 4

CIgG* (x 0 % (w/W) )

FIG. 6. Calculated values ofAF715. - kflrgGTot)c,*,t, for thedifferent reaction times with the experimental data from set I, versus

CWpo. Symbols o, a, a, *, and v correspond to the reaction at 2, 3,4, 4.5, and 5 hr, respectively. Dashed line shows the amount oflabeled molecules that replaced the desorbed ones (0.06 pgcm2; seeFig. 5). X, Additional experiments performed according to set I withonly labeled molecules (see text).

covered with unlabeled molecules is brought in contact withthe labeled ones.Such a process cannot be observed when unlabeled pro-

teins are introduced after the same dilution steps, as in set II.Only the release of labeled proteins can be observed in thatcase; this included a pure desorption (independent of thepresence of additional proteins, see above) contribution of0.06 pgcm-2 which should also occur in set I. Some of theadditional adsorption after the adsorption-rinse-adsorptioncycle would replace this desorbed protein amount, but thehigher additional adsorption, with a plateau value of 0.125gCm-2 as seen in Fig. 6, suggests, if our assumption is

correct, that the plateau value of the isotherm is increased bya supplementary amount of about 0.06 rgcm-2. This was infact observed in two additional experiments performed as inset I but by adsorbing labeled IgG both during the initialadsorption period and during the adsorption-exchange step.These two additional experiments were performed with Coo*= (4.0 ± 0.3) x 10-2% (wt/wt) in the plateau region. The totalamount of adsorbed material was measured as in set II fromthe difference between the activity of the homogeneoussolution and the supernatant after centrifugation. In the firstexperiment, the measurement was performed after 4 hr; asupplementary adsorbed amount of 0.06 Wgcm-2 was in factfound (Fig. 6). In a second kinetic experiment (Fig. 7) we

0--C)

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aI*0

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0.5

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6

FIG. 7. Kinetics of supplementary adsorption that takes placeafter the four removal-dilution-redispersion-centrifugation steps.Labeled IgG molecules at an initial concentration of (3.5 ± 0.1) x10-2% (wt/wt) were first adsorbed onto the surface of the particles,and after the dilution steps, labeled proteins at (4.0 ± 0.3) x 10-(wt/wt) were added. Dashed line corresponds to the mean adsorbedvalue after the first adsorption step: 0.54 pg-cm2.

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followed the supplementary adsorption in the interval 2-5 hr.The readsorbed amount remained constant in this time in-terval, confirming that the further adsorption observed in setI after 2 hr was indeed protein exchange. The additionaladsorbed amount (Fig. 6), behaving like the direct adsorptionisotherm, could be the consequence of an interfacial rear-rangement taking place in the adsorbed layer during theremoval-dilution-redispersion-centrifugation steps. If onesubtracts from the data of experimental set I (for a reactiontime equal to 4 hr; see Fig. 5) this additional adsorbed amount(data from Fig. 6 minus the 0.06 Mg.cm-2 attributed to thereplacement of desorbed molecules), one finds a linear vari-ation of the exchanged amount (total amount minus theadditional adsorption contribution) with the bulk concentra-tion (Fig. 8). The linear regression by least-squares fit (cor-relation coefficient, 0.92) intercepts the y axis at 0.045gMcm-2, a value which is close to the one found previouslyin the pure desorption experiment.The straight line confirms moreover that the exchange

process is of first order with respect to the molecules in thesolution. The slope of the straight line must again be equal tok.cf(r,,To)-t (with t = 4 hr). The experimentally deducedvalue of (2.9 ± 0.8) x 10-5 cm-hr-' found for kgf(rWTot) isin good agreement with that deduced from the kinetic data(set 1), (2.3 ± 0.4) x 10-5 cm-hr-'. Moreover, the fixed andthe released amounts involved in the exchange process areequal within experimental precision.From these sets of experiments one can conclude that we

have described the exchange process for the system IgG/IgGon latex particles. We have also demonstrated (and to ourknowledge this has never been done before for proteins) thatsuch an exchange process can be modeled by a chemicalreaction of order 1 with respect to the bulk molecules, whichcorresponds to a kinetic law given by Eq. 1. The apparent

X 0.15

4i0

0Cd

X 0.000 1 2 3 4

CIgG* ( x 102 % (w/w) )

FIG. 8. Amount of IgG* exchanged with the unlabeled IgG-saturated latex particles after 4 hr of reaction, versus cwo*, calcu-lated after subtraction of the additional adsorbed quantities (valuesabove 0.06 Wcm-2 in Fig. 6) from the data plotted in Fig. 5. Solidline shows least-squares fit for the data of set I at 4 hr of reaction.

reaction rate constant has also been evaluated and is of theorder of (2.3 ± 0.4) x i0o cm hr-1. These experiments donot allow us to determine the order of the reaction withrespect to the adsorbed protein molecules, as has been donefor polymers (2). Such a determination would require labelingonly part of the adsorbed proteins and performing similarexperiments as in set II. The data in Fig. 5 clearly demon-strate that the accuracy ofthe results would be insufficient todraw any absolute conclusion about the order of this reac-tion. However, it is likely that the order of the reaction isequal to 1 with respect to the adsorbed molecules. If oneassumes such a second-order law, the kinetic equation for theadsorption process becomes

dAr~/dt = ke rigc,rotC-cgc*. [2]The kinetic constant k4 can then be estimated at (5.8 ± 1.0) x103 litermol-1 hr-1 for a rg value of 0.60 pgcm-2 (we used0.60 pgcm-2 instead of 0.54 Wcm-2 since we have shownthat the plateau value of the isotherm increases after the fourremoval-dilution-redispersion-centrifugation steps).

We thank H. Zins for typing the manuscript and E. Mann forcritical reading and improvement of the English text.

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