- Email: [email protected]
Sticking coefficients of adsorbing proteins
577
Sticking coefficients of adsorbing proteins Daniel R. Weaver and William G. Pitt Department
of Chemical
Engineering,
Brigham
Young University,
Provo, Utah 84602, USA
The protein sticking coefficient, 4, the fraction of collisions that result in adsorption, is a function of the molecular interactions between the protein and the surface. A random walk and diffusionto-capture model was used to describe the kinetics of protein adsorption. The assumption of a constant sticking coefficient leads to a first-order model of the kinetics. A solution of the problem of adsorption from a semi-infinite medium with first-order kinetics at the boundary was obtained by numerical simulation on the computer. The results of the computer simulations match the time dependence observed experimentally. A correlation was developed to estimate C#J from experimental data. I#Jhas been found to be in the range 10-5-10-* for several protein adsorption kinetic studies reported in the literature. Keywords: Received
Protein adsorption,
11 November
diffusion,
1991; revised 8 January
The adsorption of proteins at interfaces has been shown’ to be a complex phenomena which includes the diffusion of the protein particle through the aqueous solution and the collision and interaction of the protein at the interface. One important goal in the study of protein adsorption has been the measurement of the affinity between the protein and the surface. The affinity has been broadly defined as a measure of the interaction between the protein and the surface. It has been suggested by Horbett and Brash’ that one measure of the affinity is the protein sticking coefficient. The sticking coefficient, denoted 4, is the fraction of the collisions between the protein and an available surface site that result in adsorption. It has been hypothesized by Horbett and Brash, and it is the hypothesis of this study, that the sticking coefficient can be deduced from kinetic measurements under conditions near the diffusion limit. Diffusion-controlled adsorption of protein from nonflowing solution has often been modelled by3
where cs is the concentration of adsorbed protein, cs is the initial (bulk) protein concentration, D is the diffusion coefficient, and t is time. Equation 1 has several shortcomings which will be discussed. However, it has been used by several investigators of protein adsorption kinetics because experimental data match the functional form of Equation 1. It has been frequently observed3-’ that, initially, the amount adsorbed increases in proportion to t”‘. However, in many cases the initial observed slope dcsld(t”‘) has been found to be lower than 2cB(Dln)“’ predicted by Equation Z. It has Correspondence
0
1992
to Professor
W.G. Pitt.
Butterworth-Heinemann Ltd
0142-9612/92/090577-06
sticking
coefficients,
1992; accepted
9 January
1992
also been frequently observed that after a period of time the adsorption rate decreases sharply due to surface filling. The departure of the initial slope from the diffusion limit has been attributed to a sticking coefficient that is not unity. Physical mechanisms such as energy barriers and reversible adsorption have been proposed to explain the departure of the sticking coefficient from unity. In the present study, the nature and consequences of the departure of the slope from the diffusion limit is investigated within the framework of a model for the sticking coefficient. To facilitate the investigation, a parameter Q is defined to be the ratio of the experimentally observed slope to the slope of the theoretical diffusion limit
Q, the dimensionless observed slope, will be shown to be a measure of the departure of the experiment from completely diffusion-controlled conditions. The goal of this study included four main aspects: 1. To determine if kinetic limitations coupled with diffusion limitations could produce this functional t”* dependence with the accompanying decrease in the slope. 2. To determine how the sticking coefficient, @, correlates with the slope Q. 3. To develop a model or correlation to calculate the sticking coefficient from kinetic measurements. 4. To evaluate experimental data and determine an order of magnitude estimate of protein sticking coefficients. Biomaterials
1992, Vol. 13 No. 9
578
Sticking
coefficients
of adsorbing
D.R. Weaver and W.G. Pitt
proteins:
protein molecules in aqueous solution. This misconception is likely to result from the fact that the equality on the right in Equation 5 can be derived from the Boltzmann distribution of velocities for ideal gases”. However, it can be derived from concepts of statistical mechanics. Berg” has discussed the validity and the use of Equation 5 for colloidal particles in liquids. A statistical mechanics derivation of Equation 5 is given in Appendix A.
Figure 1
Feller’s elastic barrier. At the surface node there is a probability @ of being adsorbed and a probability (1 - I$) of being reflected. At the other nodes the probability of a step to the right is p and the probability of a step to the left is q.
THEORY Sticking coefficient
theory
The concept of a sticking coefficient has been discussed in literature on random walks near barriers in the study of the so-called ‘elastic barrier”. The elastic barrier is depicted in Figure 1, in which the particle in solution is assumed to execute a discrete random walk on the nodes as labelled. At the interfacial node, i=s, a constant fraction, 4, of the jumps to the left from node zero result in adsorption. A constant fraction (1 - o] of these jumps result in reflection. For all of the other nodes, zero and greater, the probability that the particle steps to the right is denoted p. The probability of a step to the left is q. For an unbiased random walk, p = q = 0.5. A parameter, S, is defined to be the step size of the random walk or the distance between the nodes. Another parameter, u, is defined to be the average jump frequency. One can also define a parameter v, = Su, which can be interpreted as the average velocity of the random step. Collins and Kimball9 and Goodrich” have shown that, given the above formulation, the flux to the surface and thus the intrinsic kinetic expression can be described by
Adsorption-diffusion
theory
The development of equations to describe adsorption coupled to diffusion in a stationary fluid medium was pioneered by Smoluchowski’3 and has been discussed by others such as Ward and Tordai14, Baret” and Varoqui and Pefferkorn”. These treatments have used the mathematical formulation:
dCJ&
aX2'
at
t>o
(‘3)
t>o lim
c = cs,
t>o
(81
=
x20
(9)
X-a
Cl,=,,
cJ,=o
CB,
=
0
(10)
where c is the protein concentration, and x is the distance from the interface. Boundary condition Equation 7 requires an expression for the rate of adsorption. The intent of this study is to solve Equations 6 through 10 with Equation 4 as the rate expression, by making the approximation that the node j=O is located at x=0. To solve Equations 6-10, and 4 for a general case they were transformed into dimensionless form by defining the following variables: r
=
FcBI
2Dt
--
(111
CM
X2BX where R, is the rate of adsorption. The assumption of a constant sticking coefficient leads to first-order irreversible kinetics. One can modify Equation 3 to allow for surface filling by including a term to account for the fraction of available surface sites
cM
CCC cB
pcs cM
where 0 is the fraction of the surface sites that are filled. The interest in the present study is in a discrete random walk where the step size is on the scale of a molecular collision with the surface. In such a case, the average velocity of the random walk would be approximately equal to the root-mean-squared velocity of the protein molecules in solution v,
c
( v,)l12 =
(E)“’
(5)
where m is the molecular mass of the protein. There is a common misconception that Equation 5 would not be a valid estimate of the root-mean-squared velocity of Biomaterials
1992, Vol. 13 No. 9
The parameter @ is a dimensionless reaction rate constant and is a measure of the rate of adsorption compared to diffusion. The parameter 8 was taken to be equal to the fraction of a simple monolayer 8=cs/cM. Equations 6-10, combined with Equation 4 become respectively
ac a% dt--ax2' dlac
r>o
(16)
~=dx~x~o=~clx~o~l-el~
r>o
(171
lim X-a
r>O
(181
c = 1,
Sticking
coefficients
Cl,=,
of adsorbing
proteins:
DR. Weaver and W.G. Pitt -_
xzo
= 1,
I-lr=O=O
(191 (20)
The analytical solution to Equations 16-20 is known for the case where the [l - 01 term is not included and is discussed by Collins and Kimball9 and by Goodrich”. An approximate analytical solution has been proposed by Varoqui and Pefferkorn’~ with the (1 - Sl term included. However, their solution is valid only under the assumption that the desorption rate is high or that the surface coverage is small. These assumptions limit their model such that it cannot be applied to most protein adsorption kinetic data.
NUMERICAL STUDY In the present work equations 16-20 were solved numerically using an explicit finite difference method17. The first forward difference was used to approximate the time derivatives, the first central difference was used to approximate the second derivative and second forward difference was used to approximate the first derivative with respect to distance. In solving the finite difference equations the solution was unstable unless the coefficient that multiplied the node of interest was greater than or equal to zero. Applying this stability criterion to Equations 16-20 required that
and 3Ar l-ZAX2-bX
@AIT I
20
where, Ahr and AX are the dimensionless time and distance steps respectively. Equations 22 and 22 place a limit on AX for a given value of Ar. The numerical solution was completed by choosing a value of Ar and then calculating the size of AX using Equatjon 23 or 22. The size of Ar that was chosen varied from about 3 X 10e6 to 3 X lo-’ depending on the value of 0. For large values of @ very small values of A r were required to maintain accuracy. For most of the computer runs approximately one million time steps were used. The requirements of boundary condition Equstjon 18 were met by using an X array large enough so that C at large X was always equal to unity. A Convex 210 Mini Super Computer was used to perform the calculations. The analytical solution of Collins and Kimball9 provided a standard with which to test the numerical solution. When the numerical simulation was performed without the (1 - 0) term the results agreed exactly with the analytical solution of Collins and Kimball for @ = 10, 1, 0.1,and 0.01. Based on the perfect agreement between the analytical and the numerical solution the numerical method was considered accurate and capable of producing valid results in the present study.
Figure2 Surface concentration as a function of time”* (dimensionless) for @ ranging from 0.01 to 100. The * represents the inflection point of each curve. The dashed line represents the diffusion-limited case predicted by Equation 1
(see text).
DISCUSSION The numerical solution The numerical solution of Equations 16-20 is shown in Figure 2, in which I is plotted as a function of r*“. The solution is shown for several different values of Q,. The * in Figure 2 denotes the inflection point of each curve. The numerical solution curves are linear with respect to r112for values of @ = 0.6 or greater. These solution curves match the behaviour observed experimentally. As Q, increases the solution approaches but does not exceed the diffusion limit (dashed line Figure 2). To determine and compare the effect of the (1 - 0) term, the Collins-Kimball solution was plotted alongside the numerical solution in Figure 3. Clearly, the (1 - 0) term causes an early departure from the Collins-Kimball solution,
Application to experimental
data
Since the dimensionless slope, Q, has been frequently observed to be lower than the diffusion limit, it is useful to know the relationship between @ and Q, The results of the numerical simulation were used to make a correlation and thus define this relationship. The goal in making this correlation was to provide a tool for the experimenter to calculate a sticking coefficient. Thus the correlation was made between the slope dcsld(t”‘] of the linear section of each curve and the value of Q?that was used as input into the numerical simulation. The correlation is shown in Figure 4. To estimate the slope of each curve certain arbitrary choices had to be made. There are several reasonable methods to estimate the slope in the linear section such as the slope at a point such as IYs= 0.5, or the slope at the inflection point, or by a least-squares fit through a section of the data. Figure 4 was constructed by correlating the slope at the inflection point because this method required the fewest arbitrary choices and Biomaterials 1992,Vol.13 No. 9
560
Sticking coefficients of adsorbing proteins: D.R. Weaver and W.G. Pitt 100
80
60 8
40
20
2.00
4.00
6.00
8.00
o3
10.00
0.00
0.20
0.40
0.60
0.80
1 .oo
Q Figure 3 Comparison of the numerical solution (solid lines) to the Collins-Kimball solution (dashed lines) for Q,equal to 1, 0.1, and 0.01.
was the best representation of the linear section of the curves. In any case, the form of the @ - Q relationship was the same for all methods tested. Figure 4 and Equation 15 were used to estimate values of @ and # from values of Q reported in the literature3* 5-7. Values of D and m were obtained from the respective references or from Refs 6 and 18, except that for vitronectinD was estimated to be 5.12 X IO-’ cm’/s. The results are shown in Table 2. In three cases, the experimental values of Q were reported to be unity. In these cases the values of Q, were assumed to be 100, since the @= 100 case was indistinguishable from the diffusion limit. The results in Table 2 indicate high values of @ for the following protein-surface systems: IgG 9.0 on silicone rubber, albumin on silicone rubber, and albumin on silica at high pH. The results also indicate a relatively low value of @ for cz2 - h4 on silicone rubber, The values of Q, in Tahoe 1 are in the range 1O-5-1O-8 Table 1
Figure4
9, as a function of Q.
indicating that the rate of collision rate of accumulation or adsorption the relationship ~,=~vACl.=o
(231
R,, the rate of collision, was estimated to be 1Ol7 molecules cm-’ s-’ for a protein with mol wt 100 000 and with c IX= o = 0.05 mg/cm3. With such a large collision rate and a sticking coefficient of 10m5 a monolayer of 0.2 ,ug/cm’ would be filled within minutes. In performing the calculations of the sticking coefficients, it was assumed that q=lJ2 in ~~ua~iun 4. This assumption, however, does not lead to large uncertainty in the value of @. This conclusion is based on an analysis of the distributions, given by Goodrich”, for the random walk of particles near a partially reflective barrier. It is our observation that g could not deviate by more than 170 from a value of 112, thus leading to an un~e~ainty in Cp of 1%. This unce~ainty is small
Estimated values of Cp,(b and C 1x = o,min
Protein
Surface
Albuming~c
Silica
Albumin’ b Albumi@ IgG 9.0’ IgG 7.8’ IgG 6.5’ Q*-Mf (z2-Mf
Silica SR SR SR SR SR z:
Transferrin’ Fibronectine Fibronectin* Vitronecti# Vitronectind Vitronectind Vitronectind
C”B __
Silica NPTES PS %PS OXPS
_.._
CL
Q
1992,
Vol.
13 NO. 9
@
cpx 108
--___-.--
0.026
0.13a
0.3
0.02 0.005 0.0044 0.0044 0.0044 0.005 0.0025
0.13= 0.14 0.30 0.26 0.21 0.18 0.18
0.0025 0.05 0.01 0.03 0.01 0.03
0.14 o.lga 0.2a 0.2a 0.15a 0.15a
CIX
0.22
8.8
0.8
1.0 1.0 0.60 0.60 0.46 0.29
100 100 1.5 1.5 0.64 0.21
3079 715 283 4.9 6.1 ::;6
::: 0.0 0.5 0.5 0.6 0.8
0.77 0.13 0.90 0.43 0.25 0.4 0.32
3.9 0.04 11 0.55 0.15 0.45 0.26
14.2 56Y 8.7 3.9 5.2 9.1
z: 0:2 0.6 0.8
Bestimated; bpH 4.8; ‘pH 4.0; ‘Ret. 6; %ef. 5: ‘Ret. 3; 9Ref. 7; hmg/ml: ‘pglcm”. SR, silicone rubber; NPTES. N-pentyltriethoxysilane; PS, polystyrene: OXPS. oxidized pOiystyrefle.
Biomaterials
is large relative to the at the interface. Using
= 0. min
Stickinq coefficients of adsorbinn proteins: D.R. Weaver and W.G. Pitt
compared with uncertainties introduced in experimental measurements. Also, it was assumed that va=( v~)~‘~ according Equation 5. This assumption is likely to be correct to within an order of magnitude. The conclusion is based on the definition” of the diffusion coefficient D=S2ui2 which, combined with the definition of va, gives v, = 2DIS. It is reasonable to assume that the mean free path for a protein molecule in aqueous solution is less [perhaps much less) than 0.1 nm and that a random walk step size of S = 0.1 nm would be uncorrelated. In this case, one can estimate that v,=lOO cm/s for a small protein with a diffusion coefficient of 5 X 10e7 and a step size of 0.1 nm. For the proteins in T&e 1 the values of v, were calculated to be in the range 100-600 cm/s using Equation 5. Intuition based on consideration of the physical system suggests that there are more collisions per second of a protein with an interface than there are jumps per second of the same protein across 0.1 nm. Therefore, it was concluded that the use of Equation 5 is approximately correct. The
relationship
581
protein concentration at the liquid interface is assumed to be depleted immediately to and remain constant at zero. The theoretical expression for the observed kinetics in this case is Equation 2. In the second solution, the reaction-limiting case, the interface concentration is assumed to remain constant equal to ca. The theoretical expression for the observed kinetics in this case would be the time integral of the rate expression with the interface concentration held constant equal to cs. Both cases are discussed adequately by Schaaf and Dejardin’l. In the third solution, the rate of adsorption is an approximate balance between diffusion and reaction. In this case, one could assume that the interface concentration is rapidly depleted to some other non-zero value, ct, that remains constant cl,=,
= c1
t>o
(241
With this new boundary condition Equations 6-10 can also be solved” and the analogous relationship to Equation 1 becomes
between cf, and Q
Several authors have been interested in the relationship between the probability of adsorption, @, and the observed slope Q. It has been commonly assumed in previous literature that Q is equal to Q4’7*1Q.Such an assumption was inherent in the calculation of values for the ‘Arrhenius’ activation energy4. 7. This assumption is incorrect and Collins and Kimball’ in 1949 discussed that such a misconception had been previously madez’. Figure 4 shows that the relationship between Q, and Q [thus the relationship between 0 and Q) is non-linear and has the appearance of a hyperbolic function, The function is asymptotic near unity. The consequence of the asymptotic behaviour is that upis not necessarily near unity when the experimental adsorption process appears completely diffusion controlled. For example, the @= 100 curve in Figure 2 is indistinguishable from the diffusion limit. Yet, with this value of a=100 and the typical values for a protein system (m = 100 000, v, = 500 cm/s, c,=0.2~g/cm2,cs=0.05 mg/ml, andD=2.5 X 10e7 cm2/s] one can estimate that @ is approxiamtely 2.5 x 10~~. Thus, the difference between a system with d, = 1 and a system with Q = 2 X 10m5 could not be discerned experimentally. That slopes have been observed which are lower than the theoretical diffusion limit is an indication that the sticking coefficients of proteins are lower than
Equation 25 sets forth the requirement for adsorption kinetics which are linear in time”‘: the interfacial concentration must be a constant not equal to cu. Equation 25 predicts identical time dependence to Equation 1 with a slope that is lower by a factor of Equations 24 and 25 provide another tcB - c,)/c,. explanation for the decrease in slope that is observed experimentally. The time behaviour of the interfacial concentration generated by the numerical solution supports this argument. Figure 5 is a plot of both r and the interface concentration, C 1y = o, versus rl” for @= 100, depicting how r and C 1x = o are related. There are three zones of interest shown in Figure 5. Zone 1 is the initial non-linear Zone
1
Zone 2
Zone
3
r /-
10-5.
The observed tl” kinetics The results of this study suggest two explanations for the experimental observation of the initial t”‘kinetics with a slope that is lower than the theoretical diffusion contro13‘. The first is that the interface concentration reaches a pseudo-steady state. The second is that surface filling is a phenomena which occurs in protein adsorption. These conclusions have been deduced from the following two arguments. The first argument is based on three solutions of the problem of adsorption from semi-infinite medium that require the assumption of a constant interface concentration. In the first solution, the diffusion-limiting case, the
Figure 5 Surface end interfacial concentration as a function of time”* (dimensionless) for Q, = 100. See text for description of each zone.
Biomaterials
1992, Vol. 13 No. 9
582
Sticking
section of the r[r”‘) curve where the curvature is concave upward: zone 2 is the central linear section; and zone 3 is the final section where the curvature is concave downward. In zone 1, I is not linear because of the rapid depletion of C (x = o, In zone 2, C (x = o has reached a pseudo-steady state or is relatively constant, and r(~l’~) is linear, as suggested by Equations 24 and 25. In zone 3, surface filling causes Cix = o to rise and r( rl”) is not linear. The second argument results from an analysis of the long-time behaviour of the Collins-Kimball solution9 which does not account for surface filling. It can be shown that the interfacial concentration predicted by Collins-Kimball solution is zero at long times. Without surface filling, the surface concentration is not linear with t”’ and the slope at long times approaches 2cs(D/ nl I”, the diffusion limit, regardless of the value of the reaction rate constant. Without surface filling the interface concentration would never reach a non-zero pseudo-steady state. The combination of surface filling and partial diffusion control causes the experimental data to appear linear in t”‘.
The extent of reaction
control
An analysis of the interface concentration data, generated numerically, leads to the definition of another parameter that can be used to evaluate the extent of reaction control. A plot of the interface concentration, Clx = s, versus r*” for several values of @ is shown in Figure 6. Initially C lx = o drops to a minimum value and then rises again approaching unity. Under completely reactioncontrolled conditions, C 1x = ,, would not be depIeted but would remain constant and equal to unity. The minimum value for each curve, C (x = o,min is an indication of the extent of reaction control or the extent to which the adsorption is dominated by the reaction. The lower the value Of CIX=O.min$the less the adsorption is dominated by the reaction. Figures 4 and 6 were used to make a correlation between Q and C 1x = oTmin which is plotted as the solid
1.00
coefficients
of adsorbing
proteins:
D.R. Weaver and W.G. Pitt
0.25 -
0.00
0.25
0.50
0.75
1.00
Q Figure7 Minimum interface concentration as a function of Q (dimensionless). The solid line was estimated from the results of the computer simulation. The dashed line represents Equation 26.
curve in Figure 7, A second approximate correlation between Q and C (x = *, mincan be developed by assuming that the minimum interface concentration is equal to cI according to Equation 24. In this approximate case, Q would be equal to [ca - ci)/ce, the ratio of Equations 25 and I, and
CIx=o,min= 11- Ql
(26)
The calculation of C 1x = *, minby this method is indicated as the dashed line in Figure 7. Figures 4 and 7 combined, indicate the relationship between C 1x = o minand the reaction rate constant @. The solid curve in $igure 7 was used to estimate a value of in Table 2. For the CIX=O,min for the experiments fibronectin on silica experiment, the interface concentration was always within 90% of the initial (bulk] concentration indicating approximately the extent of reaction control.
Inadequacies
of Equation
2
The inadequacies in the use of Equation 3 to describe protein adsorption are evident. First of all, Equation 1 does not account for surface filling which has been shown to be an important aspect of protein adsorption kineticsz3. Secondly, a functionality of t”’ does not necessarily indicate diffusion control. The adsorption process in zone 2 is not completely diffusion-controlled unless the minimum value of C 1x = o drops to zero. The existence of a pseudo-steady state concentration at the interface implies that the rates of adsorption and diffusion are approximately balanced in this zone. Third,~qua~ion 2 provides little insight into the intrinsic kinetics.
Polynomial Figure6 Interfacial concentration as a function (dimensionless) for @ ranging from 0.01 to 100. Biomaterials
1992,
Vol. 13 NO. 9
of time”*
fit of Figure 4
To make Figure4 simpler to use, the data have been plotted in logarithmic coordinates in Figure 8. Also, it
Stickino
coefficients
of adsorbina
proteins:
CB
c1 CS CM
D k m RA RC
t &)
112
“,
I
,
I
I,,
2
1
6
8
Z lJ 1
Q Figure 8
The parameters @ and @provide a measure with which to compare different protein-surface studies. a, the dimensionless reaction rate constant, can be estimated directly from experimental kinetic measurements of the observed slope, Q, using Figure 4. The probability of adsorption or the sticking coefficient, 0, can be calculated from estimates of @ using Equation 25. It is not even approximately correct to assume that the probability of adsorption is equal to the measured slope, Q. Measured values of Q that have been reported in the literature have been in the range of 0.1< Q < 1. and such an assumption has led to the conclusion that 0.1 < @ < 1. Our estimates based on previous measurements of the observed slope indicate that, for many protein-surface systems, the probability of adsorption is in the range 1O-5-1O-8. Also, because of the asymptotic relationship @ and Q, the difference between a system with @= 1 and a system with 0 = 1 X 10m5 cannot be experimentally determined. 4 is a function of energy barriers, stearic hindrance, electrostatic repulsion, the strength of the protein-surface interaction, and other factors which decrease the probability of adsorption. It is expected that careful measurements of @Jwill provide insight to these physical phenomena. TABLE OF NOMENCLATURE
ACKNOWLEDGEMENTS Financial support and Company.
Protein concentration Protein concentration face x = 0 (mg/cm3)
(mg/cm”) evaluated at the inter-
for this work was provided
by Eli Lilly
REFERENCES J.D.+ in Surface and Inter-facial Aspects of Polymers, Vol. 2 (Ed. J.D. Andrade), Plenum Press, New York, USA, 1985, p 1 Horbett, T.A. and Brash. J.L., in Proteins at Interfaces, (Eds J.L. Brash and T.A. Horbett), American Chemical Society, Washington DC, USA, 1986 Young, B.R.. Pitt, W.G. and Cooper, S.L., J. Colloid Interface Sci. 1988, 125, 246 Van Dulm, P. and Norde, W., J. Colloid Interface Sci. 1983, 91, 248 Iwamoto, G.K., Winterton. L.C., Stoker, R.S., Van Wagenen, R.A.. Andrade. J.D. and Mosher. D.F., Andrade,
Biomedical
J. Colloid
Variables
variables
Protein concentration Protein concentration evaluated at the interface X = 0 Minimum protein concentration evaluated at the interface Concentration of the adsorbed protein Probability of a step to the right Probability of a step to the left Slop dI’/d( ,l”) Time Distance from the interface Reaction rate constant Sticking coefficient Fraction of the surface area that is occupied Time step Distance step
CONCLUSIONS
C
Protein initial concentration (mg/cm3) Protein concentration at the interface (mg/cm3) Concentration of adsorbed protein (mg/cm’) Concentration of one monolayer (mg/cm’) Diffusion coefficient of protein (cm’/s) Boltzmann’s constant (erg/molec/K) Molecule weight (g/molec) Rate of adsorption (mg/(cm’ s)) Rate of collision (mg/(cm’ s]) Time [s) Absolute temperature (K) Root mean squared velocity of the molecule (cm/s) Average velocity of random walk step (cm/s) Distance from the interface [cm) Average jump size of random walk [cm) Average jump frequency of random walk (s-7
Dimensionless
@ as a function of Q.
has been found that a plot of @ versus l/(1 - Q) is nearly linear. Therefore, polynomials in l/(1 - Q) and Q/ (1 - Q] have been fitted to the data in Figure 4. The polynomials and their applicable range are listed in Appendix B.
Cl,=0
583
D.R. Weaver and W.G. Pitt
Interface
Sci. 1985,
106, 459
Pitt, W.G., Fabrizius-Holman. D.J., Mosher, D.F. and Cooper, S.L., J. Colloid Interface Sci. 1989, 129, 231 Hlady, V.. Reinecke, D.R. and Andrade, J.D., J. ColJoid Interface Sci. 1986, 111, 555 Biomaterials
1992. Vol. 13 No. 9
584
Sticking Feller, W., An Introduction to Probability Theory and it’s Applications, Vol. I, 3rd Edn, John Wiley, New York, USA, 1966, p 343 Collins, F.C. and Kimball, G.E., I. Colloid Sci. 1949, 4, 425 Goodrich, F.C., I. Chem. Phys. 1954, 22, 586 Alberty, R.A. and Daniel% F., Physical Chemistry, 5th Edn, John Wiley, New York, USA, 1979, p 461 Berg, H.C., Random Walks in Biology, Princeton University Press, Princeton, NJ, USA, 1983, p 5 Smoluchowski, M.V., Z. Phys. Chem. 1917, 92, 129 Ward, A.F.H. and Tordai, L., J. Chem. Phys. 1946, 14, 453 Baret. J.F., J. Phys. Chem. 1968, 72, 2755 Varoqui, R. and Pefferkorn, E., J. Colloid Inferface Sci. 1986,109,520 Richtmeyer, R.D. and Morton, K.W., DifferenceMethods for Initial Value Problems, Wiley Interscience, New York, USA, 1967 Young, B.R., Protein adsorption on polymeric biomaterials and its role in thrombogenesis, PhD Thesis, University of Wisconsin, Madison, WI, USA, 1984 Pitt, W.G., Protein adsorption on polyurethanes, PhD Thesis, University of Wisconsin, Madison, WI, USA, 1987 Sveshnikoff, B., Acta Physicochim. URSS 1935, 3, 257 Schaaf, P. and Dejardin, Ph., Colloids Surfaces 1987,24, 239 Carslaw, H.S. and Jaeger, JC.. Conduction of i-feat in Solids, Oxford University Press, London, UK, 1947 Giroux, T.A. and Cooper, S.L., J. Colloid Interface Sci. 1991, 146, 179 Reed, T.A. and Gubbins, K.E., Applied Statistical Mechanics, McGraw Hill, New York, USA, 1973, P 58
8
9 10 11 12 13 14 15 16 17
18
19
20 21 22 23 24
Appendix
mechanical
systemz4 are
frequently assumed to partition independently so that the canonical partition function, Q, is a product of the partition functions for the individual modes such as vibration, rotation, translation, etc. The canonical partition function is also commonly represented semi-classically, where the translational degrees of freedom, and the potential energy in the configuration of the molecules are described by classical mechanics, and the internal degrees of freedom (vibration, rotation, etc.) are described by quantum mechanics. For a two-component system the canonical partition function would be Q
QlintQ2intZ
_
____~
I\dNl AdN2 1
2
N
1’
I N
2’
I
(AlI
where the subscripts 1 and 2 denote the two components, the subscript ‘int’ denotes the internal degrees of freedom, Z is the configurational partition function, AdN’ is the partition function for the translational motion of each molecule, and d is the dimensionality of the translational motion, The classical representation of the
Biomaterials
of adsorbing
translational temperatures Aj
=
J
proteins:
DR.
Weaver
and W.G. Pitt
motion is considered to be valid for above 50 K, in which case A is equal to h2
PW
2rrmikT
where the subscript j denotes the component, h is Planck’s constant, mj is the molecular mass, k is Boltzmann’s constant and T is the absolute temperature. The internal energy, U, of the system is
Since the partition function, Q, is a product of terms, the internal energy is a sum of terms. The portion of U which is attributable to the translational molecular motion of each component is obtained by differentiation of AdN’ u,,=~NikT
t-44)
where the subscript ‘t’ denotes the translational portion of the sum. If the translational energy of each component is equated to the kinetic energy of all of the molecules of that component then Ni
c
fmjvfL
=sN,kT
(A51
L=l
Thus the root-mean-squared velocity in one dimension for each component of a statistical mechanical system is given by Equation A5. This derivation is not limited to any particular phase of matter, and thus can be applied to colloidal molecules in aqueous solution. This result was first deduced by Einstein in 1905.
A
states of a statistical
The energy
coefficients
1992, Vol. 13 No. 9
Appendix
B
The following polynomials 4 to within 3%.
fit the numerical
data
in
Figure
@ = 0.83215
Y4 - 1.5877 Y3 + 1.7067 Y2 + 0.00869
0 < Q < 0.2 @ = -0.24505Z3
+ 1.5780Z2
- 2.1698Z
0.2 < Q < 0.5 @ = -0.0082083
+ 0.81127 U341
Z2 + 1.3674 Z - 1.8914
0.5 & Q < 0.9 Q, = -0.0017663
Y
U33)
U351
Z2 + 1.1385 Z - 0.087618
0.9 S Q < 0.99
u36)
Sticking coefficients of adsorbing proteins Daniel R. Weaver and William G. Pitt Department
of Chemical
Engineering,
Brigham
Young University,
Provo, Utah 84602, USA
The protein sticking coefficient, 4, the fraction of collisions that result in adsorption, is a function of the molecular interactions between the protein and the surface. A random walk and diffusionto-capture model was used to describe the kinetics of protein adsorption. The assumption of a constant sticking coefficient leads to a first-order model of the kinetics. A solution of the problem of adsorption from a semi-infinite medium with first-order kinetics at the boundary was obtained by numerical simulation on the computer. The results of the computer simulations match the time dependence observed experimentally. A correlation was developed to estimate C#J from experimental data. I#Jhas been found to be in the range 10-5-10-* for several protein adsorption kinetic studies reported in the literature. Keywords: Received
Protein adsorption,
11 November
diffusion,
1991; revised 8 January
The adsorption of proteins at interfaces has been shown’ to be a complex phenomena which includes the diffusion of the protein particle through the aqueous solution and the collision and interaction of the protein at the interface. One important goal in the study of protein adsorption has been the measurement of the affinity between the protein and the surface. The affinity has been broadly defined as a measure of the interaction between the protein and the surface. It has been suggested by Horbett and Brash’ that one measure of the affinity is the protein sticking coefficient. The sticking coefficient, denoted 4, is the fraction of the collisions between the protein and an available surface site that result in adsorption. It has been hypothesized by Horbett and Brash, and it is the hypothesis of this study, that the sticking coefficient can be deduced from kinetic measurements under conditions near the diffusion limit. Diffusion-controlled adsorption of protein from nonflowing solution has often been modelled by3
where cs is the concentration of adsorbed protein, cs is the initial (bulk) protein concentration, D is the diffusion coefficient, and t is time. Equation 1 has several shortcomings which will be discussed. However, it has been used by several investigators of protein adsorption kinetics because experimental data match the functional form of Equation 1. It has been frequently observed3-’ that, initially, the amount adsorbed increases in proportion to t”‘. However, in many cases the initial observed slope dcsld(t”‘) has been found to be lower than 2cB(Dln)“’ predicted by Equation Z. It has Correspondence
0
1992
to Professor
W.G. Pitt.
Butterworth-Heinemann Ltd
0142-9612/92/090577-06
sticking
coefficients,
1992; accepted
9 January
1992
also been frequently observed that after a period of time the adsorption rate decreases sharply due to surface filling. The departure of the initial slope from the diffusion limit has been attributed to a sticking coefficient that is not unity. Physical mechanisms such as energy barriers and reversible adsorption have been proposed to explain the departure of the sticking coefficient from unity. In the present study, the nature and consequences of the departure of the slope from the diffusion limit is investigated within the framework of a model for the sticking coefficient. To facilitate the investigation, a parameter Q is defined to be the ratio of the experimentally observed slope to the slope of the theoretical diffusion limit
Q, the dimensionless observed slope, will be shown to be a measure of the departure of the experiment from completely diffusion-controlled conditions. The goal of this study included four main aspects: 1. To determine if kinetic limitations coupled with diffusion limitations could produce this functional t”* dependence with the accompanying decrease in the slope. 2. To determine how the sticking coefficient, @, correlates with the slope Q. 3. To develop a model or correlation to calculate the sticking coefficient from kinetic measurements. 4. To evaluate experimental data and determine an order of magnitude estimate of protein sticking coefficients. Biomaterials
1992, Vol. 13 No. 9
578
Sticking
coefficients
of adsorbing
D.R. Weaver and W.G. Pitt
proteins:
protein molecules in aqueous solution. This misconception is likely to result from the fact that the equality on the right in Equation 5 can be derived from the Boltzmann distribution of velocities for ideal gases”. However, it can be derived from concepts of statistical mechanics. Berg” has discussed the validity and the use of Equation 5 for colloidal particles in liquids. A statistical mechanics derivation of Equation 5 is given in Appendix A.
Figure 1
Feller’s elastic barrier. At the surface node there is a probability @ of being adsorbed and a probability (1 - I$) of being reflected. At the other nodes the probability of a step to the right is p and the probability of a step to the left is q.
THEORY Sticking coefficient
theory
The concept of a sticking coefficient has been discussed in literature on random walks near barriers in the study of the so-called ‘elastic barrier”. The elastic barrier is depicted in Figure 1, in which the particle in solution is assumed to execute a discrete random walk on the nodes as labelled. At the interfacial node, i=s, a constant fraction, 4, of the jumps to the left from node zero result in adsorption. A constant fraction (1 - o] of these jumps result in reflection. For all of the other nodes, zero and greater, the probability that the particle steps to the right is denoted p. The probability of a step to the left is q. For an unbiased random walk, p = q = 0.5. A parameter, S, is defined to be the step size of the random walk or the distance between the nodes. Another parameter, u, is defined to be the average jump frequency. One can also define a parameter v, = Su, which can be interpreted as the average velocity of the random step. Collins and Kimball9 and Goodrich” have shown that, given the above formulation, the flux to the surface and thus the intrinsic kinetic expression can be described by
Adsorption-diffusion
theory
The development of equations to describe adsorption coupled to diffusion in a stationary fluid medium was pioneered by Smoluchowski’3 and has been discussed by others such as Ward and Tordai14, Baret” and Varoqui and Pefferkorn”. These treatments have used the mathematical formulation:
dCJ&
aX2'
at
t>o
(‘3)
t>o lim
c = cs,
t>o
(81
=
x20
(9)
X-a
Cl,=,,
cJ,=o
CB,
=
0
(10)
where c is the protein concentration, and x is the distance from the interface. Boundary condition Equation 7 requires an expression for the rate of adsorption. The intent of this study is to solve Equations 6 through 10 with Equation 4 as the rate expression, by making the approximation that the node j=O is located at x=0. To solve Equations 6-10, and 4 for a general case they were transformed into dimensionless form by defining the following variables: r
=
FcBI
2Dt
--
(111
CM
X2BX where R, is the rate of adsorption. The assumption of a constant sticking coefficient leads to first-order irreversible kinetics. One can modify Equation 3 to allow for surface filling by including a term to account for the fraction of available surface sites
cM
CCC cB
pcs cM
where 0 is the fraction of the surface sites that are filled. The interest in the present study is in a discrete random walk where the step size is on the scale of a molecular collision with the surface. In such a case, the average velocity of the random walk would be approximately equal to the root-mean-squared velocity of the protein molecules in solution v,
c
( v,)l12 =
(E)“’
(5)
where m is the molecular mass of the protein. There is a common misconception that Equation 5 would not be a valid estimate of the root-mean-squared velocity of Biomaterials
1992, Vol. 13 No. 9
The parameter @ is a dimensionless reaction rate constant and is a measure of the rate of adsorption compared to diffusion. The parameter 8 was taken to be equal to the fraction of a simple monolayer 8=cs/cM. Equations 6-10, combined with Equation 4 become respectively
ac a% dt--ax2' dlac
r>o
(16)
~=dx~x~o=~clx~o~l-el~
r>o
(171
lim X-a
r>O
(181
c = 1,
Sticking
coefficients
Cl,=,
of adsorbing
proteins:
DR. Weaver and W.G. Pitt -_
xzo
= 1,
I-lr=O=O
(191 (20)
The analytical solution to Equations 16-20 is known for the case where the [l - 01 term is not included and is discussed by Collins and Kimball9 and by Goodrich”. An approximate analytical solution has been proposed by Varoqui and Pefferkorn’~ with the (1 - Sl term included. However, their solution is valid only under the assumption that the desorption rate is high or that the surface coverage is small. These assumptions limit their model such that it cannot be applied to most protein adsorption kinetic data.
NUMERICAL STUDY In the present work equations 16-20 were solved numerically using an explicit finite difference method17. The first forward difference was used to approximate the time derivatives, the first central difference was used to approximate the second derivative and second forward difference was used to approximate the first derivative with respect to distance. In solving the finite difference equations the solution was unstable unless the coefficient that multiplied the node of interest was greater than or equal to zero. Applying this stability criterion to Equations 16-20 required that
and 3Ar l-ZAX2-bX
@AIT I
20
where, Ahr and AX are the dimensionless time and distance steps respectively. Equations 22 and 22 place a limit on AX for a given value of Ar. The numerical solution was completed by choosing a value of Ar and then calculating the size of AX using Equatjon 23 or 22. The size of Ar that was chosen varied from about 3 X 10e6 to 3 X lo-’ depending on the value of 0. For large values of @ very small values of A r were required to maintain accuracy. For most of the computer runs approximately one million time steps were used. The requirements of boundary condition Equstjon 18 were met by using an X array large enough so that C at large X was always equal to unity. A Convex 210 Mini Super Computer was used to perform the calculations. The analytical solution of Collins and Kimball9 provided a standard with which to test the numerical solution. When the numerical simulation was performed without the (1 - 0) term the results agreed exactly with the analytical solution of Collins and Kimball for @ = 10, 1, 0.1,and 0.01. Based on the perfect agreement between the analytical and the numerical solution the numerical method was considered accurate and capable of producing valid results in the present study.
Figure2 Surface concentration as a function of time”* (dimensionless) for @ ranging from 0.01 to 100. The * represents the inflection point of each curve. The dashed line represents the diffusion-limited case predicted by Equation 1
(see text).
DISCUSSION The numerical solution The numerical solution of Equations 16-20 is shown in Figure 2, in which I is plotted as a function of r*“. The solution is shown for several different values of Q,. The * in Figure 2 denotes the inflection point of each curve. The numerical solution curves are linear with respect to r112for values of @ = 0.6 or greater. These solution curves match the behaviour observed experimentally. As Q, increases the solution approaches but does not exceed the diffusion limit (dashed line Figure 2). To determine and compare the effect of the (1 - 0) term, the Collins-Kimball solution was plotted alongside the numerical solution in Figure 3. Clearly, the (1 - 0) term causes an early departure from the Collins-Kimball solution,
Application to experimental
data
Since the dimensionless slope, Q, has been frequently observed to be lower than the diffusion limit, it is useful to know the relationship between @ and Q, The results of the numerical simulation were used to make a correlation and thus define this relationship. The goal in making this correlation was to provide a tool for the experimenter to calculate a sticking coefficient. Thus the correlation was made between the slope dcsld(t”‘] of the linear section of each curve and the value of Q?that was used as input into the numerical simulation. The correlation is shown in Figure 4. To estimate the slope of each curve certain arbitrary choices had to be made. There are several reasonable methods to estimate the slope in the linear section such as the slope at a point such as IYs= 0.5, or the slope at the inflection point, or by a least-squares fit through a section of the data. Figure 4 was constructed by correlating the slope at the inflection point because this method required the fewest arbitrary choices and Biomaterials 1992,Vol.13 No. 9
560
Sticking coefficients of adsorbing proteins: D.R. Weaver and W.G. Pitt 100
80
60 8
40
20
2.00
4.00
6.00
8.00
o3
10.00
0.00
0.20
0.40
0.60
0.80
1 .oo
Q Figure 3 Comparison of the numerical solution (solid lines) to the Collins-Kimball solution (dashed lines) for Q,equal to 1, 0.1, and 0.01.
was the best representation of the linear section of the curves. In any case, the form of the @ - Q relationship was the same for all methods tested. Figure 4 and Equation 15 were used to estimate values of @ and # from values of Q reported in the literature3* 5-7. Values of D and m were obtained from the respective references or from Refs 6 and 18, except that for vitronectinD was estimated to be 5.12 X IO-’ cm’/s. The results are shown in Table 2. In three cases, the experimental values of Q were reported to be unity. In these cases the values of Q, were assumed to be 100, since the @= 100 case was indistinguishable from the diffusion limit. The results in Table 2 indicate high values of @ for the following protein-surface systems: IgG 9.0 on silicone rubber, albumin on silicone rubber, and albumin on silica at high pH. The results also indicate a relatively low value of @ for cz2 - h4 on silicone rubber, The values of Q, in Tahoe 1 are in the range 1O-5-1O-8 Table 1
Figure4
9, as a function of Q.
indicating that the rate of collision rate of accumulation or adsorption the relationship ~,=~vACl.=o
(231
R,, the rate of collision, was estimated to be 1Ol7 molecules cm-’ s-’ for a protein with mol wt 100 000 and with c IX= o = 0.05 mg/cm3. With such a large collision rate and a sticking coefficient of 10m5 a monolayer of 0.2 ,ug/cm’ would be filled within minutes. In performing the calculations of the sticking coefficients, it was assumed that q=lJ2 in ~~ua~iun 4. This assumption, however, does not lead to large uncertainty in the value of @. This conclusion is based on an analysis of the distributions, given by Goodrich”, for the random walk of particles near a partially reflective barrier. It is our observation that g could not deviate by more than 170 from a value of 112, thus leading to an un~e~ainty in Cp of 1%. This unce~ainty is small
Estimated values of Cp,(b and C 1x = o,min
Protein
Surface
Albuming~c
Silica
Albumin’ b Albumi@ IgG 9.0’ IgG 7.8’ IgG 6.5’ Q*-Mf (z2-Mf
Silica SR SR SR SR SR z:
Transferrin’ Fibronectine Fibronectin* Vitronecti# Vitronectind Vitronectind Vitronectind
C”B __
Silica NPTES PS %PS OXPS
_.._
CL
Q
1992,
Vol.
13 NO. 9
@
cpx 108
--___-.--
0.026
0.13a
0.3
0.02 0.005 0.0044 0.0044 0.0044 0.005 0.0025
0.13= 0.14 0.30 0.26 0.21 0.18 0.18
0.0025 0.05 0.01 0.03 0.01 0.03
0.14 o.lga 0.2a 0.2a 0.15a 0.15a
CIX
0.22
8.8
0.8
1.0 1.0 0.60 0.60 0.46 0.29
100 100 1.5 1.5 0.64 0.21
3079 715 283 4.9 6.1 ::;6
::: 0.0 0.5 0.5 0.6 0.8
0.77 0.13 0.90 0.43 0.25 0.4 0.32
3.9 0.04 11 0.55 0.15 0.45 0.26
14.2 56Y 8.7 3.9 5.2 9.1
z: 0:2 0.6 0.8
Bestimated; bpH 4.8; ‘pH 4.0; ‘Ret. 6; %ef. 5: ‘Ret. 3; 9Ref. 7; hmg/ml: ‘pglcm”. SR, silicone rubber; NPTES. N-pentyltriethoxysilane; PS, polystyrene: OXPS. oxidized pOiystyrefle.
Biomaterials
is large relative to the at the interface. Using
= 0. min
Stickinq coefficients of adsorbinn proteins: D.R. Weaver and W.G. Pitt
compared with uncertainties introduced in experimental measurements. Also, it was assumed that va=( v~)~‘~ according Equation 5. This assumption is likely to be correct to within an order of magnitude. The conclusion is based on the definition” of the diffusion coefficient D=S2ui2 which, combined with the definition of va, gives v, = 2DIS. It is reasonable to assume that the mean free path for a protein molecule in aqueous solution is less [perhaps much less) than 0.1 nm and that a random walk step size of S = 0.1 nm would be uncorrelated. In this case, one can estimate that v,=lOO cm/s for a small protein with a diffusion coefficient of 5 X 10e7 and a step size of 0.1 nm. For the proteins in T&e 1 the values of v, were calculated to be in the range 100-600 cm/s using Equation 5. Intuition based on consideration of the physical system suggests that there are more collisions per second of a protein with an interface than there are jumps per second of the same protein across 0.1 nm. Therefore, it was concluded that the use of Equation 5 is approximately correct. The
relationship
581
protein concentration at the liquid interface is assumed to be depleted immediately to and remain constant at zero. The theoretical expression for the observed kinetics in this case is Equation 2. In the second solution, the reaction-limiting case, the interface concentration is assumed to remain constant equal to ca. The theoretical expression for the observed kinetics in this case would be the time integral of the rate expression with the interface concentration held constant equal to cs. Both cases are discussed adequately by Schaaf and Dejardin’l. In the third solution, the rate of adsorption is an approximate balance between diffusion and reaction. In this case, one could assume that the interface concentration is rapidly depleted to some other non-zero value, ct, that remains constant cl,=,
= c1
t>o
(241
With this new boundary condition Equations 6-10 can also be solved” and the analogous relationship to Equation 1 becomes
between cf, and Q
Several authors have been interested in the relationship between the probability of adsorption, @, and the observed slope Q. It has been commonly assumed in previous literature that Q is equal to Q4’7*1Q.Such an assumption was inherent in the calculation of values for the ‘Arrhenius’ activation energy4. 7. This assumption is incorrect and Collins and Kimball’ in 1949 discussed that such a misconception had been previously madez’. Figure 4 shows that the relationship between Q, and Q [thus the relationship between 0 and Q) is non-linear and has the appearance of a hyperbolic function, The function is asymptotic near unity. The consequence of the asymptotic behaviour is that upis not necessarily near unity when the experimental adsorption process appears completely diffusion controlled. For example, the @= 100 curve in Figure 2 is indistinguishable from the diffusion limit. Yet, with this value of a=100 and the typical values for a protein system (m = 100 000, v, = 500 cm/s, c,=0.2~g/cm2,cs=0.05 mg/ml, andD=2.5 X 10e7 cm2/s] one can estimate that @ is approxiamtely 2.5 x 10~~. Thus, the difference between a system with d, = 1 and a system with Q = 2 X 10m5 could not be discerned experimentally. That slopes have been observed which are lower than the theoretical diffusion limit is an indication that the sticking coefficients of proteins are lower than
Equation 25 sets forth the requirement for adsorption kinetics which are linear in time”‘: the interfacial concentration must be a constant not equal to cu. Equation 25 predicts identical time dependence to Equation 1 with a slope that is lower by a factor of Equations 24 and 25 provide another tcB - c,)/c,. explanation for the decrease in slope that is observed experimentally. The time behaviour of the interfacial concentration generated by the numerical solution supports this argument. Figure 5 is a plot of both r and the interface concentration, C 1y = o, versus rl” for @= 100, depicting how r and C 1x = o are related. There are three zones of interest shown in Figure 5. Zone 1 is the initial non-linear Zone
1
Zone 2
Zone
3
r /-
10-5.
The observed tl” kinetics The results of this study suggest two explanations for the experimental observation of the initial t”‘kinetics with a slope that is lower than the theoretical diffusion contro13‘. The first is that the interface concentration reaches a pseudo-steady state. The second is that surface filling is a phenomena which occurs in protein adsorption. These conclusions have been deduced from the following two arguments. The first argument is based on three solutions of the problem of adsorption from semi-infinite medium that require the assumption of a constant interface concentration. In the first solution, the diffusion-limiting case, the
Figure 5 Surface end interfacial concentration as a function of time”* (dimensionless) for Q, = 100. See text for description of each zone.
Biomaterials
1992, Vol. 13 No. 9
582
Sticking
section of the r[r”‘) curve where the curvature is concave upward: zone 2 is the central linear section; and zone 3 is the final section where the curvature is concave downward. In zone 1, I is not linear because of the rapid depletion of C (x = o, In zone 2, C (x = o has reached a pseudo-steady state or is relatively constant, and r(~l’~) is linear, as suggested by Equations 24 and 25. In zone 3, surface filling causes Cix = o to rise and r( rl”) is not linear. The second argument results from an analysis of the long-time behaviour of the Collins-Kimball solution9 which does not account for surface filling. It can be shown that the interfacial concentration predicted by Collins-Kimball solution is zero at long times. Without surface filling, the surface concentration is not linear with t”’ and the slope at long times approaches 2cs(D/ nl I”, the diffusion limit, regardless of the value of the reaction rate constant. Without surface filling the interface concentration would never reach a non-zero pseudo-steady state. The combination of surface filling and partial diffusion control causes the experimental data to appear linear in t”‘.
The extent of reaction
control
An analysis of the interface concentration data, generated numerically, leads to the definition of another parameter that can be used to evaluate the extent of reaction control. A plot of the interface concentration, Clx = s, versus r*” for several values of @ is shown in Figure 6. Initially C lx = o drops to a minimum value and then rises again approaching unity. Under completely reactioncontrolled conditions, C 1x = ,, would not be depIeted but would remain constant and equal to unity. The minimum value for each curve, C (x = o,min is an indication of the extent of reaction control or the extent to which the adsorption is dominated by the reaction. The lower the value Of CIX=O.min$the less the adsorption is dominated by the reaction. Figures 4 and 6 were used to make a correlation between Q and C 1x = oTmin which is plotted as the solid
1.00
coefficients
of adsorbing
proteins:
D.R. Weaver and W.G. Pitt
0.25 -
0.00
0.25
0.50
0.75
1.00
Q Figure7 Minimum interface concentration as a function of Q (dimensionless). The solid line was estimated from the results of the computer simulation. The dashed line represents Equation 26.
curve in Figure 7, A second approximate correlation between Q and C (x = *, mincan be developed by assuming that the minimum interface concentration is equal to cI according to Equation 24. In this approximate case, Q would be equal to [ca - ci)/ce, the ratio of Equations 25 and I, and
CIx=o,min= 11- Ql
(26)
The calculation of C 1x = *, minby this method is indicated as the dashed line in Figure 7. Figures 4 and 7 combined, indicate the relationship between C 1x = o minand the reaction rate constant @. The solid curve in $igure 7 was used to estimate a value of in Table 2. For the CIX=O,min for the experiments fibronectin on silica experiment, the interface concentration was always within 90% of the initial (bulk] concentration indicating approximately the extent of reaction control.
Inadequacies
of Equation
2
The inadequacies in the use of Equation 3 to describe protein adsorption are evident. First of all, Equation 1 does not account for surface filling which has been shown to be an important aspect of protein adsorption kineticsz3. Secondly, a functionality of t”’ does not necessarily indicate diffusion control. The adsorption process in zone 2 is not completely diffusion-controlled unless the minimum value of C 1x = o drops to zero. The existence of a pseudo-steady state concentration at the interface implies that the rates of adsorption and diffusion are approximately balanced in this zone. Third,~qua~ion 2 provides little insight into the intrinsic kinetics.
Polynomial Figure6 Interfacial concentration as a function (dimensionless) for @ ranging from 0.01 to 100. Biomaterials
1992,
Vol. 13 NO. 9
of time”*
fit of Figure 4
To make Figure4 simpler to use, the data have been plotted in logarithmic coordinates in Figure 8. Also, it
Stickino
coefficients
of adsorbina
proteins:
CB
c1 CS CM
D k m RA RC
t &)
112
“,
I
,
I
I,,
2
1
6
8
Z lJ 1
Q Figure 8
The parameters @ and @provide a measure with which to compare different protein-surface studies. a, the dimensionless reaction rate constant, can be estimated directly from experimental kinetic measurements of the observed slope, Q, using Figure 4. The probability of adsorption or the sticking coefficient, 0, can be calculated from estimates of @ using Equation 25. It is not even approximately correct to assume that the probability of adsorption is equal to the measured slope, Q. Measured values of Q that have been reported in the literature have been in the range of 0.1< Q < 1. and such an assumption has led to the conclusion that 0.1 < @ < 1. Our estimates based on previous measurements of the observed slope indicate that, for many protein-surface systems, the probability of adsorption is in the range 1O-5-1O-8. Also, because of the asymptotic relationship @ and Q, the difference between a system with @= 1 and a system with 0 = 1 X 10m5 cannot be experimentally determined. 4 is a function of energy barriers, stearic hindrance, electrostatic repulsion, the strength of the protein-surface interaction, and other factors which decrease the probability of adsorption. It is expected that careful measurements of @Jwill provide insight to these physical phenomena. TABLE OF NOMENCLATURE
ACKNOWLEDGEMENTS Financial support and Company.
Protein concentration Protein concentration face x = 0 (mg/cm3)
(mg/cm”) evaluated at the inter-
for this work was provided
by Eli Lilly
REFERENCES J.D.+ in Surface and Inter-facial Aspects of Polymers, Vol. 2 (Ed. J.D. Andrade), Plenum Press, New York, USA, 1985, p 1 Horbett, T.A. and Brash. J.L., in Proteins at Interfaces, (Eds J.L. Brash and T.A. Horbett), American Chemical Society, Washington DC, USA, 1986 Young, B.R.. Pitt, W.G. and Cooper, S.L., J. Colloid Interface Sci. 1988, 125, 246 Van Dulm, P. and Norde, W., J. Colloid Interface Sci. 1983, 91, 248 Iwamoto, G.K., Winterton. L.C., Stoker, R.S., Van Wagenen, R.A.. Andrade. J.D. and Mosher. D.F., Andrade,
Biomedical
J. Colloid
Variables
variables
Protein concentration Protein concentration evaluated at the interface X = 0 Minimum protein concentration evaluated at the interface Concentration of the adsorbed protein Probability of a step to the right Probability of a step to the left Slop dI’/d( ,l”) Time Distance from the interface Reaction rate constant Sticking coefficient Fraction of the surface area that is occupied Time step Distance step
CONCLUSIONS
C
Protein initial concentration (mg/cm3) Protein concentration at the interface (mg/cm3) Concentration of adsorbed protein (mg/cm’) Concentration of one monolayer (mg/cm’) Diffusion coefficient of protein (cm’/s) Boltzmann’s constant (erg/molec/K) Molecule weight (g/molec) Rate of adsorption (mg/(cm’ s)) Rate of collision (mg/(cm’ s]) Time [s) Absolute temperature (K) Root mean squared velocity of the molecule (cm/s) Average velocity of random walk step (cm/s) Distance from the interface [cm) Average jump size of random walk [cm) Average jump frequency of random walk (s-7
Dimensionless
@ as a function of Q.
has been found that a plot of @ versus l/(1 - Q) is nearly linear. Therefore, polynomials in l/(1 - Q) and Q/ (1 - Q] have been fitted to the data in Figure 4. The polynomials and their applicable range are listed in Appendix B.
Cl,=0
583
D.R. Weaver and W.G. Pitt
Interface
Sci. 1985,
106, 459
Pitt, W.G., Fabrizius-Holman. D.J., Mosher, D.F. and Cooper, S.L., J. Colloid Interface Sci. 1989, 129, 231 Hlady, V.. Reinecke, D.R. and Andrade, J.D., J. ColJoid Interface Sci. 1986, 111, 555 Biomaterials
1992. Vol. 13 No. 9
584
Sticking Feller, W., An Introduction to Probability Theory and it’s Applications, Vol. I, 3rd Edn, John Wiley, New York, USA, 1966, p 343 Collins, F.C. and Kimball, G.E., I. Colloid Sci. 1949, 4, 425 Goodrich, F.C., I. Chem. Phys. 1954, 22, 586 Alberty, R.A. and Daniel% F., Physical Chemistry, 5th Edn, John Wiley, New York, USA, 1979, p 461 Berg, H.C., Random Walks in Biology, Princeton University Press, Princeton, NJ, USA, 1983, p 5 Smoluchowski, M.V., Z. Phys. Chem. 1917, 92, 129 Ward, A.F.H. and Tordai, L., J. Chem. Phys. 1946, 14, 453 Baret. J.F., J. Phys. Chem. 1968, 72, 2755 Varoqui, R. and Pefferkorn, E., J. Colloid Inferface Sci. 1986,109,520 Richtmeyer, R.D. and Morton, K.W., DifferenceMethods for Initial Value Problems, Wiley Interscience, New York, USA, 1967 Young, B.R., Protein adsorption on polymeric biomaterials and its role in thrombogenesis, PhD Thesis, University of Wisconsin, Madison, WI, USA, 1984 Pitt, W.G., Protein adsorption on polyurethanes, PhD Thesis, University of Wisconsin, Madison, WI, USA, 1987 Sveshnikoff, B., Acta Physicochim. URSS 1935, 3, 257 Schaaf, P. and Dejardin, Ph., Colloids Surfaces 1987,24, 239 Carslaw, H.S. and Jaeger, JC.. Conduction of i-feat in Solids, Oxford University Press, London, UK, 1947 Giroux, T.A. and Cooper, S.L., J. Colloid Interface Sci. 1991, 146, 179 Reed, T.A. and Gubbins, K.E., Applied Statistical Mechanics, McGraw Hill, New York, USA, 1973, P 58
8
9 10 11 12 13 14 15 16 17
18
19
20 21 22 23 24
Appendix
mechanical
systemz4 are
frequently assumed to partition independently so that the canonical partition function, Q, is a product of the partition functions for the individual modes such as vibration, rotation, translation, etc. The canonical partition function is also commonly represented semi-classically, where the translational degrees of freedom, and the potential energy in the configuration of the molecules are described by classical mechanics, and the internal degrees of freedom (vibration, rotation, etc.) are described by quantum mechanics. For a two-component system the canonical partition function would be Q
QlintQ2intZ
_
____~
I\dNl AdN2 1
2
N
1’
I N
2’
I
(AlI
where the subscripts 1 and 2 denote the two components, the subscript ‘int’ denotes the internal degrees of freedom, Z is the configurational partition function, AdN’ is the partition function for the translational motion of each molecule, and d is the dimensionality of the translational motion, The classical representation of the
Biomaterials
of adsorbing
translational temperatures Aj
=
J
proteins:
DR.
Weaver
and W.G. Pitt
motion is considered to be valid for above 50 K, in which case A is equal to h2
PW
2rrmikT
where the subscript j denotes the component, h is Planck’s constant, mj is the molecular mass, k is Boltzmann’s constant and T is the absolute temperature. The internal energy, U, of the system is
Since the partition function, Q, is a product of terms, the internal energy is a sum of terms. The portion of U which is attributable to the translational molecular motion of each component is obtained by differentiation of AdN’ u,,=~NikT
t-44)
where the subscript ‘t’ denotes the translational portion of the sum. If the translational energy of each component is equated to the kinetic energy of all of the molecules of that component then Ni
c
fmjvfL
=sN,kT
(A51
L=l
Thus the root-mean-squared velocity in one dimension for each component of a statistical mechanical system is given by Equation A5. This derivation is not limited to any particular phase of matter, and thus can be applied to colloidal molecules in aqueous solution. This result was first deduced by Einstein in 1905.
A
states of a statistical
The energy
coefficients
1992, Vol. 13 No. 9
Appendix
B
The following polynomials 4 to within 3%.
fit the numerical
data
in
Figure
@ = 0.83215
Y4 - 1.5877 Y3 + 1.7067 Y2 + 0.00869
0 < Q < 0.2 @ = -0.24505Z3
+ 1.5780Z2
- 2.1698Z
0.2 < Q < 0.5 @ = -0.0082083
+ 0.81127 U341
Z2 + 1.3674 Z - 1.8914
0.5 & Q < 0.9 Q, = -0.0017663
Y
U33)
U351
Z2 + 1.1385 Z - 0.087618
0.9 S Q < 0.99
u36)