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Studies on enzyme systems at a solid-liquid interface. II. The kinetics of adsorption and reaction
Studies on Enzyme Systems at a Solid-Liquid Interface. II. The Kiqetics of Adsorption and Reaction Hans J. Trurnitl From
the Physicochemical Branch, Toxicology Laboratories, Army Chemical
Received November
Division,
Chemical
Corps
Medical
Cenler, Maryland 24, 1953
The first paper in this series (1) reported on experimental data obtained with a new instrument, the recording ellipsometer,2 on the’ reaction between chymotrypsin (ChTr) and bovine serum albumin (BSA) at a liquid-solid interface. Deposited layers of surface-denatured BSA were the solid part, of the interface. The ChTr was dissolved in buffer solution forming the liquid phase. The results were presented by arbitrarily defining the slope of the straight part of the reaction curves as reaction velocity and by showing this reaction velocity as a function of buffer concentration, pH, temperature, enzyme concentration, and substrate-layer thickness. In this way a large body of experimental data could be arranged in a simple but rather empirical manner. An attempt was made to analyze the curves in terms of formal reaction kinetics. Also, some preliminary data on ChTr adsorption were reported. Meanwhile it was realized that diffusion is the governing factor in this reaction system. A new series of adsorption and reaction experiments, with and without stirring, were carried out, and the analysis of the results shows that Nernst’s theory of reactions in heterogeneous systems (2) leads to a quantitatively satisfying explanation of the adsorption curves. Furthermore, the initial phase of the reaction curves can be interpreted on this basis. METHODS The recording ellipsometer is essentially a normal ellipsometer combined with photoelectric cell, amplifier, and recorder. Its mode of operation has been described 1 With the technical assistance of Mrs. Willie Mae Lawson. * U. S. Patent 2,666,355. 176
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in two previous publications (1,3). The theory of the ellipsometer proper has been given by Rothen (4) and by Rothen and Hanson (5,6) based on the pioneer work of Drude (7) and others.3 The actual instrumentation for the work t,o be described in this paper consisted of the following combination: The light source was a 6-v,, 18-amp. tungsten ribbon bulb powered over a stabilizing transformer and a stepdown transformer. A condenser lens formed an image of this source on the entrance slit of a Gaertner monochromator (model 2231) with Hroca prism. The light emerging from the exit slit of the monochromator (X = 580 rnp) entered the ellipsometer proper in such a manner that the exit slit of t,he monochromator was t,he entrance slit of the collimator. A Gaertner Polarizing Spectrometer (model 1,111) acted as ellipsometer. A special adapter connected the open exit end of its telescope to a photomultiplier cell of a Photovolt Multiplier Photometer (model 520-M). This instrument, by special arrangement with the manufacturer,” was fitt,ed with a second search unit, acting as balancing cell, which was exposedthrough gray filters and a variable set of Polaroid screens-to part of the ellipsometer light beam split off before its reflection on the test slide. The output signal of the photomet,er was fed into a potentiometer-type Brown Recorder (model Y153C) with a O-100 lincnr scale. For adjustment, and visual observation of the test slide, a microscope eyepiece with a small 90” deflecting prism attached in front could be inserted into a perpendicular side tube near the end of the telesope tube of the ellipsometer, A special field lens with light stop was mounted in t,he telescope tube in order to trl,t,ain an image of the test slide in the eyepiece. ,4n adjustable horizontal slit K:LS mounted between the test slide and the compensator. The I)uff’er-containing glass cell (with optical-quality walls) was mounted in a special clamp holder on the prism table of the ellipsometer. The test slide could be rotated around a vertical axis, lying in its frontal plane, and could also be moved up and down in a reproducible manner Tvit,h a micrometer screw. This micrometer in turn was attached to a vertical rack-and-pinion device to lift the test slide out, of solution and later t)o bring it back accurately into its previous position. This procedure was necessary in experiments without stirring (see below). A rot,ary type helical glass stirrer (about 500 r.p.m.) was used5 in this work. At this stirring rate, the thickness of the undisturbed water layer becomes nearly independent of stirring rat,e (see below). The glass cell was encased in a heat-insulating wall with two windows provided for the lxtssage of the light beam. Water from a thermoregulated bath could l)e passed through a coiled silver tube inserted in the glass cell. The temperature of the cell liquid was measured thermoelectrically. The stability of the combined apparatus under working condition is such that with a conditioned test slide (see below) in distilled w:tt,er, the recorder pen does not deviate more than an equivalent of 2~0.5 A. within 30 minutes. The photometer output is adjusted so that one angstrom unit corresponds to about one scale di3 See Ref. (6) for the older 1it)erature. 4 I wish to thank Dr. Brewer, Photovolt Corporation, Kcw York Cit,y, for his special efforts in this matter. 5 Therefore the adsorption and reaction velocities reported in t)his paper are higher than those given in Ref. (I), where a less efficient st,irrer was used.
178
HANS J. TRURNIT
vision on the recording paper. The precise calibration factor for each experiment is determined by moving the test slide in a vertical direction so that light from successive steps can pass through the horizontal slit which is set to about s the height of a barium stearate step. The corresponding recorder pen positions then indicate increments in thickness of 49-50 A. We use always the value 50 A. because ours are relative rather than absolute measurements. These calibration readings are reproducible within a fraction of 1 A. on the same slide. The relative error in our measurements and records is not so much determined by the electrical and optical stability of the instrument but by variations of the concentrations and age of the protein solutions. To avoid waste and prolonged standing of the enzyme solution, only very small amounts of enzyme (several milligrams) were weighed out on a torsion balance with relatively low accuracy. With very dilute solution, adsorption of protein on glass walls further decreases the accuracy. If a group of identical experiments is carried out one after the other with a series of slides which are prepared simultaneously and with carefully pipetted equal amounts of ChTr solution from the same master solution, the values of adsorption velocities are reproducible within a small fraction of 1 A&in. If various amounts of ChTr solution from one master solution are used in such a series, then the increase in adsorption velocity is proportional to the enzyme concentration, again within a small fraction of 1 A./min. (see Table IV). However, if different master solutions are used as ChTr source, the scattering of the angstrom/minute values may be as large as l-2 A./min. due to an inaccuracy in ChTr concentration (see Tables I and II). The use of a microbalance and of a microburet will remedy this situation. We have also observed that a freshly prepared ChTr solution in distilled water has a slightly lower activity during the first half hour than during the next few hours-even if kept at 4°C. between sample taking. Therefore all solutions used were between 30 min. and 6 hr. old. ADSORPTION
EXPERIMENTS
The procedure was the same as described previousiy. A test slide with or without three BSA double layers on its barium stearate base was placed into the 40-ml. cell of the ellipsometer. The cell was filled with buffer or distilled water, the liquid being stirred constantly. After calibrating the test slide, ChTr was added to the buffer. The resulting adsorption process was automatically recorded. Adsorption of ChTr takes place under these conditions on Langmuir-Blodgett slides with or without a conditioning layer of uranyl ions (conditioned and unconditioned slides) over a large range of buffer concentrations. With BSA-coated slides (standard slides), the buffer concentration had to be kept very low in order to prevent enzyme activity which would result in a decrease of the BSA layer thickness and distort the adsorption curve. Adsorption becomes noticeable a few seconds or more after adding the ChTr solution, depending on its concentration, and takes place at a constant rate, the magnitude of which depends on the concentration in the cell. This straight part of the curve lasts until Xi-20 A. are adsorbed. Thereafter, the curve gradually loses slope until it reaches its final level of constant adsorption (Fig. 1).
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179
I 2 3. 4 5 6 min FIG. 1. Recorded adsorption curves for ChTr. The unconditioned test slides were inserted in 0.01 A4 Verona1 buffer, pH 8.1 at 20°C. Then an amount of ChTr tiolution was added which gave in 40 ml. buffer the concentration indicated in the diagram. The arrangement of the curves with respect to the t.ime axis is such that time zero for each curve is the moment where it begins to leave the zero line. The preceding time interval between the injection of ChTr and this time zero (fraction of a second to several seconds) increases with decreasing ChTr concentration. The t.ime element zero indicates the moment at which the enzyme molecules actually arrive to a measurable extent at the surface after diffusing through the undisturbed water layer (see below). This initial arrival rate is actually an exponential process [see Eq. (3)], and the very onset of the adsorption curves therefore should be gradual and not instantaneous as in the diagram. But the response time (2 sec. for full scale motion of pen) and sensitivity of the recording system :Lre insufficient to show this very short initial phase except with extremely low ChTr concentrat,ions. With ChTr concentrations between 1 and 10 pg./ml., the maximal adsorption is bet,ween 30 and 35 A., corresponding to one monolayer, and with higher concentrations it may reach values of up to 60 A.6 However, after one layer of ChTr molecules has been adsorbed from a solution of low or moderate concentrat,ion (I-10 pg./ml.), no amount of ChTr added thereafter to the solution will increase the thickness of the adsorbed layer more than a few angstroms. The layers adsorbed on unconditioned or conditioned slides or on standard 6 The theoretical value for the thickness of one monolayer of ChTr is 31.2 A. if the molecules are visualized as cubes of the specific volume 0.73 and the molecular weight is taken as 25,000.
180
HANS
J.
TRURNIT
slides in low buffer concentrations cannot be removed from the solid surface by replacing the ChTr solution with protein-free buffer solution of the same concentration. This adsorption therefore is not an equilibrium phenomenon. On standard slides and with higher buffer concentrations (0.01 M), no adsorption TABLE Adsorption
Velocify
(Unconditioned Buffer
of ChTr
I
as Function
of Buffer
Concentration
slides, Verona1 buffer pH 8.0, temp. 2O”C., ChTr concentration 5 pg./ml.)
concn. M
Adsorption velocity A.jfh.
3 xs10-5
8.8
1 x-10-4
7.6 8.4 7.3 6.7
3 x
10-4
1 x 10-3 x 10-a 1 x 10-2 3 x 10-t 3
8.0 6.7
II
TABLE
Adsorption
Velocity
(Unconditioned
of ChTr
as Function
of pH
slides, Verona1 buffer 0.01 M, temp. 2O”C., ChTr concentration 5 ag./ml.) Adsorption velocity A.JPP&.
PH
7.3 7.3 6.7 7.7 7.3 7.7 7.8
6.9
7.2 7.5 7.7 7.9 8.3 8.7
TABLE
III
of Tenaperafure (Conditioned slides, Verona1 buffer 0.0003 M, pH 8.1) (Adsorption velocity in A./m&.)
Adsorption
Velocity
of ChTr
as Function
Temperature,
in “C. AE
ChTr concentration 9.6
17.6
27.0
rg.Jml. 3
3.4
4.2
-
10
10.3
-
20
28.5
36.5
Cd.JVdL?
18.0 -
4800 5400 5100
ES’ZYME
SYSTEMS.
TABLE: ,4 dsorptr’on
T’elocitt~
of C%T’r
and
y-Globulin
(Unconditioned Concentration (0 pg./wrl. ChTr, veronal 0.01 M, pH
buffer: 8.1
slides;
phosphate 0.01 M, pHj.6
181
IV ns Function
of Their
temp. 20°C.) Adsorption velocity (2)
1.0 2.0 4.0 6.0 23.0
(2)/(0
1.11 2.86 6.02 X.70 11 .!)
10.0
17.5 26.G 32.6 52.1
5.0 10.0 15.0
Concentration
A jmb.
15.0 20.0 30.0
r-Globulin, buffer:
IT
1.00 3.70 5 00
1.41 1.43 1.51 1.45 1.49
1.77 I.63 ---1.75 1 .73 0.38 0.37 0.39 OT
gamba Globulin
FIG. 2. The adsorption cwncent.ration. The data shown in Fig. 1. -,-Globulin
velocity of ChTr and for ChTr are obtained was used in 1)hosphat.c
y-globulin as function of thei] from the experimental curves ljuffer 0.01 ;1/, 1111 5.6 :It 2OY’.
182
HANS
J.
TRURNIT
TABLE V Combined’Adsorption and Reaction Experiments ChTr was first adsorbed from 0.0005 M Verona1 buffer on standard slides. After reaching a certain layer thickness (col. 2) the buffer concentration was increased to 0.01 M. The resulting reaction velocity is given in col. 3. Column 4 shows reaction velocities from corresponding experiments without preadsorption (see also Fig. 3). ChTr concn. ha.lmJ.
0.1 0.2 0.5 1.0 1.0 1.5 2.0 2.0 4.0 4.0 7.0 8.0 12.0 20.0 30.0 60.0
Thickness of adsorbed ChTr layer A.
1 1 2 5 8 13 16 21 25 30 42 41 50 56 48 47
Reaction velocity A.lmin.
2.1 3.3 6.2 16 27 30 29 27 26 27 32 29 31 18 19 20
Reaction velocity without preadsorption A./mis.
4.0 9.2 9.2 14 14 18 18 25
can be observed in the ChTr concentration range from 0 to 15 pg./ml. The experimental curve shows only substrate removal. With higher ChTr concentrations the curves show initial adsorption with following increasingly dominant substrate removal [see Fig. 1 in Ref. (l)]. The slope of the initial straight part of the curve is called adsorption velocity and measured in angstroms per minute. The influence of changing buffer concentration, pH, temperature, and protein concentration is presented in Tables I-IV and in Fig. 2. Part of the results given in Table IV and Fig. 2 will be used as a basis for the calculations in the following chapters. In a separate set of experiments, ChTr was first adsorbed on standard slides at low buffer concentration. Then strong buffer was added, so that the ChTr could react with the underlying BSA. The resulting reaction velocities may be compared with those from experiments of the usual type. (See Table V and Fig. 3.) The results show that with preadsorption the reaction velocities are not a function of the ChTr concentration in solution but rather of the amount of ChTr adsorbed on the substrate. This holds for the adsorption range from 1 to 10 A. Further increase of the thickness of the adsorbed layer does not result in a higher reaction velocity. With more than 50 A. adsorption and high ChTr concentrations, the reaction velocity declines to lower values again. The reaction curves after
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II
preadsorption are straight lines in the beginning; whereas, the usual reaction curves have a parabolic onset (see Fig. 3). These data show that maximum velocity of substrate removal occurs when approximately one third of the slide surface is covered at any given moment with enzyme molecules in molecular contact with the substrate.
X + IO 0 - IO 20 30
40 + IO 0 - IO 20 30 40
50
i
20
+to
’
mma/ml
1 t
\
0 - IO 20
30 40
50 0.
4
6
12
16
20
24
mln
FIG. 3. Reaction curves obtained in four experiments with preadsorption of ChTr and in three corresponding experiments without preadsorption (usual type). The ordinate-level zero marks the original position of the substrate-liquid interface. In the experiments with preadsorption the cell was filled with 0.0005 M veronal buffer. Enzyme was added at the time marked by arrow No. 1. The enzyme concentration is given at the beginning of each set of curves. The enzyme is adsorbed with a speed depending on it’s concentration (between arrows Nos. 1 and 2). This adsorption is then interrupted (at arrow No. 2) by adding strong buffer solution, giving a buffer concentration of 0.01 hf. Thereby the enzyme is desorbed and at the same time activated and removes substrate from the test-slide surface. For comparison three curves are shown from experiments with the same enzyme concentrations as in the first, third, and fourth experiments, but with 0.01 M buffer already in the cell prior to adding the enzyme. The maximum reaction velocities in angstroms/min. are given in round figures near each curve.
184
HANS J. TRURNIT
The data for adsorption velocity are well reproducible if the rate of stirring is kept constant. In buffer concentrations higher than 0.05 molar, the shape of the adsorption curve changes : the first straight part becomes shorter with increasing buffer concentration, and in 0.1 M buffer the curve is parabolic. This deviation from the normal behavior will be discussed later. DFP-inactivated7 ChTr has the same adsorption velocity as ChTr under otherwise comparable conditions. If the buffer concentration is increased after DFP-inactivated ChTr has been adsorbed on a standard slide as monolayer from distilled water or weak buffer, only twothirds of the adsorbed material is desorbed. Active ChTr under those conditions is completely desorbed. The slope of the adsorption curves as well as the reaction curves decreases rapidly if during the experiment the stirrer is stopped. It comes back to its previous value immediately after stirring is resumed. The main result of these various measurements and observations is that within a certain concentration range the adsorption velocity is directly proportional to the ChTr concentration.* This indicates that diffusion plays a dominant role in this process at the solid-liquid interface. In the following chapters, therefore, an attempt will be made to analyze the adsorption data and part of the reaction data on the basis of the diffusion theory. DIFFUSION
AND ADSORPTION
VELOCITY
IN EXPERIMENTS
WITH STIRRING
Noyes and Whitney (8) and later Bruner and Tolloczka (9) showed that the dissolution rate of a solid in a well-stirred liquid is proportional to the coefficient of diffusion D of the solute, to the geometrical surface of the solid exposed S, and to the difference between the solubility CO of the solid in the pure solvent and the concentration of the solid in the liquid at time t. dn = kDS(co - c) dt They deduced from these experiments that (a) the solid is surrounded by a saturated undisturbed layer of solution through which the solvent molecules have to diffuse towards the solid, (b) stirring does not cause any commotion in this layer, and (c) the thickness d of this layer decreases with increasing rate of stirring but only to a mininum value which is approached asymptotically. Nernst (2) generalized these ideas in his theory of reaction kinetics in heterogeneous systems. He states that whenever the primary physical or chemical reaction at the interface is rapid as compared to the diffusion 7 See also Table II in Ref. (1). 8 The faster increase of adsorption velocity in the concentration 10 pg./ml. (see Table IV) will be discussed at a later date.
range beyond
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185
through the adherent layer, interfacial reaction rates as a whole governed by diffusion. The quantitative formulation for t,his is: G?-L~Sc: dt
are
(11
d
where dn over-all reaction rate ilt = J> = coefficient of diffusion S = surface area C = concentration d = t,hickness of the undisturbed This becomes identical with
&k’s
layer
first law of diffusion:Y
!?=L)sn’c dt where
dc - = concentration
dX
(2)
dx
gradient,
if we assume (a) a linear concentration gradient across cl, (b) c to be zero at, the wall at all times, and (c) x to be identical with d. With water as a solvent and with many different interfacial reactions, d has always been found (10) to be in the neighborhood of 3 X 1O-3 cm. if vigorous stirring was employed. These ideas will now be applied to the adsorption process in our system (see Fig. 4). The solid surface may be at z = 0; t*he thickness of the thin undisturbed water layer may be d. For z values beyond d, the concentration of the solute (ChTr) is ci at all times. At r = 0, that, is at the surface proper, c shall be zero at all times. Between x = 0 and x = d, c shall be zero at, t = 0 and after a short initial period the cowentration gradient shall be const,ant, or
Under these conditions
it is permissible
to ident~ify dn/dt with
the ad-
o Hereafter the term S will be omitted in the application of Fick’s first lnw PA{. (1 j tJec:tusr all calculations will employ absolute units and the area referred will be always 1 sq. cm. ?z or A are therefore given in g/q. cm.
01 to
186
HANS
J.
TRURNIT
sorption velocity dA/dt and to apply Eq. (1). It will be correct for the time interval during which the adsorption curve (Fig. 1) has a constant slope, that is, as Iong as we can assume the concentration in the immediate vicinity of the solid surface to be zero. However, we have to be certain that the time interval mentioned above between t = 0 and the establishement of a constant concentration gradient between x = 0 and 2 = d is actually short as compared to the straight line interval of the adsorption curve. Otherwise, Eq. (1) is not applicable. This may be decided by inserting experimental data into the solution of the equation for diffusion through a flat as adapted to the boundary conditions of our system: c =
X sin?
~+~~~
G;
Xexp(-@$!)I
[
The question then is how fast the concentration c at a given value of x, e.g., the half-way point 2 = d/2 (Fig. 4) approaches its final value cJ2. For this half-way point the concentration is
e=;+- : g (-y
X sin?
(3)
X exp(-@$$)
By inserting for D the diffusion coefficient of ChTr Dzo = 10 X 10-’ sq. cm./sec. and for d the value 3 X lO+ cm. into the exponential term of Eq. (3),we fmd for t = 5 sec.:
:: *ILL c=-- ei
0.0035s
2
C
Cl -r
.. ,..: : : _--I ; : : i;. ' ,. i .:: .:' : ! ... .: .A.
0 -+* 6 P
FIG.
Cl
6
-x 4. Diffusion process near interface.
(For explanation
see text.)
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187
II
This means that after 5 sec. the concentration at the half-way point is already within 0.35 y0 of its final value. This period of 5 sec. is very short as compared to the period of linear adsorption rate observed in experiments with ChTr concentrations below 10 pg./ml. We therefore have:
where dA/dt is measured in g./sq. cm./sec., D in sq. cm./sec., ci in g./‘ml., and d in cm. If this equation describes the adsorption data correctly one should find that dA/dt is directly proportional to the concentration. That this is actually the case is shown in Fig. 2. One should further expect to find d of the same order of magnitude as given in the literature if experimental values for dA/dt, D, and ci are inserted in Eq. (4). For a ChTr concentratjion of 10U6 g./ml., n-e find for 20°C. (see Fig. 2) d,Z/dt = 1.46 A./min. Using 0.73 as partial specific volume of ChTr. dA -zz dt
6. ‘;(‘:
73 lo-’
= 3.31 X 10-l’ g./sq. rm.,!sec.
For D20 we find in the literature polated for infinite dilution). Therefore d = loa
(11) 10.2 X 1W7 sq. cm./sec.
’ lo-’ ’ lo+ 3.31 x 10-10
= 3 08 x 1O-3 cm .
This value is in excellent agreement with the older above. Finally we should expect that dA/dt is proportional of diffusion of the substance used. Experiments with bovine plasma (Armour and Company) (see Fig. 2) y-globulin’0 gave for a concentration of 1O-6 g./ml. : dA - = 0.38 A./min. dt
= 6. “;(“:
(extra-
data mentioned to the coefficient Fraction II from which is mainly
72 X low8 = 8.8 X lo-l1 g./sq. cm./sec.
188
HANS J. TRURNIT
If this value is used together with the value for d as calculated before, we obtain from Eq. (4): D
20
= 3.08 X lo+ x 8.8 x lo-l1 = 2.7 X lo-’ sq. cm./sec. 10-G
This value for DUOis about 30 % smaller than the data found in the literature (around 4 X lo-‘). But considering the impurity of the material and the relatively simple and crude way in which this value has been obtained it sustains our argument. It seems,therefore, permissible to use Nernst’s theory as a physical interpretation of the results of these adsorption experiments. We propose to use this simple type of experiment for the determination of diffusion coefficients of adsorbable substances. By improving the apparatus it should be possible to make d an instrument constant which could be empirically determined by calibration with a substance with a wellknown diffusion coefficient. The advantages of this method would be the possibility of working with extremely dilute solutions and the fact that values can be obtained within 5-10 min. CONNECTION
BETWEEN
ADSORPTION VELOCITY
RATE
AND REACTION
In sufficiently strong buffer (0.01 M) ChTr is not adsorbed to a detectable degree if its bulk concentration does not exceed 10-15 pg./ml. Therefore one cannot speak in a strict sense of an adsorption rate in discussing reaction velocities. Arrival rate would be a better texm. As a matter of convenience, however, we will continue to use the term adsorption. Even if on standard slides under reaction conditions (that is in strong buffer) “visible” adsorption does not occur, it is known from experiments on unconditioned slides (Table I) that the higher buffer concentration does not change the adsorption rate. In order to react with the substrate, the ChTr molecules have to come into molecular contact with it.” But their contact with the substrate surface under reaction conditions is obviously lessintimate or so frequently interrupted that they do not participate sufficiently in the optical interference phenomenon and are in this sensenot part of the solid phase.12Nevertheless, 11Unless we assume the existence of long-range forces. 12 It may well be that whether an enzyme molecule is part of the liquid phase, or part of the solid phase (more loosely or more strongly adsorbed) makes the difference between its active and inactive state in stronger or weaker buffer, respectively.
t’he rate of arrival of ChTr molecules in t’he molecular vicinity of t,hc surface \rill be t,he same function of concentrat~ion 8,s shown in Fig. 2. (111t’his basis we make t’he simple aud obvious assumption that, initia.lly the reac*tioir rate at any tdme is directly proportional t,o the amount d4 of OhTr present, at, t,he slide surface at t,his moment.. Tntegra.tiou of Eq. (4) shows that during the initial adsorption period A = k,c&
(5)
\\-here .I is t)he amount adsorbed (g.jsq. cm.) aft#er time 2 (sec.), Ii,, is I),/d (cm./scc.), and c; the bulk concent’rat’ion (g/w.). The rracbion rate dT,/dt (loss per time elrment) should therefore 1~: ClLlch = k,k,c,t
((9
where k, is a reaction const,ant with t’he dimension of reciprocal seconds. Ry int’egration we obt,ain :
Thus, the shape of the reaction curve in this initial phase should lw parabolic. ,I careful evaluation of reaction curves observed with enzyme concent,rations of 2, 1, and 0.2 pg./ml. shows that this is actually the case. One example is given in Fig. 5. Therefore t*he assumption on which this calculat,ion is based seemsto be justified. The dat’a given in Fig. 5 are expressed iI1 angst,roms and minutes. 111 absolute u&s one obtains from hhe experiment,al data of Fig. 5 and from Eqs. (S), (6), and (7): I;,
=
3.32
X
IO-”
cm./sw.
and lc,
=
2.56
x
10-Z
SW--l
if the concentration is given in g./cc., and if the partial specific volume of BSA is taken as 0.73. It follows that 1 g. of ChTr in contact with the solid surface removes 2.56 X 1O-2g. BSA/sec. With the molecular weights of ChTr and BSA taken as 25,000 and 68,000, respectively, this means that on the average one molecule of enzyme removes from the surface 9.4 X IOP substratcl molecules/set.
190
HANS
At 6
I
J.
TRURNIT
I
I
I
, I
I
/ \
4 +2 0 -2 4 6 6 IO 12 14 I6 16
FIG. 5. Reaction as related to adsorption. The upper curve is from an adsorption experiment with 1 pg. ChTr/ml. in 0.0005 M Verona1 buffer, pH 8.1, on an unconditioned slide. A reaction experiment on a standard slide with the same ChTr concentration but in 0.01 M buffer gave a curve indicated by the series of circles below the zero line. From this curve the value of k, in Eq. (7) was calculated and then the curve drawn which is represented by Eq. (7). This theoretical curve follows very closely the experimental curve.
This is as far as the experimental evidence permits an interpretation. Any further speculation about the number of bonds split per enzyme molecule per second would be unfounded because we do not know how far the splitting process has to disintegrate the denatured BSA in order to make the split products leave the surface. Also, we cannot estimate the proportion of activated enzyme molecules even if we know that the Arrhenius coefficient for the over-all reaction is in the neighborhood of 8000 cal./mole [see (l)] and that about 5000 cal./mole hereof (see Table III) are accounted for by the energy of diffusion. The reason is that the splitting of different types of bonds requires different activation energies. As an average value we would obtain exp (-3000/586) = 0.006 = 0.6% for the fraction of activated molecules. The parabolic course of the reaction curves continues until all but about two layers of substrate are removed. This is independent of the total numbers of substrate layers originally present on the slide. At this point the curve has an inflection point and thereafter gradually approaches a horizontal course. Either the substrate layers at the bottom of the pile are not as easily digested as the higher layers (influence of the uranyl ions?) or the split products are more tightly bound (for the same reason?) or the attack of the enzyme from the beginning must be visualized more as an irregular “corrosion,” which acts faster in some parts of the surface than in others. The onset of the deviation from the parabolic course
ENZYME would then the bottom
indicate that of the pile.
at this
SYSTEMS.
moment
EXPERIMENTS
the
enzyme
WITHOUT
191
11 has reached
in some
spots
STIRRING
I. Adsorption
A series of experiments was made without stirring. They were carried out as follows: The slide was inserted into buffer solution for calibration and t.hen withdrawn; enzyme was added and the solution thoroughly stirred for half a minute. The solution was then allowed to stand quiet for a few minutes. The test slide was then brought back into the solution in its original position. The adsorption curves obtained under these conditions are very closely horizontal half paraboIas as shown in Fig. 6. Sometimes they deviate from the parabolic shape during the first few minutes due to a disturbance caused by inserting the slide. But generally during the increase in thickness of the adsorbed layer which corresponds A 30
0
IO
20
30
40
SOmin
FIG. 6. Adsorption curves from experiments without stirring. As in Fig. 5, the course of the experimental curves is indicated by circles. The drawn-out curves represent Eq. (9). Curve a: Adsorption of ChTr (5 pg./ml.) on unconditioned slide in 0.01 M Verona1 buffer, pH 8.0, at 27°C. Curve b: Adsorption of -t-globulin (10 pg./ml.) on unconditioned slides in 0.01 M phosphate buffer, pH 5.6, at 27°C. At 30 min. the theoretical value for A with ChTr is 19.5 A. (see text). The csperimental value is 23 A. With r-globulin the theoretical value at 30 min. (using l&r = 4.81 X 10-r sq. cm./sec.) is 24.4 A.; the experiment,al value is 21 A. .4ssuming that the experimental value were higher than the theoretical one by the same percentage as in the case of ChTr, we would obtain 29 A. for the experimental value. The actual value of 21 A. shows the same discrepancy of about 30% as in the experiments with stirring (see above).
192
HANS
J.
TRURNIT
to the straight part of the adsorption curve obtained in experiments with stirring, they are perfectly parabolic. Thereafter they leave the parabolic course and fade slowly out into a horizontal line. This corresponds to the upper end of the curves from experiments with stirring (Fig. 1). In order to explain the parabolic shape as such and to compare the quantitative data from these experiments with those to be expected from the theory, we proceed as follows: Our system, without stirring, should obey the law of diffusion in a semi-infinite cylinder with the following boundary conditions : c=Oatz=O c=ciatz>O c = f(x) at 2 > 0
for all t fort
= 0
for t > 0
where c is the concentration at any x or t, ci is the initial concentration, x is the distance from the boundary, and t is the time. For this case we find : E = --& ~‘e+
cl.2= erf(x);
[z = *]
(8)
The curve representing erf(z) is plotted in Fig. 7 for 2dfi = 1, or z = 5. The area above the erf(z) curve (marked F in Fig. 7) represents the amount A’ of material adsorbed on the surface at a time which satisfies the condition 2di% = 1. This area F is the product of a distance 2 (which represents a volume over 1 sq. cm. of surface) and a concentra-
FIG. 7. The curve representing erf(z). See text.
ENZYME
SYSTEMS.
tion c. Mechanical integration of F with a planimeter 0.57. This value is very close to l/d;. Thus for .z = x, and for c = c;, we have:
and for z = x/26
193
II
yields the value
we obtain:
A=xXci=ciLX22/iSi. G
(‘3)
01'
A = 2ci This is the equation of a parabola. It perimental curve correctly and permits sorbed after time t. For ci = 5 X 1P6 sq. cm./sec. and for 1800 sec. (see Fig. 6)
A = 1O-5
.m
describes the shape of the CSpredicting the amount A adg/ml. and DS = 12.3 X 1P we obtain:
= 26.5 X 10d8 g./sq. cm.
This corresponds to 26.5 X 0.736 X 1C8 = 19.5 A. adsorption. The experimental value (Fig. 6) is 23 A. The difference is probably due to the presence of some convection in the solution.13 Equation (9) may also be obtained in a more conventional way: By differentiation we obtain from Eq. (8) : 13 As mentioned above, these experiments were carried out in a very simple way. If adequate precautions were taken to prevent convection currents the esperimental figures would be 611 closer to the theoretical figures. Another precaution which must be taken in this type of experiment is to prevent the pickup of denatured surface protein while the slide is inserted into the solution. This can be accomplished by filling i.he cell almost to the rim and covering all but a narrow slit of the liquid surface with glass slides. The test slide is then inserted through this slit. However, this has to be done slowly in order to prevent commotion in the liquid as much as possible. Therefore it is not possible to prevent pickup of surfact-denatured protein completely. The zero level after insertion (that is f he axis of the adsorption parabola) will therefore be slightly higher on the recording paper than during calibration. Its position can later be estahlishcd by finding a value for CLwhich yields a constant value for h- in the parabolic equation: r = k(y - 0)2, if scvernl coordinated pairs of s :~nd !/ of the ntisorl)tion p:~rnlml:t are tested.
194
HANS
J.
TRURNIT
This is the concentration gradient at any given time and distance from the surface. In order to determine the rate of arrival of diffusing molecules at the surface we are only interested in the value of dc/ds in the immediate vicinity of the surface, that is, for z -+ 0. We may therefore neglect the exponential term, which becomes unity for 2 = 0, and insert the first term as concentration gradient in Fick’s first law [Eq. (a)]. We obtain:
or by transformation
: dA dt = ci
(10)
which is identical with Eq. (9). This formal derivation of Eq. (9) in contrast to the previous one does not reveal the physical meaning of the term l/l/n. cln the other hand it introduces l/g ?r in a mathematically correct sense. In the first derivation the value of the area F (Fig. 7) was only assumed to be l/G. Therefore both these derivations combined, that is the identity of Eqs. (9) and (ll), constitute a mathematical proof that the area above the erf(z) curve has the simple value l/l/?r.14 2. Reaction The theoretical shape of the initial part of reaction curves from experiments without stirring can be derived in t#he same way as before for experiments with stirring. Again we assume that the reaction rate dL/dt at any time is proportional to the amount of enzyme present at the interface at that moment. This amount is given by Eq. (11). 14 The author has not been able to find a statement to this effect in the literature. An attempt to calculate the value of F by developing the integral in Eq. (8) into a series and integrate term by term failed because the convergence of this series is too slow.
ENZYME
SYSTEMS.
195
II
Thus we have
!z! at=k’r ’ c pfi x 2c.
By integration
we obtain the equation
L=iXk:XCi
of the reaction
d- $ x t3’2
(12) curve
(13)
In Fig. 8 this curve (C) is drawn for ci = 5 X IO+ g/ml. and A’, = 2.56 X 1CP sec.-l, the same value as obtained from experiments with stirring. The actual experiment performed with this concentSration gave a curve which coincides accurately with curve B. B is drawn from Eq. (13) using the same value of ci and a value for k’, which is 3.1 times smallcl than k, . We find, therefore, that t.he shape of the actual reaction curves in experiments without stirring can be predicted from the theory, but that the reaction const.ant is about 4 Gmes smaller than the one on csperiA
20
+10 0 - IO
20
30 i 40 an -FIG. 8. Adsorption and reaction curves from experiments is the same curve as curve a in Fig. 6. The circles along curve action curve in 0.01 M buffer with 5 pg. ChTr/ml. For further explanation see text.
without stirring. ;I B represent the re-
196
HANS
J.
TRURNIT
ments with stirring. A difference was to be expected because the removal of split products in this case is obviously slower. DISCUSSION
It was clear at the beginning of this work that the various theories developed for the kinetics of enzyme reactions in solution could not be applied to a system where one of the two reaction partners, in this case TABLE
Governing the Initial
Equations
VI
Phase and and Reaction
Initial
the Steady
State
phase
Adsorption
With stirring (Figs. 1,3,
5)
dA vn = -dt
=
Reaction
D 2 x
c
A=;xcxt
Without (Figs.
stirring 6, 8)
of Adsorption
dL ?I+ = x
= k, x
L=k,X$XcXtZ
--= dA ‘a - dt
A = 2
Steady
state
Adsorption
Preadsorption
k, , k: , k! A A’ L c t D d
Reaction
dA --so v0 - dt
= = = = = = = =
A = constant
= A’
A = constant
= f(c)
L=k;XA’xt < A’
Reaction constants. Amount adsorbed. Maximum effective amount in contact Amount removed. Initial enzyme concentration. Time. Coefficient of diffusion of enzyme. Thickness of adherent water layer.
L = kf x j(c)t
with
interface.
D ;i-xcxt
ENZYME
SYSTEMS.
II
197
the substrate, is not part of the liquid phase. In fact, the work was undertaken in order to obtain somebasic experimental data which would permit the development of the kinetics for this type of interfacial reaction. By introducing diffusion as the rate-limiting factor, it has been possible to derive a set of equations which describe in a satisfactory manner the initial phase of adsorption and reaction in experiments with and without stirring. The possibility of investigating the reaction in its very beginning, before secondary effects change the reaction rat,e, permits one t,o express at least part, of the over-all react,ion constant in physicbal terms. This part is the adsorption constant k a , which is D/d with stirring and t/Bz without stirring. Table VI shows the set of equations systematically arranged. These equations hold for adsorption as long as the adsorption curve is a straight line or, in other words, as long as cz=o = 0, and for reaction they are valid until an optimal amount of enzyme (A’ x 10 A.) is present at the interface. From this moment on, the reaction rate remains constant. But in experiments with stirring, the point of inflection on the reaction curves comes always before this critical amount A’ has accllmulated, that is, if slides with three double layers of substrate are used. In experiments without stirring or with preadsorption of enzyme and \\-ith stirring, the steady state may actually be observed for a short while, before secondary processes change the pattern. Cnder steady-state conditions, the reaction velocity is independent of time and concentration. But if the critical amount A’ cannot be reached because the total amount of enzyme in the solution is too small, the reaction velocit’y even under steady-state conditions depends on the concentration. This is where ow system begins to resemble the usual reaction conditions in solut,ion. One of the basic laws in bulk enzyme kinetics is the proportionality between reaction rate and enzyme concentration during the steady state. The same proportionality occurs in the reaction equations for the initial phase (Table VI). This is another way of stating that during this initial interval, A has not yet reached its optimum value A’. In order to determine the basic physical components ivhich are &II hidden in the reaction co&ant X;, , similar experiment,s u-ill have to 1~ made with a simple substrate. If only one well-defined split, prodllct leaves the surface, it may be possible to apply the diffusion theory to t,his reverse process and to calculate a reaction constant which is idrnticsal with the turnover number, that is, if we define turnover number hy thc~
198
HANS
J.
TRURNIT
number of bond-splitting acts which an enzyme molecule performs per second. In our case, however, we may tentatively define the turnover number as the number of substrate molecules removed each second from the surface by one enzyme molecule. If we apply this to the results of experiments with optimal preadsorption (Table V), we may say that before activating the enzyme with strong buffer, about one-third of the surface is covered with stable enzyme-substrate complexes. Their number is roughly 3 X 1Ol2 complexes/sq. cm. The corresponding reaction velocity is about 30 A./min. = 0.5 A./set. = 6.9 X leg g./sq. cm./sec. This rate is equal to about 6 X lOLo substrate molecules/sq. cm./sec. Each enzyme molecule therefore removes about 2 X 1e2 substrate molecules/set. This turnover number has twice the value as the one derived in a previous chapter [following Eq. (?)I from experiments without preadsorption. The fact that we find two different turnover numbers, depending on how the complex formation is accomplished, seems to offer the possibility of investigating the splitting mechanism more closely. The parabolic course of the reaction curves in experiments with stirring explains in a simple way the increase in reaction velocity with increasing number of substrate layers, which has been described previously [see Fig. 11 in Ref. (l)]. It could not be explained at the time, Now it becomes clear that with increasing substrate pile thickness the inflection point moves downward along the parabola; that is, the tangent to the curve at this point becomes steeper and steeper. The slope of this tangent was called reaction velocity in the preceding paper. The connection between the basic concept of Michaelis and Menten and the theory developed in this paper will be discussed in a later publication. At the same time more will have to be said about the physical meaning of the undisturbed water layer. It should be emphasized that the recording ellipsometer technique, because of the role diffusion plays in it, permits one to study the initial phases of enzyme reactions in detail over periods of minutes. TJnder the conditions of bulk reactions, extremely rapid methods have to be employed to achieve the same purpose. Aside from these theoretical considerations there is a practical aspect. The recording ellipsometer seems to provide the possibility of measuring diffusion coefficients rapidly and in extremely dilute solutions. Table VI shows that with stirring D is directly proportional to dA/dt which may be read from the curves. The instrumental problem is to make d suffi-
ENZYME
SYSTEMS. II
199
cient,ly constant. Without stirring D is proportional to t’he squaw of dA/dt or A, This means it is much more sensitive against errors made in determining &l/c& or A. The instrumental problem in this case is to avoid disturbances in the solution. The first method permifs readings to br made sooner than the second one. SUMMARY
The kinetics of adsorption of chymotrypsin (ChTr) from solution onto I solid surface has been studied experimentally and theoret’ically. 01~this l)usis the initial phase of the reaction of ChTr with surface-denatured bovine serum albumin at’ the liquid-solid interfaw can be dewribcd qualitatively and quant it,atively. Experiment, and theory show furthermore that the recording ellipsomctcr may he used to measure coefficients of diffusion rapidly and in VSt,remely dilute solutions. It is cmphasieed that interfacial enzyme reaction st,udies of this t,ype permit one to investigat’e t’he initial reaction phase more easily and in tiner detail t’han studies of bulk reactions, where very rapid met’hodshave to be employed. REFERENCES II. J., Arch. Hiochem. and Biophys. 47, 261 (I!K<). W., 2. ph!/sik. (‘hem. 47, 52 (1904). T,nyers,” American Associa3. TRVRNIT, H.J., “Symposium on Monomolecular tion for the Advanremcnt of Science (A.A.A.S.) Jlrcting. Philadelphia, Pa., Dec., 1951. 1954 A..i.A.S. Monograph. 4. ROTHES, A.. Rev. Sci. Znslr. 16, 26 (1945). 5. Itmms, A., AND HANSOS. ,lf., Rev. Sci. Instr. 19, 889 (1948). 6. ROTJI~, A., AND I~AZISON~ l’r., Rev. sci. znstr. 20, 66 (1949). I.
TRZ-RSIT,
2.
SERSST,
7.
Dams,
N.
SOYES,
A. A.,
I'.,
Ann.
9.
BRUKER,
I,.,
AND
AND
Physik W'HITX.EY.
Tor.1,orzs.4.
11. C’hem.
36, 865
W. R., Z. physik. ST., Z. physik.
of R,eact.ions in Solution,“2nd Ed. Oxford, 1947. 11. SEURATH, H., AND BAILEY, K., “The Proteins,” Vol. 1, part A. Bcademir Press, New York, 1953. For the basic mathematical equations and for the values of crf(z) the following lmks have been tied: .JOST, W., “Diffusion in Solids, Liquids, Gases.” Xratfemic Press, New York. 1952. J~ARRER, R. M., “Diffusion In and Through Solids.” University Press, Cambridge, 1941. (-~.~RsLAw, H. S., AND JdEGER, J. C., “Conduction of freat in Solids.” Clarrndon Press, Oxford, 1947. 10. MOELWYN-HUGHES,
5:. A., “The Kinetics
(1889). C’hcm. 23, 689 (1897). Chem. 36, 288 (1900).
the Physicochemical Branch, Toxicology Laboratories, Army Chemical
Received November
Division,
Chemical
Corps
Medical
Cenler, Maryland 24, 1953
The first paper in this series (1) reported on experimental data obtained with a new instrument, the recording ellipsometer,2 on the’ reaction between chymotrypsin (ChTr) and bovine serum albumin (BSA) at a liquid-solid interface. Deposited layers of surface-denatured BSA were the solid part, of the interface. The ChTr was dissolved in buffer solution forming the liquid phase. The results were presented by arbitrarily defining the slope of the straight part of the reaction curves as reaction velocity and by showing this reaction velocity as a function of buffer concentration, pH, temperature, enzyme concentration, and substrate-layer thickness. In this way a large body of experimental data could be arranged in a simple but rather empirical manner. An attempt was made to analyze the curves in terms of formal reaction kinetics. Also, some preliminary data on ChTr adsorption were reported. Meanwhile it was realized that diffusion is the governing factor in this reaction system. A new series of adsorption and reaction experiments, with and without stirring, were carried out, and the analysis of the results shows that Nernst’s theory of reactions in heterogeneous systems (2) leads to a quantitatively satisfying explanation of the adsorption curves. Furthermore, the initial phase of the reaction curves can be interpreted on this basis. METHODS The recording ellipsometer is essentially a normal ellipsometer combined with photoelectric cell, amplifier, and recorder. Its mode of operation has been described 1 With the technical assistance of Mrs. Willie Mae Lawson. * U. S. Patent 2,666,355. 176
ENZYME
SYSTEMS.
II
177
in two previous publications (1,3). The theory of the ellipsometer proper has been given by Rothen (4) and by Rothen and Hanson (5,6) based on the pioneer work of Drude (7) and others.3 The actual instrumentation for the work t,o be described in this paper consisted of the following combination: The light source was a 6-v,, 18-amp. tungsten ribbon bulb powered over a stabilizing transformer and a stepdown transformer. A condenser lens formed an image of this source on the entrance slit of a Gaertner monochromator (model 2231) with Hroca prism. The light emerging from the exit slit of the monochromator (X = 580 rnp) entered the ellipsometer proper in such a manner that the exit slit of t,he monochromator was t,he entrance slit of the collimator. A Gaertner Polarizing Spectrometer (model 1,111) acted as ellipsometer. A special adapter connected the open exit end of its telescope to a photomultiplier cell of a Photovolt Multiplier Photometer (model 520-M). This instrument, by special arrangement with the manufacturer,” was fitt,ed with a second search unit, acting as balancing cell, which was exposedthrough gray filters and a variable set of Polaroid screens-to part of the ellipsometer light beam split off before its reflection on the test slide. The output signal of the photomet,er was fed into a potentiometer-type Brown Recorder (model Y153C) with a O-100 lincnr scale. For adjustment, and visual observation of the test slide, a microscope eyepiece with a small 90” deflecting prism attached in front could be inserted into a perpendicular side tube near the end of the telesope tube of the ellipsometer, A special field lens with light stop was mounted in t,he telescope tube in order to trl,t,ain an image of the test slide in the eyepiece. ,4n adjustable horizontal slit K:LS mounted between the test slide and the compensator. The I)uff’er-containing glass cell (with optical-quality walls) was mounted in a special clamp holder on the prism table of the ellipsometer. The test slide could be rotated around a vertical axis, lying in its frontal plane, and could also be moved up and down in a reproducible manner Tvit,h a micrometer screw. This micrometer in turn was attached to a vertical rack-and-pinion device to lift the test slide out, of solution and later t)o bring it back accurately into its previous position. This procedure was necessary in experiments without stirring (see below). A rot,ary type helical glass stirrer (about 500 r.p.m.) was used5 in this work. At this stirring rate, the thickness of the undisturbed water layer becomes nearly independent of stirring rat,e (see below). The glass cell was encased in a heat-insulating wall with two windows provided for the lxtssage of the light beam. Water from a thermoregulated bath could l)e passed through a coiled silver tube inserted in the glass cell. The temperature of the cell liquid was measured thermoelectrically. The stability of the combined apparatus under working condition is such that with a conditioned test slide (see below) in distilled w:tt,er, the recorder pen does not deviate more than an equivalent of 2~0.5 A. within 30 minutes. The photometer output is adjusted so that one angstrom unit corresponds to about one scale di3 See Ref. (6) for the older 1it)erature. 4 I wish to thank Dr. Brewer, Photovolt Corporation, Kcw York Cit,y, for his special efforts in this matter. 5 Therefore the adsorption and reaction velocities reported in t)his paper are higher than those given in Ref. (I), where a less efficient st,irrer was used.
178
HANS J. TRURNIT
vision on the recording paper. The precise calibration factor for each experiment is determined by moving the test slide in a vertical direction so that light from successive steps can pass through the horizontal slit which is set to about s the height of a barium stearate step. The corresponding recorder pen positions then indicate increments in thickness of 49-50 A. We use always the value 50 A. because ours are relative rather than absolute measurements. These calibration readings are reproducible within a fraction of 1 A. on the same slide. The relative error in our measurements and records is not so much determined by the electrical and optical stability of the instrument but by variations of the concentrations and age of the protein solutions. To avoid waste and prolonged standing of the enzyme solution, only very small amounts of enzyme (several milligrams) were weighed out on a torsion balance with relatively low accuracy. With very dilute solution, adsorption of protein on glass walls further decreases the accuracy. If a group of identical experiments is carried out one after the other with a series of slides which are prepared simultaneously and with carefully pipetted equal amounts of ChTr solution from the same master solution, the values of adsorption velocities are reproducible within a small fraction of 1 A&in. If various amounts of ChTr solution from one master solution are used in such a series, then the increase in adsorption velocity is proportional to the enzyme concentration, again within a small fraction of 1 A./min. (see Table IV). However, if different master solutions are used as ChTr source, the scattering of the angstrom/minute values may be as large as l-2 A./min. due to an inaccuracy in ChTr concentration (see Tables I and II). The use of a microbalance and of a microburet will remedy this situation. We have also observed that a freshly prepared ChTr solution in distilled water has a slightly lower activity during the first half hour than during the next few hours-even if kept at 4°C. between sample taking. Therefore all solutions used were between 30 min. and 6 hr. old. ADSORPTION
EXPERIMENTS
The procedure was the same as described previousiy. A test slide with or without three BSA double layers on its barium stearate base was placed into the 40-ml. cell of the ellipsometer. The cell was filled with buffer or distilled water, the liquid being stirred constantly. After calibrating the test slide, ChTr was added to the buffer. The resulting adsorption process was automatically recorded. Adsorption of ChTr takes place under these conditions on Langmuir-Blodgett slides with or without a conditioning layer of uranyl ions (conditioned and unconditioned slides) over a large range of buffer concentrations. With BSA-coated slides (standard slides), the buffer concentration had to be kept very low in order to prevent enzyme activity which would result in a decrease of the BSA layer thickness and distort the adsorption curve. Adsorption becomes noticeable a few seconds or more after adding the ChTr solution, depending on its concentration, and takes place at a constant rate, the magnitude of which depends on the concentration in the cell. This straight part of the curve lasts until Xi-20 A. are adsorbed. Thereafter, the curve gradually loses slope until it reaches its final level of constant adsorption (Fig. 1).
ENZYME
SYSTEMS.
II
179
I 2 3. 4 5 6 min FIG. 1. Recorded adsorption curves for ChTr. The unconditioned test slides were inserted in 0.01 A4 Verona1 buffer, pH 8.1 at 20°C. Then an amount of ChTr tiolution was added which gave in 40 ml. buffer the concentration indicated in the diagram. The arrangement of the curves with respect to the t.ime axis is such that time zero for each curve is the moment where it begins to leave the zero line. The preceding time interval between the injection of ChTr and this time zero (fraction of a second to several seconds) increases with decreasing ChTr concentration. The t.ime element zero indicates the moment at which the enzyme molecules actually arrive to a measurable extent at the surface after diffusing through the undisturbed water layer (see below). This initial arrival rate is actually an exponential process [see Eq. (3)], and the very onset of the adsorption curves therefore should be gradual and not instantaneous as in the diagram. But the response time (2 sec. for full scale motion of pen) and sensitivity of the recording system :Lre insufficient to show this very short initial phase except with extremely low ChTr concentrat,ions. With ChTr concentrations between 1 and 10 pg./ml., the maximal adsorption is bet,ween 30 and 35 A., corresponding to one monolayer, and with higher concentrations it may reach values of up to 60 A.6 However, after one layer of ChTr molecules has been adsorbed from a solution of low or moderate concentrat,ion (I-10 pg./ml.), no amount of ChTr added thereafter to the solution will increase the thickness of the adsorbed layer more than a few angstroms. The layers adsorbed on unconditioned or conditioned slides or on standard 6 The theoretical value for the thickness of one monolayer of ChTr is 31.2 A. if the molecules are visualized as cubes of the specific volume 0.73 and the molecular weight is taken as 25,000.
180
HANS
J.
TRURNIT
slides in low buffer concentrations cannot be removed from the solid surface by replacing the ChTr solution with protein-free buffer solution of the same concentration. This adsorption therefore is not an equilibrium phenomenon. On standard slides and with higher buffer concentrations (0.01 M), no adsorption TABLE Adsorption
Velocify
(Unconditioned Buffer
of ChTr
I
as Function
of Buffer
Concentration
slides, Verona1 buffer pH 8.0, temp. 2O”C., ChTr concentration 5 pg./ml.)
concn. M
Adsorption velocity A.jfh.
3 xs10-5
8.8
1 x-10-4
7.6 8.4 7.3 6.7
3 x
10-4
1 x 10-3 x 10-a 1 x 10-2 3 x 10-t 3
8.0 6.7
II
TABLE
Adsorption
Velocity
(Unconditioned
of ChTr
as Function
of pH
slides, Verona1 buffer 0.01 M, temp. 2O”C., ChTr concentration 5 ag./ml.) Adsorption velocity A.JPP&.
PH
7.3 7.3 6.7 7.7 7.3 7.7 7.8
6.9
7.2 7.5 7.7 7.9 8.3 8.7
TABLE
III
of Tenaperafure (Conditioned slides, Verona1 buffer 0.0003 M, pH 8.1) (Adsorption velocity in A./m&.)
Adsorption
Velocity
of ChTr
as Function
Temperature,
in “C. AE
ChTr concentration 9.6
17.6
27.0
rg.Jml. 3
3.4
4.2
-
10
10.3
-
20
28.5
36.5
Cd.JVdL?
18.0 -
4800 5400 5100
ES’ZYME
SYSTEMS.
TABLE: ,4 dsorptr’on
T’elocitt~
of C%T’r
and
y-Globulin
(Unconditioned Concentration (0 pg./wrl. ChTr, veronal 0.01 M, pH
buffer: 8.1
slides;
phosphate 0.01 M, pHj.6
181
IV ns Function
of Their
temp. 20°C.) Adsorption velocity (2)
1.0 2.0 4.0 6.0 23.0
(2)/(0
1.11 2.86 6.02 X.70 11 .!)
10.0
17.5 26.G 32.6 52.1
5.0 10.0 15.0
Concentration
A jmb.
15.0 20.0 30.0
r-Globulin, buffer:
IT
1.00 3.70 5 00
1.41 1.43 1.51 1.45 1.49
1.77 I.63 ---1.75 1 .73 0.38 0.37 0.39 OT
gamba Globulin
FIG. 2. The adsorption cwncent.ration. The data shown in Fig. 1. -,-Globulin
velocity of ChTr and for ChTr are obtained was used in 1)hosphat.c
y-globulin as function of thei] from the experimental curves ljuffer 0.01 ;1/, 1111 5.6 :It 2OY’.
182
HANS
J.
TRURNIT
TABLE V Combined’Adsorption and Reaction Experiments ChTr was first adsorbed from 0.0005 M Verona1 buffer on standard slides. After reaching a certain layer thickness (col. 2) the buffer concentration was increased to 0.01 M. The resulting reaction velocity is given in col. 3. Column 4 shows reaction velocities from corresponding experiments without preadsorption (see also Fig. 3). ChTr concn. ha.lmJ.
0.1 0.2 0.5 1.0 1.0 1.5 2.0 2.0 4.0 4.0 7.0 8.0 12.0 20.0 30.0 60.0
Thickness of adsorbed ChTr layer A.
1 1 2 5 8 13 16 21 25 30 42 41 50 56 48 47
Reaction velocity A.lmin.
2.1 3.3 6.2 16 27 30 29 27 26 27 32 29 31 18 19 20
Reaction velocity without preadsorption A./mis.
4.0 9.2 9.2 14 14 18 18 25
can be observed in the ChTr concentration range from 0 to 15 pg./ml. The experimental curve shows only substrate removal. With higher ChTr concentrations the curves show initial adsorption with following increasingly dominant substrate removal [see Fig. 1 in Ref. (l)]. The slope of the initial straight part of the curve is called adsorption velocity and measured in angstroms per minute. The influence of changing buffer concentration, pH, temperature, and protein concentration is presented in Tables I-IV and in Fig. 2. Part of the results given in Table IV and Fig. 2 will be used as a basis for the calculations in the following chapters. In a separate set of experiments, ChTr was first adsorbed on standard slides at low buffer concentration. Then strong buffer was added, so that the ChTr could react with the underlying BSA. The resulting reaction velocities may be compared with those from experiments of the usual type. (See Table V and Fig. 3.) The results show that with preadsorption the reaction velocities are not a function of the ChTr concentration in solution but rather of the amount of ChTr adsorbed on the substrate. This holds for the adsorption range from 1 to 10 A. Further increase of the thickness of the adsorbed layer does not result in a higher reaction velocity. With more than 50 A. adsorption and high ChTr concentrations, the reaction velocity declines to lower values again. The reaction curves after
ENZYME
SYSTEMS.
183
II
preadsorption are straight lines in the beginning; whereas, the usual reaction curves have a parabolic onset (see Fig. 3). These data show that maximum velocity of substrate removal occurs when approximately one third of the slide surface is covered at any given moment with enzyme molecules in molecular contact with the substrate.
X + IO 0 - IO 20 30
40 + IO 0 - IO 20 30 40
50
i
20
+to
’
mma/ml
1 t
\
0 - IO 20
30 40
50 0.
4
6
12
16
20
24
mln
FIG. 3. Reaction curves obtained in four experiments with preadsorption of ChTr and in three corresponding experiments without preadsorption (usual type). The ordinate-level zero marks the original position of the substrate-liquid interface. In the experiments with preadsorption the cell was filled with 0.0005 M veronal buffer. Enzyme was added at the time marked by arrow No. 1. The enzyme concentration is given at the beginning of each set of curves. The enzyme is adsorbed with a speed depending on it’s concentration (between arrows Nos. 1 and 2). This adsorption is then interrupted (at arrow No. 2) by adding strong buffer solution, giving a buffer concentration of 0.01 hf. Thereby the enzyme is desorbed and at the same time activated and removes substrate from the test-slide surface. For comparison three curves are shown from experiments with the same enzyme concentrations as in the first, third, and fourth experiments, but with 0.01 M buffer already in the cell prior to adding the enzyme. The maximum reaction velocities in angstroms/min. are given in round figures near each curve.
184
HANS J. TRURNIT
The data for adsorption velocity are well reproducible if the rate of stirring is kept constant. In buffer concentrations higher than 0.05 molar, the shape of the adsorption curve changes : the first straight part becomes shorter with increasing buffer concentration, and in 0.1 M buffer the curve is parabolic. This deviation from the normal behavior will be discussed later. DFP-inactivated7 ChTr has the same adsorption velocity as ChTr under otherwise comparable conditions. If the buffer concentration is increased after DFP-inactivated ChTr has been adsorbed on a standard slide as monolayer from distilled water or weak buffer, only twothirds of the adsorbed material is desorbed. Active ChTr under those conditions is completely desorbed. The slope of the adsorption curves as well as the reaction curves decreases rapidly if during the experiment the stirrer is stopped. It comes back to its previous value immediately after stirring is resumed. The main result of these various measurements and observations is that within a certain concentration range the adsorption velocity is directly proportional to the ChTr concentration.* This indicates that diffusion plays a dominant role in this process at the solid-liquid interface. In the following chapters, therefore, an attempt will be made to analyze the adsorption data and part of the reaction data on the basis of the diffusion theory. DIFFUSION
AND ADSORPTION
VELOCITY
IN EXPERIMENTS
WITH STIRRING
Noyes and Whitney (8) and later Bruner and Tolloczka (9) showed that the dissolution rate of a solid in a well-stirred liquid is proportional to the coefficient of diffusion D of the solute, to the geometrical surface of the solid exposed S, and to the difference between the solubility CO of the solid in the pure solvent and the concentration of the solid in the liquid at time t. dn = kDS(co - c) dt They deduced from these experiments that (a) the solid is surrounded by a saturated undisturbed layer of solution through which the solvent molecules have to diffuse towards the solid, (b) stirring does not cause any commotion in this layer, and (c) the thickness d of this layer decreases with increasing rate of stirring but only to a mininum value which is approached asymptotically. Nernst (2) generalized these ideas in his theory of reaction kinetics in heterogeneous systems. He states that whenever the primary physical or chemical reaction at the interface is rapid as compared to the diffusion 7 See also Table II in Ref. (1). 8 The faster increase of adsorption velocity in the concentration 10 pg./ml. (see Table IV) will be discussed at a later date.
range beyond
ENZYME
SYSTEMS.
II
185
through the adherent layer, interfacial reaction rates as a whole governed by diffusion. The quantitative formulation for t,his is: G?-L~Sc: dt
are
(11
d
where dn over-all reaction rate ilt = J> = coefficient of diffusion S = surface area C = concentration d = t,hickness of the undisturbed This becomes identical with
&k’s
layer
first law of diffusion:Y
!?=L)sn’c dt where
dc - = concentration
dX
(2)
dx
gradient,
if we assume (a) a linear concentration gradient across cl, (b) c to be zero at, the wall at all times, and (c) x to be identical with d. With water as a solvent and with many different interfacial reactions, d has always been found (10) to be in the neighborhood of 3 X 1O-3 cm. if vigorous stirring was employed. These ideas will now be applied to the adsorption process in our system (see Fig. 4). The solid surface may be at z = 0; t*he thickness of the thin undisturbed water layer may be d. For z values beyond d, the concentration of the solute (ChTr) is ci at all times. At r = 0, that, is at the surface proper, c shall be zero at all times. Between x = 0 and x = d, c shall be zero at, t = 0 and after a short initial period the cowentration gradient shall be const,ant, or
Under these conditions
it is permissible
to ident~ify dn/dt with
the ad-
o Hereafter the term S will be omitted in the application of Fick’s first lnw PA{. (1 j tJec:tusr all calculations will employ absolute units and the area referred will be always 1 sq. cm. ?z or A are therefore given in g/q. cm.
01 to
186
HANS
J.
TRURNIT
sorption velocity dA/dt and to apply Eq. (1). It will be correct for the time interval during which the adsorption curve (Fig. 1) has a constant slope, that is, as Iong as we can assume the concentration in the immediate vicinity of the solid surface to be zero. However, we have to be certain that the time interval mentioned above between t = 0 and the establishement of a constant concentration gradient between x = 0 and 2 = d is actually short as compared to the straight line interval of the adsorption curve. Otherwise, Eq. (1) is not applicable. This may be decided by inserting experimental data into the solution of the equation for diffusion through a flat as adapted to the boundary conditions of our system: c =
X sin?
~+~~~
G;
Xexp(-@$!)I
[
The question then is how fast the concentration c at a given value of x, e.g., the half-way point 2 = d/2 (Fig. 4) approaches its final value cJ2. For this half-way point the concentration is
e=;+- : g (-y
X sin?
(3)
X exp(-@$$)
By inserting for D the diffusion coefficient of ChTr Dzo = 10 X 10-’ sq. cm./sec. and for d the value 3 X lO+ cm. into the exponential term of Eq. (3),we fmd for t = 5 sec.:
:: *ILL c=-- ei
0.0035s
2
C
Cl -r
.. ,..: : : _--I ; : : i;. ' ,. i .:: .:' : ! ... .: .A.
0 -+* 6 P
FIG.
Cl
6
-x 4. Diffusion process near interface.
(For explanation
see text.)
ENZYME
SYSTEMS.
187
II
This means that after 5 sec. the concentration at the half-way point is already within 0.35 y0 of its final value. This period of 5 sec. is very short as compared to the period of linear adsorption rate observed in experiments with ChTr concentrations below 10 pg./ml. We therefore have:
where dA/dt is measured in g./sq. cm./sec., D in sq. cm./sec., ci in g./‘ml., and d in cm. If this equation describes the adsorption data correctly one should find that dA/dt is directly proportional to the concentration. That this is actually the case is shown in Fig. 2. One should further expect to find d of the same order of magnitude as given in the literature if experimental values for dA/dt, D, and ci are inserted in Eq. (4). For a ChTr concentratjion of 10U6 g./ml., n-e find for 20°C. (see Fig. 2) d,Z/dt = 1.46 A./min. Using 0.73 as partial specific volume of ChTr. dA -zz dt
6. ‘;(‘:
73 lo-’
= 3.31 X 10-l’ g./sq. rm.,!sec.
For D20 we find in the literature polated for infinite dilution). Therefore d = loa
(11) 10.2 X 1W7 sq. cm./sec.
’ lo-’ ’ lo+ 3.31 x 10-10
= 3 08 x 1O-3 cm .
This value is in excellent agreement with the older above. Finally we should expect that dA/dt is proportional of diffusion of the substance used. Experiments with bovine plasma (Armour and Company) (see Fig. 2) y-globulin’0 gave for a concentration of 1O-6 g./ml. : dA - = 0.38 A./min. dt
= 6. “;(“:
(extra-
data mentioned to the coefficient Fraction II from which is mainly
72 X low8 = 8.8 X lo-l1 g./sq. cm./sec.
188
HANS J. TRURNIT
If this value is used together with the value for d as calculated before, we obtain from Eq. (4): D
20
= 3.08 X lo+ x 8.8 x lo-l1 = 2.7 X lo-’ sq. cm./sec. 10-G
This value for DUOis about 30 % smaller than the data found in the literature (around 4 X lo-‘). But considering the impurity of the material and the relatively simple and crude way in which this value has been obtained it sustains our argument. It seems,therefore, permissible to use Nernst’s theory as a physical interpretation of the results of these adsorption experiments. We propose to use this simple type of experiment for the determination of diffusion coefficients of adsorbable substances. By improving the apparatus it should be possible to make d an instrument constant which could be empirically determined by calibration with a substance with a wellknown diffusion coefficient. The advantages of this method would be the possibility of working with extremely dilute solutions and the fact that values can be obtained within 5-10 min. CONNECTION
BETWEEN
ADSORPTION VELOCITY
RATE
AND REACTION
In sufficiently strong buffer (0.01 M) ChTr is not adsorbed to a detectable degree if its bulk concentration does not exceed 10-15 pg./ml. Therefore one cannot speak in a strict sense of an adsorption rate in discussing reaction velocities. Arrival rate would be a better texm. As a matter of convenience, however, we will continue to use the term adsorption. Even if on standard slides under reaction conditions (that is in strong buffer) “visible” adsorption does not occur, it is known from experiments on unconditioned slides (Table I) that the higher buffer concentration does not change the adsorption rate. In order to react with the substrate, the ChTr molecules have to come into molecular contact with it.” But their contact with the substrate surface under reaction conditions is obviously lessintimate or so frequently interrupted that they do not participate sufficiently in the optical interference phenomenon and are in this sensenot part of the solid phase.12Nevertheless, 11Unless we assume the existence of long-range forces. 12 It may well be that whether an enzyme molecule is part of the liquid phase, or part of the solid phase (more loosely or more strongly adsorbed) makes the difference between its active and inactive state in stronger or weaker buffer, respectively.
t’he rate of arrival of ChTr molecules in t’he molecular vicinity of t,hc surface \rill be t,he same function of concentrat~ion 8,s shown in Fig. 2. (111t’his basis we make t’he simple aud obvious assumption that, initia.lly the reac*tioir rate at any tdme is directly proportional t,o the amount d4 of OhTr present, at, t,he slide surface at t,his moment.. Tntegra.tiou of Eq. (4) shows that during the initial adsorption period A = k,c&
(5)
\\-here .I is t)he amount adsorbed (g.jsq. cm.) aft#er time 2 (sec.), Ii,, is I),/d (cm./scc.), and c; the bulk concent’rat’ion (g/w.). The rracbion rate dT,/dt (loss per time elrment) should therefore 1~: ClLlch = k,k,c,t
((9
where k, is a reaction const,ant with t’he dimension of reciprocal seconds. Ry int’egration we obt,ain :
Thus, the shape of the reaction curve in this initial phase should lw parabolic. ,I careful evaluation of reaction curves observed with enzyme concent,rations of 2, 1, and 0.2 pg./ml. shows that this is actually the case. One example is given in Fig. 5. Therefore t*he assumption on which this calculat,ion is based seemsto be justified. The dat’a given in Fig. 5 are expressed iI1 angst,roms and minutes. 111 absolute u&s one obtains from hhe experiment,al data of Fig. 5 and from Eqs. (S), (6), and (7): I;,
=
3.32
X
IO-”
cm./sw.
and lc,
=
2.56
x
10-Z
SW--l
if the concentration is given in g./cc., and if the partial specific volume of BSA is taken as 0.73. It follows that 1 g. of ChTr in contact with the solid surface removes 2.56 X 1O-2g. BSA/sec. With the molecular weights of ChTr and BSA taken as 25,000 and 68,000, respectively, this means that on the average one molecule of enzyme removes from the surface 9.4 X IOP substratcl molecules/set.
190
HANS
At 6
I
J.
TRURNIT
I
I
I
, I
I
/ \
4 +2 0 -2 4 6 6 IO 12 14 I6 16
FIG. 5. Reaction as related to adsorption. The upper curve is from an adsorption experiment with 1 pg. ChTr/ml. in 0.0005 M Verona1 buffer, pH 8.1, on an unconditioned slide. A reaction experiment on a standard slide with the same ChTr concentration but in 0.01 M buffer gave a curve indicated by the series of circles below the zero line. From this curve the value of k, in Eq. (7) was calculated and then the curve drawn which is represented by Eq. (7). This theoretical curve follows very closely the experimental curve.
This is as far as the experimental evidence permits an interpretation. Any further speculation about the number of bonds split per enzyme molecule per second would be unfounded because we do not know how far the splitting process has to disintegrate the denatured BSA in order to make the split products leave the surface. Also, we cannot estimate the proportion of activated enzyme molecules even if we know that the Arrhenius coefficient for the over-all reaction is in the neighborhood of 8000 cal./mole [see (l)] and that about 5000 cal./mole hereof (see Table III) are accounted for by the energy of diffusion. The reason is that the splitting of different types of bonds requires different activation energies. As an average value we would obtain exp (-3000/586) = 0.006 = 0.6% for the fraction of activated molecules. The parabolic course of the reaction curves continues until all but about two layers of substrate are removed. This is independent of the total numbers of substrate layers originally present on the slide. At this point the curve has an inflection point and thereafter gradually approaches a horizontal course. Either the substrate layers at the bottom of the pile are not as easily digested as the higher layers (influence of the uranyl ions?) or the split products are more tightly bound (for the same reason?) or the attack of the enzyme from the beginning must be visualized more as an irregular “corrosion,” which acts faster in some parts of the surface than in others. The onset of the deviation from the parabolic course
ENZYME would then the bottom
indicate that of the pile.
at this
SYSTEMS.
moment
EXPERIMENTS
the
enzyme
WITHOUT
191
11 has reached
in some
spots
STIRRING
I. Adsorption
A series of experiments was made without stirring. They were carried out as follows: The slide was inserted into buffer solution for calibration and t.hen withdrawn; enzyme was added and the solution thoroughly stirred for half a minute. The solution was then allowed to stand quiet for a few minutes. The test slide was then brought back into the solution in its original position. The adsorption curves obtained under these conditions are very closely horizontal half paraboIas as shown in Fig. 6. Sometimes they deviate from the parabolic shape during the first few minutes due to a disturbance caused by inserting the slide. But generally during the increase in thickness of the adsorbed layer which corresponds A 30
0
IO
20
30
40
SOmin
FIG. 6. Adsorption curves from experiments without stirring. As in Fig. 5, the course of the experimental curves is indicated by circles. The drawn-out curves represent Eq. (9). Curve a: Adsorption of ChTr (5 pg./ml.) on unconditioned slide in 0.01 M Verona1 buffer, pH 8.0, at 27°C. Curve b: Adsorption of -t-globulin (10 pg./ml.) on unconditioned slides in 0.01 M phosphate buffer, pH 5.6, at 27°C. At 30 min. the theoretical value for A with ChTr is 19.5 A. (see text). The csperimental value is 23 A. With r-globulin the theoretical value at 30 min. (using l&r = 4.81 X 10-r sq. cm./sec.) is 24.4 A.; the experiment,al value is 21 A. .4ssuming that the experimental value were higher than the theoretical one by the same percentage as in the case of ChTr, we would obtain 29 A. for the experimental value. The actual value of 21 A. shows the same discrepancy of about 30% as in the experiments with stirring (see above).
192
HANS
J.
TRURNIT
to the straight part of the adsorption curve obtained in experiments with stirring, they are perfectly parabolic. Thereafter they leave the parabolic course and fade slowly out into a horizontal line. This corresponds to the upper end of the curves from experiments with stirring (Fig. 1). In order to explain the parabolic shape as such and to compare the quantitative data from these experiments with those to be expected from the theory, we proceed as follows: Our system, without stirring, should obey the law of diffusion in a semi-infinite cylinder with the following boundary conditions : c=Oatz=O c=ciatz>O c = f(x) at 2 > 0
for all t fort
= 0
for t > 0
where c is the concentration at any x or t, ci is the initial concentration, x is the distance from the boundary, and t is the time. For this case we find : E = --& ~‘e+
cl.2= erf(x);
[z = *]
(8)
The curve representing erf(z) is plotted in Fig. 7 for 2dfi = 1, or z = 5. The area above the erf(z) curve (marked F in Fig. 7) represents the amount A’ of material adsorbed on the surface at a time which satisfies the condition 2di% = 1. This area F is the product of a distance 2 (which represents a volume over 1 sq. cm. of surface) and a concentra-
FIG. 7. The curve representing erf(z). See text.
ENZYME
SYSTEMS.
tion c. Mechanical integration of F with a planimeter 0.57. This value is very close to l/d;. Thus for .z = x, and for c = c;, we have:
and for z = x/26
193
II
yields the value
we obtain:
A=xXci=ciLX22/iSi. G
(‘3)
01'
A = 2ci This is the equation of a parabola. It perimental curve correctly and permits sorbed after time t. For ci = 5 X 1P6 sq. cm./sec. and for 1800 sec. (see Fig. 6)
A = 1O-5
.m
describes the shape of the CSpredicting the amount A adg/ml. and DS = 12.3 X 1P we obtain:
= 26.5 X 10d8 g./sq. cm.
This corresponds to 26.5 X 0.736 X 1C8 = 19.5 A. adsorption. The experimental value (Fig. 6) is 23 A. The difference is probably due to the presence of some convection in the solution.13 Equation (9) may also be obtained in a more conventional way: By differentiation we obtain from Eq. (8) : 13 As mentioned above, these experiments were carried out in a very simple way. If adequate precautions were taken to prevent convection currents the esperimental figures would be 611 closer to the theoretical figures. Another precaution which must be taken in this type of experiment is to prevent the pickup of denatured surface protein while the slide is inserted into the solution. This can be accomplished by filling i.he cell almost to the rim and covering all but a narrow slit of the liquid surface with glass slides. The test slide is then inserted through this slit. However, this has to be done slowly in order to prevent commotion in the liquid as much as possible. Therefore it is not possible to prevent pickup of surfact-denatured protein completely. The zero level after insertion (that is f he axis of the adsorption parabola) will therefore be slightly higher on the recording paper than during calibration. Its position can later be estahlishcd by finding a value for CLwhich yields a constant value for h- in the parabolic equation: r = k(y - 0)2, if scvernl coordinated pairs of s :~nd !/ of the ntisorl)tion p:~rnlml:t are tested.
194
HANS
J.
TRURNIT
This is the concentration gradient at any given time and distance from the surface. In order to determine the rate of arrival of diffusing molecules at the surface we are only interested in the value of dc/ds in the immediate vicinity of the surface, that is, for z -+ 0. We may therefore neglect the exponential term, which becomes unity for 2 = 0, and insert the first term as concentration gradient in Fick’s first law [Eq. (a)]. We obtain:
or by transformation
: dA dt = ci
(10)
which is identical with Eq. (9). This formal derivation of Eq. (9) in contrast to the previous one does not reveal the physical meaning of the term l/l/n. cln the other hand it introduces l/g ?r in a mathematically correct sense. In the first derivation the value of the area F (Fig. 7) was only assumed to be l/G. Therefore both these derivations combined, that is the identity of Eqs. (9) and (ll), constitute a mathematical proof that the area above the erf(z) curve has the simple value l/l/?r.14 2. Reaction The theoretical shape of the initial part of reaction curves from experiments without stirring can be derived in t#he same way as before for experiments with stirring. Again we assume that the reaction rate dL/dt at any time is proportional to the amount of enzyme present at the interface at that moment. This amount is given by Eq. (11). 14 The author has not been able to find a statement to this effect in the literature. An attempt to calculate the value of F by developing the integral in Eq. (8) into a series and integrate term by term failed because the convergence of this series is too slow.
ENZYME
SYSTEMS.
195
II
Thus we have
!z! at=k’r ’ c pfi x 2c.
By integration
we obtain the equation
L=iXk:XCi
of the reaction
d- $ x t3’2
(12) curve
(13)
In Fig. 8 this curve (C) is drawn for ci = 5 X IO+ g/ml. and A’, = 2.56 X 1CP sec.-l, the same value as obtained from experiments with stirring. The actual experiment performed with this concentSration gave a curve which coincides accurately with curve B. B is drawn from Eq. (13) using the same value of ci and a value for k’, which is 3.1 times smallcl than k, . We find, therefore, that t.he shape of the actual reaction curves in experiments without stirring can be predicted from the theory, but that the reaction const.ant is about 4 Gmes smaller than the one on csperiA
20
+10 0 - IO
20
30 i 40 an -FIG. 8. Adsorption and reaction curves from experiments is the same curve as curve a in Fig. 6. The circles along curve action curve in 0.01 M buffer with 5 pg. ChTr/ml. For further explanation see text.
without stirring. ;I B represent the re-
196
HANS
J.
TRURNIT
ments with stirring. A difference was to be expected because the removal of split products in this case is obviously slower. DISCUSSION
It was clear at the beginning of this work that the various theories developed for the kinetics of enzyme reactions in solution could not be applied to a system where one of the two reaction partners, in this case TABLE
Governing the Initial
Equations
VI
Phase and and Reaction
Initial
the Steady
State
phase
Adsorption
With stirring (Figs. 1,3,
5)
dA vn = -dt
=
Reaction
D 2 x
c
A=;xcxt
Without (Figs.
stirring 6, 8)
of Adsorption
dL ?I+ = x
= k, x
L=k,X$XcXtZ
--= dA ‘a - dt
A = 2
Steady
state
Adsorption
Preadsorption
k, , k: , k! A A’ L c t D d
Reaction
dA --so v0 - dt
= = = = = = = =
A = constant
= A’
A = constant
= f(c)
L=k;XA’xt < A’
Reaction constants. Amount adsorbed. Maximum effective amount in contact Amount removed. Initial enzyme concentration. Time. Coefficient of diffusion of enzyme. Thickness of adherent water layer.
L = kf x j(c)t
with
interface.
D ;i-xcxt
ENZYME
SYSTEMS.
II
197
the substrate, is not part of the liquid phase. In fact, the work was undertaken in order to obtain somebasic experimental data which would permit the development of the kinetics for this type of interfacial reaction. By introducing diffusion as the rate-limiting factor, it has been possible to derive a set of equations which describe in a satisfactory manner the initial phase of adsorption and reaction in experiments with and without stirring. The possibility of investigating the reaction in its very beginning, before secondary effects change the reaction rat,e, permits one t,o express at least part, of the over-all react,ion constant in physicbal terms. This part is the adsorption constant k a , which is D/d with stirring and t/Bz without stirring. Table VI shows the set of equations systematically arranged. These equations hold for adsorption as long as the adsorption curve is a straight line or, in other words, as long as cz=o = 0, and for reaction they are valid until an optimal amount of enzyme (A’ x 10 A.) is present at the interface. From this moment on, the reaction rate remains constant. But in experiments with stirring, the point of inflection on the reaction curves comes always before this critical amount A’ has accllmulated, that is, if slides with three double layers of substrate are used. In experiments without stirring or with preadsorption of enzyme and \\-ith stirring, the steady state may actually be observed for a short while, before secondary processes change the pattern. Cnder steady-state conditions, the reaction velocity is independent of time and concentration. But if the critical amount A’ cannot be reached because the total amount of enzyme in the solution is too small, the reaction velocit’y even under steady-state conditions depends on the concentration. This is where ow system begins to resemble the usual reaction conditions in solut,ion. One of the basic laws in bulk enzyme kinetics is the proportionality between reaction rate and enzyme concentration during the steady state. The same proportionality occurs in the reaction equations for the initial phase (Table VI). This is another way of stating that during this initial interval, A has not yet reached its optimum value A’. In order to determine the basic physical components ivhich are &II hidden in the reaction co&ant X;, , similar experiment,s u-ill have to 1~ made with a simple substrate. If only one well-defined split, prodllct leaves the surface, it may be possible to apply the diffusion theory to t,his reverse process and to calculate a reaction constant which is idrnticsal with the turnover number, that is, if we define turnover number hy thc~
198
HANS
J.
TRURNIT
number of bond-splitting acts which an enzyme molecule performs per second. In our case, however, we may tentatively define the turnover number as the number of substrate molecules removed each second from the surface by one enzyme molecule. If we apply this to the results of experiments with optimal preadsorption (Table V), we may say that before activating the enzyme with strong buffer, about one-third of the surface is covered with stable enzyme-substrate complexes. Their number is roughly 3 X 1Ol2 complexes/sq. cm. The corresponding reaction velocity is about 30 A./min. = 0.5 A./set. = 6.9 X leg g./sq. cm./sec. This rate is equal to about 6 X lOLo substrate molecules/sq. cm./sec. Each enzyme molecule therefore removes about 2 X 1e2 substrate molecules/set. This turnover number has twice the value as the one derived in a previous chapter [following Eq. (?)I from experiments without preadsorption. The fact that we find two different turnover numbers, depending on how the complex formation is accomplished, seems to offer the possibility of investigating the splitting mechanism more closely. The parabolic course of the reaction curves in experiments with stirring explains in a simple way the increase in reaction velocity with increasing number of substrate layers, which has been described previously [see Fig. 11 in Ref. (l)]. It could not be explained at the time, Now it becomes clear that with increasing substrate pile thickness the inflection point moves downward along the parabola; that is, the tangent to the curve at this point becomes steeper and steeper. The slope of this tangent was called reaction velocity in the preceding paper. The connection between the basic concept of Michaelis and Menten and the theory developed in this paper will be discussed in a later publication. At the same time more will have to be said about the physical meaning of the undisturbed water layer. It should be emphasized that the recording ellipsometer technique, because of the role diffusion plays in it, permits one to study the initial phases of enzyme reactions in detail over periods of minutes. TJnder the conditions of bulk reactions, extremely rapid methods have to be employed to achieve the same purpose. Aside from these theoretical considerations there is a practical aspect. The recording ellipsometer seems to provide the possibility of measuring diffusion coefficients rapidly and in extremely dilute solutions. Table VI shows that with stirring D is directly proportional to dA/dt which may be read from the curves. The instrumental problem is to make d suffi-
ENZYME
SYSTEMS. II
199
cient,ly constant. Without stirring D is proportional to t’he squaw of dA/dt or A, This means it is much more sensitive against errors made in determining &l/c& or A. The instrumental problem in this case is to avoid disturbances in the solution. The first method permifs readings to br made sooner than the second one. SUMMARY
The kinetics of adsorption of chymotrypsin (ChTr) from solution onto I solid surface has been studied experimentally and theoret’ically. 01~this l)usis the initial phase of the reaction of ChTr with surface-denatured bovine serum albumin at’ the liquid-solid interfaw can be dewribcd qualitatively and quant it,atively. Experiment, and theory show furthermore that the recording ellipsomctcr may he used to measure coefficients of diffusion rapidly and in VSt,remely dilute solutions. It is cmphasieed that interfacial enzyme reaction st,udies of this t,ype permit one to investigat’e t’he initial reaction phase more easily and in tiner detail t’han studies of bulk reactions, where very rapid met’hodshave to be employed. REFERENCES II. J., Arch. Hiochem. and Biophys. 47, 261 (I!K<). W., 2. ph!/sik. (‘hem. 47, 52 (1904). T,nyers,” American Associa3. TRVRNIT, H.J., “Symposium on Monomolecular tion for the Advanremcnt of Science (A.A.A.S.) Jlrcting. Philadelphia, Pa., Dec., 1951. 1954 A..i.A.S. Monograph. 4. ROTHES, A.. Rev. Sci. Znslr. 16, 26 (1945). 5. Itmms, A., AND HANSOS. ,lf., Rev. Sci. Instr. 19, 889 (1948). 6. ROTJI~, A., AND I~AZISON~ l’r., Rev. sci. znstr. 20, 66 (1949). I.
TRZ-RSIT,
2.
SERSST,
7.
Dams,
N.
SOYES,
A. A.,
I'.,
Ann.
9.
BRUKER,
I,.,
AND
AND
Physik W'HITX.EY.
Tor.1,orzs.4.
11. C’hem.
36, 865
W. R., Z. physik. ST., Z. physik.
of R,eact.ions in Solution,“2nd Ed. Oxford, 1947. 11. SEURATH, H., AND BAILEY, K., “The Proteins,” Vol. 1, part A. Bcademir Press, New York, 1953. For the basic mathematical equations and for the values of crf(z) the following lmks have been tied: .JOST, W., “Diffusion in Solids, Liquids, Gases.” Xratfemic Press, New York. 1952. J~ARRER, R. M., “Diffusion In and Through Solids.” University Press, Cambridge, 1941. (-~.~RsLAw, H. S., AND JdEGER, J. C., “Conduction of freat in Solids.” Clarrndon Press, Oxford, 1947. 10. MOELWYN-HUGHES,
5:. A., “The Kinetics
(1889). C’hcm. 23, 689 (1897). Chem. 36, 288 (1900).