Random Coil Scission Rates Determined by Time-Dependent Total Intensity Light Scattering: Hyaluronate Depolymerization by Hyaluronidase W A Y N E F. REED,' CHRISTOPHER E. REED,' and LARRY D. BYERS'

Departments of 'Physics and 'Chemistry, Tulane University, New Orleans, Louisiana 701 18

SYNOPSIS

A recently developed theory of the light scattering by random coils undergoing random scission is applied to the digestion of hyaluronate by hyaluronidase. The time dependence of the scattered light from solutions undergoing digestion was monitored. Working a t a high angle with high molecular weight hyaluronate allowed the use of a powerful approximation for determining initial velocities and the Henri-Michaelis-Menten coefficients, without explicit knowledge of the hyaluronate molecular weight, radius of gyration, second virial coefficient, or polydispersity. Effects due to a molecular weight dependent second virial coefficient and to non-Gaussian behavior are briefly considered. Assays were performed over nearly two orders of magnitude in substrate concentration. The initial velocities are compared with those obtained by a standard reducing sugar assay, which was performed on identical samples. The main advantages of the light scattering assay procedure over the more traditional assays are that many relatively high-precision data points can be quickly and automatically collected with simple apparatus, and that the technique is most sensitive for the initial period of digestion, where the other assays are least sensitive. The shapes of the scattering curves also provide evidence that hyaluronate in these solutions is not a stable double strand and that the hyaluronidase cleaves bonds randomly. The curves also indicate that enzyme deactivation occurs, which accounts for the lower velocities yielded by the slower reductimetric assay, which is measured over longer initial periods.

INTRODUCTION A recent paper' showed that the random scission of an initially monodisperse population of random coils has the remarkable property of preserving the functional form of the scattering form factor P ( u ) ,which has the well-known expression

P ( u )=

2 U

(e-"

+u

-

1)

(

) = ( 4 T n / ~sin ) (01 where originally = ( q ~2. H~~~ 2 1, s is the rms radius of gyration of the coils of the initial population, X the wavelength of the incident light, the index of refraction of the solvent, and 0

0 1990 ,John Wiley & Sons, Inc. CCC O006-HS2S/90/11- 121073-10 $04.00 Biopolymers, Vol. 30, 1073-1082 (1990)

the scattering observation angle. If a n average number of random cuts r is made per polymer molecule in a n initially monodisperse population of random coils, then the scattering form factor is still given by eq. ( 11, where the argument u is now given by u = [r uo] where u g = ( qSo)*, and So is the rms radius of gyration of the coils before digestion begins. This means that many classical light scattering results and techniques for investigating random coils can be adapted to study their scission. T h e current work uses the above result together with several approximations to determine initial velocities for the depolymerization of hyaluronate ( H A ) by hyaluronidase. Significantly, within these approximations, no accurate information concerning the polydispersity, molecular weight averages, radii of gyration, or second virial coefficients is needed. It is only necessary to know whether these parameters fall within the appropriate regimes for the approximations to be valid.

+

1073

1074

REED, REED, AND BYERS

HA is a linear polysaccharide of N-acetyl-D-glucosamine ( NAG ) and D-glucuronic acid ( GlcUA ) units with the repeating structure 0-@-NAG(1,4) -0-P-GlucUA- (1,3) . I t has the largest molecular weight and is the lowest charge density member of the general class of glycosaminoglycans. Its biological functions include the maintenance of such mechanical properties a s compressive stiffness and swelling pressure in connective tissues, controlling tissue hydration, acting as a backbone for proteoglycan aggregation, providing lubricating properties in synovial fluid and heart valves, and binding t o cell surfaces in cell-cell adhesion. Hyaluronic acid is cleaved by hyaluronidase (hyaluronate 4-glycanhydrolase, EC 3.2.1.35), a lysosoma1 endo-glycohydrolase. The p( 1 + 4 ) bond of HA is cleaved by the enzyme, leaving a reducing NAG end group. T h e activity of mammalian hyaluronidase has generally been determined by measuring one of the following changes in HA: ( l ) the decrease in viscosity, 2,3 ( 2 ) the loss of ability to form i n ~ o l u b l eor ~ -colored6 ~ salts, and ( 3 ) the increase in reducing sugar Various electrophoretic9 and chromatographic lo techniques have also been described. These methods are generally tedious and time-consuming. The light scattering assay automatically collects many relatively accurate points during the initial phases of the digestion, allowing a verification of the mechanism (random scission), and determination of initial rate constants.

---

---

MATERIALS A N D METHODS Bacterial hyaluronate ( BHA ) from Streptococcus zooepidemicus was obtained from Sigma (sodium salt form) and used without further purification. T h e protein content was determined t o be less than 0.1 % by weight." Colorimetric assay l 2 using dimethylmethylene blue gave a n upper limit on sulfated glycosaminoglycan impurities of 0.03% by weight. All the quantitative data presented in this work was obtained using the BHA. T h e BHA was stored a t about -15°C and light scattering properties for fresh solutions from the same bottle did not vary over the several months of frozen storage. By contrast, cock's comb hyaluronate (CCHA) from Sigma, purified by DEAE sephacel, gave a protein content around 1%by weight and a sulfated glycosaminoglycan content less than 0.05% by weight. Using the absorbance a t 260 nm, residual nucleic acid content was estimated to be around 0.1% by weight. Depolymerization of this hyaluro-

nate preparation never went t o completion and only qualitative results will be mentioned, in order t o illustrate potential pitfalls that can arise from samples with measurable impurities. All HA solutions were slowly stirred a t 4°C for a t least two days before use. Samples were prepared for the light scattering assay by filtering the HA and hyaluronidase solutions through 0.2-pm nylon filters and then centrifuging them a t 10,OOOX g for one hour a t room temperature. Bovine testicular hyaluronidase ( HAdase ) was obtained from Sigma (type VI-S, with a specific activity 3600-8400 N F units/mg, depending on the l o t ) . The stock enzyme solution was prepared by dissolving approximately 1mg of lyophilized powder per mL of buffer (0.1M phosphate, 0.15 M NaC1, p H 5.3) and filtering through a 0.45-p nylon membrane. Solutions were stored a t 4°C and generally used within one day. Less than 10% activity loss was found after two days. Protein concentration was estimated based on E&$ = 8.13The reductimetric assay of hyaluronidase activity (i.e., the N-acetylhexosamine assay of Reissig et al.' was carried out as described by Keleti and LedererI4). N-acetylglucosamine (Sigma) was used to construct a calibration curve (0.05-1.5 m M ) yielding a n extinction coefficient a t 595 nm of 1.42 ( t . 0 2 ) X 103Mp1cm-'. A standard (0.33 m M NAG) was included with each time point in the kinetic assay. Assays were carried out a t 25°C and generally followed for about one hour. Typical enzyme concentrations were 30-150 units/mL. The p H after complete hydrolysis was identical to the initial p H ( E 5 . 3 ) . Absorbances were measured on a Hewlett Packard model 8452A diode array spectrophotometer. T h e p H measurements were carried out on a Metrohm (Brinkman) combititrator. T h e steady state kinetic parameters, V and K , , were determined by a nonlinear regression fit to the Henri-MichaelisMenten e q ~ a t i o n . ' ~ Determinations of hyaluronate polydispersity were made on a high performance liquid chromatography ( H P L C ) system consisting of a series of Beckman columns ( T S K 4000 SW, TSK 2000 SW, and TSK P W 6 0 ) , with a Waters 410 differential refractometer detector. The pump was a Shimadzu LC-Ga, operated a t a flow rate of 0.7 mL/s.

The light Scattering Assay Method

T h e light scattering apparatus used consisted of a Coherent Innova 90-5 argon ion laser operating at 488 nm, a Thorn EM1 head-on phototube for single

photon counting, and a Brookhaven BI-2030 autocorrelator. Details of the system, its use for static and dynamic light scattering, and its calibration are given elsewhere.16 All measurements were made a t 25°C. The value of 3.96 X cm-' was used for the absolute Rayleigh ratio for toluene a t 25°C a t A = 488 nm1.l7 Differential index of refraction measurements ( d n / d c ) needed for the determination of molecular weight and second virial coefficient were made on a Bryce-Phoenix differential refractometer. It should be pointed out that in this technique a n autocorrelator is not needed, as any method of monitoring the scattered intensity as a function of time is adequate (e.g., a strip chart recorder or A / D converter board in a microcomputer). Once the major experimental problems and sources of error were understood, the light scattering assay could be carried out very quickly and efficiently. The two major problems were dust in the scattering solutions and ambiguities in the final scattering baseline level. The problem of dust affects all light scattering experiments. T h e usual precautions must be taken against it: All buffers and solutions must be made with filtered, deionized, distilled water. A 0.22 pm Millistak filter a t the outlet of a Corning Megapure I distillation unit provided adequately dust-free water. Glassware was soaked in a chromic acid bath prior to thorough rinsing and drying. Although these procedures, together with centrifugation of the sample prior to the light scattering assay, gave a n acceptably low level of dust, further improvements in data quality could be gained by performing the experiments a t relatively high angles (90"-110" ). Besides reducing the number of bad data points due to dust, high-angle measurements provided a starting uovalue high enough t o allow use of a powerful approximate expression for the light scattering intensity, which gives initial velocities without explicit knowledge of A z , molecular weight, Soor polydispersity. This important limiting case is analogous to the result of Benoit for time-independent light scattering from polydisperse random coils in the limit of large u . The theory is presented in detail elsewhere.19 T h e approximations and results are summarized here. Our simple analysis procedure closely follows the arguments used by ZimmZot o prove that for a monodisperse polymer, the baseline subtracted absolute Rayleigh scattering ratio IR can be written as

RANDOM COIL SCISSION RArES

1075

+ 2A2cQ(H)

(2)

Kc

-_ IR

P-'(u)/M

where M is the molecular weight and c the concentration by weight of the polymer. K is a cluster of constants given by (3)

Here N A is Avogadro's number and d n / d c is the differential index of refraction of the buffer /polymer system. In eq. ( 2 ) ,Q ( 8 ) is a complicated function of scattering angle t h a t depends on intermolecular interactions." By assuming that two interacting random coil molecules make contact a t only one place and by ignoring interactions within molecules, Zimm 2o found that Q ( 0 ) = 1, that P ( 8 ) is still given by eq. (1)for interacting molecules, and that A2 does not depend on molecular weight (in general A2 should depend on molecular weight, except a t the 0 point). Under the same assumptions as Zimm," we found l9 that eq. ( 2 ) still holds for random scission of initially monodisperse random coils where now M is the original molecular weight and uo = u r . In this model A 2 is also independent of molecular weight. For samples that are initially polydisperse, the Zimm-type analysis yields the basic theoretical result 19:

+

Kc

-

IR(t)

1

+ 2A2c (4)

m W o ( m ) P ( u ( mt ,) ) d m

Wo(m ) is the original normalized weight distribution of the polymers and u = uo( m ) r ( m , t ) is now a function of polymer mass. Random scission of bonds means t h a t all bonds have a n equal probability of being cut when one cut is made in the entire population. Thus the expected number of cuts r ( m ,t ) per original molecule is proportional to that molecule's original weight,

+

r ( m ) = 6mt

(5)

6

Here is the number of cuts per unit time per unit mass of polymer (i.e., a rate constant). Thus, experimental determination of allows the initial velocity in molar bonds broken per second to be expressed simply as

6

u = 6c

(6)

where c is here expressed in milligrams per milliliter.

1076

REED, REED, AND BYERS

If epuo-@ 1for the whole original distribution, then

Using

Here F ( t ) = l / ( I mt () - B ) , where I,( t) is the relative scattering intensity from the polymer solution undergoing depolymerization and B is the baseline level of relative scattering measured from the solution after the reaction is finished. [ HA] is in milligrams per milliliter. The average number of bonds broken per initial polymer molecule is

in eq. ( 7 ) yields, upon substitution into eq. ( 4 ) and retaining terms to first order in u ,

S i / M for ideal random coils, that is, y where LT is the persistence length and p the linear mass density of the coil. Hence y does not depend on polymer mass and is time independent. Equation ( 9 ) shows that if the depolymerization mechanism resembles random scission of random coils, and if 6 is constant, then KC/IR( t) increases linearly with time. Assuming that Az is independent of mass (i.e., does not change, a t least during the initial phase of depolymerization) , then Here, y

=

= LT/3p,

dt

Thus, assuming large uo for the original polymer population and mass-independent A z ,the initial velocity given by eq. ( 6 ) can be found by simply measuring the decrease in scattered intensity as a function of time. Absolute Rayleigh ratios are obtained from the ratio of the scattering by the polymer solution to that of a reference whose absolute Rayleigh ratio is known (e.g., toluene). If only relative changes in light scattering levels have been measured, the initial depolymerization velocities can also be obtained, but with a considerably larger requirement of auxiliary data. If S o , A 2 , and the initial weight average molecular weight are known, for example, by means of a Zimm plot, and the ratios M w/M , , and M Z / M , ,are known for the original population (where Mn and M z are the number average and z-average molecular weight), for example, by a n HPLC determination, then the initial velocity of bond breaking in pM bonds per hour is

Regardless of whether absolute or relative intensities are measured, the baseline scattering intensity B must be determined with great care, for if B is the true baseline and B' the measured baseline, then the ratio of the calculated initial velocity of bond breaking t o the true velocity is

1, is the scattering intensity (either relative or absolute) of the polymers before adding the enzyme. Sources of deviation between B and B' may include differences in stray light level if the scattering cell is repositioned before the baseline is measured, incomplete digestion a t the time of baseline measurement, additional reactions that change the polymer sizes other than by depolymerization (e.g., aggregate formation), inattention to dust, and drift in the light source output power. The scattered light from the solution undergoing digestion was monitored for total time periods of 15 min to 1.5 h after addition of hyaluronidase, with 300-1000 data points normally being collected. The sampling time per period ranged from 1to 10 s, with a n additional dead time of around 0.77 s present for all sampling times due to the microprocessor performing screen, disk drive, and correlator hardware functions. For BHA the final baseline value was consistently within a few percent of the scattering level of the pure buffer, as it should be if the originally very large molecules ( M W E 1.3 X lo6) are depolymerized into very small oligosaccharide fragments" by the end of the digestion. The concentration of HAdase was too low ( E 4 p g / m L ) to add significantly to the buffer scattering level. For cock's comb hyaluronate, B was always well above the buffer scattering level. The cause of this is probably a certain concentration of indigestible scatterers in the solution, as confirmed by dynamic light scattering, which showed a particle size on the order of that of the original CCHA.

RANDOM COIL SCISSION RATES

T h e limitations of the light scattering assay technique arise both from the assumptions made in arriving at eq. ( 4 ) and from the details of the experiments themselves. It is assumed that the polymers behave very nearly as ideal (Gaussian) random coils, where S 2increases as M . Deviations from this behavior are not included in the current theory. Excluded volume theory predictsz3that if A2 S 0 then, for sufficiently long polymers, S 2increases as M " , where 11 E 1.2 in the limit of high excluded volume (see Discussion). BenoitZ4suggested the following form for P ( u ) for the case where the mean square separation between two Kuhn segments p segments apart is h 2 = L2p": ( N - p ) e x p ( - q 2 L 2 p " / 6 ) dp

(14)

Here N is the number of Kuhn statistical segment lengths in the polymer, 1 is the segment length, and u = q2l2N"/6. In the limit of high u , assuming the limiting formz5of eq. ( 1 4 ) ,

P ( U ) E ~ ~ r ( ~ )-u2 -t r' ( 2 t ) ~ - * e

(15)

Calculating P ( u ( m ,t ) )by eq. (14) of Ref. 1, with eq. (15) instead of eq. (1)for P ( u ) ,eq. (4) gives

+-26 ttrr( z2 t()t +) 2 A 2 ~

(16)

where t = 1/11, r is the gamma function and p is the linear mass density of the polymer, so that the ratio of the rate, taking excluded volume into account, to the rate ignoring the effect is

T h e effect of ignoring excluded volume is thus to underestimate the velocity. For example, values of t of 0.9 and 0.83 ( p = 1.11 and 1.20, respectively) lead to underestimates of about 10 and 21%, respectively, the latter representing the upper limit of the underestimate. When there are many cuts per average original molecule, the bulk of the population is depolymerized into isotropic scatterers, and it can be seen from eqs. ( 1 4 ) and ( 1 7 ) in Ref. 1. that ZR =

KcMo(2/r2)(e-'

+ r - 1)

(18)

1077

The reciprocal 1 / I R becomes proportional to r in the limit of large r . Thus, when there are many cuts per average molecule, 1 / I R becomes linear for any linear molecule (random coil with or without excluded volume, rod, globule, worm-like chain, etc.) . This suggests that adapting the technique to lowangle laser light scattering might be useful to avoid model-dependent assumptions about the polymer's shape and excluded volume effects. It is also assumed that A2 is nearly constant and that Q ( 0 ) = 1. As discussed below, these seem to be good approximations for the hyaluronate used in this study. The experimental limitations are basically set by the absolute intensity of the light scattered by the solutions undergoing digestion and by the time scales involved. First, the theory assumes that the photodetector monitors light that has only been scattered once. Hence the scattering solution must not be turbid. On the other hand, if the scattering signal is too weak it will be difficult to achieve a good detector count rate, and the data will show both significant fluctuations in count rate and a higher degree of scatter due to dust.

RESULTS Zimm plot analysis ( a n example is given in Figure 1), using the measured value of d n / d c of 0.176 mL/ g for BHA, over the concentration range of 0.050.8 mg/mL and angular range of 28"-52" yielded a weight average molecular weight A?,, of 1.28 X lo6, a square root of the z average of S 2 of 1278 A, and a second virial coefficient A 2 of 2.96 X cm3mole/gz that is fairly insensitive to angle. Thus the

0.0

0.5

1 .o

I .5

SlN2(O/2)

+ 2.500~

2.0

2.5

Figure 1. Zimm plot for BHA. Angular range is 0 = 24"52". Concentrations shown are from 0.1 to 0.7 mg/mL.

1078

REED, REED, AND BYERS

approximation of Q ( 0 ) z 1 is good and u g > 3 for any HA with S > 716 A a t B = 90". In light of these data, the approximations used in arriving a t eq. ( 10) appear justified for BHA under the current solvent conditions. Again, none of these quantities are used in the calculation of initial velocities by eqs. ( 1 0 ) and ( 6 ) . Figure 2a shows the rate of change in the scattering intensity of a typical BHA sample ( a t 0.4 mg/ m L ) being digested by hyaluronidase a t a concentration of 25.88 units/mL. Figure 2b shows the data from Figure 2a plotted as Kc/IR, where IR is the baseline subtracted Rayleigh scattering ratio. The initial rise (first few minutes) is fitted with a straight line. T h e initial linearity is a t least consistent with the random scission of random coils (see Discussion). Using eq. ( 1 0 ) yields = 1.168 X cuts/ (dalton-s), which gives, by eq. ( 6 ) , 168.2 pM bonds broken/h. For a n HA molecule of a n initial mass of

6

7 ,

I

6

5

x 4 .0 4 C

-23

.-S

2

I 0

2

4

6

8

10

12

14

time (rnin)

.

I

.m

i4 oml

om

'1',

.

o m

.

.

o a

. . o w. . im. . ~m. .L W.

ow

C H I I 1 .

2

4

6

8

10

12

I 14

time (rnin) Figure 2. ( a ) Relative scattering intensity (in units of los photocounts/2 s sampling time) at 0 = 90" for [ BHA] = 0.4 mg/mL and [ HAdase] = 25.9 units/mL. The horizontal line at 0.75 is the finai baseline scattering level. ( b ) The data from Figure l a plotted as K c / [ , , where IR is the absolute baseline subtracted Rayleigh ratio.

,

.

11)

.

, , zw

Figure 3. Dependence of the normalized initial rate from light scattering data [ K M bonds cleaved/ h divided by enzyme concentration (units/mL) ] on hyaluronate concentration. Reactions were carried out at 25°C and pH 5.3 (0.1M Pi, 0.15M NaCl). The solid curve is the theoretical fit to the Henri-Michaelis-Menten equation based on V,,,/ET = 9.6 pM bonds cleaved/h per unit/mL enzyme and K,,, = 0.18 mg/mL.

one million (about 2500 cleavable bonds), there would be about 420 cuts per hour under these conditions. T h e initial velocities scaled linearly with enzyme concentration over the range of 10-95 units/mL tested. The velocities were also found to be independent of angle, and repeatability between runs was to better than 9%. Figure 3 shows the initial velocities of the hyaluronate digestion vs hyaluronate concentration with the corresponding fit to the standard Henri-Michaelis-Menten equation. The K, was found to be 0.18 k .03 mg/mL by this fit. This can be compared with reported values obtained under slightly different conditions: K, = 0.09 mg/mL ( p H 6.0, 37"C, 0.25M acetate, 0.15M NaCl, 0.05M potassium phosphate),26K, 0.3 mg/mL a t p H 5.0, 27"C, 0.1M acetate, 0.15M NaC1." [This is based on the reported value of K , = 0.74 ( k . 0 3 ) m M of "hexosamine units" and a dimer subunit ( NAG-GlcUA) MW = 4011 and K, N 0.6 mg/mL ( p H 5.0, 37"C, 20 m M MES, 0.1M NaC1).27The maximum velocity was found t o be 9.6 (k0.3) p M of bonds broken per hour per unit/ml of enzyme, corresponding to kcat G 7 s - I based on a specific activity of 42,000 units/ mg for the pure enzyme" and a n enzyme M , = 61,000.29 T h e K , and V m a x / E from T the reductimetric assay (0.5 mg/mL I [HA] I 2 mg/mL) were found to be 0.46 f .05 mg/mL and 4.3 k .2 pM bonds/h per unit/ mL, respectively. T h e "initial velocities" were calculated on the basis of around 6 points taken during the first hour of digestion. There is more un-

-

0' 0

,

LLO

"9,".

RANDOM COIL SCISSION RATES

certainty in K, obtained with the Reissig assay than with the light scattering assay because of the insensitivity in obtaining initial velocities a t low HA concentrations. Furthermore, the problem of enzyme deactivation, discussed below, as well a s transglycosylation, render questionable the comparison of initial velocities calculated over different time scales. While there is no difficulty in obtaining relative V,,, values with the reducing sugar ( N-acetylhexosamine) assay, it is subject t o systematic error. Thus, when a calibration curve is constructed with N-acetylgalactosamine the extinction coefficient is lower [by a factor of 2.30 (k.04) ] than when N-acetylglucosamine is used. Thus, since stereochemistry a t C4 has such a dramatic effect on color development, it is not surprising that derivatization of C-3 with GlcUA in the reducing termini of the digested HA fragments results in less efficient chromophore development than does NAG. It is interesting to compare the results with those obtained by the A2 and polydispersity corrections of eq. ( 11) . The HPLC runs yielded a polymer mass distribution very close to a log normal distribution, with the following polydispersity characteristics: M u . / M , = 1.22, M Z / M w= 1.19 and M,, = 1.1X lo6. With these ratios and the Zimm plot results, K , was found to be 0.32 (t-0.3) mg/mL, and V m a x / E Twas 9.4 (k0.3) pM bonds/h per unit/mL of enzyme. The individual initial velocities were typically within 20% of the velocities determined by the absolute scattering intensity method for BHA concentrations above about 0.2 mg/mL. The equivalent hydrodynamic diameters of the HA solutions were measured before and after the digestion. Before digestion BHA gave concentrationdependent diffusion coefficients with equivalent hydrodynamic diameters of from 400 t o 900 8, in diluting from 2.0 t o 0.025 mg/mL. After the baseline was reached, it was never possible to autocorrelate the scattered signal.

DISCUSSION Reducing sugar assays, such as that of Reissig e t al.,' have been routinely used for measuring HAdase activity. While these assays can provide reliable determinations of relative velocities, the absolute velocities are subject t o systematic errors. Furthermore, for the reductimetric assay, deactivation of enzyme makes it hard to define what a n adequately short initial period for obtaining initial velocities is. T h e systematic error results from the fact that the reducing sugar terminal is converted t o a chromo-

1079

phore3' and the rate of color development is dependent on the structure of the reducing sugar. This may differ from that of the standard. Even when a n assay is used where the chromophore is independent of the reducing sugar (e.g., ferrocyanide), kinetic factors can result in color yields that depend on the structure of the reducing sugar (e.g., the assay described by Avigad, 31 which employs the 2,4,6-tripyridyl-s-triazine chelate of ferrous ion produced by ferricyanide oxidation of the reducing sugar). Although this assay is about 12 times more sensitive toward NAG than is the Reissig assay, differences between standard reducing sugars are still observed. Thus, absorbance yields (595 n m ) are 13% lower with N-acetylgalactosamine, 16% higher with glucuronic acid, and 28% higher with 2-deoxy-2-aminoglucose when compared with NAG under standard assay conditions. Thus, in addition to insensitivity ( a t early time points) and inconvenience, these single-point reductimetric assays depend on the choice of the standard. The initial velocities obtained from the light scattering assay indicate that HA digestion by HAdase conforms well t o the simple Henri-Michaelis-Menten model. T h a t Figure 2b is initially a straight line supports the assumption that bonds are being broken at a constant rate b , a t least for the first few tens of cuts per molecule. This means t h a t it is really the concentration of unbroken scissile bonds that is t o be considered the substrate concentration, and not the concentration of polymer molecules, as this latter concentration increases as digestion proceeds, but the velocity remains constant, even for [ H A ] well below saturation. This implies that to a good approximation the kinetic parameters of hyaluronate digestion by hyaluronidase should be independent of molecular weight over a wide range. Although eq. ( 9 predicts that 1/ZR( t ) should be linear in time for hyaluronate, because uo% 1 for all weights present in the initial population, after tens of cuts 1/IR ( t ), usually begins to bend down from the initial straight line, as seen in Figures 2b and 4b. With roughly 4000 cleavable bonds per original HA, after tens of cuts there is not enough decrease in substrate concentration to affect the rate of scission of bonds. Aggregation of the HA during depolymerization is also highly unlikely since the scattering level of the final digested solution was always within a couple percent of the pure buffer scattering level. This virtually excludes the possibility of spuriously low measured baselines being the origin of the downward curve in these experiments. [Using a value of B lower than the true value will make F ( t )

1080

REED, REED, AND BYERS

curve downward. An incorrectly high value of B makes the plot curve upwards.] Nonideality of the Gaussian chains due t o excluded volume, according to eq. ( 1 6 ) , will not produce a downward curve. T h e strongest possibility is that enzyme is partially deactivated during the depolymerization, due to adsorption onto glass or other mechanisms. Scattering data taken a t different initial enzyme concentrations were plotted against initial [ HAdase ] X time (Figure 4 gives examples) according to the suggestion of S e l ~ y n . Except ~' for the initial slopes these curves are not superposable on each other (and even greater deviations are seen a t lower enzyme concentration), which is strong evidence for partial enzyme deactivation being the origin of the downward curvature for 1/ I R ( t ) . This probably also accounts for the larger velocities obtained with the light scattering assay than with the reductimetric assay, since the latter requires on the order of a n hour to collect a sufficient number of points to analyze. In fact, to investigate this possibility, straight lines were fitted to the 1/ IR( t )data for the first 600 data points of all the reactions that had a t least 600 points ( a t 2.77 s / p t ) . As expected, this produced a substantially lower value of V,,,/ET of 4.69 pM bonds/h per unit/mL, close to that obtained by the reductimetric assay. T h e lowest [ H A ] experiments were not included in this sample because they re-

1

I 6

-( r 4

2\ Y

M

0

-

2

0' 0

100

200

1

300

[HAdase] x time Figure 4. Selwyn plots, of reciprocal baseline subtracted intensity vs initial [ HAdase] X time (units: min/mL) for [ BHA] 0.3 mg/mL and 8 = 90" for ( a ) [ HAdase] = 51.8 units/mL (total real time is 5.8 min) and ( b ) [ HAdase] = 25.9 units/mL (about 0.01 mg/mL; total real time is 11.6 min). The initial slope over y intercept, i.e., the normalized initial velocity [ = u/ET], is independent of enzyme concentration. However, the nonsuperimposability of the curves over longer extents of reaction suggests enzyme inactivation during the reaction.

quired only a few minutes to digest significantly, and light scattering experiments were only run about 15 min. Therefore the value of K , yielded by this subset of data, 0.092 mg/mL, is not very meaningful. While it is discomforting to fit a straight line through obviously curved data (e.g., Figure 2b with even more curvature over twice the time scale), if half a dozen points are taken a t roughly equal intervals from such a curve with hundreds of points, then the appearance of these points is quite similar to those of the reductimetric assay. It is less visually shocking t o fit a line through a handful of such points, especially when some noise is present, as for the reductimetric assay. Another possibility for the downward curve is that A2 really does vary as some inverse power of the molecular weight. Numerical calculations using the fragment size distribution f ( x , r ) for a n initially monodisperse population undergoing random scission given by eq. ( 15) in Reed and Reed, and Cass a ~ s a ' treatment s~~ of the second virial coefficient, show that l / I Rt() does bend down from a straight line. These calculated curves, however, show most of their curvature within a few cuts and then become flatter, whereas the experimental curves are initially flat and then begin to bend down (e.g., Figures 2b and 4b) after, typically, tens of cuts per molecule. T h e Cassassa theory of A2 does not fit the data well, although its prediction of a downward curvature suggests that a more accurate theory might better explain the experimental shape. However, the existing data for the A2 dependence on weight for HA is inadequate to assert this as the origin of the effect. At this point we note that polymer theoryZ1zz3 predicts that for linear uncharged polymers if A2 > 0, then for sufficiently large polymer mass M , A2 should vary as about M-o-2.For charged polymers, with electrostatic screening by a salt solution, Odijk and Houwaart3' postulate that there are two effects of polymer charge, a stiffening of the polymer chain by mutual repulsion of neighboring charges, and a n interactionlike excluded volume for charges located far apart along the chain. If their view,34which is not universally accepted, 35 is correct, then for charged polymers, at constant salt concentration, for sufficiently large M , A2 should still vary as about M-'.'. At any rate, Cleland and Wang36 have reported light scattering and osmotic pressure data results on fractionated hyaluronate over the molecular weight range from 110,000 to 1 X lo6. Using their data we performed a linear regression to log(A2)vs log ( M )and found that A2 falls as the -0.061 rf: .074 power of M . Thus the assumption of weight-inde-

RANDOM COIL SCISSION RATES

pendent A 2 used in the analysis is reasonable in the light of such experimental uncertainties. It has been suggested t h a t HA is double stranded or self-associates in some solvents, with a minimum length for association of about 7 disaccharides reported in 0.15M NaC1.37Although these experiments do not rule out self-association, the initial linear rise of 1/ I H with time is evidence against the more simplistic model that HA is a stable double strand ( a t least in the phosphate buffer used). A simple stable double strand should break only when a cleavage on one strand occurs closer than the minimum distance for self-association to one on the other strand. If for the moment we assume the enzyme makes only single attacks a t random sites, the probability of cutting the double strand will increase quadratically with time, making the graph of 1 / I R vs time resemble a parabola. This was never observed. Similarly, the near linearity of 1/I, over a period of time in which it increases to twice or several times its original value is evidence for the (unsurprising) conclusion that bovine testicular hyaluronidase is not a pure exoenzyme. If it were, again momentarily assuming the exoenzyme t o make only single or low numbers of attacks each time, then, because of the large number of units in a BHA molecule, we can conclude that nearly all molecules would be shortened hy nearly the same fraction of their original length, and a t a nearly constant rate. Recalling that for random coils u = q 2 y m , and setting m = mo n i t , with ni constant [ eqs. ( 1) and ( 2 ) ] and Q ( 1 9 ) set equal to 1, allows one to graph the predicted I / I,{. lJsing parameters similar to those derived from the Zimm plot in Figure 1, and letting x be the fraction by which the polymer’s length has been reduced, one can graph 1/IR [ x ( t ) ] vs x. The resultant curves show only a modest rise in F above its original level, followed by a violent upturn that occurs a s x nears 1, because as x approaches 1, u approaches 0 and P ( u ) goes to 1, so that 1/ I R becomes proportional to 1 / ( 1 - x ) . Again, this is quite different from the curves in Figures 2b, 4a, and 4b. The two inferences just discussed are strengthened by the good agreement between the number of cuts found from light scatting and the number of cleavages found from the reductimetric assay when determined over similar time scales, with the respective values of v,,, 4.69 and 4.3 p M bonds/h per unit / mL. This agreement implies that HAdase cleaves only one bond a t each site of random attack. If multiple local attacks were made, V,,, measured by the reductimetric assay would be higher than the value from the light scattering assay. For if the enzyme cuts the HA a t several nearby sites, the light ~

1081

scattering increases only as much as if it is cut once. This argues against a “sticky” enzyme where HAdase acts by cleaving and “sliding.” Such a mechanism has been reported for some protein-DNA interaction~.~~ An attack a t the end of an HA molecule would decrease its light scattering much less than one that cut it in two pieces of comparable size. The good agreement between the light scattering and reductimetric measurements of scission rates thus independently confirms that the HAdase used here cannot be predominantly an exoenzyme. In the case of a simplistic stable double-stranded picture of HA, a n enzyme making multiple local attacks could often break both strands close together, yielding a linear increase of l/lRwith time. However, double strandedness as well as multiple local attacks would tend to reduce the number of cuts calculated from light scattering (which would see only gross breakages of the whole double strand) below the number of cleavages found by reductimetry, so this is unlikely. More complex scenarios involving partial self-association in HA or fleeting interactions between HA molecules39 are not ruled out by the present data, nor is stable double strandedness in other solvents.

’’

SUMMARY The introduction of a relatively simple light scattering assay for determining rate constants in the random scission of random coil polymers represents a significant advance in speed, efficiency, and accuracy. Typically, after preparing the desired hyaluronate solutions and centrifuging them for one hour, the kinetics of each solution could be determined in less than one half hour. Since many samples can be simultaneously centrifuged in a n hour, the kinetics of about 15 different samples could be determined in a n average work day. If the approximations of large uo for the initial population and mass independence of A 2can be justified ( a s they are for BHA in this case), then the minimum apparatus required for such measurements is extremely simple; a monochromatic light source ( n o t necessarily a laser), a n optical quality scattering cell, a linear photodetector (preferably a photomultiplier tube), and a strip chart recorder or other means of recording I ( t ) . The light scattering assay provides several advantages over the more tedious and time-consuming assays. Unlike single-point techniques, such as the reductimetric assay, no additional reagents (of variable stability) are required, and a n essentially con-

1082

REED, REED, AND BYERS

tinuous and accurate record of the initial phase of hyaluronic acid hydrolysis can be readily obtained over a wide range of substrate concentrations. This assay provides a convenient method for obtaining reliable initial velocity data, and thus the evaluation of the steady-state kinetic parameters of hyaluronidase. Additionally, the shape of the l / I R (t ) curve allows certain deductions to be made about the mechanism of the enzyme attack and the structure of the polymeric substrate. It should also prove useful in the purification of the enzyme and in clinical chemistry. Because the basic theory applies to random scission of random coil molecules, the assay should be applicable to a wide variety of depolymerizations. W. F. Reed and C. E. Reed acknowledge support for this work from National Science Foundation Grant DMB 8803760. L. D. Byers acknowledges support from NIH grant GM34070. We are grateful to Jeanne Franqois and Dominique Sarazin for help in obtaining the HPLC results, and to Rob Peitzsch and Li Xiao for technical assistance.

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14. Keleti, G. & Lederer, W. H. ( 1974) Handbook of M i cromethods for the Biological Sciences, Van Nostrand Reinhold Co., New York, pp. 45-46. 15. Leatherbarrow, R. T. (1987) Enzfitter, A Non-Linear Regression Data Analysis Program for the IBM-PC, Elsevier Biosoft, Cambridge. 16. Reed, C. E., Xiao L. & Reed, W. F. (1989) Biopolymers 28, 1981-2000. 17. Bender, T. M., Lewis, R. J. & Pecora, R. ( 1986) Macromolecules 19, 244-246. 18. Benoit, H. (1953) J. Polym. Sci. 11, 507-510. 19. Reed, C. E. & Reed, W. F. J. Chem. Phys., in press. 20. Zimm, B. H. (1948) J. Chem. Phys. 16, 1093-1099, 1099-1116. 21. Yamakawa, H. ( 1971) Modern Theory of Polymer Solutions, Harper & Row, New York, especially section 27. 22. Houck, J. C. & Pearce, R. H. (1957) Biochim. Biophys. Acta 25, 555-562. 23. De Gennes, P.-G. ( 1979) Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, NY. 24. Benoit, H. (1957) C R Acad. Sci. 245, 2244. 25. Loucheux, C., Weill, G. & Benoit, H. (1958) J . Chim. Phys. 540-546. 26. Friszer, B. (1964) Bull. SOC.Chim. Biol. 46, 12111221. 27. Jones, C. P. & Sawyer, R. T. (1989) Thrombosis Res. 55, 791-796. 28. Highsmith, S., Garvin, J. H. & Chipman, D. M. (1975) J . Biol. Chem. 2 5 0 , 7473-7480. 29. Borders, C. L., Jr. & Raftery, M. A. (1968) J . Biol. Chem. 243, 3756-3762. 30. Davidson, E. A. ( 1967) Carbohydrate Chemistry Holt, Rinehart and Winston, New York, pp. 393-394. 31. Avigad, G. (1975) Methods Enzymol. 41, 27-29. 32. Selwyn, M. J. (1965) Biochim. Biophys. Acta 105, 193-195. 33. Cassassa, E. F. (1962) Polymer 3, 625-638. 34. Odijk, T. & Houwaart, A. C. (1978) J. Polym. Sci. Polym. Phys. Ed. 16, 627-639. 35. Soumpasis, D. M. & Bennemann, K. H. ( 1981) Macromolecules 14, 50-54. 36. Cleland, R. L. & Wang, J. L. (1970) Biopolymers 9, 799-810. 37. Turner, R. E., Lin, P. & Cowman, M. K. (1988) Arch. Biochem. Biophys. 265, 484-495. 38. von Hipple, P. H. & Berg, 0 . G. (1989) J. Biol. Chem. 264,675-678. 39. Morris, E. R., Rees, D. A. & Welsh, E. J. (1980) Biopolymers 10, 1213-1227. Received April I I , I990 Accepted J uly 3, 1990