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WORKING PAPER

ALFRED P. SLOAN SCHOOL OF MANAGEMENT

AN EXPLORATORY ANALYSIS OF RADIOIMMUNOASSAYS DATA:

ESTABLISHING SUBTYPES OF THE HEPATITIS-B VIRUS

By

M. Anthony Wong, Jack R. Wands, Robert Marciniak,

Russell D. Brown and James Shorey

WP 1468

December 1983

MASSACHUSETTS

INSTITUTE OF TECHNOLOGY

50 MEMORIAL DRIVE

CAMBRIDGE, MASSACHUSETTS 02139

AN EXPLORATORY ANALYSIS OF RADIOIMMUNOASSAYS DATA:

ESTABLISHING SUBTYPES OF THE HEPATITIS-B VIRUS

By

M. Anthony Wong, Jack R. Wands, Robert Marciniak,

Russell D. Brown and James Shorey

WP 1^68 December 1983

An Exploratory Analysis of Radioimmunoassays Data:

Establishing Subtypes of the Hepatitis-B Virus

by

M. Anthony Wong

Sloan School of Management

Massachusetts Institute of Technology

Cambridge, MA 02139 U.S.A.

Jack R. Wands, Robert Marciniak, Russell D. Brown

Gastrointestinal Unit, Massachusetts General Hospital

and the Department of Medicine, Harvard Medical School

Boston, MA 02114 U.S.A.

and

James Shorey

Department of Medicine

University of Texas Southwestern Medical School

Dallas, TX 75325 U.S.A.

AUTHOR'S FOOTNOTES

M. Anthony Wong is Associate Professor, Sloan School of Management,

Massachusetts Institute of Technology, Cambridge, MA 02139. Jack R. Wands

is Associate Professor and R. D. Brown and R. A. Marciniak are research

associates at the Gastrointestinal Unit, Massachusetts General Hospital and

the Department of Medicine, Harvard Medical School, Boston, MA 02114, James

Shorey is Professor, Department of Medicine, University of Texas

Southwestern Medical School and the V. A. Medical Center, Dallas, TX

75325. This work was supported by Grants AA-02666 and CA-35711 from the

National Institutes of Health. JRW is the recipient of a Research Career

Development Award AA-00048 from the National Institutes of Health.

ABSTRACT

A novel approach has been developed for the analysis of hepatitis B

virus (HBV) antigenic structure which creates numerical "signatures" of HBV

viral strains. This technique employs high affinity IgM and IgG monoclonal

antibodies (anti-HBs) directed toward distinct and separate determinants on

hepatitis B surface antigen (HBsAg). Such antibodies have been used to

develop sensitive and specific radioimmunoassays for measurement of

HBsAg-associated determinants in serum. This approach also involves the use

of a statistical technique for aligning the unknown concentrations of HBsAg

present in different individuals with the same viral strain. In this paper,

we will describe the iterative procedure developed for analyzing these

radioimmunoassays data and we will also present the findings of our

exploratory analysis. In summary, this approach suggests that HBV is far

more heterogeneous than previously recognized by polyvalent anti-HBs

antibodies, and the present technique is shown to be useful in epidemiologic

studies of HBV transmission. This approach also has broader significance

for the study of subtle or major antigenic changes among other viral agents

including polio, influenza and herpes since it is not necessary to know the

concentration of virus or viral protein in complex protein mixtures.

KEY WORDS: Monoclonal antibodies; radioinmiunoassays data; least-squares;

alignment procedure; exploratory analysis.

1. INTRODUCTION

The development of monoclonal antibodies by Kohler and Milstein has led

to an explosion in biological research. (See, for example, Chisari 1984).

In this study, we will focus on the use of monoclonal antibodies to

characterize the antigenic structure of the hepatitis B virus (HBV) at the

molecular level, and demonstrate with the aid of statistical analysis, that

HBV is far more heterogeneous than previously recognized.

We (Wands, Ben-Porath and Wong, 1984) have previously described the

production and characterization of monoclonal antibodies to HBV (anti-HBs)

and hepatitis B surface antigen (HBsAg). By taking advantage of the unique

properties of these antibodies, we were able to construct highly sensitive

and specific radioimmunoassays (RIAs) for the measurement of

HBsAg-associated epitopes in serum; these RIAs and the statistical data they

generate will be described in Section 2. Since each such antibody binds to

a distinct and separate antigenic determinant on HBsAg, the antigenic

structure of a HBV viral strain is characterized by its binding activities

to a panel of monoclonal antibodies over a range of viral concentrations. A

model for the functional relationship between binding activity and viral

concentration is discussed in Section 3, and a statistical procedure is

developed for estimating this functional relationship. This development is

necessary since the concentration of virus or viral protein in serum samples

cannot be determined because they are complex protein mixtures.

In Section 4, an iterative least-squares procedure for aligning the

concentration of samples from the same HBV subtype is described. In

addition, we will report on our exploratory analysis of the functional form

of the relationship between binding activity and viral concentration.

_ 2 -

The biological significance of the present approach to characterizing the

antigenic structure of HBV subtypes will be discussed in Section 5. We will

also indicate its usefulness in epidemiologic studies of HBV transmission

and its value in studying the antigenic composition of other vital agents

including polio, influenza, and herpes.

- 3 -

2. THE RADIOIMMUNOASSAYS

The immunization protocols, cell fusion technique, growth and

cloning of hybridomas producing anti-HBs have been reported in Wands and

Zurawski (1981). The monoclonal anti-HBs antibodies have been characterized

with respect to specificity for determinants on HBsAg, antibody class and

subclass and affinity for HBsAg-associated determinants. (See, for example.

Wands et al., 1981.) Monoclonal antibodies designated 5C3(Ig G^^) , 5C11,

2C6, 1C7 and 4E8(I G, ) and 5D3, 1F8 and 2F11(I M) were selected for

8 1 g

use in this study from a library of monoclonal anti-HBs antibodies because

of their following properties: 1) several bind to all known subtypes of

HBsAg; 2) some demonstrate qualitative binding differences to major HBsAg

subtypes; 3) they recognize distinct and separate determinants on HBsAg; and

9 11

4) they possess very high affinity constants (4.8 x 10 to 4.0 x 10

liters/mole per molecule) for HBsAg-associated epitopes. (For details, see

Wands and Zurawski, 1981).

We employed 8 monoclonal "simultaneous sandwich" radioimmunoassays

(RIAs) for analysis of HBsAg-associated binding activities in various HBsAg

positive serum samples. In brief, polystyrene beads were coated with a

monoclonal IgM anti-HB designated 5D3 and the other 8 antibodies, including

5D3, were radiolabeled to a specific activity of 10-12 yCi/yg protein

with the Hunter-Bolton reagent (Bolton and Hunter, 1973). For each

antibody, the 5D3 coated beads were incubated with serial 10-fold dilutions

(lO'^; X = 0, 1, .., 5) of samples (lOOyl) of a given serum and 100 yl

of radiolabeled probe, consisting of approximately 150,000 cpm of the

- 4 -

monoclonal anti-HB, for 4 hours at 45 C. The beads were washed

extensively with distilled water and the radioactivity bound to each bead

was measured with a Packard gamma well counter.

The RIAs data generated from a serum sample was, therefore, an 6 x 8

matrix whose (ij)th element was the natural logarithm of the signal-to-noise

ratio (log S/N, defined as the median cpm bound in the experimental

samples of the serum/median cpm bound in samples of a negative control

serum), obtained when the 10 th dilution of the serum and the jth antibody

were used. In the present study, we selected 64 individuals from the United

States whose serum are representative of known HBsAg subtypes. These

specimens have been classified as aywâ€ž (subtype 2), ayw_ (subtype 3),

adw/ (subtype 6) and adw2 (subtype 7) HBsAg subtypes by RIAs using

conventional polyvalent anti-HBs antibodies. (See Shorey, Brown and Wands,

1981.)

- 5 -

3. THE STATISTICAL PROBLEM

Since each monoclonal antibody binds to a distinct and separate

antigenic determinant on HBsAg, the antigenic structure of a HBV subtype is

characterized by its binding activities to a panel of monoclonal antibodies

over a range of viral concentrations. The statistical problem involved in

establishing the antigenic structure of a HBV subtype is therefore one of

estimating the multiple response functions f^ (A= 1, 2, ..., 8) between

logp S/N and logig viral concentration (x) for the 8 selected antibodies

using serum samples from that subtype. It is hypotyhesized that each

f (x) is a smooth growth curve which increases monotonely from zero to a

unique local maximum at a high value of x and then decreases with further

increase in x because of binding interferences (see Figure 1); it should be

pointed out, however, that most serum samples do not have very high viral

concentrations .

Let y 4 d ''^ ^^^ lÂ°8e S/N value obtained for antibody A (A = 1, 2,

...,8 at log Q dilution factor d (d = 0, 1, .... 5) of the ith serum

sample (i = 1, 2, ..., n) from a HBV subtype. Then the statistical model

for the RIAs data can be expressed as follows:

A A A

^i d " ^ (Xj^ - d) + e^ , d = 0, 1, .... 5; A = 1,2,. ..,8; i = 1,2,..., n [1]

where x^ is the unknown logj^g concentration of the ith serum sample at d

-0, the e 's are the error terms with E[e ] =0 and standard derivation

r At A rA

le J = a , and the f^'s are of the same functional form. The x^'s

cannot be determined because serum samples are complex protein mixtures.

Since the x.'s are unknown, we cannot use standard curve-fitting methods

- 6 -

to estimate the f^(x)'s directly from the data; this problem is

illustrated in Figure 2a where the y ,'s obtained for the 14 subtype 6

1 ,a

samples are plotted against d, ignoring the variations of the x 's. We

have therefore developed an iterative least-squares procedure for aligning

the concentrations of serum samples from the same subtype. The alignment is

effected by first setting x, = 0, and then estimating the other x 's (i

= 2, 3, ..., n) using the statistical model given in Equation [1]. The

aligned data are subsequently used to estimate the f (x - Xj^)'s which

collectively characterizes the antigenic structure of a HBV subtype. This

iterative procedure will be described in the next section.

- 7 -

4. AN ITERATIVE ALIGNMENT PROCEDURE

The proposed procedure for aligning the unknown x. 's (1 = 1, 2, ...,

n) of serum samples from a HBV subtype consists of the following steps:

STEP 1 ( Initial Alignment ); Select a "leader" whose undiluted log, ^

viral concentration x, is set to be zero. Using the 8 piecewise linear

functions defined by the log S/N values obtained for this leader at x =

0, 1, ..., 5, the estimates of the other x's (i = 2, 3, ..., n) on this

selected scale are then obtained using a least squares method.

STEP 2 ( Curve Fitting ); Estimate the response functions f (A = 1, 2,

..., 8) from the aligned data using exploratory regression techniques. The

overall sum of squared residuals is also computed.

STEP 3 ( Updating Alignment ): Using the f^'s obtained in STEP 2, the

estimates of x.'s (i = 1, 2, ..., n) are updated. If the resulting

decrease in the overall sum of squared residuals is less than a specified

threshold, stop; otherwise, go to STEP 2.

A detailed description of the proposed alignment procedure is given in

the following, but it should be noted that this procedure only provides

estimates of the (x. - x,)'s, and not the x.'s themselves.

4.1 Initial Alignment Step

The aim of the procedure is to align the uninown viral concentrations of

the serum samples from a HBV subtype. It is equivalent to estimating the

(x. - X, )'s (i = 1, 2, ..., n), and if we set x, to be zero, then we

are only estimating the x.'s (1 = 2, ..., n). In this step, the objective

is to find initial estimates of the x.'s.

- 8 -

A A A A

Since from [1], y. , = f (x -d) + e. and each f is a smooth growth curve

increasing monotonely from zero to a unique local maximum, the following 8

piecewise linear functions, f^ (A = 1, 2, ..., 8), defined by y, ^ (d = 0,

p J- .a

1, ,,,, 5; A = 1, 2, ..,, 8) are approximations of the f 's:

f^ (x) = a^ + b^ X, d < X < d + 1; d = 0, 1, 2, ... [2]

where a, and b, is respectively the intercept and slope of the line between

AAA â€žA ,

(d.yi d) ^^^ i^+'i-, yi,d+l)Â« ^^'^ yi,d = for d > 6 because 10Â° or higher

dilutions of practically all serum samples would contain effectively no

viral protein. The x. 's are then estimated by using a least squares

method. Specifically, for each i, we find the value of x. to minimize the

sum of squared residuals

8 5

ssR. = 1 L [yt ^ - f^ (^-i - ^^^'

1 i,d p 1

A=l d=0

Since interpolation is preferred to extrapolation, the "leader" sample is

ma Y A

the one with the most . y. values among the 8 possible maxima. Using

the f^'s given in [2], it can be shown that if j < x < j + 1 (j = 0,1,2,

P i

..., 5), then the least squares estimate of x. is given by

8 J+5 , . 8 j+5 ^ ^ 8 j+5

X., . = j + A=l d=j ^ ^''^ J A=l d=j d d A=l d=j

^'J 8 j+5 , 2

I I (b^ )

A=l d=j ^

By computing x^., j = 0, 1, 2, ..., 5, and their associated SSR^

values, the least squares estimate of x can be found by identifying the

X. , . with the smallest SSR, value.

- 9 -

It should be noted that, instead of the i 's, we could have used a smooth

spline function (see, for example, Winsberg and Ramsay, 1982); but the least

squares estimate of the x 's would then have to be obtained by a numerical

optimization algorithm. We decide to use the picewise linear functions

because our objective here is to find a set of reasonably good initial esti-

mates of the X 's. Moreover, an estimation method more resistant than the

least squares procedure could have been used; instead, y values associated

1 ,a

with large least-squares residuals are identified and removed from all subse-

quent analysis. The aligned data for the 14 subtype 6 serum samples are

shown in Figure 2b, where the log S/N values are plotted against the nega-

tive aligned log, q viral concentration (-x).

4.2 Curve-Fitting Step

A A

With the aligned data, a better approximation of the f 's than the f 's

can be obtained by pooling the sample data from the same HBV subtype. Our

objective is therefore to fit a function f to the aligned data for monoclo-

clonal antibody A, and the f^'s should be of the same functional form. By

examining Figure 2b, it can be observed that the variability of log S/N is

not constant across the range of viral concentrations; as expected from the

experimental procedure, the variability is high when the concentration is low

or when log S/N is close to zero. Hence, we use a weighted least squares

procedure, in which log S/N values close to zero carry very little weight,

e

to find a function that would fit the aligned data well.

Several functional forms including the quadratic, cubic. Gamma and Gaussian

curves have been fitted to the aligned data for all known HBV subtypes, but

plots of the residuals obtained for these curves indicate that they do not

- 10-

fit the data well. As suggested by residuals analysis, the best-fitting

function is the logistic curve of the form:

3 3

f^* (x) = exp [ r 13^ x^]/a + exp [ [ B, x^ ] ) [3]

Â° k=0 ^ k=0 ^

A* JV

where f (x) = (f^ (x) + l)/9. This function fits the monotonely increas-

n n

ing part of the aligned data particularly well, but it cannot be used for

extrapolation purposes. Moreover, it should be noted that the cubic term is

not necessary to fit the aligned data for most HBV subtypes; but for those

whose serum samples have high viral concentrations, this extra term is

needed to capture the transition at the peak of the growth curves. The final

fitted curves and the aligned data for subtype 2, 3, 6 and 7 are shown in

Figure 3-6 respectively, and they are not very different from the plots

obtained after this intermediate step. The overall sum of squared residuals

n n 8 5 2

I ssR. = 1 I t [y? d- ^ ^^i-^)l

i=l ^ i=l A=l d=0 ^'^ ^ ^

is also computed for the samples from each subtype.

4.3 Updating Step

With the f^ 's, the x. 's, i=l, 2, ..., n are re-estimated in turn by

n 1

finding the value of x that would minimize

i

Q C

SSR. =1 [ [y^ ^ - f^ (x^ - d)]2

' A=l d=0 ^'^ " ^

Since the updated estimates will be in the neighborhood of the previous

estimates and SSR. is a continuous function of x. , the new estimates can be

obtained numerically. It has been found that the solutions after the first

few iterations are practically the same and hence in order to avoid excessive

computing, a threshold value T is specified so that when the difference in

-11-

Tl

J SSR is not greater than T, the iterative procedure stops; otherwise,

1-1 1

the Iterations will continue by going back to STEP 2.

Although the developed procedure Is unduly dependent on the selected

leader, our Immunoassays data are so well behaved that this dependency does

not create a problem In the present application* We have also examined the

variability of the estimated relative viral concentrations of samples from a

HBV subtype by using a jack-knife procedure. By leaving out one serum

sample at a time, n sets of estimates of relative concentrations are

obtained by the Iterative procedure* Again, since our data align very well,

these solution sets are found to be very similar. Finally, we would like to

point out that this procedure can be generalized to the case where the serum

samples have very high viral concentrations and can also be expanded to

Include other monoclonal anti-HBs antibodies*

- 12 -

5. DISCUSSION

We have described a novel approach, "signature" analysis, for the study

of hepatitis B viral antigenic structure. The cardinal feature of this

approach included: 1) the use of high affinity anti-HBs monoclonal

antibodies, 2) the construction of RIAs specific for different epitopes on

HBsAg, and 3) a statistical technique for aligning the unknown

concentrations of HBsAg present in different individuals of the same

subtype. The estimated binding curves which describe the functional

relationships between log S/N and viral concentration for a panel of

selected monoclonal antibodies are characteristic for any HBV viral strain

and the composite profile defines the antigenic composition of HBV.

"Signature" analysis suggests that HBV shows far more antigenic

heterogeneity than previously recognized by polyvalent anti-HBs antibodies.

First with respect to classic HBsAg subtypes, we find the greatest antigenic

diversity when comparing the ad^ (subtype 6 and 7) to ay (subtype 2 and 3)

classical subtype group. (See, for example. Figure 7a; note that this and

all subsequent figures are obtained by aligning the subtypes which are being

compared.) This diversity is in accord with recent sequence data (Fujiyama

et al., 1983) showing substantial changes in amino acid composition between

HBsAg subtypes adw and ayw. In constrast, when the ad and ay domains are

the same, and the difference between the w specificities are examined,

signature analysis demonstrates remarkable antigenic homogeneity among the w

domains (see, for example. Figure 7b).

Further evidence of the antigenic diversity of HBV is supported by the

demonstration of subgroups within the "classic" HBsAg subtypes. For

example, ayw- may now be divided into at least two separate subgroups. In

- 13 -

the United States, an additional subgroup (see Figure 8) has been found to

be distinct from the larger ayw_ (subtype 3) group. Moreover, we have

found that Australian aborigines are infected with a ayw^ strain which

differs substantially from aywo groups or subgroups in the United States

with respect to epitope composition and density. Therefore, "signature"

analysis may permit studies of the genetic evolution of HBS in various parts

of the world. Indeed, preliminary studies of the aywo subtype from Africa

and the Far East suggest additional viral strains distinct from those found

in the United States and Australia. Furthermore, recent investigations

suggest that vertical transmission of HBV from mother to child or horizontal

spread within families breeds true. That is, the RIAs data obtained for

serum samples from all family members infected with HBV including mother and

child align very well. Therefore, "signature" analysis will be useful in

epidemiologic studies of HBV transmission.

Finally we believe that "signature" analysis may also be of value in

studying the antigenic composition of other viral agents. For example, such

an approach may be useful in the characterization of antigenic structure of

polio, influenza, herpes, and arenavirus groups and other viral agents

propagated in tissue culture. Since "signature" analysis does not require

that the concentration of viral protein in complex protein mixtures be

known, study of antigenic similarity or differences among other viruses

would seem possible.

- 14 -

REFEEIENCES

Bolton, A.E., and Hunter, W.M. (1973), "The labeling of proteins to high

specific radioactivities by conjugation to a I^^S-cont-ai^ing acetylating

agent," Journal of Biochemistry , 133, 529-539.

Chisari, F.V. (1984), Advances in Hepatitis Research (editor), Masson

Publishing Company (to appear).

Fujiyama, A., Miyanohara, A., Nozaki, C, Yoneyama, T., Ohotomo, N. and

Matsubara, K. (1983). Nucleic Acid Research , 11, 4601-4610.

Shorey, J., Brown, R.D. and Wands, J.R. (1981), "A new analysis of HBV

subtypes by epitope identification using monoclonal anti-HBs

radioimmunoassays," Hepatology , 1, 546.

Wands, J.R., Ben-Porath, E. and Wong, M.A. (1984), "Use of monoclonal

antibodies in the study of viral hepatitis," in Advances in Hepatitis

Research (Chisori, F.V., ed.), Masson Publishing Company (to appear).

, Carlson, R.I., Schumacher, H., Zurawski, V.R., and Isselbacher,

K.J. (1981), "Immunodiagnosis of hepatitis B by epitope identification with

high affinity IgM monoclonal antibodies," Proceedings of the National

Academy of Science, USA , 78, 1214-1218.

, Wong, M.A., Shorey, J., Brown, R.D., Marciniak, R., and

Isselbacher, K.J. (1984), "Hepatitis B viral antigenic structure: signature

analysis by monoclonal radioimmunoassays," Proceedings of the National

Academy of Science, USA (to appear).

, and Zurawski, V.R. (1981), "High affinity monoclonal antibodies

to hepatitis B surface antigen (HBsAg) produced by somatic cell hybrids,'

Gastroenterology , 225-232.

Winsberg, S., and Ramsay, J.O. (1982), "Monotone splines: a family of

transformations useful for data analysis," C0MPSAT,5, 451-456.

Log _ Dilution Factor of Serum

Log S/N

Log â€ž Viral Concentration in Serum

Fig. 1. A sketch of the expected relationship between

log S/N and log â€ž viral concentration in

serum.

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ALFRED P. SLOAN SCHOOL OF MANAGEMENT

AN EXPLORATORY ANALYSIS OF RADIOIMMUNOASSAYS DATA:

ESTABLISHING SUBTYPES OF THE HEPATITIS-B VIRUS

By

M. Anthony Wong, Jack R. Wands, Robert Marciniak,

Russell D. Brown and James Shorey

WP 1468

December 1983

MASSACHUSETTS

INSTITUTE OF TECHNOLOGY

50 MEMORIAL DRIVE

CAMBRIDGE, MASSACHUSETTS 02139

AN EXPLORATORY ANALYSIS OF RADIOIMMUNOASSAYS DATA:

ESTABLISHING SUBTYPES OF THE HEPATITIS-B VIRUS

By

M. Anthony Wong, Jack R. Wands, Robert Marciniak,

Russell D. Brown and James Shorey

WP 1^68 December 1983

An Exploratory Analysis of Radioimmunoassays Data:

Establishing Subtypes of the Hepatitis-B Virus

by

M. Anthony Wong

Sloan School of Management

Massachusetts Institute of Technology

Cambridge, MA 02139 U.S.A.

Jack R. Wands, Robert Marciniak, Russell D. Brown

Gastrointestinal Unit, Massachusetts General Hospital

and the Department of Medicine, Harvard Medical School

Boston, MA 02114 U.S.A.

and

James Shorey

Department of Medicine

University of Texas Southwestern Medical School

Dallas, TX 75325 U.S.A.

AUTHOR'S FOOTNOTES

M. Anthony Wong is Associate Professor, Sloan School of Management,

Massachusetts Institute of Technology, Cambridge, MA 02139. Jack R. Wands

is Associate Professor and R. D. Brown and R. A. Marciniak are research

associates at the Gastrointestinal Unit, Massachusetts General Hospital and

the Department of Medicine, Harvard Medical School, Boston, MA 02114, James

Shorey is Professor, Department of Medicine, University of Texas

Southwestern Medical School and the V. A. Medical Center, Dallas, TX

75325. This work was supported by Grants AA-02666 and CA-35711 from the

National Institutes of Health. JRW is the recipient of a Research Career

Development Award AA-00048 from the National Institutes of Health.

ABSTRACT

A novel approach has been developed for the analysis of hepatitis B

virus (HBV) antigenic structure which creates numerical "signatures" of HBV

viral strains. This technique employs high affinity IgM and IgG monoclonal

antibodies (anti-HBs) directed toward distinct and separate determinants on

hepatitis B surface antigen (HBsAg). Such antibodies have been used to

develop sensitive and specific radioimmunoassays for measurement of

HBsAg-associated determinants in serum. This approach also involves the use

of a statistical technique for aligning the unknown concentrations of HBsAg

present in different individuals with the same viral strain. In this paper,

we will describe the iterative procedure developed for analyzing these

radioimmunoassays data and we will also present the findings of our

exploratory analysis. In summary, this approach suggests that HBV is far

more heterogeneous than previously recognized by polyvalent anti-HBs

antibodies, and the present technique is shown to be useful in epidemiologic

studies of HBV transmission. This approach also has broader significance

for the study of subtle or major antigenic changes among other viral agents

including polio, influenza and herpes since it is not necessary to know the

concentration of virus or viral protein in complex protein mixtures.

KEY WORDS: Monoclonal antibodies; radioinmiunoassays data; least-squares;

alignment procedure; exploratory analysis.

1. INTRODUCTION

The development of monoclonal antibodies by Kohler and Milstein has led

to an explosion in biological research. (See, for example, Chisari 1984).

In this study, we will focus on the use of monoclonal antibodies to

characterize the antigenic structure of the hepatitis B virus (HBV) at the

molecular level, and demonstrate with the aid of statistical analysis, that

HBV is far more heterogeneous than previously recognized.

We (Wands, Ben-Porath and Wong, 1984) have previously described the

production and characterization of monoclonal antibodies to HBV (anti-HBs)

and hepatitis B surface antigen (HBsAg). By taking advantage of the unique

properties of these antibodies, we were able to construct highly sensitive

and specific radioimmunoassays (RIAs) for the measurement of

HBsAg-associated epitopes in serum; these RIAs and the statistical data they

generate will be described in Section 2. Since each such antibody binds to

a distinct and separate antigenic determinant on HBsAg, the antigenic

structure of a HBV viral strain is characterized by its binding activities

to a panel of monoclonal antibodies over a range of viral concentrations. A

model for the functional relationship between binding activity and viral

concentration is discussed in Section 3, and a statistical procedure is

developed for estimating this functional relationship. This development is

necessary since the concentration of virus or viral protein in serum samples

cannot be determined because they are complex protein mixtures.

In Section 4, an iterative least-squares procedure for aligning the

concentration of samples from the same HBV subtype is described. In

addition, we will report on our exploratory analysis of the functional form

of the relationship between binding activity and viral concentration.

_ 2 -

The biological significance of the present approach to characterizing the

antigenic structure of HBV subtypes will be discussed in Section 5. We will

also indicate its usefulness in epidemiologic studies of HBV transmission

and its value in studying the antigenic composition of other vital agents

including polio, influenza, and herpes.

- 3 -

2. THE RADIOIMMUNOASSAYS

The immunization protocols, cell fusion technique, growth and

cloning of hybridomas producing anti-HBs have been reported in Wands and

Zurawski (1981). The monoclonal anti-HBs antibodies have been characterized

with respect to specificity for determinants on HBsAg, antibody class and

subclass and affinity for HBsAg-associated determinants. (See, for example.

Wands et al., 1981.) Monoclonal antibodies designated 5C3(Ig G^^) , 5C11,

2C6, 1C7 and 4E8(I G, ) and 5D3, 1F8 and 2F11(I M) were selected for

8 1 g

use in this study from a library of monoclonal anti-HBs antibodies because

of their following properties: 1) several bind to all known subtypes of

HBsAg; 2) some demonstrate qualitative binding differences to major HBsAg

subtypes; 3) they recognize distinct and separate determinants on HBsAg; and

9 11

4) they possess very high affinity constants (4.8 x 10 to 4.0 x 10

liters/mole per molecule) for HBsAg-associated epitopes. (For details, see

Wands and Zurawski, 1981).

We employed 8 monoclonal "simultaneous sandwich" radioimmunoassays

(RIAs) for analysis of HBsAg-associated binding activities in various HBsAg

positive serum samples. In brief, polystyrene beads were coated with a

monoclonal IgM anti-HB designated 5D3 and the other 8 antibodies, including

5D3, were radiolabeled to a specific activity of 10-12 yCi/yg protein

with the Hunter-Bolton reagent (Bolton and Hunter, 1973). For each

antibody, the 5D3 coated beads were incubated with serial 10-fold dilutions

(lO'^; X = 0, 1, .., 5) of samples (lOOyl) of a given serum and 100 yl

of radiolabeled probe, consisting of approximately 150,000 cpm of the

- 4 -

monoclonal anti-HB, for 4 hours at 45 C. The beads were washed

extensively with distilled water and the radioactivity bound to each bead

was measured with a Packard gamma well counter.

The RIAs data generated from a serum sample was, therefore, an 6 x 8

matrix whose (ij)th element was the natural logarithm of the signal-to-noise

ratio (log S/N, defined as the median cpm bound in the experimental

samples of the serum/median cpm bound in samples of a negative control

serum), obtained when the 10 th dilution of the serum and the jth antibody

were used. In the present study, we selected 64 individuals from the United

States whose serum are representative of known HBsAg subtypes. These

specimens have been classified as aywâ€ž (subtype 2), ayw_ (subtype 3),

adw/ (subtype 6) and adw2 (subtype 7) HBsAg subtypes by RIAs using

conventional polyvalent anti-HBs antibodies. (See Shorey, Brown and Wands,

1981.)

- 5 -

3. THE STATISTICAL PROBLEM

Since each monoclonal antibody binds to a distinct and separate

antigenic determinant on HBsAg, the antigenic structure of a HBV subtype is

characterized by its binding activities to a panel of monoclonal antibodies

over a range of viral concentrations. The statistical problem involved in

establishing the antigenic structure of a HBV subtype is therefore one of

estimating the multiple response functions f^ (A= 1, 2, ..., 8) between

logp S/N and logig viral concentration (x) for the 8 selected antibodies

using serum samples from that subtype. It is hypotyhesized that each

f (x) is a smooth growth curve which increases monotonely from zero to a

unique local maximum at a high value of x and then decreases with further

increase in x because of binding interferences (see Figure 1); it should be

pointed out, however, that most serum samples do not have very high viral

concentrations .

Let y 4 d ''^ ^^^ lÂ°8e S/N value obtained for antibody A (A = 1, 2,

...,8 at log Q dilution factor d (d = 0, 1, .... 5) of the ith serum

sample (i = 1, 2, ..., n) from a HBV subtype. Then the statistical model

for the RIAs data can be expressed as follows:

A A A

^i d " ^ (Xj^ - d) + e^ , d = 0, 1, .... 5; A = 1,2,. ..,8; i = 1,2,..., n [1]

where x^ is the unknown logj^g concentration of the ith serum sample at d

-0, the e 's are the error terms with E[e ] =0 and standard derivation

r At A rA

le J = a , and the f^'s are of the same functional form. The x^'s

cannot be determined because serum samples are complex protein mixtures.

Since the x.'s are unknown, we cannot use standard curve-fitting methods

- 6 -

to estimate the f^(x)'s directly from the data; this problem is

illustrated in Figure 2a where the y ,'s obtained for the 14 subtype 6

1 ,a

samples are plotted against d, ignoring the variations of the x 's. We

have therefore developed an iterative least-squares procedure for aligning

the concentrations of serum samples from the same subtype. The alignment is

effected by first setting x, = 0, and then estimating the other x 's (i

= 2, 3, ..., n) using the statistical model given in Equation [1]. The

aligned data are subsequently used to estimate the f (x - Xj^)'s which

collectively characterizes the antigenic structure of a HBV subtype. This

iterative procedure will be described in the next section.

- 7 -

4. AN ITERATIVE ALIGNMENT PROCEDURE

The proposed procedure for aligning the unknown x. 's (1 = 1, 2, ...,

n) of serum samples from a HBV subtype consists of the following steps:

STEP 1 ( Initial Alignment ); Select a "leader" whose undiluted log, ^

viral concentration x, is set to be zero. Using the 8 piecewise linear

functions defined by the log S/N values obtained for this leader at x =

0, 1, ..., 5, the estimates of the other x's (i = 2, 3, ..., n) on this

selected scale are then obtained using a least squares method.

STEP 2 ( Curve Fitting ); Estimate the response functions f (A = 1, 2,

..., 8) from the aligned data using exploratory regression techniques. The

overall sum of squared residuals is also computed.

STEP 3 ( Updating Alignment ): Using the f^'s obtained in STEP 2, the

estimates of x.'s (i = 1, 2, ..., n) are updated. If the resulting

decrease in the overall sum of squared residuals is less than a specified

threshold, stop; otherwise, go to STEP 2.

A detailed description of the proposed alignment procedure is given in

the following, but it should be noted that this procedure only provides

estimates of the (x. - x,)'s, and not the x.'s themselves.

4.1 Initial Alignment Step

The aim of the procedure is to align the uninown viral concentrations of

the serum samples from a HBV subtype. It is equivalent to estimating the

(x. - X, )'s (i = 1, 2, ..., n), and if we set x, to be zero, then we

are only estimating the x.'s (1 = 2, ..., n). In this step, the objective

is to find initial estimates of the x.'s.

- 8 -

A A A A

Since from [1], y. , = f (x -d) + e. and each f is a smooth growth curve

increasing monotonely from zero to a unique local maximum, the following 8

piecewise linear functions, f^ (A = 1, 2, ..., 8), defined by y, ^ (d = 0,

p J- .a

1, ,,,, 5; A = 1, 2, ..,, 8) are approximations of the f 's:

f^ (x) = a^ + b^ X, d < X < d + 1; d = 0, 1, 2, ... [2]

where a, and b, is respectively the intercept and slope of the line between

AAA â€žA ,

(d.yi d) ^^^ i^+'i-, yi,d+l)Â« ^^'^ yi,d = for d > 6 because 10Â° or higher

dilutions of practically all serum samples would contain effectively no

viral protein. The x. 's are then estimated by using a least squares

method. Specifically, for each i, we find the value of x. to minimize the

sum of squared residuals

8 5

ssR. = 1 L [yt ^ - f^ (^-i - ^^^'

1 i,d p 1

A=l d=0

Since interpolation is preferred to extrapolation, the "leader" sample is

ma Y A

the one with the most . y. values among the 8 possible maxima. Using

the f^'s given in [2], it can be shown that if j < x < j + 1 (j = 0,1,2,

P i

..., 5), then the least squares estimate of x. is given by

8 J+5 , . 8 j+5 ^ ^ 8 j+5

X., . = j + A=l d=j ^ ^''^ J A=l d=j d d A=l d=j

^'J 8 j+5 , 2

I I (b^ )

A=l d=j ^

By computing x^., j = 0, 1, 2, ..., 5, and their associated SSR^

values, the least squares estimate of x can be found by identifying the

X. , . with the smallest SSR, value.

- 9 -

It should be noted that, instead of the i 's, we could have used a smooth

spline function (see, for example, Winsberg and Ramsay, 1982); but the least

squares estimate of the x 's would then have to be obtained by a numerical

optimization algorithm. We decide to use the picewise linear functions

because our objective here is to find a set of reasonably good initial esti-

mates of the X 's. Moreover, an estimation method more resistant than the

least squares procedure could have been used; instead, y values associated

1 ,a

with large least-squares residuals are identified and removed from all subse-

quent analysis. The aligned data for the 14 subtype 6 serum samples are

shown in Figure 2b, where the log S/N values are plotted against the nega-

tive aligned log, q viral concentration (-x).

4.2 Curve-Fitting Step

A A

With the aligned data, a better approximation of the f 's than the f 's

can be obtained by pooling the sample data from the same HBV subtype. Our

objective is therefore to fit a function f to the aligned data for monoclo-

clonal antibody A, and the f^'s should be of the same functional form. By

examining Figure 2b, it can be observed that the variability of log S/N is

not constant across the range of viral concentrations; as expected from the

experimental procedure, the variability is high when the concentration is low

or when log S/N is close to zero. Hence, we use a weighted least squares

procedure, in which log S/N values close to zero carry very little weight,

e

to find a function that would fit the aligned data well.

Several functional forms including the quadratic, cubic. Gamma and Gaussian

curves have been fitted to the aligned data for all known HBV subtypes, but

plots of the residuals obtained for these curves indicate that they do not

- 10-

fit the data well. As suggested by residuals analysis, the best-fitting

function is the logistic curve of the form:

3 3

f^* (x) = exp [ r 13^ x^]/a + exp [ [ B, x^ ] ) [3]

Â° k=0 ^ k=0 ^

A* JV

where f (x) = (f^ (x) + l)/9. This function fits the monotonely increas-

n n

ing part of the aligned data particularly well, but it cannot be used for

extrapolation purposes. Moreover, it should be noted that the cubic term is

not necessary to fit the aligned data for most HBV subtypes; but for those

whose serum samples have high viral concentrations, this extra term is

needed to capture the transition at the peak of the growth curves. The final

fitted curves and the aligned data for subtype 2, 3, 6 and 7 are shown in

Figure 3-6 respectively, and they are not very different from the plots

obtained after this intermediate step. The overall sum of squared residuals

n n 8 5 2

I ssR. = 1 I t [y? d- ^ ^^i-^)l

i=l ^ i=l A=l d=0 ^'^ ^ ^

is also computed for the samples from each subtype.

4.3 Updating Step

With the f^ 's, the x. 's, i=l, 2, ..., n are re-estimated in turn by

n 1

finding the value of x that would minimize

i

Q C

SSR. =1 [ [y^ ^ - f^ (x^ - d)]2

' A=l d=0 ^'^ " ^

Since the updated estimates will be in the neighborhood of the previous

estimates and SSR. is a continuous function of x. , the new estimates can be

obtained numerically. It has been found that the solutions after the first

few iterations are practically the same and hence in order to avoid excessive

computing, a threshold value T is specified so that when the difference in

-11-

Tl

J SSR is not greater than T, the iterative procedure stops; otherwise,

1-1 1

the Iterations will continue by going back to STEP 2.

Although the developed procedure Is unduly dependent on the selected

leader, our Immunoassays data are so well behaved that this dependency does

not create a problem In the present application* We have also examined the

variability of the estimated relative viral concentrations of samples from a

HBV subtype by using a jack-knife procedure. By leaving out one serum

sample at a time, n sets of estimates of relative concentrations are

obtained by the Iterative procedure* Again, since our data align very well,

these solution sets are found to be very similar. Finally, we would like to

point out that this procedure can be generalized to the case where the serum

samples have very high viral concentrations and can also be expanded to

Include other monoclonal anti-HBs antibodies*

- 12 -

5. DISCUSSION

We have described a novel approach, "signature" analysis, for the study

of hepatitis B viral antigenic structure. The cardinal feature of this

approach included: 1) the use of high affinity anti-HBs monoclonal

antibodies, 2) the construction of RIAs specific for different epitopes on

HBsAg, and 3) a statistical technique for aligning the unknown

concentrations of HBsAg present in different individuals of the same

subtype. The estimated binding curves which describe the functional

relationships between log S/N and viral concentration for a panel of

selected monoclonal antibodies are characteristic for any HBV viral strain

and the composite profile defines the antigenic composition of HBV.

"Signature" analysis suggests that HBV shows far more antigenic

heterogeneity than previously recognized by polyvalent anti-HBs antibodies.

First with respect to classic HBsAg subtypes, we find the greatest antigenic

diversity when comparing the ad^ (subtype 6 and 7) to ay (subtype 2 and 3)

classical subtype group. (See, for example. Figure 7a; note that this and

all subsequent figures are obtained by aligning the subtypes which are being

compared.) This diversity is in accord with recent sequence data (Fujiyama

et al., 1983) showing substantial changes in amino acid composition between

HBsAg subtypes adw and ayw. In constrast, when the ad and ay domains are

the same, and the difference between the w specificities are examined,

signature analysis demonstrates remarkable antigenic homogeneity among the w

domains (see, for example. Figure 7b).

Further evidence of the antigenic diversity of HBV is supported by the

demonstration of subgroups within the "classic" HBsAg subtypes. For

example, ayw- may now be divided into at least two separate subgroups. In

- 13 -

the United States, an additional subgroup (see Figure 8) has been found to

be distinct from the larger ayw_ (subtype 3) group. Moreover, we have

found that Australian aborigines are infected with a ayw^ strain which

differs substantially from aywo groups or subgroups in the United States

with respect to epitope composition and density. Therefore, "signature"

analysis may permit studies of the genetic evolution of HBS in various parts

of the world. Indeed, preliminary studies of the aywo subtype from Africa

and the Far East suggest additional viral strains distinct from those found

in the United States and Australia. Furthermore, recent investigations

suggest that vertical transmission of HBV from mother to child or horizontal

spread within families breeds true. That is, the RIAs data obtained for

serum samples from all family members infected with HBV including mother and

child align very well. Therefore, "signature" analysis will be useful in

epidemiologic studies of HBV transmission.

Finally we believe that "signature" analysis may also be of value in

studying the antigenic composition of other viral agents. For example, such

an approach may be useful in the characterization of antigenic structure of

polio, influenza, herpes, and arenavirus groups and other viral agents

propagated in tissue culture. Since "signature" analysis does not require

that the concentration of viral protein in complex protein mixtures be

known, study of antigenic similarity or differences among other viruses

would seem possible.

- 14 -

REFEEIENCES

Bolton, A.E., and Hunter, W.M. (1973), "The labeling of proteins to high

specific radioactivities by conjugation to a I^^S-cont-ai^ing acetylating

agent," Journal of Biochemistry , 133, 529-539.

Chisari, F.V. (1984), Advances in Hepatitis Research (editor), Masson

Publishing Company (to appear).

Fujiyama, A., Miyanohara, A., Nozaki, C, Yoneyama, T., Ohotomo, N. and

Matsubara, K. (1983). Nucleic Acid Research , 11, 4601-4610.

Shorey, J., Brown, R.D. and Wands, J.R. (1981), "A new analysis of HBV

subtypes by epitope identification using monoclonal anti-HBs

radioimmunoassays," Hepatology , 1, 546.

Wands, J.R., Ben-Porath, E. and Wong, M.A. (1984), "Use of monoclonal

antibodies in the study of viral hepatitis," in Advances in Hepatitis

Research (Chisori, F.V., ed.), Masson Publishing Company (to appear).

, Carlson, R.I., Schumacher, H., Zurawski, V.R., and Isselbacher,

K.J. (1981), "Immunodiagnosis of hepatitis B by epitope identification with

high affinity IgM monoclonal antibodies," Proceedings of the National

Academy of Science, USA , 78, 1214-1218.

, Wong, M.A., Shorey, J., Brown, R.D., Marciniak, R., and

Isselbacher, K.J. (1984), "Hepatitis B viral antigenic structure: signature

analysis by monoclonal radioimmunoassays," Proceedings of the National

Academy of Science, USA (to appear).

, and Zurawski, V.R. (1981), "High affinity monoclonal antibodies

to hepatitis B surface antigen (HBsAg) produced by somatic cell hybrids,'

Gastroenterology , 225-232.

Winsberg, S., and Ramsay, J.O. (1982), "Monotone splines: a family of

transformations useful for data analysis," C0MPSAT,5, 451-456.

Log _ Dilution Factor of Serum

Log S/N

Log â€ž Viral Concentration in Serum

Fig. 1. A sketch of the expected relationship between

log S/N and log â€ž viral concentration in

serum.

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