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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 ayw„ (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 ''^ ^^^ l°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 „A ,

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

° 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 „ Viral Concentration in Serum



Fig. 1. A sketch of the expected relationship between
log S/N and log „ viral concentration in
serum.



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Online LibrarySloan School of ManagementAn exploratory analysis of radioimmunoassays data : establishing subtypes of the hepatitis-B virus → online text (page 1 of 2)