When two are better than one: Modeling the mechanisms of antibody mixtures

Tal Einav1, Jesse D. Bloom1,2*

${}^{1}\,$Basic Sciences Division and Computational Biology Program, Fred
Hutchinson Cancer Research Center, Seattle, WA, United States of America

${}^{2}\,$Howard Hughes Medical Institute, Seattle, WA, United States of
America

*<EMAIL_ADDRESS>
## Abstract

It is difficult to predict how antibodies will behave when mixed together,
even after each has been independently characterized. Here, we present a
statistical mechanical model for the activity of antibody mixtures that
accounts for whether pairs of antibodies bind to distinct or overlapping
epitopes. This model requires measuring $n$ individual antibodies and their
$\frac{n(n-1)}{2}$ pairwise interactions to predict the $2^{n}$ potential
combinations. We apply this model to epidermal growth factor receptor (EGFR)
antibodies and find that the activity of antibody mixtures can be predicted
without positing synergy at the molecular level. In addition, we demonstrate
how the model can be used in reverse, where straightforward experiments
measuring the activity of antibody mixtures can be used to infer the molecular
interactions between antibodies. Lastly, we generalize this model to analyze
engineered multidomain antibodies, where components of different antibodies
are tethered together to form novel amalgams, and characterize how well it
predicts recently designed influenza antibodies.

## Author summary

With the rise of new combination antibody therapeutic regimens, it is
important to understand how antibodies work together as well as individually.
Here, we investigate the specific case of monoclonal antibodies targeting a
cancer-causing receptor or the influenza virus and develop a statistical
mechanical framework that predicts the effectiveness of a mixture of
antibodies. The power of this model lies in its ability to make a large number
of predictions based on a limited amount of data. For example, once 10
antibodies have been individually characterized, our model can predict how any
of the $2^{10}=1024$ combinations will behave. This predictive power provides
ample opportunities to test our model and paves the way to expedite the design
of future therapeutics.

## Introduction

Antibodies can bind with strong affinity and exquisite specificity to a
multitude of antigens. Due to their clinical and commercial success,
antibodies are one of the largest and fastest growing classes of therapeutic
drugs [1]. While most therapies currently use monoclonal antibodies (mAbs),
mounting evidence suggests that mixtures of antibodies can behave in
fundamentally different ways [2]. There is ample precedent for the idea that
combinations of therapeutics can be extremely powerful—for instance, during
the past 50 years the monumental triumphs of combination anti-retroviral
therapy and chemotherapy cocktails have provided unprecedented control over
HIV and multiple types of cancer [3, 4], and in many cases no single drug has
emerged with comparable effects. However, it is difficult to predict how
antibody mixtures will behave relative to their constitutive parts. Often, the
vast number of potential combinations is prohibitively large to systematically
test, since both the composition of the mixture and the relative concentration
of each component can influence its efficacy [5].

Here, we develop a statistical mechanical model that bridges the gap between
how an antibody operates on its own and how it behaves in concert.
Specifically, each antibody is characterized by its binding affinity and
potency, while its interaction with other antibodies is described by whether
its epitope is distinct from or overlaps with theirs. This information enables
us to translate the molecular details of how each antibody acts individually
into the macroscopic readout of a system’s activity in the presence of an
arbitrary mixture.

To test the predictive power of our framework, we apply it to a beautiful
recent case study of inhibitory antibodies against the epidermal growth factor
receptor (EGFR), where 10 antibodies were individually characterized for their
ability to inhibit receptor activity and then all possible 2-Ab and 3-Ab
mixtures were similarly tested [6]. We demonstrate that our framework can
accurately predict the activity of these mixtures based solely on the
behaviors of the ten monoclonal antibody as well as their epitope mappings.

Lastly, we generalize our model to predict the potency of engineered
multidomain antibodies from their individual components. Specifically, we
consider the recent work by Laursen et al. where four single-domain antibodies
were assayed for their ability to neutralize a panel of influenza strains, and
then the potency of constructs comprising 2-4 of these single-domain
antibodies were measured [7]. Our generalized model can once again predict the
efficacy of the multidomain constructs based upon their constitutive
components, once a single fit parameter is inferred to quantify the effects of
the linker joining the single-domain antibodies. This enables us to
quantitatively ascertain how tethering antibodies enhances the two key
features of potency and breadth that are instrumental for designing novel
anti-viral therapeutics. Notably, our models do _not_ posit complex molecular
synergy between antibodies. Our results therefore show that many antibody
mixtures function without synergy, and hence that their effects can be
computationally predicted to expedite future experiments.

## Results

### Modeling the mechanisms of action for antibody mixtures

Consider a monoclonal antibody that binds to a receptor and inhibits its
activity. Two parameters characterize this inhibition: (1) the dissociation
constant $K_{D}$ quantifies an antibody’s binding affinity (with a smaller
value indicating tighter binding) and (2) the potency $\alpha$ relates the
activity when an antibody is bound to the activity in the absence of antibody.
A value of $\alpha=1$ represents an impotent antibody that does not affect
activity while $\alpha=0$ implies that an antibody fully inhibits activity
upon binding. As derived in S1 Text Section A, for an antibody that binds to a
single site on a receptor, the activity at a concentration $c$ of antibody is
given by

$\text{Fractional
Activity}=\frac{1+\alpha\frac{c}{K_{D}}}{1+\frac{c}{K_{D}}}.$ (1)

To characterize a mixture of two antibodies, we not only need their individual
dissociation constants and potencies but also require a model for how these
antibodies interact. When two antibodies bind to distinct epitopes, the
simplest scenario is that their ability to bind and inhibit activity is
independent of the presence of the other antibody, and hence that their
combined potency when simultaneously bound equals the product of their
individual potencies (Fig 1A). Alternatively, if the two antibodies compete
for the same epitope, they cannot both be simultaneously bound (Fig 1B).

We also define the general case of a synergistic interaction where the binding
of the first antibody alters the binding or potency of the second antibody
(Fig 1C, purple text). This definition encompasses cases where the second
antibody binds more tightly ($K_{D,\text{eff}}^{(2)}<K_{D}^{(2)}$) or more
weakly ($K_{D,\text{eff}}^{(2)}>K_{D}^{(2)}$) in the presence of the first
antibody, as well as when the potency of the second antibody may increase
($\alpha_{2,\text{eff}}>\alpha_{2}$) or decrease
($\alpha_{2,\text{eff}}<\alpha_{2}$). This also includes cases where two
epitopes slightly overlap and partially inhibit one another’s binding, and the
competitive binding model can be viewed as the extreme limit
$K_{D,\text{eff}}^{(2)}\to\infty$ where one antibody infinitely penalizes the
binding of the other.

Figure 1: Binding modes for a 2-Ab mixture. Two antibodies with concentrations
$c_{1}$ and $c_{2}$ can bind (A) independently to different epitopes or (B)
competitively to the same epitope. (C) Antibodies bind synergistically if
either the product of binding affinities ($K_{D}^{(j)}$) or potencies
($\alpha_{j}$) are altered when both antibodies bind.

While the synergistic model in Fig 1C has the merit of being highly general,
an important feature of the independent and competitive models (Fig 1A,B) is
that they predict all antibody combinations with few parameters. In both of
these latter models, once the $K_{D}^{(j)}$ and $\alpha_{j}$ of 10 antibodies
are known (which requires $2\cdot 10$ experiments) and their epitopes are
mapped ($\frac{10\cdot 9}{2}$ additional experiments), the potency of all
$2^{10}=1024$ possible mixtures of these antibodies can be predicted without
recourse to fitting. In contrast, because the synergistic model allows
arbitrary interactions between each combination of antibodies, the behavior of
a mixture exhibiting synergy cannot be predicted without actually making a
measurement on that combination to quantify the synergy.

For these reasons, in this work we focus on the two cases of independent or
competitive binding and show how we can combine both models to transform our
molecular understanding of each monoclonal antibody’s action into a prediction
of the efficacy of an antibody mixture. Deviations from our predictions
provide a rigorous way to measure antibody synergy by computing
$\frac{K_{D,\text{eff}}^{(2)}}{K_{D}^{(2)}}$ and
$\frac{\alpha_{2,\text{eff}}}{\alpha_{2}}$.

To mathematize the independent and competitive binding models, we enumerate
the possible binding states and compute their relative Boltzmann weights. The
fractional activity of each state equals the product of its relative
probability and relative activity divided by the sum of all relative
probabilities for normalization (see S1 Text Section A). When two antibodies
bind independently as in Fig 1A, this factors into the form

$\text{Fractional Activity}_{\text{(distinct
epitopes)}}=\left(\frac{1+\alpha_{1}\frac{c_{1}}{K_{D}^{(1)}}}{1+\frac{c_{1}}{K_{D}^{(1)}}}\right)\left(\frac{1+\alpha_{2}\frac{c_{2}}{K_{D}^{(2)}}}{1+\frac{c_{2}}{K_{D}^{(2)}}}\right).$
(2)

If these two antibodies compete for the same epitope as in Fig 1B, the
activity becomes

$\text{Fractional Activity}_{\text{(overlapping
epitopes)}}=\frac{1+\alpha_{1}\frac{c_{1}}{K_{D}^{(1)}}+\alpha_{2}\frac{c_{2}}{K_{D}^{(2)}}}{1+\frac{c_{1}}{K_{D}^{(1)}}+\frac{c_{2}}{K_{D}^{(2)}}}.$
(3)

These equations are readily extended to mixtures with three or more antibodies
(see S1 Text Section A).

### Antibody mixtures against EGFR are well characterized using independent
and competitive binding models

To test the predictive power of the independent and competitive binding
models, we applied them to published experiments on the epidermal growth
factor receptor (EGFR) where ten monoclonal antibodies were individually
characterized and then the activity of all 165 possible 2-Ab and 3-Ab mixtures
was measured [6]. We first use each monoclonal antibody’s response to infer
its dissociation constant $K_{D}$ and potency $\alpha$. We then utilize
surface plasmon resonance (SPR) measurements to determine which pairs of
antibodies bind independently and which compete for the same epitope. These
data enable us to use the above framework and predict EGFR activity in the
presence of any mixture.

EGFR is a transmembrane protein that activates in the presence of epidermal
growth factors. Upon ligand binding, the receptor’s intracellular tyrosine
kinase domain autophosphorylates which leads to downstream signaling cascades
central to cell migration and proliferation. Overexpression of EGFR has been
linked to a number of cancers, and decreasing EGFR activity in such tumors by
sterically occluding ligand binding has reduced the rate of cancer
proliferation [6].

Koefoed et al. investigated how a panel of ten monoclonal antibodies inhibit
EGFR activity in the human cell line A431NS [6]. They then measured how 1:1
mixtures of two antibodies or 1:1:1 mixtures of three antibodies affect EGFR
activity. All measurement were carried out at a total concentration of
$2\frac{\mu\text{g}}{\text{mL}}$, implying that each antibody was half as
dilute in the 2-Ab mixtures and one-third as dilute in the 3-Ab mixtures
relative to the monoclonal antibody measurement.

The 45 possible 2-Ab mixtures (35 binding to distinct epitopes; 10 binding to
overlapping epitopes) and the 120 possible 3-Ab mixtures (50 binding to
distinct epitopes; 70 binding to overlapping epitopes) were assayed for their
ability to inhibit EGFR activity. Fig 2A shows the experimental measurements
for mixtures of two antibodies, with the monoclonal antibody measurements
shown on the diagonal, the measured activity of 2-Ab mixtures shown on the
bottom-left and the predicted activity on the top-right. Each antibody is
labeled with its binding epitopes inferred through SPR [6], so that antibodies
binding to overlapping epitopes are predicted using Eq (3) (pairs within the
dashed gray boxes) while mixtures binding to distinct epitopes use Eq (2).

For example, antibodies #1 and #4 bind to distinct epitopes (III/C and III/B,
respectively). Hence, the predicted activity of their mixture (0.50) very
nearly equals the product of their individual activity ($0.65\times
0.75=0.49$), with the slight deviation arising because each antibody
concentration was halved in the mixture
($c_{1}=c_{2}=1\frac{\mu\text{g}}{\text{mL}}$ for the 2-Ab mixture
characterized by Eq (2), whereas the individual mAbs were measured at
$c=2\frac{\mu\text{g}}{\text{mL}}$ using Eq (1)). The predicted activity
roughly approximates the measured value 0.43 of the mixture.

Figure 2: Predicting how antibody mixtures affect the epidermal growth factor
receptor (EGFR). (A) The fractional activity of EGFR in the presence of
monoclonal antibodies (diagonal) together with the measured (bottom-left) and
predicted (top-right) activity of all 2-Ab combinations. The dashed gray boxes
enclose antibody pairs that compete for the same epitope while all other pairs
bind independently. (B) The predicted versus measured fractional activity for
all 2-Ab and 3-Ab mixtures using the same epitope mapping as in Panel A
inferred by SPR. Without the epitope map, the activity of the mixtures could
alternately be predicted by assuming that all antibodies either (C) bind
independently or (D) compete for the same epitope; in either case, the
resulting predictions fall further from the diagonal line, indicating poorer
predictive power.

On the other hand, antibodies #1 and #2 bind to the same epitope (III/C), and
hence their predicted combined activity (0.67) lies between their individual
activities (0.65 and 0.69) since both antibodies compete for the same site.
The measured activity of the mixture (0.65) closely matches the prediction of
the overlapping epitope model, but is very different than the prediction of
0.45 made by the distinct-binding model.

Fig 2B shows the measured EGFR activity in the presence of all 2-Ab and 3-Ab
mixtures is highly correlated with the predicted activity ($R^{2}=0.90$)
Notably, the predictions are made solely from the monoclonal antibody data and
epitope measurements, and do not involve any fitting of the 2-Ab or 3-Ab
measurements. The strong correlation between the predicted and measured
activities suggests that EGFR antibody mixtures can be characterized with
minimal synergistic effects in either their binding or effector functions. If
we did not have the epitope mapping through SPR and assumed that all
antibodies bound to distinct epitopes (Fig 2C, $R^{2}=0.85$) or competed for
the same epitope (Fig 2D, $R^{2}=0.86$), the resulting predictions are
slightly more scattered from the diagonal, demonstrating that properly
acknowledging which pairs of antibodies vie for the same epitope boosts the
predictive power of the model.

That said, the predictions incorporating the SPR mapping display a consistent
bias towards having a slightly lower measured than predicted activity,
suggesting that several pairs of antibody enhance one another’s binding
affinity or potency. To quantify this, if we recharacterize the activity from
the 2-Ab mixtures to a synergistic model where each $\alpha_{2,\text{eff}}$ is
fit to exactly match the data, we find an average value of
$\frac{\alpha_{2,\text{eff}}}{\alpha_{2}}=0.9$, showing that when pairs of
antibodies are simultaneously bound they typically boost their collective
inhibitory activity by $\sim$10%.

### Differentiating distinct versus overlapping epitopes using antibody
mixtures

In the previous section, we used SPR measurements to quantify which antibodies
compete for overlapping epitopes, thereby permitting us to translate the
molecular knowledge of antibody interactions into a macroscopic quantity of
interest, namely, the activity of EGFR. In this section, we do the reverse and
utilize activity measurements to categorize which subsets of antibodies bind
to overlapping epitopes. This method can be applied to model antibody mixtures
in other biological systems where SPR measurements are not readily available.

For the remainder of this section, we ignore the known epitope mappings
discerned by Koefoed et al. and consider what mapping best characterizes the
data. For example, given the individual activities of antibody #1 (0.65) and
#2 (0.69), the predicted activity of their combination (at the concentration
of $1\frac{\mu\text{g}}{\text{mL}}$ for each antibody dictated by the
experiments) would be 0.45 if they bind to distinct epitopes and 0.67 if they
bind to overlapping epitopes. Since the measured activity of this mixture was
0.65, it suggests the latter option. We note that such analysis will work best
for potent antibodies (whose individual activity is far from 1), since only in
this regime will the predictions of the distinct versus overlapping models be
significantly different. Therefore, the activity measurements of each
individual antibody would optimally be carried out at saturating
concentrations (where Eq (1) is as far from 1 as possible).

Proceeding to the other antibodies, we characterize each pair according to
whichever model prediction lies closer to the experimental measurement. To
account for experimental error, we left an antibody pair uncategorized if the
two model predictions were too close to one another (within $4\sigma=0.16$
where $\sigma$ is the SEM of the measurements) or if the experimental
measurement was close (within $1\sigma$) to the average of the two model
predictions (see S1 Text Section B).

Fig 3A shows how this analysis compares to the experimental measurement
inferred by SPR. While the model predictions are much sparser (with the
majority of antibody pairs uncategorized because the two model predictions
were too close to one another), the classifications only disagreed with the
SPR measurements in two cases (claiming that antibodies #7-8 overlap with
antibody #10; notice that antibodies #7-8 have individual activities close to
1, making them difficult to characterize).

Using these classifications, we defined unique EGFR epitopes by grouping
together any antibodies that bind to overlapping epitopes. In this way, we
split the ten antibodies into four distinct groups (antibodies #1-3, #4-5, #6,
and #7-10 indicated by the dashed gray rectangles in Fig 3A), enabling us to
distinguish which antibodies bind independently or competitively and hence
predict the activity of the 2-Ab and 3-Ab mixtures. Note that it is not the
pairwise classification between two antibodies that determines whether we
apply the distinct or competitive models, but rather these four groupings of
antibody epitopes. For example, although antibodies #7 and #8 are
uncategorized through their 2-Ab mixture, they fall within a single epitope
group and hence are considered to bind competitively. Similarly, antibody #1
and #4 are modeled as binding independently because they belong to two
distinct epitope groups. Antibody #6 is considered to be in its own epitope
group since it did not overlap with any other antibody.

Surprisingly, the results shown in Fig 3B have a coefficient of determination
$R^{2}=0.90$ that is on par with the results obtained using the SPR
measurements (Fig 2B). This suggests that there is no loss in the predictive
power of the model when an epitope mapping is inferred through activity
measurements.

Figure 3: Classifying antibody epitopes as overlapping or distinct. (A)
Comparing the experimentally measured activity to the overlapping or distinct
epitope models enables us to characterize each antibody pair (provided the two
models predict sufficiently different activities). (B) The resulting
predictions for the 2-Ab and 3-Ab mixtures have the same predictive power
($R^{2}=0.90$) as a model that relies on epitope groupings given by SPR
measurements (Fig 2B).

In summary, whether antibodies bind independently or competitively can be
determined either: (1) directly through pairwise competition experiments or
(2) by analyzing the activity of their 2-Ab mixtures in light of our two
models. When this information is combined with the potency and dissociation
constant of each antibody, the activity of an arbitrary mixture can be
predicted. The Supplementary Information contains a Mathematica program that
can analyze either form of the pairwise interactions to determine the epitope
grouping. If the characteristics of the individual antibodies are also
provided, the program can predict the activity of any antibody mixtures at any
specified ratio of the constituents.

### Multidomain antibodies boost breadth and potency via avidity

While the previous sections analyzed combinations of whole, unmodified
antibodies, we now extend our framework to connect with the rising tide of
engineering efforts that genetically fuse different antibody components to
construct multi-domain antibodies [8]. Specifically, we focus our attention on
recent work by Laursen et al. who isolated single-domain antibodies from
llamas immunized with H2 or H7 influenza hemagglutinin (HA) [7]. The four
single-domain antibodies isolated in this manner included one antibody that
preferentially binds influenza A group 1 strains ($\text{Ab}_{\text{A1}}$),
another that binds influenza A group 2 strains ($\text{Ab}_{\text{A2}}$), and
two antibodies that bind to influenza B strains ($\text{Ab}_{\text{B}}^{(1)}$
and $\text{Ab}_{\text{B}}^{(2)}$). Fig 4A,B shows data from a representative
influenza A group 1 strain (blue dot, only bound by the blue
$\text{Ab}_{\text{A1}}$), influenza A group 2 strain (green dot, only bound by
the green $\text{Ab}_{\text{A2}}$), and influenza B strain (gold dot, bound by
both of the yellow $\text{Ab}_{\text{B}}^{(1)}$ and
$\text{Ab}_{\text{B}}^{(2)}$ antibodies).

In the contexts of rapidly evolving pathogens such as influenza, two important
characteristics of antibodies are their potency and breadth. Potency is
measured by the inhibitory concentration $\text{IC}_{50}$ at which 50% of a
virus is neutralized, where a smaller $\text{IC}_{50}$ represents a better
antibody. Breadth is a measure of how many strains are susceptible to an
antibody.

In an effort to improve the potency and breadth of their antibodies, Laursen
et al. tethered together different domains using a flexible amino acid linker
(right-most columns of Fig 4A,B) and tested them against a panel of influenza
strains. To make contact with these multidomain constructs, consider a
concentration $c$ of the tethered antibody
$\text{Ab}_{\text{A1}}\text{\textendash}\text{Ab}_{\text{A2}}$. Relative to
the unbound HA state, the $\text{Ab}_{\text{A1}}$ or $\text{Ab}_{\text{A2}}$
portions of the antibody will neutralize the virus with relative probability
$\frac{c}{\text{IC}_{50,\text{A1}}}$ or $\frac{c}{\text{IC}_{50,\text{A2}}}$,
respectively. Although neutralization is mediated by antibody binding, the two
quantities may or may not be proportional [9, 10, 11], and hence we replace
dissociation constants with $\text{IC}_{50}$s in our model (see S1 Text
Section C).

Laursen et al. determined that their tethered constructs cannot intra-spike
crosslink two binding sites on a single HA trimer, but they can inter-spike
crosslink adjacent HA [7]. The linker connecting the two antibody domains
facilitates such crosslinking, since when one domain is bound the other domain
is confined to a smaller volume around its potential binding sites. This
effect can be quantified by stating that the second domain has an effective
concentration $c_{\text{eff}}$ (Fig 4C, purple), making the relative
probability of the doubly bound state
$\frac{c}{\text{IC}_{50,\text{A1}}}\frac{c_{\text{eff}}}{\text{IC}_{50,\text{A2}}}$.
Therefore, the fraction of virus neutralized by two tethered antibody domains
is given by

$\text{Fraction
Neutralized}=\frac{\frac{c}{\text{IC}_{50,\text{A1}}}+\frac{c}{\text{IC}_{50,\text{A2}}}+\frac{c}{\text{IC}_{50,\text{A1}}}\frac{c_{\text{eff}}}{\text{IC}_{50,\text{A2}}}}{1+\frac{c}{\text{IC}_{50,\text{A1}}}+\frac{c}{\text{IC}_{50,\text{A2}}}+\frac{c}{\text{IC}_{50,\text{A1}}}\frac{c_{\text{eff}}}{\text{IC}_{50,\text{A2}}}}.$
(4)

Note that this equation assumes that influenza virus is fully neutralized at
saturating concentrations of antibody ($\alpha=0$ in Eq (1), with Fraction
Neutralized analogous to $1-\text{Fractional Activity}$).

Figure 4: Tethering influenza antibodies increases breadth and potency. (A)
The influenza A antibodies $\text{Ab}_{\text{A1}}$ and $\text{Ab}_{\text{A2}}$
were tethered together to form
$\text{Ab}_{\text{A1}}\text{\textendash}\text{Ab}_{\text{A2}}$ while (B) two
influenza B antibodies formed
$\text{Ab}_{\text{B}}^{(1)}\text{\textendash}\text{Ab}_{\text{B}}^{(2)}$.
Representative data shown for an influenza A group 1 (blue), influenza A group
2 (green), and influenza B (gold) strains. Strong potency is marked by a small
$\text{IC}_{50}$ while large breadth implies that multiple strains are
controlled by an antibody. (C) Representative states of HA and their
corresponding Boltzmann weights for multidomain antibodies, where crosslinking
between adjacent spikes boosts neutralization via avidity
($c_{\text{eff}}=1400\,\text{nM}$ in Eq (5)). (D) Theoretical predictions of
the potency of all multidomain antibodies versus their measured values. The
red points denote two outlier influenza strains discussed in the text that are
not neutralized by $\text{Ab}_{\text{A1}}$ or $\text{Ab}_{\text{A2}}$
individually but are highly neutralized by their combination.

The $\text{IC}_{50}$ of the tethered construct is defined as the concentration
$c$ at which half of the virus is neutralized, which can be solved to yield

$\text{IC}_{\text{50,A1\text{\textendash}A2}}=\frac{\text{IC}_{\text{50,A1}}\,\text{IC}_{\text{50,A2}}}{c_{\text{eff}}+\text{IC}_{\text{50,A1}}+\text{IC}_{\text{50,A2}}},$
(5)

with an analogous expression holding for the
$\text{Ab}_{\text{B}}^{(1)}\text{\textendash}\text{Ab}_{\text{B}}^{(2)}$
construct. Using the measured $\text{IC}_{50}$s of
$\text{Ab}_{\text{A1}}\text{\textendash}\text{Ab}_{\text{A2}}$ and
$\text{Ab}_{\text{B}}^{(1)}\text{\textendash}\text{Ab}_{\text{B}}^{(2)}$
against the various influenza strains, we can infer the value of the single
parameter $c_{\text{eff}}=1400\,\text{nM}$ (see S1 Text Section C). This
result is both physically meaningful and biologically actionable, as it
enables us to predict the $\text{IC}_{50}$ of the tethered multidomain
antibodies against the entire panel of influenza strains. Fig 5A,B compares
the resulting predictions to the experimental measurements, where plot markers
linked by horizontal line segments indicate a close match between the
predicted and measured values.

The two tethered antibodies display unique trends that arise from their
compositions. Since the two domains in
$\text{Ab}_{\text{A1}}\text{\textendash}\text{Ab}_{\text{A2}}$ bind nearly
complementary strains, the tethered construct will increase breadth (since
this multidomain antibodies can now bind to both group 1 and group 2 strains)
but will only marginally improve potency. Mathematically, if
$\text{Ab}_{\text{A1}}$ binds tightly to an influenza A group 1 strain while
$\text{Ab}_{\text{A2}}$ binds weakly to this same strain
($\text{IC}_{\text{50,A2}}\to\infty$), their tethered construct has an
$\text{IC}_{\text{50,A1\text{\textendash}A2}}\approx\text{IC}_{\text{50,A1}}$.
Said another way,
$\text{Ab}_{\text{A1}}\text{\textendash}\text{Ab}_{\text{A2}}$ should be
approximately as potent as a mixture of the individual antibodies
$\text{Ab}_{\text{A1}}$ and $\text{Ab}_{\text{A2}}$. Note that since the
experiments could not accurately measure weak binding ($>1000\,\text{nM}$),
the predicted $\text{IC}_{50}$ for the multidomain antibodies represents a
lower bound.

On the other hand, tethering the two influenza B antibodies yields a marked
improvement in potency over either individual antibody, since both domains can
bind to any influenza B strain and boost neutralization via avidity. The
process of engineering a multivalent interaction is reminiscent of engineered
bispecific IgG [8], and adding additional domains could yield further
enhancement in potency, provided that all domains can simultaneously bind.

Figure 5: Neutralization of multidomain antibodies. (A,B) The potency of the
2-Ab constructs and their constitutive antibodies against a panel of influenza
strains. $\text{Ab}_{\text{A1}}$ primarily binds influenza A group 1 (blue),
$\text{Ab}_{\text{A2}}$ to influenza A group 2 (green), and the two
$\text{Ab}_{\text{B}}$ antibodies to influenza B strains (gold). (C) All four
antibodies were tethered to form the linear chain
$\text{Ab}_{\text{A1}}\text{\textendash}\text{Ab}_{\text{A2}}\text{\textendash}\text{Ab}_{\text{B}}^{(1)}\text{\textendash}\text{Ab}_{\text{B}}^{(2)}$
and (D) two copies of this chain were placed on an IgG backbone. The model
suggests that the two arms of the IgG are not capable of simultaneously
binding a virion. Red lines indicate two outlier influenza strains discussed
in the text that are not neutralized by $\text{Ab}_{\text{A1}}$ or
$\text{Ab}_{\text{A2}}$ individually but are highly neutralized by their
combination. Data was digitized from Figs 1 and 3 of Ref [7].

While the model is able to characterize the majority of tethered antibodies,
it also highlights some of the outliers in the data. For example, the H3N2
strains A/Panama/2007/99 and A/Wisconsin/67/05 were poorly neutralized by
either $\text{Ab}_{\text{A1}}$ or $\text{Ab}_{\text{A2}}$ ($\text{IC}_{50}\geq
1000\,\text{nM}$), but the tethered construct exhibited an
$\text{IC}_{50}=14\,\text{nM}$ and $\text{IC}_{50}=17\,\text{nM}$,
respectively, far more potent than the $300\,\text{nM}$ lower limit predicted
for both viruses (red circles in Fig 4D and red lines in Fig 5A).
Interestingly, Laursen et al. found that mixing the individual, untethered
antibodies $\text{Ab}_{\text{A1}}$ and $\text{Ab}_{\text{A2}}$ also resulted
in shockingly poor neutralization ($\text{IC}_{50}\geq 1000\,\text{nM}$),
suggesting that the tether is responsible for the increase in potency [7].
From the vantage of our quantitative model, this outlier cries out for further
investigation.

To further boost neutralization, Laursen et al. created two additional
constructs that combined all four antibody domains, the first being the linear
chain
($\text{Ab}_{\text{A1}}\text{\textendash}\text{Ab}_{\text{A2}}\text{\textendash}\text{Ab}_{\text{B}}^{(1)}\text{\textendash}\text{Ab}_{\text{B}}^{(2)}$).
Since the influenza A antibodies do not bind the influenza B strains (and vise
versa), this construct should have the same $\text{IC}_{50}$ as
$\text{Ab}_{\text{A1}}\text{\textendash}\text{Ab}_{\text{A2}}$ for the
influenza A strains and as
$\text{Ab}_{\text{B}}^{(1)}\text{\textendash}\text{Ab}_{\text{B}}^{(2)}$ for
the influenza B strains, as was found experimentally (compare the Predicted
columns in Fig 5A-C). For example, the two H3N2 strains (A/Panama/2007/99 and
A/Wisconsin/67/05) were again found to have measured $\text{IC}_{50}$s
($15\,\text{nM}$ and $23\,\text{nM}$) far smaller than their predicted lower
bound of $300\,\text{nM}$ (red squares in Fig 4D, red lines in Fig 5C).

A second construct containing all four antibody domains attached two copies of
$\text{Ab}_{\text{A1}}\text{\textendash}\text{Ab}_{\text{A2}}\text{\textendash}\text{Ab}_{\text{B}}^{(1)}\text{\textendash}\text{Ab}_{\text{B}}^{(2)}$
through an IgG backbone (Fig 5D). Since the identical domains in both arms of
this construct should be able to simultaneously bind, the new antibody should
markedly boost potency through avidity. Surprisingly, the neutralization of
this final construct was well characterized as half the $\text{IC}_{50}$ of an
individual
$\text{Ab}_{\text{A1}}\text{\textendash}\text{Ab}_{\text{A2}}\text{\textendash}\text{Ab}_{\text{B}}^{(1)}\text{\textendash}\text{Ab}_{\text{B}}^{(2)}$,
suggesting that there was no noticeable avidity and that the increase in
neutralization only arose from having twice as many antibody domains. As
above, this intriguing result presents an opportunity to both quantitatively
check experimental results and to advocate for future studies in potentially
highly promising directions. In this particular instance, it suggests that the
IgG backbone used did not permit simultaneous binding of both arms. If a
different multivalent scaffold (perhaps with greater flexibility or with
longer linkers) enabled bivalent binding of both arms, it could potentially
increase the neutralization of this construct by 100-fold as seen in the
influenza B constructs.

## Discussion

In this work, we developed a statistical mechanical model that predicts the
collective efficacy of an antibody mixture whose constituents are assumed to
bind to a single site on a receptor. Each antibody is first individually
characterized by its ability to bind the receptor (through its dissociation
constant $K_{D}$) and inhibit activity (via its potency $\alpha$) as per Eq
(1). Importantly, this implies that the activity of each monoclonal antibody
must be measured at a minimum of two concentrations in order to infer both
parameters, and additional measurements would further refine these parameter
values and the corresponding model predictions.

After each antibody is individually characterized, the activity of a
combination of antibodies will depend upon whether they bind independently to
distinct epitopes or compete for overlapping epitopes. Theoretical models
often assume for simplicity that all antibodies bind independently, and in the
contexts where this constraint can be experimentally imposed such models can
accurately predict the effectiveness of antibody mixtures [12]. Yet when the
antibody epitopes are unknown or when a large number of antibodies are
combined, it is likely that some subset of antibodies will compete with each
other while others will bind independently, which will give rise to a markedly
different response. Our model generalized these previous results to account
for antibody mixtures where arbitrary subsets can bind independently or
competitively (Eqs (2) and (3), S1 Text Section A).

We showed that in the context of the EGFR receptor, where every pairwise
interaction was measured using surface plasmon resonance, our model is better
able to predict the efficacy of all 2-Ab and 3-Ab mixtures than a model that
assumes all antibodies bind independently or competitively (Fig 2). This
suggest that mixtures of antibodies do not exhibit large synergistic effects.
More generally, similar models in the contexts of anti-cancer drug cocktails
and anti-HIV antibody mixtures also found that the majority of cases that were
described as synergistic could instead be characterized by an independent
binding model [13, 12]. This raises the possibility that synergy is more the
exception then the norm, and hence that simple models can computationally
explore the full design space of antibody combinations.

While it is often straightforward to measure the efficacy of $n$ individual
antibodies, it is more challenging to quantify all $\frac{n(n+1)}{2}$ pairwise
interactions and determine which antibodies bind independently and which
compete for an overlapping epitope. We demonstrated that after each antibody
is individually characterized, our model can be applied in reverse by using
the activity of 2-Ab mixtures to classify whether antibodies compete or bind
independently (Fig 3). Surprisingly, while the resulting categorizations were
much sparser than the direct SPR measurements, the classifications produced by
this method predicted the efficacy of antibody combinations with an
$R^{2}=0.90$, comparable to the predictions made using the complete SPR
results (Fig 2B). This suggests that key features of how antibodies interact
on a molecular level can be indirectly inferred from simple activity
measurements of antibody combinations.

Modern bioengineering has opened up a new avenue of mixing antibodies by
genetically fusing different components to construct multi-domain antibodies
[8]. Such antibodies can harness multivalent interactions to greatly increase
binding avidity by over 100x (e.g. comparing the $\text{IC}_{50}$s of the
A/Wisconsin/67/05 and B/Harbin/7/94 strains of the 4-fused domains on an IgG
backbone in Fig 5D to the corresponding $\text{IC}_{50}$s for the individual
antibody domains in Panels A and B). For such constructs, the composition of
the linker can heavily influence the ability to multivalently bind and
neutralize a virus [14, 11], although Laursen et al. surprisingly found little
variation when they modified the length of their amino acid linker (see Table
S11 in Ref [7]). Another curious feature of their system was that placing
their linear 4-domain antibody (Fig 5C) on an IgG backbone (Fig 5D) only
resulted in a 2x decrease in $\text{IC}_{50}$, suggesting that the two “arms”
of the IgG could not simultaneously bind. We would expect that a different
backbone that allows both arms to simultaneously bind would markedly increase
the neutralization potency of this construct. In this way, quantitatively
modeling these multidomain antibodies can guide experimental efforts to design
more potent constructs.

To close, we mention that two possible avenues of future work. First, although
our model classifies antibody epitopes as either distinct or overlapping, SPR
measurements indicate that there is a continuum of possible interactions. It
would be fascinating to translate this more nuanced level of interaction into
more precise dissociation constants when two antibodies are bound. Second,
while our model focused on mixtures of antibodies, it can be applied equally
well to small molecule drugs where the number of distinct combinations may be
prohibitively large to measure experimentally but straightforward to explore
computationally.

## Methods

The coefficient of determination used to quantify how well the theoretical
predictions matched the experimental measurements (Fig 2B-D, Fig 3B, Fig 4D)
was calculated using

$R^{2}=1-\frac{\sum_{j=1}^{n}\left(y_{\text{measured}}^{(j)}-y_{\text{predicted}}^{(j)}\right)^{2}}{\sum_{j=1}^{n}\left(y_{\text{data}}^{(j)}\right)^{2}}$
(6)

where $y_{\text{measured}}$ and $y_{\text{predicted}}$ represent a vector of
the measured and predicted activities for the $n$ mixtures analyzed. In Fig
4D, we computed the $R^{2}$ of $\log_{10}(\text{activity})$ to prevent the
largest activities from dominating the result (since the $\text{IC}_{50}$
values span multiple decades).

Data from the EGFR antibody mixtures was obtained by digitizing Ref [6] Fig S1
using WebPlotDigitizer [15]. Data for the influenza multidomain antibodies was
obtained from the authors of Ref [7].

The EGFR antibody epitopes experimentally characterized through SPR (Fig 3A,
bottom-left) were categorized as overlapping if the average of the two
antibody measurements (with preincubation by either antibody) were $>50$ and
as distinct if the average was $<50$.

The original nomenclature for the antibodies used in Koefoed et al. and
Laursen et al. are given in S1 Text Table S1.

##  Appendix A Characterizing Antibodies Targeting the Receptor Tyrosine
Kinase EGFR

###  A.1 Monoclonal Antibody Binding to a Receptor

In this section, we explain the states and weights notation used to develop
the equilibrium statistical mechanical models used in this work. As a focus,
we consider the case of an antibody binding to a single site on a receptor as
shown in Fig S1A. We assume that the concentration of the antibody far exceeds
that of the receptor, so that one binding event will not noticeably affect the
concentration $c$ of free antibody.

The receptor can exist in two states where it is either unbound or bound to
the antibody, where the relative probability of the bound state compared to
the unbound state equals $\frac{c}{K_{D}}$, where $K_{D}$ is the dissociation
constant of the receptor-antibody binding [16]. The (normalized) probability
of each state is given by its relative weight divided by the sum of the
relative weights of all states. For example, the probability that the receptor
is bound to the antibody is shown to be the standard
$\frac{\frac{c}{K_{D}}}{1+\frac{c}{K_{D}}}$ sigmoidal response. As expected,
the receptor will always be unbound in the absence of antibody ($c=0$), while
the receptor will always be bound at saturating concentrations of antibody
($c\to\infty$).

Each receptor state also has a relative activity (i.e. the activity of the
receptor when it is in this state). In the context of EGFR, where the
fractional activity is measured relative to the receptor in the absence of
antibody, the relative activity of the unbound state is by definition equal to
1. As in the main text, we define the activity of the bound receptor to be
$\alpha$, where $\alpha=0$ implies that the receptor is completely inactive
when the antibody is bound and $\alpha=1$ represents the opposite limit where
antibody binding does not inhibit the receptor’s activity. A value of
$0<\alpha<1$ represents an antibody that partially inhibits activity upon
binding.

The activity of the receptor in each state is given by the product of its
relative activity and the normalized probability of that state. Lastly, the
average activity of the receptor is given by the sum of its activity in each
state,

$\text{Activity}=\frac{1+\alpha\frac{c}{K_{D}}}{1+\frac{c}{K_{D}}}.$ (S1)

Figure S1: A statistical mechanical model of an antibody binding to a
receptor. (A) The relative probability and relative activity of each receptor
state enable us to derive the average activity of the receptor. (B) This
method gives the same result of the dynamics model shown by the multiple
rates, provided that the system is in steady state.

As a quick aside, we note out that all the models considered in this work
analyze binding reactions that quickly reach steady, and hence the equilibrium
model derived above will should accurately describe such systems. Eq (S1) can
also be derived by considering the dynamics of the system shown in the rates
diagram Fig S1B. The unbound receptor (R) will switch to the bound state
(R–Ab) at the rate $ck_{\text{on}}$ (recall that $c$ is the concentration of
antibody) and subsequently unbind at a rate $k_{\text{off}}$. Hence, the
system is governed by the two differential equations

$\displaystyle\frac{d[\text{R}]}{dt}$
$\displaystyle=k_{\text{off}}[\text{R}\text{\textendash}\text{Ab}]-ck_{\text{on}}[\text{R}]$
(S2) $\displaystyle\frac{d[\text{R}\text{\textendash}\text{Ab}]}{dt}$
$\displaystyle=ck_{\text{on}}[\text{R}]-k_{\text{off}}[\text{R}\text{\textendash}\text{Ab}].$
(S3)

Since the total amount of receptor
$[\text{R}_{\text{tot}}]=[\text{R}]+[\text{R}\text{\textendash}\text{Ab}]$ is
fixed, these two equations are equivalent, and either one can be solved to
yield the dynamics of the system, namely,

$[R]=c_{1}e^{-\left(k_{\text{off}}+ck_{\text{on}}\right)t}+[\text{R}_{\text{tot}}]\frac{k_{\text{off}}}{k_{\text{off}}+ck_{\text{on}}}.$
(S4)

$c_{1}$ is fixed by the initial concentration of free receptor $[R]$ at $t=0$,
but regardless of its value, we see that the exponential term will shrink to
zero at a time scale of $\frac{1}{k_{\text{off}}+ck_{\text{on}}}$, after which
the system will be in steady state with
$[R]=[\text{R}_{\text{tot}}]\frac{k_{\text{off}}}{k_{\text{off}}+ck_{\text{on}}}$.
Defining the dissociation constant
$K_{D}\equiv\frac{k_{\text{off}}}{k_{\text{on}}}$, the probability that any
receptor will be unbound in steady state is given by

$\frac{[R]}{[\text{R}_{\text{tot}}]}=\frac{1}{1+\frac{c}{K_{D}}}$ (S5)

as found in Fig S1A.

To close, we note that antibodies typically have a
$K_{D}=10^{-12}\text{\textendash}10^{-8}\,\text{M}$, and assuming a diffusion-
limited on rate
$k_{\text{on}}=10^{8}\text{\textendash}10^{-9}\,\frac{1}{\text{M}\cdot\text{s}}$,
this implies that
$k_{\text{off}}=10^{-3}\text{\textendash}10^{1}\,\frac{1}{\text{s}}$.
Therefore, an upper bound (in the limit of little antibody) for the time scale
it takes for such a system to reach steady state is
$10^{-1}\text{\textendash}10^{3}\,\text{s}$, which is the time before
experimental measurements should be conducted. Since antibodies are typically
preincubated for even longer periods during experiments, an equilibrium model
should be valid in all of the case studies we consider in this work.

###  A.2 The Fractional Activity of 3-Ab Mixtures

As described in the main text, the fractional activity of 2-Ab mixtures is
given by Eq 2 if the two antibodies bind to distinct epitopes and Eq 3 if they
bind to overlapping epitopes. These equations are straightforward to extend to
mixtures with multiple antibodies by drawing all of the statistical weights
and Boltzmann weights for the mixture (analogous to Fig 1) and then computing
the fractional activity as per Section A.1.

For example, a mixture of three antibodies all binding to distinct epitopes
would give rise to

$\text{Fractional Activity}_{\text{(1,2,3 distinct
epitopes)}}=\left(\frac{1+\alpha_{1}\frac{c_{1}}{K_{D}^{(1)}}}{1+\frac{c_{1}}{K_{D}^{(1)}}}\right)\left(\frac{1+\alpha_{2}\frac{c_{2}}{K_{D}^{(2)}}}{1+\frac{c_{2}}{K_{D}^{(2)}}}\right)\left(\frac{1+\alpha_{3}\frac{c_{3}}{K_{D}^{(3)}}}{1+\frac{c_{3}}{K_{D}^{(3)}}}\right).$
(S6)

If antibodies 1 and 2 bind to an overlapping epitope but antibody 3 binds to a
distinct epitope, then

$\text{Fractional Activity}_{\text{(1,2 overlapping epitopes; 3 distinct
epitope)}}=\left(\frac{1+\alpha_{1}\frac{c_{1}}{K_{D}^{(1)}}+\alpha_{2}\frac{c_{2}}{K_{D}^{(2)}}}{1+\frac{c_{1}}{K_{D}^{(1)}}+\frac{c_{2}}{K_{D}^{(2)}}}\right)\left(\frac{1+\alpha_{3}\frac{c_{3}}{K_{D}^{(3)}}}{1+\frac{c_{3}}{K_{D}^{(3)}}}\right).$
(S7)

If all three antibodies binds to overlapping epitopes, the fractional activity
becomes

$\text{Fractional Activity}_{\text{(1,2 overlapping epitopes; 3 distinct
epitope)}}=\frac{1+\alpha_{1}\frac{c_{1}}{K_{D}^{(1)}}+\alpha_{2}\frac{c_{2}}{K_{D}^{(2)}}+\alpha_{3}\frac{c_{3}}{K_{D}^{(3)}}}{1+\frac{c_{1}}{K_{D}^{(1)}}+\frac{c_{2}}{K_{D}^{(2)}}+\frac{c_{3}}{K_{D}^{(3)}}}.$
(S8)

###  A.3 Characterizing Ten EGFR Monoclonal Antibodies from Koefoed 2011

Koefoed et al. investigated how a panel of ten monoclonal antibodies inhibit
EGFR by measuring the protein’s activity at multiple antibody concentrations
in the human cell line HN5 [6]. By fitting these titration curves to Eq 1, we
can infer the dissociation constant $K_{D}$ between each antibody and EGFR as
well as the potency $\alpha$ of each antibody in the HN5 cell line (Fig
S2A,B). For each curve, the $K_{D}$ corresponds to the midpoint of the curve
(halfway between its minimum and maximum activity values) while $\alpha$
represents the activity at saturating antibody concentration.

Koefoed et al. found that the majority (7/10) of these antibodies reduced EGFR
activity below 20% at saturating concentrations, and since mixtures of
antibodies would likely further decrease activity their potency would be
difficult to accurately measure. To that end, Koefoed et al. switched to the
A431NS cell line that is partially resistant to EGFR antibodiess where they
remeasured EGFR activity for all ten monoclonal antibodies as well as mixtures
of two or three Abs (with 1:1 and 1:1:1 ratios, respectively). Each
measurement was performed at a mixture concentration of
$2\frac{\mu\text{g}}{\text{mL}}$, implying that each antibody was half as
dilute in the 2-Ab mixtures and one-third as dilute in the 3-Ab mixtures
relative to the monoclonal antibody measurement.

Figure S2: Inferring the model parameters for the 10 monoclonal antibodies in
Koefoed 2011. (A) Activity of EGFR in the presence of ten antibodies in the
HN5 cell line. (B) The inferred dissociation constant ($K_{D}$) and fractional
activity in the presence of saturating antibody ($\alpha$) in each cell line.
(C) Fractional EGFR activity in the presence of
$2\frac{\mu\text{g}}{\text{mL}}$ of each antibody in the A431NS cell line.
Data reproduced from Ref [6] Figure 2B,C.

We assume that the antibody-EGFR binding interaction is identical in the
A431NS cell line, and hence that these same $K_{D}$ parameters characterizes
these antibodies in that cell line. Hence, by using the additional measurement
of each antibody’s potency at $2\frac{\mu\text{g}}{\text{mL}}$ in the A431NS
cell line (Fig S2C), we can infer the potency $\alpha$ of each antibody in the
A431NS cell line (Fig S2B). These parameters are sufficient to predict how any
mixture (at any concentration and ratio) will behave in the HN5 and A431NS
cell lines. Note that all data presented in the main text correspond to the
A431NS cell line.

###  A.4 Comparing the HN5 and A431NS Cell Lines from Koefoed 2011

Koefoed et al. measured the potency of 176 mixtures in the A431NS cell line
but only 55 mixtures in the HN5 cell line. Fig S3 extends our analysis to both
the A431NS and HN5 cell lines. While the majority of mixtures have very little
predicted and measured activity ($\lesssim 0.2$), approximately 13 outliers
fall outside this range and appear to be poorly predicted.

While the coefficient of determination $R^{2}=0.61$ is significantly lower for
this cell line, we note that: (1) there are far fewer data points in this cell
line and (2) that our $R^{2}$ definition places more importance on points with
larger predicted or measured fractional activity, and hence these few outliers
have a disproportionate effect. That said, it remains unknown whether with
more data our model would be as successful in the HN5 cell line. Another open
question is why some antibody mixtures had inhibitory effects in one cell line
but exacerbating effects in the other (e.g. the mixture of
$\text{Ab}_{3}+\text{Ab}_{10}$ resulted in 1.22 fractional activity of EGFR in
the A431NS cell line but 0.19 fractional activity in the HN5 cell line).

Figure S3: Predicting the potency of antibody mixtures on EGFR in different
cell lines. Our model predictions versus experimental measurements for EGFR
activity in the (A) A431NS and (B) HN5 cell lines.

###  A.5 Separating the 2-Ab and 3-Ab Predictions for EGFR Antibody Mixtures

Fig S4 separates the 2-Ab and 3-Ab mixture predictions from the three models
in Fig 2. More specifically, Panels A and B of Fig S4 show the predictions for
combinations of two and three antibodies using the epitope mappings produced
by SPR (see captions on the diagonal of Fig 2A; there are four EGFR epitopes
bound by antibodies #1-3, #4, #5-6, and #7-10, respectively).

Without this SPR data, we could have alternately assumed that antibodies all
bind independently (Fig S4C,D) or that all antibodies vie for the same epitope
(Fig S4E,F). Either of these models generate slightly worse predictions, as
exhibited by their lower coefficients of determination $R^{2}$.

Figure S4: Separate predictions for 2-Ab and 3-Ab mixtures. (A,B) Predictions
using the epitope mappings produced using SPR. (C,D) Predictions assuming that
all antibodies bind independently. (E,F) Predictions assuming that all
antibodies bind competitively.

###  A.6 Original Antibody Nomenclature

Table S1 shows the antibodies considered in this work were named in the
original works of Refs. [6] and [7]. In our work, we indexed the EGFR
antibodies by their potency in the A431NS cell line and labeled the influenza
antibodies by the viral group that they most effectively neutralized.

Koefoed 2011 Antibody Nomenclature |  | Laursen 2018 Antibody Nomenclature
---|---|---
…..This Work….. | Original Work |  | …..This Work….. | Original Work
$\text{Ab}_{1}$ | 1565 |  | $\text{Ab}_{\text{A1}}$ | SD38
$\text{Ab}_{2}$ | 1320 |  | $\text{Ab}_{\text{A2}}$ | SD36
$\text{Ab}_{3}$ | 1024 |  | $\text{Ab}_{\text{B}}^{(1)}$ | SD83
$\text{Ab}_{4}$ | 992 |  | $\text{Ab}_{\text{B}}^{(2)}$ | SD84
$\text{Ab}_{5}$ | 1277 |  |  |
$\text{Ab}_{6}$ | 1030 |  |  |
$\text{Ab}_{7}$ | 1284 |  |  |
$\text{Ab}_{8}$ | 1347 |  |  |
$\text{Ab}_{9}$ | 1260 |  |  |
$\text{Ab}_{10}$ | 1261 |  |  |
Table S1: Matching the antibody nomenclature used in this work with the names
in the original manuscript.

##  Appendix B Characterizing Distinct versus Overlapping EGFR Epitopes

In this section, we describe in more detail how we can use activity data from
the 2-Ab mixtures to classify which subsets of antibodies bind to the same
epitopes. A basic classification scheme would categorize two antibodies as
binding to distinct epitopes if Eq 2 predicted their mixture’s activity better
than Eq 3; otherwise, the two antibodies would be categorized as binding to
overlapping epitopes.

However, this simple classification scheme does not account for the
uncertainty that arises from experimental noise. As an extreme case, if the
activities of a mixtures are measured at extremely small concentrations, then
the fractional activity will be $\approx 1$ for all mixtures (as will be the
predictions from both the distinct and overlapping models), and this
classification scheme would only be fitting the noise. Hence, it is best to
measure the activity of each mixture at saturating antibody concentrations
where the signal-to-noise of the system will be greatest.

We determined from Koefoed et al. (Figure S1) that the standard error of the
mean (SEM) of their activity measurements was $\sigma=0.04$, and we proceed to
incorporate this uncertainty into our categorization scheme using a simple
threshold model. More specifically, we add two components to the
classification scheme: (1) As shown in Fig S5A, activity measurements that
fall within $\sigma$ of the midpoint of the two model predictions are left
unclassified, since experimental noise could easily lead to such points being
incorrectly classified. (2) If the two model predictions lie sufficiently
close (within $4\sigma$) to one another, then the uncertainty (from both the
measurement and the model predictions) make it difficult to distinguish
between the two models with certainty.

As mentioned above, for potent antibodies that are measured at saturating
concentrations, the difference between the independent and overlapping binding
models will be large, making it easier to definitively classify antibody
epitopes. Koefoed et al. measured their 2-Ab combinations at a total
concentration of $2\,\frac{\mu\text{g}}{\text{mL}}$ (and hence
$1\,\frac{\mu\text{g}}{\text{mL}}$ for each antibody), and Fig S2A suggest
that while this is close to saturating concentration for the majority of
antibodies, increasing the concentration by 10x may have allowed more pairs of
antibodies to be definitively categorized.

Figure S5: Accounting for uncertainty in the classification scheme. (A) 2-Ab
combinations whose activity falls within $\sigma$ of the midpoint between the
two model predictions will be labeled as “unclassified,” since the difference
in distance from the measurement to either prediction is less than the
experimental error. (B) In addition, when the two model predictions are less
than $4\sigma$ different, any measurement is labeled as “unclassified” because
measurement error could result in overlap with both model predictions.

##  Appendix C Characterizing Multidomain Antibodies

###  C.1 Relating Influenza Neutralization to Binding

In this section, we discuss how the microscopic dissociation constants of two
tethered antibodies (shown in Fig 4C) relate to influenza viral
neutralization. We begin by considering a single antibody with an effective
dissociation constant $K_{D}^{(1)}$ that quantifies its avidity to the virus
[17]. The probability that this antibody at concentration $c$ will be bound to
a virion is given by

$p_{\text{bound}}=\frac{\frac{c}{K_{D}^{(1)}}}{1+\frac{c}{K_{D}^{(1)}}}.$ (S9)

For influenza virus with $N\approx 400$ hemagglutinin (HA) trimers, the number
of bound trimers is given by $N_{\text{bound}}=Np_{\text{bound}}$.

The relationship between viral binding and neutralization remain unclear [18].
It has been suggested that $\sim 50$ HA trimers are required to infect a cell
[17]. However, an IgG bound to one trimer may sterically preclude neighboring
HA from binding. It has been proposed that neutralization is a sigmoidal
function of neutralization (see Figure S1 of Ref [10]),

$\text{Fraction
Neutralized}=\frac{N^{h}+N_{50}^{h}}{N^{h}}\frac{\left(Np_{\text{bound}}\right)^{h}}{\left(Np_{\text{bound}}\right)^{h}+N_{50}^{h}},$
(S10)

where $N_{50}$ is the number of bound trimers required to reduce infectivity
to 50%, $h$ is a Hill coefficient, and the prefactor assures that the fraction
neutralized ranges from 0 (in the absence of antibody) to 1 (in the presence
of saturating antibody).

In the absence of data for our influenza strain of interest, we will assume
$h=1$ in the following analysis. This enables us to rewrite Eq (S10) as

$\text{Fraction
Neutralized}=\frac{\frac{c}{\text{IC}_{50}^{(1)}}}{1+\frac{c}{\text{IC}_{50}^{(1)}}},$
(S11)

where we have defined the inhibitory concentration of antibody at which 50% of
the virus is neutralized

$\text{IC}_{50}^{(1)}=\frac{N_{50}}{N+N_{50}}K_{D}^{(1)}.$ (S12)

For example, if the virus is 50% neutralized when $N_{50}=100$ trimers are
bound, the midpoint of a viral neutralization curve would occur at roughly
$\nicefrac{{1}}{{5}}$ the antibody concentration required to bind 50% of the
trimers, as has been observed for some influenza antibodies (see Figure 2 of
Ref [9]).

We now consider the tethered two-domain antibody shown in Fig 4C. Denote the
antibody concentration as $c$, the effective dissociation constants of its two
domains as $K_{D}^{(1)}$ and $K_{D}^{(2)}$, and the effective concentration
when both domains simultaneously bind as $\tilde{c}_{\text{eff}}$ (which we
will shortly relate to the $c_{\text{eff}}$ in the antibody neutralization
given in Eq 4). The probability that this antibody is bound to a virion is
given by

$p_{\text{bound}}=\frac{\frac{c}{K_{D}^{(1)}}+\frac{c}{K_{D}^{(2)}}+\frac{c}{K_{D}^{(1)}}\frac{\tilde{c}_{\text{eff}}}{K_{D}^{(1)}}}{1+\frac{c}{K_{D}^{(1)}}}.$
(S13)

As above, we assume that neutralization is related to the binding probability
through Eq (S10) with Hill coefficient $h=1$, which upon substituting Eq (S13)
yields

$\text{Fraction
Neutralized}=\frac{\frac{c}{\text{IC}_{50,\text{A1}}}+\frac{c}{\text{IC}_{50,\text{A2}}}+\frac{c}{\text{IC}_{50,\text{A1}}}\frac{c_{\text{eff}}}{\text{IC}_{50,\text{A2}}}}{1+\frac{c}{\text{IC}_{50,\text{A1}}}+\frac{c}{\text{IC}_{50,\text{A2}}}+\frac{c}{\text{IC}_{50,\text{A1}}}\frac{c_{\text{eff}}}{\text{IC}_{50,\text{A2}}}}$
(S14)

where we have defined the $\text{IC}_{50}$s of both antibodies using Eq (S12)
as well as the rescaled effective concentration

$c_{\text{eff}}=\frac{N_{50}}{N+N_{50}}\tilde{c}_{\text{eff}}.$ (S15)

Therefore, in the case where $h=1$ the functional form of neutralization (Eq
(S14)) is identical to the probability that an HA trimer is bound (Eq (S13)),
with the dissociation constants and the effective concentration rescaled by
$\frac{N_{50}}{N+N_{50}}$.

To put these results into perspective, we note that many viruses are covered
in spikes (analogous to influenza HA) that enable them to bind and fuse to
their target cells [18], and hence the sigmoidal dependence between viral
binding and neutralization is likely widely applicable. However, HIV is a
clear exception, since each virion has an average of 14 envelope spikes [19].
In that context, neutralization is roughly proportional to the number of bound
spikes so that $\text{IC}_{50}\approx K_{D}$ [20].

###  C.2 Assuming Different Antibody Constructs have Distinct Effective
Concentrations

In the main text, we quantified the boost in avidity from tethering two
antibodies using the effective concentration $c_{\text{eff}}=1400\,\text{nM}$
by using least-squares regression to minimize the (log) predicted
$\text{IC}_{50}$ for each tethered construct binding to all strains. This
effective concentration depends on the distance between binding sites on a
virion, and hence the tethered
$\text{Ab}_{\text{A1}}\text{\textendash}\text{Ab}_{\text{A2}}$ construct may
have a different $c_{\text{eff}}$ when binding to influenza A group 1 and
group 2 strains, and
$\text{Ab}_{\text{B}}^{(1)}\text{\textendash}\text{Ab}_{\text{B}}^{(2)}$ may
have yet another effective concentration when binding to the influenza B
strains.

Fig S6 shows the best-fit $c_{\text{eff}}$ for each of these cases. While this
plot suggests that there are differences between each tethered antibody and
influenza strain, we note that there is very limited data to infer such values
(e.g., there are 7, 7, and 5 data points in the influenza A1, A2, and B
groups, respectively; note that we ignore the H3N2 outlier strains
A/Panama/2007/99 and A/Wisconsin/67/05 discussed in the main text). That said,
incorporating this fine-grained level of modeling could further boost the
accuracy of modeling efforts and is worth pursuing as more data is gathered.

Finally, we mention that Laursen et al. measured the efficacy of
$\text{Ab}_{\text{A1}}\text{\textendash}\text{Ab}_{\text{A2}}$ with different
linkers of length 18 amino acids ($\sim 63\text{\AA}$), 38 amino acids ($\sim
133\text{\AA}$), and 60 amino acids ($\sim 210\text{\AA}$) against the four
influenza strains H1N1 A/California/07/09, H1N1 A/Puerto Rico/8/34-MA, H5N1
A/Vietnam/1194/04, and H3N2 A/Wisconsin/67/05 (see Ref [7] Table S11). They
found very little difference between the $\text{IC}_{50}$ of each construct,
which might naively suggest that the length of the linker does not matter.
However, since these four strains were all negligibly inhibited by
$\text{Ab}_{\text{A2}}$ ($\text{IC}_{50}\geq 1000\,\text{nM}$; see Fig 5A),
this domain cannot meaningfully contribute to bivalent binding, so that
$\text{Ab}_{\text{A1}}\text{\textendash}\text{Ab}_{\text{A2}}$ would be
expected to be as potent as $\text{Ab}_{\text{A1}}$ irrespective of linker
length. On the other hand, the length of the linker should matter when both
Abs in a multidomain antibody can bind, as is the case for
$\text{Ab}_{\text{A1}}\text{\textendash}\text{Ab}_{\text{A2}}$ binding to the
three influenza A group 2 strains with $\text{IC}_{50}<1000\,\text{nM}$ and
for $\text{Ab}_{\text{B}}^{(1)}\text{\textendash}\text{Ab}_{\text{B}}^{(2)}$
binding to the five influenza B strains.

Figure S6: Inferring $c_{\text{eff}}$ between every tethered construct binding
to each group of influenza virus. The best-fit effective concentration of
$\text{Ab}_{\text{A1}}\text{\textendash}\text{Ab}_{\text{A2}}$ against the
influenza A group 1 and group 2 strains, together with the effective
concentration of
$\text{Ab}_{\text{B}}^{(1)}\text{\textendash}\text{Ab}_{\text{B}}^{(2)}$
binding to influenza B.

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