Stochastic modeling of antibody binding predicts programmable migration on antigen patterns

Abstract

Viruses and bacteria commonly exhibit spatial repetition of surface molecules that directly interface with the host immune system. However the complex interaction of patterned surfaces with immune molecules containing multiple binding domains is poorly understood. We developed a pipeline for constructing mechanistic models of antibody interactions with patterned antigen substrates. Our framework relies on immobilized DNA origami nanostructures decorated with precisely placed antigens. The results revealed that antigen spacing is a spatial control parameter that can be tuned to influence antibody residence time and migration speed. The model predicts that gradients in antigen spacing can drive persistent, directed antibody migration in the direction of more stable spacing. These results depict antibody-antigen interactions as a computational system wherein antigen geometry constrains and potentially directs antibody movement. We propose that this form of molecular programmability could be exploited during co-evolution of pathogens and immune systems or in the design of molecular machines.

1. Introduction

Due to their multiple binding domains, immunoglobulin molecules like the bivalent IgG antibody exhibit complex interactions with multivalent antigens, i.e. clusters of multiple copies of molecules or molecular domains occurring on the order of 1-30 nm separation distances. Multivalent interactions enhance the stability of binding interactions by enabling simultaneous attachment of multiple ligands, increasing the magnitude of the apparent affinity or Gibbs free energy of multivalent binding, also called functional affinity in favor of the term “avidity”, and extending the residence times of bound antibodies [1,2,3].

Many pathogenic surfaces exhibit spatial repetition at length scales relevant to antibody multivalence. Viral capsid proteins undergo self-assembly into periodic patterns [4], and some neutralizing antibodies achieve their high affinity and neutralization capability through bivalence [5,6]. Self-assembling crystalline arrays of surface-layer (S-layer) proteins, the outermost structure on many bacteria and archaea, are a major contact point between pathogen and host [7] and are implicated as mediators of innate [8] and adaptive immunity [9,10]. Their repetitive organization may be integral to their immunological role, as their removal from bacterial surfaces was seen to reduce immune response [11]. Multivalence is also likely an important factor during affinity maturation of antibodies and thus in vaccine design [12,13,14].

Antibody interaction with patterned surfaces presents a challenge for both experimental control and mathematical modeling as it is a many-bodied problem occurring on timescales of seconds to minutes. Such systems are too computationally expensive for full-atom molecular simulation. Models treating antibodies and antigens as abstract binding and non-binding units have been the most successful at capturing relevant dynamics, and have historically treated multivalence as a function of ligand coating density whereby multivalence emerges statistically as the average nearest-neighbor distance between ligands decreases [15]. More recently, coarse-grained molecular simulations have been used fruitfully to quantify the effects of co-operative binding of the antibody subunits on binding affinity [16,17]. Nevertheless a challenge of precisely calibrating such models remains due to the absence of both experimental tools to independently assess monovalent and multivalent binding dynamics as well as a pipeline to connect such data to models.

The patterned surface plasmon resonance (PSPR) technique enables measurement of binding kinetics on precise, monodisperse patterns of ligands, achieving robust control of geometry through the use of DNA origami nanostructures [18] (Fig 1 a). Herein, we demonstrate a pipeline for the automated conversion of PSPR data into a flexible, experimentally parameterized model of antibody interaction with arbitrarily complex multivalent surfaces. The model is based on a coarse-grained simplification of bivalent antibody binding to antigens as a discrete Markov process with distinct states: empty antigen, monovalent antibody-antigen complexes, and bivalent antibody-antigen complexes with transitions between these states governed by elementary rates (Fig 1 b). From this basis, the dynamics of more complex patterns of multiple antigens can be reduced (Fig 1 c) to combinations of these elementary states. A causal linkage between pattern geometry and antibody dynamics could potentially be exploited as a form of spatial programmability by either immunity or pathogens during their adversarial co-evolution. We investigate this possibility and the role of spatial tolerance, i.e. the range and impact of antigen separation distances on bivalent binding kinetics (Fig 1 d), in determining effective binding affinity, the walking speed of antibody migration on patterned surfaces, and the direction of antibody migration. Such control mechanisms might inform the development of vaccines for greater control over the affinity maturation process.

Fig. 1. Scheme for modeling binding dynamics of antibodies on multi-antigen substrates.

a: Illustration of patterning concept, whereby small molecule antigens (haptens) are arranged using short, flexible tethers at well-defined locations on DNA origami nanostructures. This enables multivalent interaction of antibodies with antigen patterns. b: Markov model of antibody binding whereby only basic binding/unbinding and bivalent interconversion processes are used to couple discrete monovalent and bivalent binding states. c: Model extension to more complex pattern geometries is accomplished by separating the system into elementary transitions between states comprising different combinations of empty and monovalently or bivalently occupied antigens. d: Pairs of antigens separated by different lengths elicit differing antibody binding kinetics due to the separation distance dependent impact of antibody structure on the chance of bivalent interconversion.

2. Results

2.1. a spatial tolerance model

We developed a model parameterization pipeline based on a progressive fitting of the transient SPR profiles for first monovalent and then bivalent binding processes in order to reduce degrees of freedom at each stage of fitting. In the first stage, we used either rabbit anti-digoxygenin IgG or mouse-derived anti-digoxygenin IgG1 targeting digoxygenin-decorated DNA origamis with a single-cycle-kinetics program whereby progressively higher concentrations of antibody were exposed to the immobilized antigen substrate (Fig 2 a and b). This program was performed with a 1-antigen configuration (Fig 2 c, e) in order to parameterize our Markov model (Methods Section 4.8) by relating the SPR signal to the average occupancy Φ defined as the number of antibodies per structure averaged according to the prevalence of each possible state (Methods Section 4.4). This yields respective association and dissociation rates k₁ = 1.93 ± 0.05 × 10⁷M⁻¹s⁻¹ and k₋₁ = 5.28 ± 0.07 × 10⁻⁴s⁻¹ as well as a monovalent dissociation constant K_D1 = 2.7 ± 0.11 × 10⁻¹¹ M defined as the ratio of the dissociation to association rates. We parameterized interconversion between monovalent and bivalent states by fixing the previously determined monovalent parameters and fitting the model to experiments involving multiple adjacent antigens (Methods Section 4.10). We fitted the model by adjusting K_D2 or the interconversion constant defined by the ratio of reverse and forward interconversion rates. For structures configured with 2 antigens separated by 14.3 ± 1.2 nm, we find K_D2 = 8 ± 6 × 10⁻³ (Fig 2 d, f).

Fig. 2. A progressive fitting pipeline for obtaining a parameterized model from minimal experimental data.

a and b: Concentration versus time plot of antibody solution exposed to patterned antigen substrates in a single cycle kinetics PSPR experiment. c and d: Experimental binding kinetics data (black line) of 1 antigen and 2 antigen configurations respectively, superimposed over the occupancy calculated from the parameterized model. Model occupancy is divided and colored according to state, with height corresponding to the state’s contribution to total antibody occupancy per structure. e and f: 1 and 2 antigen configuration respective transient state probabilities stratified according to model prediction, colored and stacked to satisfy the normalization condition whereby all probabilities add to 1. Legends list either all or the top 5 most prevalent states σ_i along with their corresponding occupancy ϕ_i and probabilities p_i at the end of their respective run. g: Interconversion constants (red points) plotted versus 2-antigen configuration separation distance x and the fitted spatial tolerance model (blue line). Blue error bars indicate model fits due to 1 standard error of the mean away from a mean input 1-antigen run, propagated to K_D2 values (vertical black error bars). Horizontal black error bars denote spatial uncertainty defined according to [44]. Vertical black error bars denatoe uncertainty due to 1 standard error of the mean variation in input data. h: Goodness of fit characterization of spatial tolerance model. Red points show an apparent random dispersion of K_D2 points minus a moving average, while blue shows dispersion of model values subtracted from K_D2 at each point. i: Sensitivity of model’s minimum to shifted values of 1-antigen input data by number of standard error away from a mean input run.

Applying progressive fitting to PSPR runs with structures patterned with 2 adjacent antigens of varied separation distances, we found the internal conversion process to vary accordingly. Small and large separation distances correspond to reduced bivalence, i.e. larger K_D2. We constructed a phenomenological equation (Methods Section 4.7) modeling the interconversion constant (Fig 2 g). The model is composed of a logistic tension-term representing the reduced bivalence at large separation distances and an exponential compression-term representing the penalty to bivalence observed at extremely close separation distances, a characteristic that has been substantiated by recent biosensing applications [19]. In our model, the interconversion constant is thus a function of adjacent antigen separation distance with the form

$$K_{D2}=\frac{K_{D2}^{\mathit{max}}}{1+e^{-{\alpha }_t\left(x-{\ell }_t\right)}}+K_{D2}^{\mathit{max}}{e}^{-{\alpha }_c\left(x-{\ell }_c\right)},$$ (1)

where ℓ_t and ℓ_c are characteristic lengths defining the scale of the tensile and compressive terms, α_t is the sharpness of the tensile penalty, α_c is the decay parameter of compressive penalty to bivalence, and $K_{D2}^{\mathit{max}}$ is the value of K_D2 at which contributions of bivalence to binding dynamics are vanishingly small. We found that when appropriately fitted (Fig 2 h), this model predicts a theoretical min(K_D2) located at ≈ 10.6 nm separation distance with an approximately 1 nm uncertainty due to expected random shift in 1-antigen input data (Fig 2 i), whereas the experimental datapoint with lowest K_D2 is located at ≈ 15. nm. This result is in agreement with a study by Zhang et al[20] who estimated optimal epitope separation distance using DNA origami and atomic force microscopy, though we note here that the curve is likely to differ between isotype, species, and possibly even clones due to angular variation in the epitope-paratope bond.

2.2. steady state and transient analysis

In order to determine the dependence of system bivalence on solution phase concentration, we used the parameterized model to obtain the steady state probability distributions for a range of solution phase concentrations (Fig 3 a). This revealed concentration regimes of differing dominant states: empty, bivalent, and saturated monovalent at respectively low, medium, and high solution phase concentrations. Entropic maxima occur at transitions between these domains (Fig 3 b), and transitions in bivalent and monovalent contributions to chemical potential occur in accordance with the transition from bivalent to saturated monovalent regimes (Fig 3 c) (Methods Section 4.11).

Fig. 3. De novo simulation with parameterized kinetics.

a: Stationary distributions colored by state of a 2-antigen system (14 nm separation) for a range of solution phase antibody concentrations demonstrating clear regions of predominantly empty, bivalent single antibody occupancy, and monovalent two antibody occupancy regimes connected by smooth transition regions. b: Distribution of state entropic contributions to free energy for a range of solution phase antibody concentrations. c: Chemical potential contributions at equilibrium from each state for a range of concentrations. (legend shows top 5 most abundant states) d: Result of a blind test with SPR signal in RU predicted for a trimeric 7.2 × 14.3 × 16.0 nm antigen configuration and known concentration intervals overlaid with the raw experimental (red) run. e: Occupancy of the predicted run stratified by state. f: Transient state probability distribution for the trimeric antigen configuration. g: The transient evolution of monovalent (blue) and bivalent (red) contributions of antibodies to the average occupancy or number of antibodies per structure in the trimeric system. The cross between the two lines demonstrates transition between regimes by changing the concentration. Legends list all or the 5 most prevalent states σ_i along with their corresponding occupancy ϕ_i and probabilities p_i at the end of their respective run or contributions to entropy in the case of (e).

In addition to simulating de novo patterns’ steady state properties, the model enables us to simulate transient dynamics of hypothetical systems with arbitrary geometries and arbitrary timing in the introduction of different solution phase concentrations. To validate the pipeline, we used the model parameterized with 1 and 2-antigen data (Fig 2 g) to create a blind a priori prediction of the evolution of a higher order system with 3 antigens arrayed in a 7.2 × 14.3 × 16.0 nm right triangle and then check its correspondence with an experimental trajectory (Fig 3 d), and additional validation is shown in Supplementary Figure 7. We found that experimental trajectories closely conformed to predictions. The model provides access to the individual contributions of states to the signal through their occupancy (Fig 3 e) and relative pro-portions (Fig 3 f) enabling us to construct a narritive explanation for the observed dynamics. We see for example in the final stage when concentration was set to zero, that while total occupancy falls and that occupancy contributions from monovalently bound antibodies decreased, bivalent state contributions counterintuitively increased. This indicates that high concentrations in the penultimate stage had inverted the system to favor monovalent-dominated saturation states that subsequently transitioned to unsaturated bivalent states as sites became available (Fig 3 g).

2.3. experiments on repetitive antigen patterns

In order to explore the potential for pattern-based control and programmability of antibody dynamics, we wished to model the dynamics of larger systems with greater relevance to periodic pathogenic surfaces. For larger systems, complete enumeration of states scales poorly with increasing numbers of adjacent antigens. We developed a Markov Chain Monte Carlo implementation of the model (Methods Section 4.12) to sample trajectories that converges to state probabilities with large sample numbers. Rather than enumerating all system states (i.e. combinations of antibodies and binding modes on a structure and possible transitions), the system performs a random walk through the large state space, computing at any point in time its rate of escape into neighboring states. We then examined collections of individual trajectories for such systems to understand their average behavior. Specifically, we examined the role of repetitive antigen spacing in simple 1D arrays.

Antigens arranged according to a spacing gradient in the range of 10 - 22 nm separation distances (i.e. the interval of steepest slope in Equation 1) elicit asymmetric accumulation toward the narrow spaced end of the array (Fig 4 a). This system also exhibited individual walking trajectories that tend toward the narrow-spaced end (Fig 4 b), asymmetric velocity (Fig 4 c), and asymmetric net displacement (Fig 4 d) according to the direction of the gradient. The mechanism for this locomotion is that of a biased random walk, whereby at any point in time, a bivalently bound antibody has a random chance to dislodge one of its paratopes and then reassociate either with the same epitope or an adjacent one. Differential spacing between adjacent epitopes will lead to a statistical preference for more stable spacings with a lower interconversion ratio K_D2.

Fig. 4. Manipulation of antibody movement through choice of antigen pattern geometry.

a: Cumulative antibody residence times as a function of antibody location on 1D antigen gradients oriented with increasing spacing (top) and decreasing spacing (bottom) b: Random walk trajectories of antibodies tracked from their initial landing locations on 1D antigen gradients with increasing (top) and decreasing (bottom) spacing gradients. c: Histograms of antibody velocity on 1D antigen gradients of increasing (left) and decreasing (right) spacing distance. d: Net displacement of antibodies tracked from their initial binding location on 1D antigen gradients of increasing (left) and decreasing (right) spacing gradients. (Sample size for c and d: 200 simulations, 10000 seconds each) e: Histogram of antibody residence times on uniform 1D antigen array with wide 22 nm spacing. f: Histogram of antibody residence times on uniform 1D antigen array with narrow 10 nm spacing. g: Histogram of antibody displacements tracked from their initial binding locations for uniform 1D antigen arrays with wide (magenta) and narrow (blue) spacings. (Sample size for e,f, and g: 100 simulations, 10000 seconds each)

Antibodies binding to 1D arrays with uniform spacing exhibited divergent residence times, with antibodies spending less cumulative time on 22 nm spaced arrays (Fig 4 e) than those of narrow 10 nm spaced arrays (Fig 4 f). The migration speeds of antibodies on high strain-inducing arrays are greater than those of low-strain inducing arrays, and a comparison of net displacement shows that antibodies moved further from their initial binding location on widely spaced arrays relative to narrowly spaced ones (Fig 4 g).

3. Discussion

Repeating epitope patterns are present in many viruses as coat proteins [21,22,23,24] and in bacteria as S-layer proteins [25,26,27] often with a high degree of symmetry or geometric periodicity. Such repetitive, quasi-crystalline patterns have been recognized as a marker of foreignness corresponding to major enhancements of IgG response compared to unorganized substrates [28]. Investigators have observed both monovalent and bivalent antibody binding to such periodic viral surfaces [29,24] and binding enhancement due to bivalence is a recognized factor in determining both immune pathogen-recognition and neutralization capability [30,31]. A high-speed atomic force microscopy study by Preiner et al. [32] captured real-time bipedal locomotion of antibodies on reconstituted pathogenic surfaces with periodic patterns of epitopes.

Preiner et al. proposed that antibody locomotion is enabled by the strain induced during bivalent binding as antibodies accommodate the geometry of their target antigens, weakening the bond and triggering a bipedal step. Our results agree and indicate that precisely tuned spacing on repetitive antigen patterns would have a major impact on the strength of bivalent bonds, and furthermore that differences in adjacent antigen spacings will statistically drive migration, as antibodies randomly move until becoming immobilized in states with minimal strain.

One limitation of our model is that torsional flexibility of the hinge region [6] is not accounted for. This is due to the design of the DNA origami nanostructure substrates in which hapten antigens are tethered by short, flexible spacers with rotational freedom. Future studies could explore rotational spatial tolerance as well as degrees of freedom in the Z-direction by including additional terms in Equation 1 calibrated with PSPR data that systematically modulates relative epitope orientations or employ structures that incorporate pathogenic protein antigens[33,34,35,36] for more realistic structural complexity and physiological relevance.

An additional limitation of our model is the potential for over-extrapolation in increasingly complex systems, with any inaccuracies in spatial tolerance model or parameterization quality subject to propagation. We predicted that long-range gradients of differential spacings could be used to establish persistent directed migration of antibodies on a surface, and we propose that PSPR and the progressive fitting pipeline presented here should be used in future studies to test predictions experimentally. Firstly, such designs should be possible using DNA origami, and secondly the stratification of states obtained by model fitting to convoluted binding data might enable one to measure the spatial distribution of antibodies on gradient structures.

Repetitive antigen arrays have been important in vaccine design [37,38]. Evolution of protective antibodies against malaria was shown to be dependent on a repetitive motif [39], and bacteria are known to interfere with antibody binding such as through Fc targeting to prevent opsonization [40]. The apparent importance of spatial organization in immunological signalling suggests a role for non-equilibrium spatial phenomena such as that studied here, and we might expect antigen organization itself to be under selective pressure during host-pathogen co-evolution. We suggest that the mechanism of stochastic walking predicted here might explain some of the pressures guiding the pathogen epitope organization, and such a mechanism might be exploited in the rational design of vaccines.

An inversion of this energy landscape phenomenon pertains to laterally mobile antigens such as the spike proteins in viral lipid envelopes. Mobile antigens would be expected to accommodate bivalent binding by laterally diffusing to achieve the minimum interconversion ratio K_D2. Amitai et al derived a theoretical affinity optimum for mobile spike proteins that depends on their surface density, arguing that intermediate densities invoke the greatest immune response, and that the low spike density characteristic of HIV is key to its immune evasion [41]. In this respect, a spatial tolerance model and experimental parameterization pipeline could aid vaccine development by informing design choices meant to elicit a precise immune response, e.g. immunostimulatory virus-like particles [35,42]. Our pipeline could also be used to dissect the complex state spaces of bi- tri- or tetra- specific antibodies which are recently being developed for thera-peutic and biosensing applications [43]. We expect these molecules to exhibit complex binding behaviors, especially as many are engineered with non-Fc-based tethering regions of various flexibilities and lengths.

The capacity for emergent dynamics and programmable behavior makes antibody-antigen interactions a subject of greater potential complexity than previously thought. Experimentally parameterized modeling provides a reality-grounded sandbox for discovery, and we anticipate that future modeling pipelines coupled to other experimental technologies will bear fruit as this subject continues to be explored.

4. Methods

4.1. Statistics and Reproducibility

Error assessment in Fig 2 g was performed using a boostrapping method whereby mean and the mean plus or minus 1 standard error of the mean input data was used to form central, upper, and lower inputs propagated to obtain individual output points shown in the figure, with upper and lower vertical error bars corresponding to outputs produced by the upper and lower inputs respectively. Goodness of fit for the phenomenological spatial tolerance function was characterized using an adapted chi-square metric described in Methods Section 4.7. Assessment of model robustness and predictive potential in Fig 3 and Supplementary Figures 7 and 8 was performed using a blinded test whereby author ITH performed model parameterization and prediction of experimental SPR curves for an untested 3-antigen triangular structure for 3 independent replicates with different corresponding bound structure amounts while experimental test data obtained by authors AS and IS was withheld until predictions had been submitted for comparison.

4.2. Computational methods overview

Some methods and explanation may be found in Shaw et al [18]. However, in the following methods material, we emphasize the original developments of this work including the following: a minimal parameterization pipeline for Markov models sensitive to arbitrary antigen spacing distances and its experimental validation; phenomenological model with analytical equation describing spatial tolerance as a continuous function; application of the model and fitting pipeline to two different antibodies, one derived from rabbit and the other mouse; a systematic approach to determining the conversion factor between SPR response units (RU) to that of the number of bound antibodies per structure; steady state analysis and the determination of thermodynamic quantities from equilibrium state probability distributions; a Markov Chain Monte Carlo random walk variant of the model which can be used to simulate larger systems with too many connected antigens to be feasible for enumerative approaches.

Briefly, the pipeline is executed in three parts. The first part requires the empirical estimation of the maximum SPR response due to saturation of antibody binding sites, based on the measured signal due to origami structures binding to the surface and a standard curve constructed to relate structure binding to maximum SPR response. This information is used as a conversion factor to relate the SPR signal for a given experiment to an average quantity of antibodies bound per structure. The second aspect of the pipeline is the fitting of a continuous time Markov Chain model to the SPR binding data of both single antigen and two-antigen structures over a range of separation distances. This enables the construction of a parameterized analytical spatial tolerance equation that is used to predict the binding kinetics for arbitrary separation distances and antigen geometry. The third part of the pipeline entails the deployment of this fitted model for predictive purposes. Steady state properties of a given system geometry are simulated by determining the distribution of states when net flux between states is zero. Large systems are simulated using a Markov Chain Monte Carlo simulation that generates many individual trajectories of single antibodies walking on a user-specified pattern geometry. Pseudocode and mathematical explanation can be found in the following subsections where this approach is described in more detail.

4.3. Model assumptions, constraints

Assume a coarse-grained model of binding states that equates all physical states in which antibodies are bound by one arm as monovalent, and which equates all physical states in which antibodies bound by two arms as bivalent.

Assume a fixed amount of bound structures that does not change with time. i.e.

$$\frac{d{R}_{struc\text{t}}}{dt}\approx 0$$ (2)

where R_struct is the SPR signal due to structures, and

$$\frac{d{n}_{struct}}{dt}\approx 0$$ (3)

where n_struct is the molar amount of structures.

The system has an IgG reservoir that is large compared to the available binding surface and thus has an effectively fixed concentration, i.e.

$$\frac{d{c}_{Ab}}{dt}\approx 0$$ (4)

where c_Ab is the concentration of solution phase antibody.

4.4. Conversion from SPR signal R_ab to bound antibody n_ab

Consider first the simple 1-1 interaction of an antibody analyte that binds and unbinds to a structure containing a single antigen ligand.

$$\sigma \_\underset{k-1}{\overset{c_{Ab}{k}_1}{\rightleftharpoons }}{\sigma }_{Ab}$$ (5)

where σ_ is the state corresponding to an onoccupied structure, σ_Ab is the state corresponding to a structure with a bound antibody, and k₁ and k₋₁ are the association and dissociation rates respectively.

We may work in terms of molar quantities rather than concentrations or surface densities, as the dimensions of the system do not change

$$n_{struct}=n_{-}+n_{Ab},$$ (6)

or the molar amount of structures both occupied and unoccupied [mol], where n_Ab is the number of bound antibody-structure complexes and n_ is the number of unoccupied structures and both where n_Ab and n_ are functions of t.

State probabilities are therefore:

$$p_{-}=\frac{n_{-}}{n_{struct}}$$ (7)

and

$$p_{Ab}=\frac{n_{Ab}}{n_{struct}}$$ (8)

corresponding to the unoccupied and 1-antibody-occupied states respectively.

We define also an occupancy, the number of antibodies that are bound to a single structure for a given state. For simple, 1 antigen structures, this value is zero for the empty state and 1 for the bound state or

$${\varphi }_{-}=0$$ (9)

and

$${\varphi }_{Ab}=1$$ (10)

The average occupancy is a macroscopic description of the state of the system comprising N states, or the average fraction of bound antibodies per structure.

$$\Phi ={\displaystyle \sum _{i=1}^N{p}_i{\varphi }_i}$$ (11)

For the case of a 1 antigen structure system, this becomes

$$\Phi =p_{Ab}\cdot 1+p_{-}\cdot 0=\frac{n_{Ab}}{n_{struct}}$$ (12)

In a 1-1 binding model, change in SPR signal is proportional to the amount of bound material or in other words the change in molar amount of structures with occpupied binding sites n_Ab.

$$R_{Ab}=n_{Ab}{\xi }_{Ab}$$ (13)

where ξ_Ab is a conversion factor corresponding to the expected change in SPR response signal per mole of bound antibodies.

The rate of change of occupied sites is equal to the rate of conversion of unoccupied sites via binding events minus the rate of conversion of occupied sites via unbinding events.

$$\frac{d{n}_{Ab}}{dt}=k_1{c}_{Ab}{n}_{-}-k_{-1}{n}_{Ab}$$ (14)

The SPR signal after the structure binding step is proportional to the molar of amount of bound structure

$$R_{struct}=n_{struct}{\xi }_{struct}$$ (15)

where ξ_struct is a conversion factor correspond to the expected change in SPR response signal per mole of structures.

Substitute RU-based expressions of molar amounts into Equation 6:

$$n_{-}=\frac{R_{struct}}{{\xi }_{struct}}-\frac{R_{\text{Ab}}}{{\xi }_{Ab}}$$ (16)

Substituting RU-based expressions of molar amounts into Equation 14 yields

$$\frac{d{n}_{Ab}}{dt}=\frac{1}{{\xi }_{Ab}}\frac{d{R}_{Ab}}{dt}=k_1{c}_{Ab}\left(\frac{R_{struct}}{{\xi }_{struct}}-\frac{R_{Ab}}{{\xi }_{Ab}}\right)-k_{-1}\left(\frac{R_{Ab}}{{\xi }_{ab}}\right)$$ (17)

which simplifies to

$$\frac{d{R}_{Ab}}{dt}=\frac{{\xi }_{Ab}{k}_1{c}_{Ab}{R}_{struct}}{{\xi }_{struct}}-k_1{c}_{Ab}{R}_{Ab}-k_{-1}{R}_{Ab}$$ (18)

Since we have gathered both conversion constants into one term in Equation 18, we define now the occupancy signal factor

$${\xi }_{\ast }\equiv \frac{{\xi }_{Ab}}{{\xi }_{struct}}$$ (19)

or the dimensionless ratio of molar conversion factors: bound-antibody relative to structure.

Note by rearrangement the relationship to average occupancy - i.e. the occupancy signal factor is the ratio of occupancy in terms of SPR signal to that of molar quantities.

$${\xi }_{\ast }=\frac{R_{Ab}}{n_{Ab}}\frac{n_{struct}}{R_{struct}}=\frac{R_{Ab}}{R_{struct}}{\left(\frac{n_{Ab}}{n_{struct}}\right)}^{-1}$$ (20)

$$=\frac{R_{Ab}}{R_{struct}}{\Phi }^{-1}$$ (21)

Substituting ξ_* we then arrive at the expression for the rate of change in SPR signal with respect to time as a function of structure-binding signal and antibody-binding signal:

$$\frac{d{R}_{Ab}}{dt}={\xi }_{\ast }{k}_1{c}_{Ab}{R}_{struct}-k_1{c}_{Ab}{R}_{Ab}-k_{-1}{R}_{Ab}$$ (22)

In the case of a monovalent structure (1 antigen available for binding) at the point of maximal saturation when average occupancy is unitary (Φ = 1), the molar quantities of bound antibody and structures are equal:

$$n_{Ab}^{\mathit{max}}=n_{struct}$$ (23)

where $n_{Ab}^{\mathit{max}}$ is the maximum number of moles of antibody that can bind to the system.

Under maximal saturation conditions, the monovalent occupancy signal factor then reduces to

$${\xi }_{\ast }=\frac{R_{Ab}^{\mathit{max}}}{R_{struct}}\cdot 1$$ (24)

where $R_{Ab}^{\mathit{max}}$ is the maximum SPR response signal due saturation of antibody binding sites.

This relationship is then used to produce a standard curve from monovalent structure binding data in order to obtain the linear relationship:

$$R_{Ab}^{\mathit{max}}=R_{struct}{\xi }_{\ast }$$ (25)

where an empirically determined ξ_* enables one to estimate the SPR signal corresponding to an occupancy of 1 antibody per structure from the R_struct signal. This is useful for structures with valency greater than 1 and whose binding kinetics do not obey simple 1-1 equations. Since $R_{Ab}^{\mathit{max}}$ on a multivalent structure will not resemble that of the monovalent 1-1 system, we refer to this conversion factor obtained from the monovalent $R_{Ab}^{\mathit{max}}$ as $R_{A{b}^{mono}}$, i.e. an SPR signal to antibody number conversion factor:

$$R_{A{b}^{\text{mono}}}\approx R_{Ab}^{\mathit{max}},V_{struct}=1,$$ (26)

where V_struct is the valency.

In such cases, we obtain the average occupancy using the estimated $R_{Ab}^{\mathit{max}}$ from the linear regression.

$$\Phi =f\left(R_{Ab}\right)=\frac{n_{Ab}}{n_{struct}}=\frac{R_{Ab}}{R_{Ab}^{max}}$$ (27)

4.5. Empirical estimation of R_abmax

Thus, we obtain a standard curve used to convert the SPR signals for arbitrary structure configurations by empirically determining the correlation between structure binding signals and the max signals corresponding to saturated monovalent (1 antigen) structures, enabling conversion from SPR signal to occupancy in the absence of a well-understood binding model provided knowledge of the structure binding signal.

The structure bound signal (Supplementary Figure 1 a) is taken to be the difference between signals before and after structures are flowed over the chip and allowed to bind.

Guess values of the parameters k₁, k₋₁, and ξ_* are supplied to a numerical minimization of the autocorrelation of residuals between experimental and theoretical curves for the 4th order Runga Kutta approximation of Equation 22, i.e. the function $\frac{d{R}_{Ab}}{dt}=f\left(R_{Ab}\right)$ recursively approximated according to the formula

$$R_{Ab,t+1}=R_{Ab,t}+\frac{h}{6}\left({\kappa }_{n1}+2{\kappa }_{n2}+2{\kappa }_{n3}+{\kappa }_{n4}\right)$$ (28)

where h is a small timestep and the constituent terms have the form

$${\kappa }_{n1}=f\left(R_{Ab,t}\right)$$ (29)

$${\kappa }_{n2}=f(R_{Ab,t}+\frac{h}{2}{\kappa }_{n1})$$ (30)

$${\kappa }_{n3}=f(R_{Ab,t}+\frac{h}{2}{\kappa }_{n2})$$ (31)

$${\kappa }_{n4}=f\left(R_{Ab,t}+h{\kappa }_n\right)$$ (32)

For each monovalent run (Supplementary Figure 1 b and c for rabbit and mouse antibodies respectively) with a unique value of R_struct, a projected value of $R_{Ab}^{\mathit{max}}$ is computed with Equation 25. By fitting the monovalent models to 1-1 kinetics, we obtain the rate constants that allows computational prediction of $R_{Ab}^{\mathit{max}}$(Supplementary Figure 1 d and e for rabbit and mouse antibodies respectively) in the absence of experimental saturation conditions. This enables us to make a standard curve to adjust $R_{Ab}^{\mathit{mono}}$ according to R_struct in the absence of a 1-1 $R_{Ab}^{\mathit{max}}$(Supplementary Figure 1 f and g for rabbit and mouse antibodies respectively). We use this value as a conversion factor, enabling us to convert SPR response units into the number of antibodies per structure (Supplementary Figure 1 h and i for rabbit and mouse antibodies respectively). By knowing R_struct, we can estimate this conversion factor for non-trivial antigen configurations where the multivalence influences the ease of reaching a saturation value corresponding to $R_{Ab}^{\mathit{max}}$.

4.6. Equilibrium characterization with dissociation constants

The equilibrium dissociation constant concisely describes the relationship between analyte and ligand, and provides a good basis for comparison between systems across experimental conditions in which dynamic behavior can vary significantly. Given a model of the process, we can derive a formula for the equilibrium dissociation constant by solving the system of equations. For a 1-1 process we have At steady state:

$$\frac{d{R}_{Ab}}{dt}=0$$ (33)

$$k_1{c}_{Ab}{R}_{Ab}^{\mathit{max}}=k_1{c}_{Ab}{R}_{A{b}^{eq}}+k_{-1}{R}_{A{b}^{eq}}$$ (34)

rearranging yields and

$$k_1{c}_{Ab}{R}_{A{b}^{\mathit{max}}}=\left(k_1{c}_{Ab}+k_{-1}\right)R_{A{b}^{eq}}$$ (35)

and

$$R_{A{b}^{eq}}\left(\frac{k_1}{k_{-1}}{c}_{Ab}+1\right)=\frac{k_1}{k_{-1}}{c}_{Ab}{R}_{Ab}^{\mathit{max}}$$ (36)

For a 1-1 monovalent model, the dissociation constant is

$$K_D=\frac{k_{-1}}{k_1}$$ (37)

thus, at equilibrium, the SPR signal is

$$R_{A{b}^{eq}}=\frac{c_{Ab}{R}_{Ab}^{\mathit{max}}}{K_D+c_{Ab}}$$ (38)

Empirical measurement of the dissociation constant is obtained by determining the equilibrium binding signals at multiple concentrations and fitting the linearized form of Equation 38 or

$$\frac{1}{R_{A{b}^{eq}}}=\frac{K_D}{c_{Ab}{R}_{Ab}^{\mathit{max}}}+\frac{1}{R_{Ab}^{\mathit{max}}}$$ (39)

where R_Ab^eq is the steady state SPR signal due to bound antibody.

The equilibrium dissociation constant is a good descriptive parameter which captures the essential dynamics concisely.

From the dissociation constant, we know the occupancy:

$${\Phi }_{eq}=\frac{R_{A{b}^{eq}}}{R_{Ab}^{max}}=\frac{c_{Ab}}{K_D+c_{Ab}}$$ (40)

where Φ_eq is the expected occupancy at steady state.

Such a concise description is desireable for complex structures as well. However difficulty arises in the case of multivalent structures which no longer exhibit simple 1-1 dynamics. One approach is to simply approximate the dynamics with a 1-1 model and obtain an apparent dissociation constant.

For the only modestly more complicated bivalent system, we can derive the relationship between an apparent dissociation constant and a complete model with two dissociation constants to describe the multiple processes taking place.

In the case of the 2-antigen structure, there are N = 5 total states, corresponding to an empty structure (σ_), two states with 1 monovalently occupied antigen each (σ_Ab_ and σ__Ab), one state with both antigens bivalently occupied by one antibody (σ_.Ab.), and a state with both antigens monovalently occupied by antibodies (σ_AbAb).

First, the reaction system can be represented according to the diagram in Figure 1 or the set of reactions below:

$${\sigma }_{\_,\_}\underset{k-1}{\overset{c_{Ab}{k}_1}{\rightleftharpoons }}{\sigma }_{\text{Ab},\_};{\sigma }_{\_,\_}\underset{k-1}{\overset{c_{Ab}{k}_1}{\rightleftharpoons }}{\sigma }_{\_,\text{Ab}}$$ (41)

$${\sigma }_{\text{Ab},\_}\underset{k-1}{\overset{c_{Ab}{k}_1}{\rightleftharpoons }}{\sigma }_{\text{Ab},\text{Ab}};{\sigma }_{\_,\text{Ab}}\underset{k-1}{\overset{c_{Ab}{k}_1}{\rightleftharpoons }}{\sigma }_{\text{Ab},\text{Ab}}$$ (42)

$${\sigma }_{\text{Ab},\_}\underset{k-2}{\overset{k_2}{\rightleftharpoons }}{\sigma }_{.\text{Ab}.};{\sigma }_{\_,\text{Ab}}\underset{k-2}{\overset{k_2}{\rightleftharpoons }}{\sigma }_{.\text{Ab}.}$$ (43)

We have two dissociation constants respectively for the processes of monovalent binding and bivalent interconversion.

$$K_{D1}=\frac{k_{-1}}{k_1}$$ (44)

$$K_{D2}=\frac{k_{-2}}{k_2}$$ (45)

The system can be represented with a system of differential equations:

$$\frac{d{p}_{{\sigma }_{--}}}{dt}=-2c_{Ab}{k}_1{p}_{{\sigma }_{--}}+k_{-1}{p}_{{\sigma }_{Ab\_}}+k_{-1}{p}_{{\sigma }_{\_Ab}}$$ (46)

$$\frac{d{p}_{{\sigma }_{Ab\_}}}{dt}=c_{Ab}{k}_1{p}_{{\sigma }_{\_\_}}-c_{Ab}{k}_1{p}_{{\sigma }_{Ab\_}}-k_2{p}_{{\sigma }_{A{b}_{\_}}}-k_{-1}{p}_{{\sigma }_{Ab\_}}+k_{-1}{p}_{{\sigma }_{AbAb}}+k_{-2}{p}_{{\sigma }_{.Ab.}}$$ (47)

$$\frac{d{p}_{{\sigma }_{\_Ab}}}{dt}=c_{Ab}{k}_1{p}_{{\sigma }_{\_\_}}-c_{Ab}{k}_1{p}_{{\sigma }_{\_Ab}}-k_2{p}_{{\sigma }_{\_Ab}}-k_{-1}{p}_{{\sigma }_{-Ab}}+k_{-1}{p}_{{\sigma }_{AbAb}}+k_{-2}{p}_{{\sigma }_{.Ab.}}$$ (48)

$$\frac{d{p}_{{\sigma }_{.Ab.}}}{dt}=k_2{p}_{{\sigma }_{A{b}_{\_}}}+k_2{p}_{{\sigma }_{-Ab}}\_2k_{-2}{p}_{{\sigma }_{.Ab}.}$$ (49)

$$\frac{d{p}_{{\sigma }_{AbAb}}}{dt}=c_{Ab}{k}_1{p}_{{\sigma }_{Ab\_}}+c_{Ab}{k}_1{p}_{{\sigma }_{-Ab}}\_2k_{-1}{p}_{{\sigma }_{AbAb}}$$ (50)

where p_{σ_} p_{σ Ab_} p_{σ_Ab} p_σ.Ab. p_{σ AbAb} the probabilities of each of the five states in the bivalent systems, subject to the normalization condition:

$$p_{{\sigma }_{\_}}+p_{{\sigma }_{A{b}_{\_}}}+p_{{\sigma }_{\_Ab}}+p_{{\sigma }_{.Ab.}}+p_{{\sigma }_{AbAb}}=1$$ (51)

Given knowledge of the constituent equilibrium constants, we can in the simple case of the bivalent system, solve for the apparent dissociation constant as a function of the microconstants. This is, in effect, specifying a certain equilbrium value predicted on the basis of the complete bivalent model, and assuming instead that it is the result of 1-1 kinetics. However for multiple concentrations, the equilibrium will not shift proportionately, thus the apparent binding constant is a function of the concentration from which the equilibrium value is derived, making its value depent on the conditions rather than serving as a concise description of the system as a whole.

The bivalent system has, at equilibrium, the condition that the rate of change of each of its states is zero

$${\left(\frac{d{\sigma }_{\_}}{dt}\right)}_{eq}={\left(\frac{d{\sigma }_{A{b}_{\_}}}{dt}\right)}_{eq}={\left(\frac{d{\sigma }_{\_Ab}}{dt}\right)}_{eq}={\left(\frac{d{\sigma }_{.Ab.}}{dt}\right)}_{eq}={\left(\frac{d{\sigma }_{AbAb}}{dt}\right)}_{eq}=0$$ (52)

This condition plus the normalization conditions allows us to solve for the equilibrium concentrations of each of the species in terms of rate constants and the fixed solution concentration of analyte antibody.

$$p_{{\sigma }_{\_}}^{eq}=\frac{k_{-1}^2{k}_{-2}}{c_{Ab}^2{k}_1^2{k}_{-2}+c_{Ab}{k}_1{k}_2{k}_{-1}+2c_{Ab}{k}_1{k}_{-1}{k}_{-2}+k_{-1}^2{k}_{-2}}$$ (53)

$$p_{{\sigma }_{A{b}_{\_}}}^{eq}=\frac{c_{Ab}{k}_1{k}_{-1}{k}_{-2}}{c_{Ab}^2{k}_1^2{k}_{-2}+c_{Ab}{k}_1{k}_2{k}_{-1}+2c_{Ab}{k}_1{k}_{-1}{k}_{-2}+k_{-1}^2{k}_{-2}}$$ (54)

$$p_{{\sigma }_{\_Ab}}^{eq}=\frac{c_{Ab}{k}_1{k}_{-1}{k}_{-2}}{c_{Ab}^2{k}_1^2{k}_{-2}+c_{Ab}{k}_1{k}_2{k}_{-1}+2c_{Ab}{k}_1{k}_{-1}{k}_{-2}+k_{-1}^2{k}_{-2}}$$ (55)

$$p_{\sigma .Ab.}^{eq}=\frac{c_{Ab}{k}_1{k}_2{k}_{-1}}{c_{Ab}^2{k}_1^2{k}_{-2}+c_{Ab}{k}_1{k}_2{k}_{-1}+2c_{Ab}{k}_1{k}_{-1}{k}_{-2}+k_{-1}^2{k}_{-2}}$$ (56)

$$p_{{\sigma }_{AbAb}}^{eq}=\frac{c_{Ab}^2{k}_1^2{k}_{-2}}{c_{Ab}^2{k}_1^2{k}_{-2}+c_{Ab}{k}_1{k}_2{k}_{-1}+2c_{Ab}{k}_1{k}_{-1}{k}_{-2}+k_{-1}^2{k}_{-2}}$$ (57)

We can combine states according to their correponding occupancy - i.e. the number of antibodies that the state contributes to the overall signal due to bound antibody, where

$$p_{{\sigma }_{\_}}+p_{{\sigma }_{A{b}_{\_}}}+p_{{\sigma }_{\_Ab}}+p_{{\sigma }_{.Ab.}}+p_{{\sigma }_{AbAb}}=1$$ (58)

The probabilistic definition of occupancy is the expectation value of state occupancy. Each state has a corresponding integer occupancy associated with the number of antibodies bound to the structure in that state as well as a respective probability of that state at any point in time. The equilibrium occupancy is thus the average occupancy of all the states weighted by their equilibrium probabilities.

$${\Phi }_{eq}={\displaystyle \sum _{i=1}^N{p}_{{\sigma }_i}{\varphi }_{{\sigma }_i}=p_{{\sigma }_{\_}}^{eq}\cdot 0+(p_{{\sigma }_{A{b}_{\_}}}^{eq}+p_{{\sigma }_{\_Ab}}^{eq}+p_{{\sigma }_{.Ab.}}^{eq})\cdot 1+p_{{\sigma }_{AbAb}}^{eq}\cdot 2}$$ (59)

substituting Equations 53 through 57, we arrive at

$${\Phi }_{eq}=\frac{c_{ab}{k}_1\left(2c_{ab}{k}_1{k}_{-2}+k_{-1}\left(k_2+2k_{-2}\right)\right)}{c_{ab}^2{k}_1^2{k}_{-2}+c_{ab}{k}_1{k}_2{k}_{-1}+2c_{ab}{k}_1{k}_{-1}{k}_{-2}+k_{-1}^2{k}_{-2}}$$ (60)

Apparent dissociation constant

Taking the 2-antigen system equilibrium occupancy from Equation 60 and applying it to the equilibrium occupancy in terms of the 1-1 dissociation constant Equation 40 can be used to solve for an apparent equilibrium dissociation constant of the form

$$K_{Dapp}=\frac{c_{Ab}\left(1-{\Phi }_{eq}\right)}{{\Phi }_{eq}}=\frac{k_{-2}\left(k_{-1}^2-c_{ab}^2{k}_1^2\right)}{k_1\left(2c_{ab}{k}_1{k}_{-2}+k_2{k}_{-1}+2k_{-1}{k}_{-2}\right)}$$ (61)

This constant is a value that would be obtained from a 1-1 fit to an equilibrium SPR value that arose from the 2-antigen kinetics. Rearranging and substituting Equations 44 and 45 into Equation 61, the formula simplifies to

$$K_{Dapp}=\frac{-K_{D2}\left(c_{Ab}^2-K_{D1}^2\right)}{2c_{Ab}{K}_{D2}+K_{D1}+2K_{D1}{K}_{D2}}$$ (62)

which we may note is a function of concentration, and which has a root at the critical value when $c_{Ab}{K}_{D2}^2=K_{D1}^2$, i.e. the point at which average equilibrium occupancy greater than 1 is expected in the 2-antigen system, and rendering impossible any 1-1 kinetic description.

Rearrangement of Equation 62 enables us to determine the interconversion constant from an apparent dissociation constant provided that we know the monovalent binding constant.

$$K_{D2}=\frac{K_{Dapp}{K}_{D1}}{-c_{Ab}^2+K_{D1}^2-2K_{Dapp}{c}_{Ab}-2K_{Dapp}{K}_{D1}}$$ (63)

At concentrations where c_Ab ≪ K_D1, the relationship between $K_{D2}^2$ and $K_{Dapp}^2$ is relatively constant (Supplementary Figure 2). Note that this is only valid for PSPR data with a 2-antigen topology of a single separation distance.

4.7. Mathematical description of spatial tolerance

Spatial tolerance refers the favorability of bivalent antibody binding according to the spatial distribution of the 2 adjoining antigens. Some antibodies stretch and compress more than others leading to a greater chance of entering and remaining in a bivalent state. In our model, we propose that the monovalent binding step occurs separately from the bivalent binding step, and that it is purely dependent on the solution phase concentration and the epitope-paratope binding affinity. Spatial tolerance therefore is a property of the interconversion step from monovalent to bivalent states and the reverse process from bivalent back to monovalent. For antigens separated by very small distances, electrostatic repulsion in response to compression and steric hindrance within the IgG molecule occurs, penalizing conversion to bivalent binding and/or favoring unbinding back to monovalent states. Conversely, at larger separation distances, the molecule must stretch to accommodate the gap, again penalizing conversion to bivalence and/or favoring conversion back to monovalence.

Spatial tolerance is a description of the landscape of this tradeoff - the breadth of the favorable region in between extremes that is conducive to bivalent binding, the sharpness and degree of symmetry of the transitions to monovalent preference at close and far separations. Progressive fitting allows us to obtain K_D2 for a single 2-antigen system provided that we have determined K_D1 for a 1-antigen system, for which we take a mean run of multiple 1-antigen runs (n = 6) with plus or minus 1 and 2 standard error of the mean away from the mean run as uncertainty intervals (Supplementary Figure 3 a and i for rabbit and mouse antibodies respectively). Determining K_D2 for different antigen separation distances gives us an empirical basis for spatial tolerance. We can model spatial tolerance phenomenologically with an equation for determining the interconversion constant K_D2 as a function of the separation distance between two antigens x:

$$K_{D2}=K_{D2-\mathit{\text{tensile}}}+K_{D2-\mathit{\text{compression}}},$$ (64)

where K_{D2−compression}(Supplementary Figure 3 b and j for rabbit and mouse antibodies respectively) and K_D2−tensile(Supplementary Figure 3 c and k for rabbit and mouse antibodies respectively) are respectively exponential and logistic terms. These model separately the decrease in interconversion due to tensile stretch of the molecule at increasing distances and that due to the onset of exluded volume, electrostatic repulsion, or steric hindrance caused by compression of the molecule to bridge close distances.

The tensile term is built from a logistic function and has the form:

$$K_{D2-\mathit{\text{tensile}}}=\frac{K_{D2}^{\mathit{\text{max}}}}{1+e^{-{\alpha }_t\left(x-{\ell }_t\right)}}$$ (65)

where $K_{D2}^{\mathit{\text{max}}}$ is an upper limit of the value of K_D2, α_t is the logistic growth rate or steepness with which the tensile penalty grows at increasing separation distances and has units of inverse length, and l_t is the value of the midpoint of the sigmoidal curve which can be thought of as a characteristic length that defines the scale below which favorable interconversion will occur and above which the function approaches minimal interconversion.

The exponential compressive term has the form:

$$K_{D2-\mathit{\text{compression}}}=K_{D2}^{\mathit{\text{max}}}{e}^{-{\alpha }_c\left(x-{\ell }_c\right)},$$ (66)

where α_c is the exponential decay rate which has units of inverse length, and l_c is another characteristic length parameter with units of length. The model is subject to the constraint l_c < l_t.

The combined expression yields Equation 1 which predicts the interconversion constant as a function of separation distance (Supplementary Figure 3 d, e and l,m for rabbit and mouse antibodies respectively). Uncertainty represented with vertical error bars is due to variation in the 1-antigen input data that has been propagated to obtain different K_D2 values fitted with correspondingly different K_D1 values as constraints. Horizontal error bars represent uncertainty in the separation distance of protruding sites on DNA origami nanostructures, estimated according to the method employed by Reuss et al [44].

This can be converted to an effective or apparent dissociation constant as if a 1-1 model on the basis of the bivalent model’s prediction of equilibrium occupancy (Supplementary Figure 3 f and n for rabbit and mouse antibodies respectively) - see Section 4.6. Propagation of 1-antigen input data uncertainties yields slightly different parameterizations of the model due to the shifted K_D2 values, and thus we see a corresponding shift in the minimum of the function where bivalent binding is strongest (Supplementary Figure 3 g and o for rabbit and mouse antibodies respectively).

To assess the goodness of fit, we employ a chi-squared metric whereby expected error is computed by projecting K_D2 onto a straight axis by subtracting a 3 point moving average. Random noise in the data should be approximately Gaussian-distributed about the moving average, and we can thus compute a standard error of the mean E(x) from this straightened profile of the data. Supplementary figures 3 h and p for rabbit and mouse antibodies respectively compare the distribution of K_D2 minus moving average (red) and model prediction (blue), a dispersion which should be Gaussian/random if appropriately fitted. The chi squared metric is:

$${\chi }^2={\displaystyle \sum _x\frac{{(K_{D2-obs}(x)-K_{D2-pred}(x))}^2}{E(x)}}$$ (67)

where K_D2−obs(x) is the observed interconversion value at a distance x and K_D2−pred(x) is that predicted by the model. A good chi-squared metric should be neither much less than 1.000 (indicating over-fitting) nor much greater than 1.000 (indicating poor fit).

4.8. Markov model of arbitrary antigen pattern geometries

For the binding kinetics of multi-antigen patterns of systems of sizes on the order of 2-8 adjacent antigens, we employ a fully enumerative Markov chain model based on a complete transfer matrix, i.e. all possible states and transitions of the system. The antigen pattern itself is modeled as a discrete network of antigen sites with a Euclidean distance matrix

$$\text{D}=\left(\begin{array}{ccccc}0& d_{1,2}& d_{1,3}& \dots & d_{1,N}\\ d_{2,1}& 0& d_{2,3}& \dots & d_{2,N}\\ d_{3,1}& d_{3,2}& 0& \dots & d_{3,N}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ d_{N,1}& d_{N,2}& d_{N,3}& \dots & 0\end{array}\right)$$ (68)

This matrix can be simplified by applying a cutoff d_crit above which antigens are considered too far apart to be neighbors. This reduces the number of possible states, eliminating those that are so unfavorable as to be negligible.

A single state σ_i of the system is defined as a set of antigens, their status (empty, monovalently occupied, bivalently occupied) and a pointer indicating of which bivalent-status antigens are linked to each other. The state space of a system is the set of all states that a structure in the system can assume, i.e. 𝕊 := {σ₀, σ₁, … σ_N}. The set of states are thus all the possible configurations of empty, monovalently bound, and bivalently bound antibodies given the constraints of the pattern geometry (Supplementary Figure 4).

Each state is linked to adjacent states by elementary transitions, i.e. the change in status of individual antibodies comprising the state. Those transitions are either the concentration-dependent addition or the subtraction of a single antibody to the system via monovalent binding or unbinding:

$${\sigma }_i\underset{k_{-1}}{\overset{c_{Ab}{k}_1}{\rightleftharpoons }}{\sigma }_j$$ (69)

or a bivalent interconversion event where a monovalently bound antibody binds to an adjacent antigen site, changing its status to bivalently bound and vice versa:

$${\sigma }_i\underset{k_{-2}}{\overset{k_2}{\rightleftharpoons }}{\sigma }_j$$ (70)

Not all states are necessarily connected. An adjacency matrix describes which states are connected by transitions.

$${\text{A}}_{\text{i},\text{j}}=\{\begin{array}{ll}1\hfill & ,\text{if}\exists \text{connection}\text{between}{\sigma }_i\text{and}{\sigma }_j\hfill \\ 0\hfill & ,\text{otherwise}\hfill \end{array}$$ (71)

The system parameters are the set of zero order transition rates {λ₁ = c_Abk₁, λ₋₁ = k₋₁, λ₂ = k₂, λ₋₂ = k₋₂. The multi-antigen-antibody system is thus fully described by the continuous time Markov model (𝕊,Λ) defined as its set of states and its corresponding transition rate matrix of the form:

$$\mathbf{\Lambda }=\{\begin{array}{ll}\lambda (i,j)\hfill & ,\text{if}{A}_{i,j}=1\hfill \\ 0\hfill & ,\text{if}{A}_{i,j}=0\hfill \end{array}$$ (72)

Automated enumeration of states and their connections in systems of arbitrary antigen pattern geometry is accomplished using an implementation of the breadth-first search (BFS) algorithm. The algorithm searches for edges between adjacent states and assigns the appropriate elementary rate process. A queue of neighboring states is made upon the visitation of any state. One-by-one, the algorithm visits each state in the queue, populating it with additional states when they are discovered, and skipping the addition of states that have already been visited. The algorithm thus is characterized by an initial expansion phase of the queue followed by a systematic reduction of the queue until all states have been visited, and the queue becomes empty. This exhaustive enumeration is deterministic, and enables us to assemble a complete transition matrix regardless of antigen geometry. However as the number of adjacent antigens grows, the number of combinations increases dramatically, thus for larger systems, a sampling based approach must be used instead.

Supplementary Algorithm 1 describes the process by which states and transitions are discovered starting from a single starting state. Here, states are distinguished by the status of each of the sites ζ_k in the pattern, being either empty, monovalently occupied, or bivalently occupied and connected to another adjacent site ζ_s. Colored text is used to separate the different classes of transition.

The time complexity of breadth first search can be expressed as O(|V| + |E|) where |V| and |E| are the number of vertices and edges respectively. In the case of antigen patterns, the former correspond to the number of antigens in the pattern. The latter corresponds to the number of adjacent pairwise connections that are possible between two antigens. This is determined by the bivalent flexibility of the antibody in question, and as a rule of thumb we could say that antigens further than 25 nm apart are not close enough for bivalent bonds to form.

4.9. Transient (non-equilibrium) dynamics of enumerative PSPR models

The continuous time Markov model enables us to compute transient evolution of the system. The probability distribution

$$\text{p}(t)=\left(\begin{array}{c}{p}_0(t)\\ p_1(t)\\ ⋮\\ p_N(t)\end{array}\right)$$ (73)

is a vector whose elements p_i(t) are the probabilities of the respective system states σ₀, σ₁, … σ_N at time t. A uniform probability distribution would, for example, represent equal probabilities of finding a structure in any one of the possible states. Or another example is at the start of a single cycle kinetics PSPR run, when the initial condition p(t₀) is that of a distribution where p_(t₀) = 1 for the state σ_ corresponding to an empty structure and p_i(t₀) = 0 for all other states.

The transient evolution of state probabilities is computed from an intitial condition using the linear system of Chapman-Kolmogorov differential equations:

$$\text{p}(t+\Delta t)=\text{p}(t)\cdot \text{Q}$$ (74)

making use of an infinitessimal generator matrix Q which is obtained from the rate matrix and used to determine the relative rates at which state probabilities change with incremental time.

$${\text{Q}}_{i,j}=\{\begin{array}{ll}{\Lambda }_{i,j}\hfill & ,\text{for}i\ne j\hfill \\ -{\displaystyle \sum }_{i\ne j}{\Lambda }_{i,j}\hfill & ,\text{for}i=j\hfill \end{array}$$ (75)

The infinitessimal generator is then used to compute the change in state probability distribution going from one time point to the next by the matrix exponential formula:

$$\text{p}(t+\Delta t)=\text{p}(t)\cdot e^{\text{Q}\cdot \Delta t}=\text{p}(t){\displaystyle \sum _{\eta =0}^{\infty }\frac{{(\text{Q}\cdot \Delta t)}^{\eta }}{\eta !}}$$ (76)

where η is the computation’s depth of recursion - the higher the more accurate, and Δt is an incremental advancement in time. Due to numerical instability of this solution, we employ the uniformized discrete time Markov chain method of Fox and Glynn in order to stably compute Equation 76 [45]. The continuous Markov model (𝕊, Λ) is approximated by a discrete model (𝕊, U) by renormalizing the generator matrix with respect to the fastest outgoing rate or the uniformization rate q:

$$\text{U}≔\text{I}+\frac{\text{Q}}{q},q\ge ma{x}_i\left\{\right|{\text{Q}}_{i,i}\left|\right\}$$ (77)

where I is the identity matrix.

Supplementary Algorithm 2 shows how this matrix is generated in practice.

Equation 76 becomes the approximation

$$\text{p}(t+\Delta t)=\text{p}(t)\cdot e^{\text{Q}\cdot \Delta t}=\text{p}(t)\cdot e^{q(\text{U}-\text{I})\cdot \Delta t}=\text{p}(t)\cdot e^{q\text{U}\Delta t}{e}^{-q\text{I}\Delta t}=\text{p}(t)e^{-q\Delta t}\cdot e^{q\text{U}\Delta t}$$ (78)

The matrix exponential is then approximated with the following Taylor series expansion:

$$\text{p}(t+\Delta t)=\text{p}(t)\cdot e^{q\Delta t}\cdot {\displaystyle \sum _{\eta =0}^{\infty }\frac{{(q\Delta t\text{U})}^{\eta }}{\eta !}}={\displaystyle \sum _{\eta =0}^{\infty }\frac{e^{-q\Delta t}{(q\Delta t)}^{\eta }}{\eta !}\cdot \text{p}(t)\cdot {\text{U}}^{\eta }}$$ (79)

Using Equation 79, we can stably compute the transient evolution of a system from an initial condition. The system entropy can by computed by

$$S=-k_B{\displaystyle \sum _{{\sigma }_i\in \mathbb{S}}{p}_i(t)\text{ln}(p_i(t))}$$ (80)

Supplementary Algorithm 3 shows how the probability distributions at different time points are computed from an initial condition.

4.10. Fitting continuous time Markov models to PSPR data using autocorrelation of residuals

Using Equation 79 to compute the transient probability distribution of the system, we are able to also compute the occupancy at each time point using the definition from Equation 11. The system occupancy is thus a function of time of the form

$$\Phi (t)={\displaystyle \sum _{I=1}^N{p}_i(t){\varphi }_i}$$ (81)

The continuous time Markov model is fitted to experimental data by comparing occupancies computed on the basis of Equations 79 and 81 with that of occupancy computed from normalizing PSPR data via Equation 27 and we can see that the theoretical curve either correctly or incorrectly fits the experimental data depending on the parameterization (Supplementary Figure 5 a and Supplementary Figure 6 a). Residuals (Supplementary Figure 5 b and Supplementary Figure 6 b) are computed by

$$e(t)=\Phi \left(R_{Ab}\right)-\Phi \left(t,\mathbb{S},\text{U}\right)$$ (82)

While fitting by minimizing the sum of squared residuals can be used to obtain acceptable model parameterizations, we used residual analysis with autocorrelation to improve the robustness of fitting and reduce systematic mis-parameterization by making fits more sensitive to divergence in curve shapes. We compute an absolute, average autocorrelation over a fixed interval kΔt with k = 50 by :

$${\rho }_{e,e}\left(t,t+k\Delta t\right)=\frac{\sqrt{{\left({{{\displaystyle \sum }}^{\text{}}}_k\text{e}\cdot \left|\overrightarrow{v}\right|\right)}^2}}{k}$$ (83)

where e(t, k)=[e(t), e(t+Δt), … e(t + kΔt)], $\overrightarrow{v}=[0,1,\dots k]$, and $\left|\overrightarrow{v}\right|$ is the conjugate of $\overrightarrow{v}$. The objective function min(ε) numerically minimized to obtain fits to experimental PSPR data is then the sum squared residual vector weighted by its autocorrelation vector:

$$ℰ={\displaystyle \sum _{t=0}^{t_f-k+1}{\rho }_{e,e}(t,t+k)\sqrt{e{(t)}^2}}$$ (84)

This provides an error function sensitive to sustained divergence of model and experimental data (Supplementary Figure 6 c) even if the two curves cross paths, and like summing the residuals provides a low value when alignment is good (Supplementary Figure 5 c).

We performed cross validation of Markov model fitting by parameterizing based on experimental data from various antigen patterns (all with fixed nearest-neighbor separation distances between antigens to remove the complication of separation distance dependence of the binding kinetics). The rate parameters derived from these training data were then fixed and the model was applied to other patterns as a limited test of extrapolation of a parameterized model to different antigen pattern geometries. Absolute sums of residuals (Supplementary Figure 7 a) and absolute sums of residuals weighted by autocorrelation (Supplementary Figure 7 b) show that the best-performing models were those that are the most complex and exhibiting bivalence such as the hexagonal and pentagonal configurations in the last two rows. This suggests that downward extrapolation in pattern complexity is more viable than upward.

To validate the spatial tolerance model, we conducted a blinded test whereby an a priori prediction was made using the spatial tolerance parameters from Figure 2f. This was done with the rabbit anti-DIG IgG antibody. The data used for prediction consisted entirely of the averaged 1-antigen data and the series of 2-antigen varied separation distance data used to parameterize the model and the structure-binding data to determine the conversion factor from RU to Ab/struct and vice versa. Thus no data with structures configured with more than 2 antigens was used for parameterization. To perform the test, we chose to predict the evolution of a trimeric 7.2 × 14.3 × 16.0 configuration with the same single-cycle kinetics protocol of timed, known concentration injections used for other runs in this study (Supplementary Figure 7 c). Predictions were made first by computing the expected occupancy values on a per-structure basis. These were then converted into SPR response units (RU) by multiplying them with the R_ABmax value determined from structure binding curve and standard curve from Supplementary Figure 1. Experimental results withheld until predictions were made were then revealed and compared with the theoretical curves each done in triplicate with respective structure-binding data used for each one (Supplementary Figure 7 d).

An important question to consider is how sensitive the stratified state probability distribution predictions are to error in the antibody concentration. As we have shown the distribution is dependent on concentration and timing, however the answer is likely quite complex. As different phases, like those shown in Figure 3 a-c, are structured in a complex manner in terms of their concentration intervals and shape of transitions between phases. To get only a very rough idea of the sensitivity though, what we have done for the triangle structure is to change each of the concentrations by factors of 0.5, 0.9, 1.1, and 2.0 (Supplementary Figure 8 a) to see how this affects both the resulting transient probability distribution (Supplementary Figure 8 b) and the relative correspondence of the predicted curve to that of the experimental (Supplementary Figure 8 c). What we see here is that the probability distribution is fairly robust, with the top 5 most represented states remaining unchanged in all 5 conditions, albeit their relative ranking has changed slightly, e.g. going from 1.1x to 2x.

4.11. Determination of thermodynamic properties

We can obtain equilibrium probabilities from the uniformized CTMC first by simulation out to long time scales at fixed solution concentration until probabilities cease to change:

$$\begin{array}{ll}{\text{p}}^{\ast }=\underset{t\to \infty }{\text{lim}}\text{p}(t),\hfill & {\left(\frac{d}{dt}\text{p}(t)\right)}_{c_{Ab}}=0\hfill \end{array}$$ (85)

We can determine the steady state probability distribution more expediently on the basis of the infinitessimal generator matrix, Equation 75, solving numerically for the probability distribution which, when multiplied with the generator matrix produces a vector of zeros, meaning that there is zero change from one moment to the next, subject to the normalization condition whereby all probabilities must sum to 1. I.e. the steady state probability distribution is the solution to the matrix equation

$$\begin{array}{ll}\mathbf{\text{p}}\cdot \mathbf{\text{Q}}=\mathbf{0},\hfill & {\displaystyle \sum _{{\sigma }_i\in \mathbb{S}}{p}_i=1}\hfill \end{array}$$ (86)

The multi-antigen structure in the context of a PSPR experiment is an open system, freely allowed to exchange particles with the large external reservoire connected to it. With (T, V, c_Ab) held constant, the system (an antigen patterned structure) will approach a minimum free energy at steady state by exchanging antibodies with the bath, obeying the Boltzmann distribution law

$$p_i^{\ast }=\frac{e^{-E_i/k_{B}T}}{{\displaystyle \sum _{{\sigma }_i\in \mathbb{S}}{e}^{-E_i/k_{B}T}}}=\frac{e^{-E_i/k_{B}T}}{\mathbb{Z}}$$ (87)

where ℤ is a grand canonical partition function which predicts equilibrium at a grand potential free energy minimum dΦ(T, V, μ_Ab) = 0 with chemical potential μ_Ab = −k_bT ln c_Ab, and E_ip_i = μ_Ab+μ_monon_mono + μ_bivn_biv are state energies determined by the environmental potential due to solution phase antibody concentration as well as the individual potentials of antibody monovalent and bivalent bonds populating the state, n_mono monovalent bonds and n_biv bivalent bonds with respective chemical potentials μ_mono and μ_biv. The value of ℤ and the state energies are solved numerically for a given steady state probability distribution, making it possible for us to determine thermodynamic quantities.

We can obtain thermodynamic quantities such as the solution concentration dependent equilibrium grand potential free energy via:

$${\Phi }^{\ast }=-k_{B}T\text{ln}\mathbb{Z}$$ (88)

The equilibrium probabilities enable us to calculate relative potential differences via

$$\frac{p_i^{\ast }}{p_j^{\ast }}=e^{-\left(E_i-E_j\right)/k_{B}T}$$ (89)

This makes it possible to compute chemical potentials of monovalent and bivalent bonds, for example from the basic 2-antigen system of a fixed separation, by comparing equilibrium probabilities in states that differ by exactly 1 bond of a given type. This is equivalent to obtaining the change in free energy via the dissociation constant for that process.

$${\mu }_{0-1}=E_{mono}-E_{empty}={\left(\frac{\partial \Phi }{\partial N_{mono}}\right)}_{T,P,Nbiv}=-k_{B}T\text{ln}\frac{{\displaystyle \sum }_{n_{mono}}{p}_{mono}^{\ast }}{{\displaystyle \sum }_{n_{empty}}{p}_{empty}^{\ast }}+k_{B}T\text{ln}{c}_{Ab}=-k_{B}T\text{ln}\frac{n_{mono}}{n_{empty}{K}_{D1}}$$ (90)

$${\mu }_{1-2}=E_{biv}-E_{mono}={\left(\frac{\partial \Phi }{\partial N_{biv}}\right)}_{T,P,Nempty}=-k_{B}T\text{ln}\frac{{\displaystyle \sum }_{n_{biv}}{p}_{biv}^{\ast }}{{\displaystyle \sum }_{n_{\mathit{\text{mono}}}}{p}_{mono}^{\ast }}=-k_{B}T\text{ln}\frac{n_{biv}}{n_{mono}{K}_{D2}}$$ (91)

where n_empty, n_mono, and n_biv are the respective degeneracies of empty, monovalently 1-occupied, and bivalently 1-occupied states. For the 2 antigen system, these values are respectively 1, 2, and 1. For the rabbit IgG, anti-digoxygen model and a separation distance of 15 nm, the chemical potentials are μ₀₋₁ = 1.805 × 10⁻²⁰ J / particle and μ₁₋₂ = 8.889 × 10⁻²¹ J/particle.

We can also obtain a stand-alone bivalent binding chemical potential such that talleying the number of monovalent and bivalent molecules in a state would yield the potential of that state.

$${\mu }_{0-2}=E_{biv}-E_{empty}={\mu }_{0-1}+{\mu }_{1-2}={\left(\frac{\partial \Phi }{\partial N_{biv}}\right)}_{T,P,Nmono}=-k_{B}T\text{ln}\frac{{\displaystyle \sum }_{n_{biv}}{p}_{biv}^{\ast }}{p_{empty}^{\ast }}+k_{B}T\text{ln}{c}_{Ab}$$ (92)

This gives us μ₀₋₂ = 2.693 × 10⁻²⁰ J / particle for the 15 nm separation. By this method, the distance-dependent K_D2 model of Equation 1 can be converted to a chemical potential curve.

4.12. Markov chain Monte Carlo version of model

Systems with larger numbers of bivalent connections have many states and transitions, and the fully enumerative CTMC does not scale well. For these systems, we use a Markov Chain Monte Carlo (MCMC) sampling approach whereby only local states are computed throughout the trajectory of a single system. Multiple trajectories are sampled to approach and approximate the probability distributions that would otherwise be computed to high precision for the enumerative method. Instead of computing the flux of state probabilities over fixed intervals of time, we instead computed Poisson intervals (dwell times) of states according to the rates of processes that each state is subject to. For any particular state σ_i there is a set of adjacent states σ_j ∀ j such that A_i,j ≠ 0.

The exit rate of that state is then the summation of all outgoing rates:

$${\lambda }_{exit}={\displaystyle \sum _{\forall j}{\lambda }_{i,j}|{\text{A}}_{\text{i},\text{j}}\ne 0}$$ (93)

and the corresponding dwell time in that state comes from the exponential cumulative distribution function:

$$\tau =\frac{\mathit{\text{ln}}(1/p)}{{\lambda }_{exit}}$$ (94)

where p is the probability that a transition to a neighboring state occurs within the time interval, and the dwell time τ is a random variable that we may sample by choosing random values of p from the interval [0, 1].

The choice of state given that a transition will occur is then a matter of the relative rates λ_i,j for the different states σ_j, with each state having a probability $p_j=\frac{{\lambda }_{i,j}}{{\lambda }_{exit}}$ of becoming the destination state.

A single iteration thus starts with an initial state, followed by enumeration of each of the adjacent states, and random sampling to determine both dwell time and destination after the transition. Simulation involves performing multiple of these random walks while keeping track of the occupancy and state distribution over a discretized timeline.

Supplementary Algorithm 4 shows how this random walk MCMC approach is used to simulate antibody dynamics by computing dwell times and local state-to-state transitions.

4.13. Experimental methods

Some experimental data used in this work was presented previously as supplementary control data in [18] and was not analyzed further than a basic fitting to a standard 1-1 model. In the current work, we have used the data to develop a more accurate mechanistic model, a pipeline for constructing such models from a minimal dataset, an in-depth physical characterization framework, and de novo simulation that go beyond previous work.

Reagents

Oligonucleotides (unmodified and digoxigenin-modified) were purchased from IDT (Belgium) in 96-well plates format. Chemicals (NaCl, KCl, MgCl2, Tris-HCl, EDTA, PEG800, NaOH, KOAc, KOH, NaOAc) for buffer preparation were purchased from Sigma-Aldrich. Rabbit anti-DIG IgG (no. 9H27L19) was purchased from Thermo Scientific. Streptavidin was purchased from Sigma-Aldrich and mouse anti-DIG IgG1 (no. ab420) was purchased from Abcam. Phosphate-buffered saline 1M stock solution was purchased from Sigma-Aldrich. BIAcore consumables (CM3 sensor chip, HBS-EP running buffer) were purchased from GE Healthcare. Amicon centrifugal filters with 100 kDa MWCO were purchased from Merk Millipore.

Origami design and production

The 18-helix bundle DNA origami nanotube was designed with caDNAno[46], using the honeycomb lattice. This structure has been characterized earlier[18,34,33,47]. In short: the p7560 scaffold was extracted from M13 phage, and the 18-helix bundle DNA nanotube was folded under the following condition: 20 nM scaffold, 100 nM each staple oligonucleotide, 13 mM MgCl₂, 5 mM Tris pH 8.5, 1 mM EDTA. The mixture was subjected to heat denaturation at 80°C for 5 min followed by a slow cooling ramp from 80°C to 60°C over 20 min and 60°C to 24°C over 14 hr. The excess staples were removed by ultrafiltration with Amicon 100 kDa MWCO filters. The wash buffer used was 1 × PBS supplemented with 10 mM MgCl₂.

Pattern surface plasmon resonance protocol

The BIAcore t200 instrument from GE Healthcare was used for surface plasmon resonance experiments. The running buffer used in all experiments is HBS-EP supplemented with 10 mM MgCl₂. The flowrate used for all kinetic experiments is 30 μl/min. Streptavidin was diluted to a final concentration of 10 μg/ml in 10 mM NaOAc pH 4.5 and chemically attached to the CM3 sensor chip with NHS/EDC coupling, using the standard protocol from GE Healthcare. Anchor oligonucleotides containing a 3’ biotin modification were diluted to 200 nM in 1 HBS-EP running buffer and was injected over the surface for 20 min followed by washing of nonspecifically bound oligos by injecting 50 mM NaOH for 5 mins. The DNA nanostructures were diluted to 5 nM and injected over the surface for 20 mins followed by washing with running buffer for 10 mins. Antibodies were diluted to various concentration in running buffer ranging from 0.025 nM to 0.5 nM. The single cycle kinetics injection method was used to sequentially inject the antibody solution over the surface, started from the lowest concentration, the contact time for each concentration is 5 min. After the final antibody injection, the dissociation curve was recorded for 15 min. The immobilized DNA nanostructures and bound antibodies were removed from the surface by injecting 50 mM NaOH for 5 mins and then the surface is ready for the next round of experiment. The t200 evaluation software was used initially to fit the acquired data, for this we used a 1:1 Langmiur binding model to fit the data and estimate the k_a, k_d, K_D and antibody binding capacity - see Supplementary section 6 on the apparent dissociation constant.

Run design

The parameterization pipeline first requires a dosing scheme for the single-cycle kinetics program, i.e. the timing and concentrations of staged antibody injections into the system. Since different antibody-antigen systems are going to exhibit different kinetic profiles, we believe the following considerations could help inform the initial choice of dosing scheme.

First of all, approximate knowledge of K_D1 is probably the best starting point. Some antibody suppliers report an in-house measured K_D(K_D1 by our terminology) or else report those values published by researchers who used their product, and this value can also be determined with a single ELISA experiment [48]. Knowing this, it is possible to choose a dosing scheme that elicits both monovalent (K_D1 dominated) kinetics and bivalent (K_D2 dominated) kinetics. K_D1 dominated kinetics occur when the magnitude of K_D1/c_Ab is smaller than K_D2, i.e. at high relative concentrations. Whereas K_D2 dominated effects are most apparent at low concentrations or during the dissociation phase when concentration is set to zero. While K_D2 is molecule-specific, this value is probably subject to less variation among the commonly used isotypes. So even if K_D2 is unknown at first, it may be a reasonable starting guess to assume that it is similar to the values found in our study, i.e. on the order of 10⁻² around optimal separation distances. Thus framing a dosing series based on a supplier’s reported K_D1 value and concentrations that span a range where relative K_D1/c_Ab to K_D2 goes through an inversion is likely to capture a useful range of kinetics well suited to parameterizing the model. The following concentrations and injection timings were used for most experiments in our study: time points (sec): 0, 84, 384, 475, 775, 866, 1166, 1257, 1557, 1656, 1956 concentrations (nM): 0, 0.025, 0, 0.05, 0, 0.1, 0, .25, 0, .5, 0.

5. Data availability

The raw experimental data used to produce the results of this study including can be found at https://github.com/Intertangler/spatial_tolerance/tree/master/data_repository and via Zenodo[49] under the subfolder “Data repository”. Data is licensed under the GNU General Public License. Source data for generating figures 2 through 4 is also available as supplementary data. In addition to raw data, the Github/Zenodo repository contains the Jupyter notebooks detailing the generation of our results including intermediate data and figures and plots are posted in ready-to-run form for reproduction.

6. Code availability

All code used to produce the results of this study including installation, demonstration, and result reproduction instructions are available on GitHub https://github.com/Intertangler/spatial_tolerance and Zenodo[49]. Code is licensed under the GNU General Public License.