Abstract
Viruses and bacteria commonly exhibit spatial repetition of the 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 the 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 where antigen geometry constrains and potentially directs the antibody movement. We propose that this form of molecular programmability could be exploited during the coevolution of pathogens and immune systems or in the design of molecular machines.
Main
Due to their multiple binding domains, immunoglobulin molecules like the bivalent immunoglobulin G (IgG) antibody exhibit complex interactions with multivalent antigens, that is, clusters of multiple copies of molecules or molecular domains occurring at separation distances of the order of 1–30 nm. Multivalent interactions enhance the stability of binding interactions by enabling the 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 the bound antibodies^{1,2,3}.
Many pathogenic surfaces exhibit spatial repetition at length scales relevant to antibody multivalence. Viral capsid proteins undergo selfassembly into periodic patterns^{4}, and some neutralizing antibodies achieve their high affinity and neutralization capability through bivalence^{5,6}. Selfassembling crystalline arrays of surfacelayer (Slayer) proteins—the outermost structure on many bacteria and archaea—are a major contact point between the pathogen and host^{7} and are implicated as the 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 the immune response^{11}. Multivalence is also probably an important factor during the 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 manybodied problem occurring on the timescales of seconds to minutes. Such systems are too computationally expensive for fullatom molecular simulation. Models treating antibodies and antigens as abstract binding and nonbinding units have been the most successful at capturing the relevant dynamics, and have historically treated multivalence as a function of ligand coating density where multivalence emerges statistically as the average nearestneighbor distance between the ligands decreases^{15}. More recently, coarsegrained molecular simulations have been fruitfully used to quantify the effects of the cooperative binding of the antibody subunits on binding affinity^{16,17}. Nevertheless, a challenge of precisely calibrating such models remains due to the absence of experimental tools to independently assess monovalent and multivalent binding dynamics as well as a pipeline to connect such data to the models.
The patterned surface plasmon resonance (PSPR) technique enables the measurement of binding kinetics on precise, monodisperse patterns of ligands, achieving a robust control of geometry through the use of DNA origami nanostructures^{18} (Fig. 1a). Here 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 coarsegrained 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. 1b). From this basis, the dynamics of more complex patterns of multiple antigens can be reduced (Fig. 1c) to combinations of these elementary states. A causal linkage between the pattern geometry and antibody dynamics could be potentially exploited as a form of spatial programmability by either immunity or pathogens during their adversarial coevolution. We investigate this possibility and the role of spatial tolerance, that is, the range and impact of antigen separation distances on bivalent binding kinetics (Fig. 1d), in determining the effective binding affinity, walking speed of antibody migration on patterned surfaces and direction of antibody migration. Such control mechanisms might inform the development of vaccines for greater control over the affinity maturation process.
Results
Spatial tolerance model
We developed a model parameterization pipeline based on a progressive fitting of the transient surface plasmon resonance (SPR) profiles first for monovalent and then for bivalent binding processes to reduce the degrees of freedom at each stage of fitting. In the first stage, we used either rabbit antidigoxygenin (DIG) IgG or mousederived antiDIG IgG1targeting DIGdecorated DNA origamis with a singlecycle kinetics program where progressively higher concentrations of antibody were exposed to the immobilized antigen substrate (Fig. 2a,b). This program was performed with a oneantigen configuration (Fig. 2c,e) to parameterize our Markov model (‘Markov model of arbitrary antigen pattern geometries’ section) 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 (‘Conversion from SPR signal R_{Ab} to bound antibody n_{Ab}’ section). This yields the respective association and dissociation rates k_{1} = 1.93 ± 0.05 × 10^{7} M^{−1} s^{−1} and k_{−1} = 5.28 ± 0.07 × 10^{−4} s^{−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 the interconversion between the monovalent and bivalent states by fixing the previously determined monovalent parameters and fitting the model to experiments involving multiple adjacent antigens (‘Fitting continuoustime Markov models to PSPR data using autocorrelation of residuals’ section). We fitted the model by adjusting K_{D2} or the interconversion constant defined by the ratio of the reverse and forward interconversion rates. For structures configured with two antigens separated by 14.3 ± 1.2 nm, we find K_{D2} = 8 ± 6 × 10^{−3} (Fig. 2d,f).
By applying progressive fitting to PSPR runs with structures patterned with two adjacent antigens of varied separation distances, we found the internal conversion process to vary accordingly. Small and large separation distances correspond to reduced bivalence, that is, larger K_{D2}. We constructed a phenomenological equation (‘Mathematical description of spatial tolerance’ section) modeling the interconversion constant (Fig. 2g). 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
where ℓ_{t} and ℓ_{c} are the characteristic lengths defining the scale of the tensile and compressive terms, respectively; α_{t} is the sharpness of the tensile penalty; α_{c} is the decay parameter of compressive penalty to bivalence; and \({K}_{\mathrm{D}2}^{\mathrm{max}}\) is the value of K_{D2} at which the contributions of bivalence to binding dynamics are vanishingly small. We found that when appropriately fitted (Fig. 2h), this model predicts a theoretical min(K_{D2}) located at ~10.6 nm separation distance with an approximately 1.0 nm uncertainty due to an expected random shift in the oneantigen input data (Fig. 2i), whereas the experimental datapoint with the lowest K_{D2} is located at ~15.0 nm. This result is in agreement with another study^{20} in which the optimal epitope separation distance is estimated using DNA origami and atomic force microscopy, although 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.
Steadystate and transient analyses
To determine the dependence of system bivalence on solutionphase concentration, we used the parameterized model to obtain the steadystate probability distributions for a range of solutionphase concentrations (Fig. 3a). This revealed concentration regimes of differing dominant states: empty, bivalent and saturated monovalent at low, medium and high solutionphase concentrations, respectively. The entropic maxima occur at transitions between these domains (Fig. 3b), and the transitions in bivalent and monovalent contributions to chemical potential occur in accordance with the transition from bivalent to saturated monovalent regimes (Fig. 3c; ‘Determination of thermodynamic properties’ section).
In addition to simulating de novo patterns’ steadystate properties, the model enables us to simulate the transient dynamics of hypothetical systems with arbitrary geometries and arbitrary timing in the introduction of different solutionphase concentrations. To validate the pipeline, we used the model parameterized with one and twoantigen data (Fig. 2g) to create a blind a priori prediction of the evolution of a higherorder system with three antigens arrayed in a 7.2 × 14.3 × 16.0 nm right triangle and then check its correspondence with an experimental trajectory (Fig. 3d); additional validation is shown in Supplementary Fig. 7. We found that the experimental trajectories closely conformed to the predictions. The model provides access to the individual contributions of states to the signal through their occupancy (Fig. 3e) and relative proportions (Fig. 3f), enabling us to construct a narrative explanation for the observed dynamics. We see, for example, in the final stage when the concentration was set to zero, as the total occupancy decreased and occupancy contributions from monovalently bound antibodies decreased, bivalentstate contributions counterintuitively increased. This indicates that higher concentrations in the penultimate stage inverted the system to favor monovalentdominated saturation states that subsequently transitioned into unsaturated bivalent states as the sites became available (Fig. 3g).
Experiments on repetitive antigen patterns
To explore the potential for patternbased control and programmability of antibody dynamics, we modeled the dynamics of larger systems with greater relevance to periodic pathogenic surfaces. For larger systems, a complete enumeration of states scales poorly with increasing numbers of adjacent antigens. We developed a Markov chain Monte Carlo (MCMC) implementation of the model (‘MCMC version of the model’ section) to sample the trajectories that converge to state probabilities with large sample numbers. Rather than enumerating all the system states (that is, combinations of antibodies and binding modes on a structure and possible transitions), the system performs a random walk through the large state space, computing its rate of escape into neighboring states at any point in time. We then examined the collections of individual trajectories for such systems to understand their average behavior. Specifically, we examined the role of repetitive antigen spacing in simple onedimensional (1D) arrays.
Antigens arranged according to a spacing gradient in the range of 10–22 nm separation distances (that is, the interval of the steepest slope in equation (1)) elicit asymmetric accumulation towards the narrowspaced end of the array (Fig. 4a). This system also exhibited individual walking trajectories that tend towards the narrowspaced end (Fig. 4b), asymmetric velocity (Fig. 4c) and asymmetric net displacement (Fig. 4d) according to the direction of the gradient. The mechanism for this locomotion is that of a biased random walk, where 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 the adjacent epitopes leads to a statistical preference for more stable spacings with a lower interconversion ratio K_{D2}.
Antibodies binding to 1D arrays with uniform spacing exhibited divergent residence times, with antibodies spending less cumulative time on 22nmspaced arrays (Fig. 4e) than those of narrow 10nmspaced arrays (Fig. 4f). The migration speeds of antibodies on highstraininducing arrays are greater than those of lowstraininducing arrays, and a comparison of the net displacement shows that antibodies moved further from their initial binding location on widely spaced arrays relative to the narrowly spaced ones (Fig. 4g).
Discussion
Repeating epitope patterns are present in many viruses as coat proteins^{21,22,23,24} and in bacteria as Slayer proteins^{25,26,27}, often with a high degree of symmetry or geometric periodicity. Such repetitive, quasicrystalline patterns have been recognized as a marker of foreignness corresponding to major enhancements of IgG response compared with unorganized substrates^{28}. Investigators have observed both monovalent and bivalent antibody binding to such periodic viral surfaces^{24,29}, and binding enhancement due to bivalence is a recognized factor in determining both immune pathogen recognition and neutralization capability^{30,31}. A highspeed atomic force microscopy study^{32} captured the realtime bipedal locomotion of antibodies on reconstituted pathogenic surfaces with periodic patterns of epitopes.
The authors of this study 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; furthermore, differences in adjacent antigen spacings 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 considered. 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 the PSPR data that systematically modulate 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 overextrapolation in increasingly complex systems, with any inaccuracies in spatial tolerance model or parameterization quality subject to propagation. We predicted that longrange gradients of differential spacings could be used to establish the 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 experimentally test the predictions. First, such designs should be possible using DNA origami; second, 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}. The 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 Fc targeting to prevent opsonization^{40}. The apparent importance of spatial organization in immunological signaling suggests a role for nonequilibrium spatial phenomena such as those studied here, and we might expect antigen organization itself to be under selective pressure during host–pathogen coevolution. 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 via lateral diffusion to achieve the minimum interconversion ratio K_{D2}. Another study 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 lowspikedensity characteristic of human immunodeficiency virus is the 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, for example, immunostimulatory viruslike particles^{35,42}. Our pipeline could also be used to dissect the complex state spaces of bi, tri or tetraspecific antibodies that are recently being developed for therapeutic and biosensing applications^{43}. We expect these molecules to exhibit complex binding behaviors, especially as many are engineered with nonFcbased 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 realitygrounded 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.
Methods
Statistics and reproducibility
Error assessment (Fig. 2g) was performed using a boostrapping method in which the mean and mean ± one standard error of the mean input data were used to form the central, upper and lower inputs propagated to obtain the individual output points shown in the figure, with the upper and lower vertical error bars corresponding to the outputs produced by the upper and lower inputs, respectively. Goodness of fit for the phenomenological spatial tolerance function was characterized using an adapted chisquared metric (‘Mathematical description of spatial tolerance’ section). The assessment of model robustness and predictive potential (Fig. 3 and Supplementary Figs. 7 and 8) was performed using a blinded test in which I.T.H. performed the model parameterization and prediction of experimental SPR curves for an untested threeantigen triangular structure for three independent replicates with different corresponding bound structure amounts, whereas the experimental test data obtained by A.S. and I.S. were withheld until the predictions were submitted for comparison.
Overview of computational methods
Some methods and explanations may be found elsewhere^{18}. However, in the following, we emphasize the original developments of this work including the following: a minimal parameterization pipeline for Markov models sensitive to arbitrary antigen spacings 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 from rabbit and the other, mouse; a systematic approach to determining the conversion factor between the SPR response units (RUs) to that of the number of bound antibodies per structure; steadystate analysis and determination of thermodynamic quantities from equilibriumstate probability distributions; a randomwalk MCMC variant of the model that 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 the saturation of antibodybinding sites, based on the measured signal due to origami structures binding to the surface and a standard curve constructed to relate structure binding to the 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 continuoustime Markov chain model to the SPR binding data of both oneantigen and twoantigen 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. The steadystate properties of a given system geometry are simulated by determining the distribution of states when the net flux between the states is zero. Large systems are simulated using an MCMC simulation that generates many individual trajectories of single antibodies walking on a userspecified pattern geometry. Mathematical explanation can be found in the following sections where this approach is described in more detail.
Model assumptions and constraints
We assume a coarsegrained model of binding states that equates all the physical states in which antibodies are bound by one arm as monovalent and which equates all the physical states in which antibodies bound by two arms as bivalent.
We assume a fixed amount of bound structures that does not change with time t, that is,
where R_{struct} is the SPR signal due to the structures and
where n_{struct} is the molar amount of structures.
The system has an IgG reservoir that is large compared with the available binding surface and thus has an effectively fixed concentration, that is,
where c_{Ab} is the concentration of the solutionphase antibody.
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.
where σ_ is the state corresponding to an unoccupied structure; σ_{Ab} is the state corresponding to a structure with a bound antibody; and k_{1} 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, that is,
or we may work in terms of the molar amount of structures, both occupied and unoccupied (units of mol), where n_{Ab} is the number of bound antibodystructure complexes and n_ is the number of unoccupied structures; n_{Ab} and n_ are functions of t.
Therefore, the state probabilities are
and
corresponding to the unoccupied and oneantibodyoccupied states, respectively.
We also define occupancy as the number of antibodies that are bound to a single structure for a given state. For simple oneantigen structures, this value is zero for the empty state and 1 for the bound state, namely,
and
respectively. 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.
For the case of the oneantigen structure, this becomes
In a 1–1 binding model, a change in the SPR signal is proportional to the amount of bound material or, in other words, a change in the molar amount of structures with occupied binding sites n_{Ab}.
where ξ_{Ab} is a conversion factor corresponding to the expected change in the SPR response signal per mole of bound antibodies.
The rate of change of the occupied sites is equal to the rate of conversion of unoccupied sites via binding events minus the rate of the conversion of occupied sites via unbinding events.
The SPR signal after the structurebinding step is proportional to the molar amount of bound structures, that is,
where ξ_{struct} is a conversion factor corresponding to the expected change in the SPR response signal per mole of structures.
Substituting RUbased expressions of molar amounts into equation (6), we get
Substituting RUbased expressions of molar amounts into equation (14) yields
which simplifies to
Since we have gathered both conversion constants into one term in equation (18), we now define the occupancy signal factor as
or the dimensionless ratio of molar conversion factors: bound antibody relative to the structure.
Note, by rearrangement, the relationship to average occupancy—that is, the occupancy signal factor—is the ratio of occupancy (in terms of SPR signal) to the molar quantities.
Substituting ξ_{*}, we then arrive at the expression for the rate of change in the SPR signal with respect to time as a function of the structurebinding signal and antibodybinding signal:
In the case of a monovalent structure (one antigen available for binding) at the point of maximum saturation when the average occupancy is unitary (Φ = 1), the molar quantities of the bound antibody and structures are equal:
where \({n}_{\mathrm{Ab}}^{\mathrm{max}}\) is the maximum number of moles of antibody that can bind to the system.
Under the maximum saturation conditions, the monovalent occupancy signal factor then reduces to
where \({R}_{\mathrm{Ab}}^{\mathrm{max}}\) is the maximum SPR response signal due to the saturation of antibodybinding sites.
This relationship is then used to produce a standard curve from monovalent structurebinding data to obtain the following linear relationship:
where an empirically determined ξ_{*} enables one to estimate the SPR signal corresponding to an occupancy of one antibody per structure from the R_{struct} signal. This is useful for structures with valence greater than 1 and whose binding kinetics do not obey simple 1–1 equations. Since \({R}_{\mathrm{Ab}}^{\mathrm{max}}\) on a multivalent structure will not resemble that of the monovalent 1–1 system, we refer to this conversion factor obtained from monovalent \({R}_{\mathrm{Ab}}^{\mathrm{max}}\) as \({R}_{\mathrm{Ab}^\mathrm{mono}}\), that is, an SPR signal to antibody number conversion factor:
where V_{struct} is the valence.
In such cases, we obtain the average occupancy using the estimated \({R}_{\mathrm{Ab}}^{\mathrm{max}}\) from linear regression.
Empirical estimation of \({R}_{\mathrm{Ab}}^{\mathrm{max}}\)
Thus, we obtain a standard curve used to convert the SPR signals for arbitrary structure configurations by empirically determining the correlation between the structurebinding signals and the maximum signals corresponding to saturated monovalent (oneantigen) structures, enabling conversion from SPR signal to occupancy in the absence of a wellunderstood binding model, provided the structurebinding signal is known.
The structurebound signal (Supplementary Fig. 1a) is taken to be the difference between the signals before and after the structures are flowed over the chip and allowed to bind. Estimates of the parameters k_{1}, k_{−1} and ξ_{*} are supplied to a numerical minimization of the autocorrelation of residuals between the experimental and theoretical curves for the fourthorder Runge–Kutta approximation of equation (22), that is, the function \(\frac{\mathrm{d}{R}_{\mathrm{Ab}}}{\mathrm{d}t}=f({R}_{\mathrm{Ab}})\) recursively approximated according to the formula
where h is a small timestep and the constituent terms have the form
For each monovalent run (Supplementary Fig. 1b,c for rabbit and mouse antibodies, respectively) with a unique value of R_{struct}, a projected value of \({R}_{\mathrm{Ab}}^{\mathrm{max}}\) is computed using equation (25). By fitting the monovalent models to 1–1 kinetics, we obtain the rate constants that allow the computational prediction of \({R}_{\mathrm{Ab}}^{\mathrm{max}}\) (Supplementary Fig. 1d,e for rabbit and mouse antibodies, respectively) in the absence of experimental saturation conditions. This enables us to make a standard curve to adjust \({R}_{\mathrm{Ab}}^{\mathrm{mono}}\) according to R_{struct} in the absence of a 1–1 \({R}_{\mathrm{Ab}}^{\mathrm{max}}\) (Supplementary Fig. 1f,g for rabbit and mouse antibodies, respectively). We use this value as a conversion factor, enabling us to convert the SPR RUs into the number of antibodies per structure (Supplementary Fig. 1h,i for rabbit and mouse antibodies, respectively). By knowing R_{struct}, we can estimate this conversion factor for nontrivial antigen configurations where the multivalence influences the ease of reaching a saturation value corresponding to \({R}_{\mathrm{Ab}}^{\mathrm{max}}\).
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 the systems across experimental conditions in which the dynamic behavior can vary substantially. 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 the following:
at the steady state,
Rearranging equation (34) yields
and
For a 1–1 monovalent model, the dissociation constant is
Thus, at equilibrium, the SPR signal is
An 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
where \({R}_{\mathrm{Ab}^{\mathrm{eq}}}\) is the steadystate SPR signal due to the bound antibody.
The equilibrium dissociation constant is a good descriptive parameter that concisely captures the essential dynamics.
From the dissociation constant, we know the occupancy as
where Φ_{eq} is the expected occupancy at the steady state.
Such a concise description is desirable for complex structures as well. However, the difficulty arises in the case of multivalent structures that 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 twoantigen structure, there are N = 5 total states: one empty structure (σ_{__}), two states with one 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 Fig. 1 or the set of reactions below:
We have the two dissociation constants for the processes of monovalent (K_{D1}) binding and bivalent (K_{D2}) interconversion as follows.
The system can be represented with a system of differential equations as follows:
where \({p}_{{\sigma }_{{{{\_}}}{{{\_}}}}}\), \({p}_{{\sigma }_{\mathrm{Ab}\_}}\,\), \({p}_{{\sigma }_{\_\mathrm{Ab}}}\,\), \({p}_{{\sigma }_{.\mathrm{Ab}.}}\,\) and \({p}_{{\sigma }_{\mathrm{Ab}\mathrm{Ab}}}\) are the probabilities of each of the five states in the bivalent systems, subject to the following normalization condition:
Given the 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 equilibrium value predicted on the basis of the complete bivalent model, and assuming instead that it is the result of the 1–1 kinetics. However, for multiple concentrations, the equilibrium will not shift proportionately; therefore, the apparent binding constant is a function of the concentration from which the equilibrium value is derived, making its value dependent 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, that is,
This condition, in addition to 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 the analyte antibody, as follows:
We can combine the states according to their corresponding occupancy—that is, the number of antibodies that the state contributes to the overall signal due to the bound antibody, where
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:
Substituting equations (53) through (57), we arrive at
Apparent dissociation constant
Taking the equilibrium occupancy of the twoantigen system 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
This constant is a value that would be obtained from a 1–1 fit to an equilibrium SPR value that arose from the twoantigen kinetics. Rearranging and substituting equations (44) and (45) into equation (61), the formula simplifies to
which is a function of concentration and has a root at the critical value when \({c}_{\mathrm{Ab}}{K}_{\mathrm{D}2}^{2}={K}_{\mathrm{D}1}^{2}\), that is, the point at which the average equilibrium occupancy greater than 1 is expected in the twoantigen system, and rendering any 1–1 kinetics description as impossible.
The rearrangement of equation (62) enables us to determine the interconversion constant from an apparent dissociation constant provided that we know the monovalent binding constant.
At concentrations where c_{Ab} ≪ K_{D1}, the relationship between \({K}_{\mathrm{D}2}^{2}\) and \({K}_{\mathrm{Dapp}}^{2}\) is relatively constant (Supplementary Fig. 2). Note that this is only valid for PSPR data with a twoantigen topology of a single separation distance.
Mathematical description of spatial tolerance
Spatial tolerance refers to the favourability of bivalent antibody binding according to the spatial distribution of the two 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 solutionphase concentration and the epitope–paratope binding affinity. Spatial tolerance, therefore, is a property of the interconversion step from the 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 the conversion to bivalent binding and/or favouring unbinding back to the monovalent states. Conversely, at larger separation distances, the molecule must stretch to accommodate the gap, again penalizing the conversion to bivalence and/or favouring 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, 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 twoantigen system provided that we have determined K_{D1} for a oneantigen system, for which we take a mean run of multiple oneantigen runs (n = 6) with ±one and two standard errors of the mean away from the mean run as uncertainty intervals (Supplementary Fig. 3a,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 phenomenologically model spatial tolerance with an equation for determining the interconversion constant K_{D2} as a function of the separation distance between two antigens x:
where K_{D2compression} (Supplementary Fig. 3b,j for rabbit and mouse antibodies, respectively) and K_{D2tensile} (Supplementary Fig. 3c,k for rabbit and mouse antibodies, respectively) are the exponential and logistic terms, respectively. These separately model the decrease in interconversion due to the tensile stretch of the molecule at increasing distances and that due to the onset of excluded 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 following form:
where \({K}_{\mathrm{D}2}^{\mathrm{max}}\) 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 that can be thought of as a characteristic length that defines the scale below which favorable interconversion occurs and above which the function approaches minimal interconversion.
The exponential compressive term has the form:
where α_{c} is the exponential decay rate that 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 Fig. 3d,e for rabbit antibodies and Supplementary Fig. 3l,m for mouse antibodies). Uncertainty represented with vertical error bars is due to variation in the oneantigen input data that has been propagated to obtain different K_{D2} values fitted with correspondingly different K_{D1} values as constraints. The horizontal error bars represent uncertainty in the separation distance of protruding sites on DNA origami nanostructures, estimated according to the method employed elsewhere^{44}.
This can be converted into an effective or apparent dissociation constant as a 1–1 model on the basis of the bivalent model’s prediction of equilibrium occupancy (Supplementary Fig. 3f,n for rabbit and mouse antibodies, respectively, and the ‘Apparent dissociation constant’ section). The propagation of oneantigen 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 the strongest (Supplementary Fig. 3g,o for rabbit and mouse antibodies, respectively).
To assess the goodness of fit, we employ a chisquared metric where the expected error is computed by projecting K_{D2} onto a straight axis by subtracting a threepoint 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 Fig. 3h,p for rabbit and mouse antibodies, respectively, compare the distribution of K_{D2} minus the moving average (red) and model prediction (blue), a dispersion that should be Gaussian/random, if appropriately fitted. The chisquared metric is
where K_{D2obs}(x) is the observed interconversion value at a distance x and K_{D2pred}(x) is that predicted by the model. A good chisquared metric should be neither much less than 1.000 (indicating overfitting) nor much greater than 1.000 (indicating poor fit).
Markov model of arbitrary antigen pattern geometries
For the binding kinetics of multiantigen 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, that is, all the 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 as follows:
This matrix can be simplified by applying a cutoff d_{crit} above which the antigens are considered too far apart to be neighbors. This reduces the number of possible states, eliminating those that are so unfavorable that they can be considered negligible.
A single state σ_{i} of the system is defined as a set of antigens, their status (empty, monovalently occupied or bivalently occupied) and a pointer indicating which bivalentstatus antigens are linked to each other. The state space of a system is the set of all the states that a structure in the system can assume, namely, \({\mathbb{S}}:=\{{\sigma }_{0},{\sigma }_{1},\ldots {\sigma }_{N}\}\). The set of states are, thus, all the possible configurations of empty, monovalently bound and bivalently bound antibodies given the constraints of pattern geometry (Supplementary Fig. 4).
Each state is linked to adjacent states by elementary transitions, that is, the change in status of individual antibodies comprising the state. Those transitions are either the concentrationdependent addition or the subtraction of a single antibody to the system via monovalent binding or unbinding:
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:
Not all the states are necessarily connected. An adjacency matrix describes which states are connected by transitions.
The system parameters are the set of zeroorder transition rates {λ_{1} = c_{Ab}k_{1}, λ_{−1} = k_{−1}, λ_{2} = k_{2}, λ_{−2} = k_{−2}}. A multiantigen–antibody system is, thus, fully described by the continuoustime Markov model \(({\mathbb{S}},{{{\bf{\Lambda }}}})\) defined as the set of states and its corresponding transition rate matrix of the form:
The automated enumeration of states and their connections in systems of arbitrary antigen pattern geometry are accomplished using an implementation of the breadthfirst search algorithm. The algorithm searches for edges between adjacent states and assigns the appropriate elementary rate process. A queue of neighboring states is made on 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 the 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 the antigen geometry. However, as the number of adjacent antigens grows, the number of combinations increases dramatically; thus, for larger systems, a samplingbased approach must be used instead.
Supplementary Algorithm 1 describes the process by which the 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, as well as connected to another adjacent site ζ_{s}. The colored text is used to separate the different classes of transition.
The time complexity of the breadthfirst search can be expressed as O(∣V∣ + ∣E∣), where ∣V∣ and ∣E∣ are the numbers of vertices and edges, respectively. In the case of antigen patterns, the former correspond to the number of antigens in the pattern. The latter correspond 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; as a rule of thumb, we could say that antigens further than 25 nm apart are not close enough for bivalent bonds to form.
Transient (nonequilibrium) dynamics of enumerative PSPR models
The continuoustime Markov model enables us to compute the transient evolution of a system. The probability distribution
is a vector whose elements _{pi}(t) are the probabilities of the respective system states σ_{0}, σ_{1},…, 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. Another example is at the start of a singlecycle kinetics PSPR run, when the initial condition p(t_{0}) is that of a distribution where p_{_}(t_{0}) = 1 for the state σ_{_} corresponding to an empty structure and _{pi}(t_{0}) = 0 for all the other states.
The transient evolution of state probabilities is computed from an initial condition using the linear system of Chapman–Kolmogorov differential equations:
which uses an infinitessimal generator matrix Q obtained from the rate matrix and used to determine the relative rates at which state the probabilities change with incremental time.
The infinitessimal generator is then used to compute the change in state probability distribution going from one timepoint to the next by the matrix exponential formula:
where η is the computation’s depth of recursion—the higher it is, the more accurate the result is; Δt is an incremental advancement in time. Due to the numerical instability of this solution, we employ the uniformized discretetime Markov chain method of Fox and Glynn to stably compute equation (76) (ref. ^{45}). The continuoustime Markov model \(({\mathbb{S}},{{{\bf{\Lambda }}}})\) is approximated by a discrete model \(({\mathbb{S}},{{{\bf{U}}}})\) by renormalizing the generator matrix with respect to the fastest outgoing rate or the uniformization rate q:
where I is the identity matrix.
Supplementary Algorithm 2 shows how this matrix is generated in practice.
Equation (76) becomes the approximation
The matrix exponential is then approximated with the following Taylor series expansion.
Using equation (79), we can stably compute the transient evolution of a system from an initial condition.
The system entropy can by computed as
where k_{B} is the Boltzmann constant.
Supplementary Algorithm 3 shows how the probability distributions at different timepoints are computed from an initial condition.
Fitting continuoustime Markov models to PSPR data using autocorrelation of residuals
By using equation (79) to compute the transient probability distribution of the system, we are able to also compute the occupancy at each timepoint using the definition from equation (11). The system occupancy is, thus, a function of time of the form
The continuoustime Markov model is fitted to the experimental data by comparing occupancies computed on the basis of equations (79) and (81) with that of the occupancy computed from the normalizing PSPR data via equation (27); evidently, the theoretical curve either correctly or incorrectly fits the experimental data depending on the parameterization (Supplementary Figs. 5a and 6a). The residuals (Supplementary Figs. 5b and 6b) are computed by
Although fitting by minimizing the sum of the squared residuals can be used to obtain acceptable model parameterizations, we used residual analysis with autocorrelation to improve the robustness of fitting and reduce the systematic misparameterization by making fits more sensitive to divergence in curve shapes. We compute an absolute, average autocorrelation over a fixed interval kΔt with k = 50 as
where e(t, k) = [e(t), e(t + Δt),…, e(t + kΔt)], \(\,\overrightarrow{v}=[0,1,\ldots k]\) and \( \overrightarrow{v}\) is the conjugate of \(\overrightarrow{v}\). The objective function \(\mathrm{min}({{{\mathcal{E}}}})\) numerically minimized to obtain fits to the experimental PSPR data is then the sumsquared residual vector weighted by its autocorrelation vector.
This provides an error function sensitive to sustained divergence of the model and experimental data (Supplementary Fig. 6c) even if the two curves cross paths, and like summing the residuals provides a low value when the alignment is good (Supplementary Fig. 5c).
We performed a crossvalidation of the Markov model fitting by parameterizing based on the experimental data from various antigen patterns (all with fixed nearestneighbor separation distances between the antigens to remove the complication of separationdistance 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 the extrapolation of a parameterized model to different antigen pattern geometries. The absolute sums of residuals (Supplementary Fig. 7a) and absolute sums of residuals weighted by autocorrelation (Supplementary Fig. 7b) show that the bestperforming models were those that are the most complex and exhibit bivalence such as 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 in which an a priori prediction was made using the spatial tolerance parameters (Fig. 2f). This was done with the rabbit antiDIG IgG antibody. The data used for prediction consisted entirely of the averaged oneantigen data and the series of twoantigen variedseparationdistance data used to parameterize the model and the structurebinding data to determine the conversion factor from RU to occupancy, and vice versa. Thus, no data with structures configured with more than two antigens was used for parameterization. To perform the test, we chose to predict the evolution of a trimeric 7.2 × 14.3 × 16.0 nm configuration with the same singlecycle kinetics protocol of timed knownconcentration injections used for the other runs in this study (Supplementary Fig. 7c). Predictions were made first by computing the expected occupancy values on a perstructure basis. These were then converted into SPR RUs by multiplying them with the \({R}_{\mathrm{Ab}}^{\mathrm{max}}\) value determined from the structurebinding curve and standard curve (Supplementary Fig. 1). The experimental results were withheld until predictions were made, and were then revealed and compared with the theoretical curves each done in triplicate with the respective structurebinding data used for each one (Supplementary Fig. 7d).
An important question to consider is how sensitive the stratifiedstate probability distribution predictions are to error in the antibody concentration. As shown earlier, the distribution is dependent on concentration and timing; however, the answer is probably fairly complex. This is because different phases (Fig. 3a–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 Fig. 8a) to see how this affects both the resulting transient probability distribution (Supplementary Fig. 8b) and the relative correspondence of the predicted curve to that of the experimental data (Supplementary Fig. 8c). Evidently, the probability distribution is fairly robust, with the five most represented states remaining unchanged in all the five conditions, albeit their relative ranking changes slightly, for example, going from 1.1x to 2.0x.
Determination of thermodynamic properties
We can obtain the equilibrium probabilities from uniformized continuoustime Markov chain first by simulation out to long timescales at a fixed solution concentration until the probabilities cease to change.
We can determine the steadystate probability distribution more expediently on the basis of the infinitessimal generator matrix, that is, equation (75), numerically solving 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 in which all the probabilities must sum to 1. That is, the steadystate probability distribution is the solution to the matrix equation
The multiantigen structure in the context of a PSPR experiment is an open system, freely allowed to exchange particles with the large external reservoir connected to it. With (T, V, c_{Ab}) held constant, the system (an antigenpatterned structure) will approach a minimum free energy at the steady state by exchanging antibodies with the bath, obeying the Boltzmann distribution law
where \({\mathbb{Z}}\) is a grand canonical partition function that predicts equilibrium at a grand potential freeenergy minimum dΦ(T, V, μ_{Ab}) = 0 with chemical potential µ_{Ab} = –k_{B}Tln[c_{Ab}], and E_{i}_{pi} = μ_{Ab} + μ_{mono}n_{mono} + μ_{biv}n_{biv} are the state energies determined by the environmental potential due to solutionphase antibody concentration and individual potentials of antibody monovalent and bivalent bonds populating the state; further, n_{mono} denotes the monovalent bonds and n_{biv} denotes the bivalent bonds with chemical potentials μ_{mono} and μ_{biv}, respectively. The value of \({\mathbb{Z}}\) and the state energies are numerically solved for a given steadystate probability distribution, making it possible for us to determine the thermodynamic quantities.
We can obtain the thermodynamic quantities such as the solutionconcentrationdependent equilibrium grand potential free energy via
The equilibrium probabilities enable us to calculate the relative potential differences as
This makes it possible to compute the chemical potentials of monovalent and bivalent bonds, for example, from the basic twoantigen system of a fixed separation, by comparing the equilibrium probabilities in states that differ by exactly one bond of a given type. This is equivalent to obtaining the change in free energy via the dissociation constant for that process.
where n_{empty}, n_{mono} and n_{biv} are the degeneracies of empty, monovalently oneoccupied and bivalently oneoccupied states, respectively. For the twoantigen system, these values are 1, 2 and 1, respectively. For the rabbit antiDIG IgG model and a separation distance of 15 nm, the chemical potentials are μ_{0−1} = 1.805 × 10^{−20} J per particle and μ_{1−2} = 8.889 × 10^{−21} J per particle.
We can also obtain a standalone bivalentbinding chemical potential such that tallying the number of monovalent and bivalent molecules in a state would yield the potential of that state.
This gives us μ_{0−2} = 2.693 × 10^{−20} J per particle for 15 nm separation. By this method, the distancedependent K_{D2} model of equation (1) can be converted into a chemical potential curve.
MCMC version of the model
Systems with larger numbers of bivalent connections have many states and transitions, and the fully enumerative continuoustime Markov chain does not scale well. For these systems, we use the MCMC sampling approach in which only the 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 a higher 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 here. 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 the outgoing rates:
and the corresponding dwell time in that state comes from the exponential cumulative distribution function
where p is the probability that a transition to a neighboring state occurs within the time interval and 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 occurs is then a matter of the relative rates λ_{i,j} for the different states σ_{j}, with each state having a probability \({p}_{j}=\frac{{\lambda }_{i,j}}{{\lambda }_{\mathrm{exit}}}\) of becoming the destination state.
A single iteration, thus, starts with an initial state, followed by the enumeration of each of the adjacent states, and random sampling to determine both dwell time and destination after the transition. The simulation involves performing multiple random walks and keeping track of the occupancy and state distribution over a discretized timeline.
Supplementary Algorithm 4 shows how this randomwalk MCMC approach is used to simulate the antibody dynamics by computing the dwell times and local statetostate transitions.
Experimental methods
Some experimental data used in this work were presented previously^{18} and were not analysed 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 indepth physical characterization framework and a de novo simulation that goes beyond the previous work.
Reagents
Oligonucleotides (unmodified and DIG modified) in 96well plates were purchased from IDT. Chemicals (NaCl, KCl, MgCl_{2}, TrisHCl, EDTA, PEG800, NaOH, KOAc, KOH and NaOAc) for buffer preparation were purchased from SigmaAldrich. Rabbit antiDIG IgG (no. 9H27L19) was purchased from Thermo Scientific. Streptavidin was purchased from SigmaAldrich and mouse antiDIG IgG1 (no. ab420) was purchased from Abcam. Phosphatebuffered saline (1 M stock solution) was purchased from SigmaAldrich. BIAcore consumables (CM3 sensor chip and HBSEP (HEPES 10 mM, NaCl 150 mM, EDTA 3 mM, 0.005% Tween 20) running buffer) were purchased from GE Healthcare. Amicon centrifugal filters with 100 kDa molecular weight cutoff were purchased from Merck Millipore.
Origami design and production
The 18helix bundle DNA origami nanotube was designed with caDNAno^{46} using the honeycomb lattice. This structure has been characterized earlier^{18,33,34,47}. In short, the p7560 scaffold was extracted from the M13 phage, and the 18helix bundle DNA nanotube was folded under the following conditions: 20 nM scaffold, 100 nM each staple oligonucleotide, 13 mM MgCl_{2}, 5 mM Tris at 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 to 60 °C over 20 min and 60 to 24 °C over 14 h. The excess staples were removed by ultrafiltration with Amicon 100 kDa molecular weight cutoff filters. The wash buffer used was 1× phosphatebuffered saline supplemented with 10 mM MgCl_{2}.
PSPR protocol
A BIAcore t200 instrument (GE Healthcare) was used for the SPR experiments. The running buffer used in all the experiments is HBSEP supplemented with 10 mM MgCl_{2}. The flow rate used for all the kinetics experiments is 30 μl min^{–1}. Streptavidin was diluted to a final concentration of 10 μg ml^{–1} in 10 mM NaOAc at pH 4.5 and chemically attached to the CM3 sensor chip with Nhydroxysuccinimide and 1ethyl3(3dimethylaminopropyl)carbodiimide coupling (using the standard protocol from GE Healthcare). Anchor oligonucleotides containing a 3'biotin modification were diluted to 200 nM in 1× HBSEP running buffer and injected over the surface for 20 min followed by the washing of nonspecifically bound oligos by injecting 50 mM NaOH for 5 min. The DNA nanostructures were diluted to 5 nM and injected over the surface for 20 mins followed by washing with the running buffer for 10 min. The antibodies were diluted to various concentrations in the running buffer, ranging from 0.025 to 0.500 nM. The singlecycle kinetics injection method was used to sequentially inject the antibody solution over the surface, starting 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 min; then, the surface is ready for the next round of experiments. The t200 evaluation software was initially used to fit the acquired data; for this, we used a 1:1 Langmiur binding model to fit the data and estimate the values of k_{a}, k_{d}, the association and dissociation rates, respectively, and dissociation constant K_{D} and antibodybinding capacity (Supplementary Section 6 details the apparent dissociation constant).
Run design
The parameterization pipeline first requires a dosing scheme for the singlecycle kinetics program, that is, the timing and concentrations of the 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, an approximate knowledge of K_{D1} is probably the best starting point. Some antibody suppliers report an inhouse 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}, that is, at higher relative concentrations. However, K_{D2}dominated effects are the most apparent at lower concentrations or during the dissociation phase when the concentration is set to zero. Although K_{D2} is molecule specific, this value is probably subject to less variation among the commonly used isotypes. Therefore, even if K_{D2} is unknown at first, it may be a reasonable starting estimate to assume that it is similar to the values found in our study, that is, of the order of 10^{−2} around the 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 corresponding injection timings were used for most experiments in our study: timepoints (s): 0, 84, 384, 475, 775, 866, 1,166, 1,257, 1,557, 1,656 and 1,956; concentrations (nM): 0, 0.025, 0, 0.050, 0, 0.100, 0, 0.250, 0, 0.500 and 0.
Reporting Summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
Data availability
The raw experimental data used to produce the results of this study can be found at https://github.com/Intertangler/spatial_tolerance/tree/master/data_repository and Zenodo^{49} under the subfolder ‘Data repository’. Data are licensed under the GNU General Public License. Source data for Figs. 2–4 are provided with this paper. In addition to the 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 readytorun form for reproduction.
Code availability
All the codes 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}. The code is licensed under the GNU General Public License.
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Acknowledgements
We would like to acknowledge F. Fördős for helpful discussions. We would like to acknowledge support from A. W. Stiftelsen (M190547) to I.T.H. and the Knut and Alice Wallenberg foundation (KAW 2017.0114 and KAW 2017.0276) and from the European Research Council ERC (GA no. 724872) to B.H.
Funding
Open access funding provided by Royal Institute of Technology.
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Contributions
I.T.H. conceived and implemented the core model and carried out the computational experiments. A.S. and I.S. obtained and preprocessed the PSPR data. I.T.H. and V.S. developed and tested the fitting pipeline and the MCMC simulation. I.T.H., A.S. and B.H. conceived the study concept. I.T.H., A.S., V.S., I.S. and B.H. wrote the manuscript.
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Nature Computational Science thanks Pontus Nordenfelt, Philippe Robert and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Handling editor: Ananya Rastogi, in collaboration with the Nature Computational Science team. Peer reviewer reports are available.
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Supplementary Information
Supplementary Figs. 1–8 and Algorithms 1–4.
Source data
Source Data Fig. 2
Data used for results presented in Fig. 2.
Source Data Fig. 3
Data used for results presented in Fig. 3.
Source Data Fig. 4
Data used for results presented in Fig. 4.
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Hoffecker, I.T., Shaw, A., Sorokina, V. et al. Stochastic modeling of antibody binding predicts programmable migration on antigen patterns. Nat Comput Sci 2, 179–192 (2022). https://doi.org/10.1038/s4358802200218z
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DOI: https://doi.org/10.1038/s4358802200218z
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