# Patent

## Title
Model identification and analysis of bivalent analytes using surface plasmon resonance

## Abstract

Methods, software, systems, and apparatuses that can identify bivalent reaction mechanisms using surface plasmon resonance (SPR) are provided. Methods, software, systems, and apparatuses that can identify SPR sensorgrams that fit a bivalent analyte model are also provided. A method can include recording multiple SPR sensorgrams with an analyte at different concentrations, fitting each sensorgram with a single exponential function with exponents, determining the exponents for each sensorgram and R2 values for each sensorgram, and plotting R2 versus analyte concentration and determining if an optimal concentration exists.

## Background
BACKGROUND OF THE INVENTION
Surface plasmon resonance (SPR) is a widely used, affinity based, label-free biophysical technique to investigate biomolecular interactions. The information gained using SPR can be applied in a multitude of ways. For example, SPR can be used in the medical and pharmaceutical industries in discovering and developing new drugs, studying protein-protein and protein-DNA interactions, and understanding the adsorption of chemical molecules. Surface plasmon resonance can also be used in the food and beverage industry to ensure safety and quality control (e.g., to test for veterinary drug residues in foodstuffs and to test for the presence of genetically modified organisms). However, applications of SPR often require a difficult process of developing models and extracting rate constants.
Obtaining accurate models and rate constants enable a better understanding of the interactions taking place, better predictions and analysis, as well as reduced computing resources. In turn, this leads to SPR being a more effective tool, regardless of the particular industry in which it is applied. Therefore, there is always a need for new and improved ways to analyze SPR data, provide more accurate models, and extract more accurate rate constants.
BRIEF SUMMARY
Embodiments of the present invention include methods, software, systems, and apparatuses that can identify bivalent reaction mechanisms. Embodiments of the present invention also include methods, software, systems, and apparatuses to identify surface plasmon resonance (SPR) sensorgrams that fit a bivalent analyte model.
Embodiments of the present invention can improve SPR analysis, obtain more accurate models and rate constants, reduce computing resources (i.e., make computers more efficient by requiring less processing power and memory) required to analyze and identify the occurrence of bivalent analyte interactions, make computer predictions simpler and more accurate, and provide better predictions of biomolecular interactions. In turn, this leads to SPR being a more effective tool, regardless of the particular industry in which it is applied.
A method for identifying a bivalent reaction mechanism or a bivalent analyte model in SPR sensorgrams can include: recording multiple surface plasmon resonance sensorgrams (i.e., a first batch) with an analyte, with the analyte having a different concentration for each sensorgram; fitting each sensorgram with a single exponential function with exponents; determining the exponents for each sensorgram and R2 values for each sensorgram; and plotting R2 values versus analyte concentration and determining if a maximum exists (i.e., determining if an optimal concentration exists). The method can further include recording a second batch of multiple surface plasmon resonance sensorgrams to perfect the method (to better determine the “optimal concentration”, better identify whether a bivalent analyte mechanism is present, and/or better understand the applicability of a bivalent analyte model).



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 shows an illustration of a bivalent analyte mechanism.
FIG. 2 shows a graph of SPR sensorgrams for ezrinAb binding to immobilized ezrin at different concentrations.
FIG. 3a is a graph showing the dependency of the sum of the exponents (σ1+σ2)s−1 as a function of ezrinAb concentration.
FIG. 3b is a graph showing the dependency of the product of the exponents (σ1*σ2)s−1 as a function of ezrinAb concentration.
FIG. 4 is a graph of R2 values versus Log(C/C0) obtained by fitting SPR association profiles at different analyte concentrations with a single exponential function.



DETAILED DESCRIPTION
Surface plasmon resonance (SPR) is a well-accepted and label-free tool to investigate and analyze biomolecular interactions, including protein-protein, protein-DNA, and protein-lipid membrane interactions. One category of reaction mechanisms that are studied using SPR is biphasic reaction mechanisms, in which more than one type of interaction is present and contributes to the total response. Neither the simplest equilibrium SPR data analysis method nor the single exponential fitting of SPR profiles can handle biphasic reaction mechanisms. Biphasic reaction models include the two-step conformation change model, the heterogeneous ligand model, the bivalent ligand model, and the bivalent analyte model. However, industry and academia have yet to develop sound methods to identify the bivalent analyte mechanism.
Embodiments of the present invention include methods, software, systems, and apparatuses that can identify bivalent reaction mechanisms. Embodiments of the present invention also include methods, software, systems, and apparatuses to identify SPR sensorgrams that fit a bivalent analyte model.
Embodiments of the present invention can improve SPR analysis, obtain more accurate models and rate constants, reduce computing resources (i.e., make computer more efficient by requiring less processing power and memory) required to analyze and identify the occurrence of bivalent analyte interactions, make computer predictions simpler and more accurate, and provide better predictions of biomolecular interactions. In turn, this leads to SPR being a more effective tool, regardless of the particular industry in which it is applied.
The bivalent analyte model involves coupled non-linear differential equations. This application proposes a unique signature for the bivalent analyte model. This signature can be used to distinguish the bivalent analyte model from other biphasic models. Finally, the proposed method will be demonstrated using in an Example using experimentally measured SPR sensorgrams.
Discussed herein is an approach to identify and analyze the bivalent analyte model that can been used to analyze SPR sensorgrams of a wide range of biomolecular interactions. Measured SPR profiles can often be fitted to different biphasic models with comparable fitting qualities. Therefore, fitting quality alone cannot identify the underlying mechanism. Disclosed herein is an approach that can identify the bivalent model without ambiguity.
FIG. 1 is an illustration demonstrating the bivalent analyte mechanism, which is represented by the following two-step process:




  
    
      
        
          
            
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where [A] represents bivalent analyte, [L] represents ligand, [AL1] represents analyte-ligand complex with one ligand, and [AL2] represents analyte-ligand complex with two ligands bound to single analyte. The ka's are the association rate constants, and kd's are the dissociation rate constants. Letting X1 be [AL1] and X2 be [AL2], the density of free ligands on the sensor chip is thus B0−X1−2X2, with B0 as the initial ligand concentration. The two-step process of FIG. 1 can then be represented by the following rate equations:

{dot over (X)}1=2ka1C(B0−X1−2X2)−kd1X1−{dot over (X)}2,  (2)

{dot over (X)}2=ka2X1(B0−X1−2X2)−(2kd2X2,  (3)

where C is the concentration of analyte.

Strictly speaking, Equation (3) is only valid when [AL1] is freely mobile in the bulk solution. When [AL1] is restricted within a layer (reaction layer) on a sensor chip, the second association rate constant (ka2) needs to be replaced by a two-dimensional (2D) rate constant, k*a2. By comparing Equations (2) and (3), it can be seen that ka1C and k*a2X1 must have the same unit of s−1. It is important to understand that the solution of the rate equations (Equations (2) and (3)) gives 2D density of ligand-analyte complex, not the SPR responses directly. In the following analysis, it will be assumed that SPR responses are proportional to the combined 2D densities X1 and X2.
When rate equations are linear differential equations, it is possible to fit SPR sensorgrams directly with solutions of rate equations. For non-linear rate equation, there is no such simplification. Additionally, non-linear differential equations, in general, have no analytical solutions. Therefore, previously proposed methods cannot be directly applied. Rewriting Equations (2) and (3) in variables Y=X1+X2 and X2, the rate equations take the form of Equations (4) and (5).

{dot over (Y)}=2ka1C(Bo−Y)−kd1Y−(2ka1C−kd1)X2,  (4)

{dot over (X)}2=k*a2(X22−Y2)+k*a2BoY−(k*a2Bo−2kd2)X2,  (5)

As expected, Equation (5) is non-linear. However, Equation (4) shows that there exists an “optimal concentration,”




  
    
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at which the rate equation for Y is independent of X2 and, therefore, is a linear differential equation with an analytical solution of single exponential function,





  
    
      
        
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The unknown constant of B0 in Equation (6) does not affect the exponent. The exponent, together with C0, determines ka1 and kd1. However, at this “optimal concentration” C0, the solution Y(t) does not depend on k*a2 or kd2. Therefore, this “optimal concentration” method does not directly obtain these two rate constants. At the optimal concentration, the SPR signal does have contributions from both X1(t) and X2(t), but the solution contains no information on the relative strength of these two components.
In practice, the SPR profiles at different analyte concentrations can be fitted individually using a single exponential function, and the fitting errors should have a minimum at the “optimal concentration.” The existence of this “optimal concentration” is a unique signature of the bivalent analyte model; it thus can be used to distinguish the bivalent-analyte model from other biphasic models. It is worthwhile to point out that this signature is discarded in any “global” fitting procedure.
A method according to an embodiment of the present invention can include recording multiple surface plasmon resonance sensorgrams (i.e., a first batch) with an analyte, with the analyte having a different concentration for each sensorgram; fitting each sensorgram with a single exponential function with exponents; determining the exponents for each sensorgram and R2 values for each sensorgram; and plotting R2 versus analyte concentration and determining if a maximum exists (i.e., determining if an optimal concentration exists). For the first batch of multiple surface plasmon resonance recordings, it is recommended that at least three recording can be conducted, but it would be beneficial to have five, seven, or even more. This is because, the more recordings you obtain, the more likely you are able to “bracket” the “optimal concentration” within the measurements (i.e., determine the optimal concentration to be within the measured range of concentrations) and predict the “optimal concentration” with greater accuracy. Of course, as discussed below, this can take the form of an iterative process, where an approximate “optimal concentration” is determined and more measurements are taken near where the “optimal concentration” appears to be. The process can consist of two, three, four, five, or more iterations, depending on the desired level of precision
Each of the multiple surface plasmon resonance sensorgrams can and generally should be conducted using equivalent chips or sensor surfaces. The single exponential function with exponents can be




  
    
      
        
          
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or an equivalent function. The method can further include determining if the maximum R2 value is over a threshold. The threshold can be, for example, 0.900, 0.910, 0.920, 0.930, 0.940, 0.950, 0.960, 0.970, 0.975, 0.980, 0.985, 0.990, 0.995, 0.998, or 0.999. If the R2 value is over the threshold, it can be concluded that a bivalent analyte reaction mechanism is present and/or a bivalent analyte reaction model is present. The method may further include determining whether the plot of R2 versus analyte concentration takes the form of a quadratic function, which, if it does, suggests that the critical concentration (C0) has been “bracketed” and a bivalent analyte mechanism is occurring or a bivalent analyte model is being demonstrated.

The method can further include fitting each sensorgram with a double exponential function,

R=D+Ee−σ1t+Fe−σ2t (association),

R=Ee−γ1t+Fe−γ2t (dissociation),

or an equivalent exponential function, with D, E, F, σ1, σ2, γ1, and γ2 being fitting parameters and wherein D=−(E+F), and plotting σ1+σ2 versus analyte concentration and determining if the plot of σ1+σ2 versus analyte concentration is non-linear. If it is determined that the plot of σ1+σ2 versus analyte concentration is non-linear, then it can be concluded that a bivalent analyte reaction mechanism is present and/or a bivalent analyte reaction model is present. The method can also include plotting σ1*σ2 versus analyte concentration and determining if the plot of σ1*σ2 versus analyte concentration is either linear or quadratic. If the plot of σ1*σ2 versus analyte concentration is either linear or quadratic, it is concluded that a bivalent analyte reaction mechanism is not present and/or a bivalent analyte reaction model is not present.

Methods, software, systems, and apparatuses of embodiments of the present invention can be used in the medical and pharmaceutical fields for drug development and analyzing protein interactions, biomolecular interactions, and DNA interactions. Methods, software, systems, and apparatuses of embodiments of the present invention can be used in the food and beverage industry for product quality control, product development, GMO testing, pesticide testing, and herbicide testing. Methods, software, systems, and apparatuses of embodiments of the present invention can also be used in environmental applications (e.g., for pollutant and remediation analysis). In addition, methods, software, systems, and apparatuses of embodiments of the present invention can be used in veterinary medicine to treat animals, diagnose diseases, or analyze biomolecular interactions. Of course, all of these applications are just examples and are not intended to limit the scope of the present invention.
The methods of the present invention can partially or completely be implemented using a computing device, having one or more processors and memory, to increase measurement prediction accuracy or to increase the efficiency of computing resources. Computing resources are conserved because less computational steps and less memory is required to provide accurate analysis and predictions.
Embodiments of the present invention provide identification and analysis of the bivalent analyte mechanism and model that can be applied to a wide range of SPR experiments. The proposed procedure can first locate an “optimal analyte concentration” by fitting the individual SPR profile at different analyte concentrations to a single exponential function. The method can be of valuable guidance for the SPR users in order to unambiguously identify and analyze the bivalent analyte mechanism and model.
The methods and processes described herein can be embodied as code and/or data. The software code and data described herein can be stored on one or more computer-readable media or machine-readable media, which may include any device or medium that can store code and/or data for use by a computer system. When a computer system reads and executes the code and/or data stored on a computer-readable medium, the computer system performs the methods and processes embodied as data structures and code stored within the computer-readable storage medium.
It should be appreciated by those skilled in the art that computer-readable media include removable and non-removable structures/devices that can be used for storage of information, such as computer-readable instructions, data structures, program modules, and other data used by a computing system/environment. A computer-readable medium includes, but is not limited to, volatile memory such as random access memories (RAM, DRAM, SRAM); and non-volatile memory such as flash memory, various read-only-memories (ROM, PROM, EPROM, EEPROM), magnetic and ferromagnetic/ferroelectric memories (MRAM, FeRAM), and magnetic and optical storage devices (hard drives, magnetic tape, CDs, DVDs); network devices; or other media now known or later developed that is capable of storing computer-readable information/data. Computer-readable media should not be construed or interpreted to include any propagating signals. A computer-readable medium of the subject invention can be, for example, a compact disc (CD), digital video disc (DVD), flash memory device, volatile memory, or a hard disk drive (HDD), such as an external HDD or the HDD of a computing device, though embodiments are not limited thereto. A computing device can be, for example, a laptop computer, desktop computer, server, cell phone, or tablet, though embodiments are not limited thereto.
The subject invention includes, but is not limited to, the following exemplified embodiments.
Embodiment 1
A method for identifying a bivalent reaction mechanism or bivalent analyte models in surface plasmon resonance (SPR) sensorgrams, the method comprising:

    
    
        recording multiple SPR sensorgrams (i.e., a first batch, preferably comprising three or more recordings) with an analyte, with the analyte having a different concentration for each sensorgram (recording of a sensorgram can include measuring the analyte concentration);
        fitting each sensorgram with a single exponential function with exponents;
        determining the exponents for each sensorgram and R2 values for each sensorgram; and
        plotting R2 versus analyte concentration and determining if a maximum exists (i.e., determining if an optimal concentration exists).
    
    


Embodiment 2
The method of Embodiment 1, wherein the multiple surface plasmon resonance sensorgrams are conducted using equivalent sensor surfaces.
Embodiment 3
The method of any of Embodiments 1 to 2, wherein the single exponential function with exponents is




  
    
      
        
          
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or an equivalent function.

Embodiment 4
The method of any of Embodiments 1 to 3, further comprising determining if the maximum R2 value is over a threshold.
Embodiment 5
The method of any of Embodiments 1 to 4, further comprising determining if the plot of R2 versus analyte concentration takes the form of a quadratic function.
Embodiment 6
The method of any of Embodiments 1 to 5, further comprising:

    
    
        fitting each sensorgram with a double exponential function,

R=D+Ee−σ1t+Fe−σ2t (association),

R=Ee−γ1t+Fe−γ2t (dissociation),

or an equivalent exponential function, with D, E, F, σ1, σ2, γ1, and γ2 being fitting parameters and with D=−(E+F); and

        plotting σ1+σ2 versus analyte concentration and determining if the plot of σ1+σ2 versus analyte concentration is non-linear.
    
    


Embodiment 7
The method of Embodiment 6, wherein if the plot of σ1+σ2 versus analyte concentration is non-linear, it is concluded that a bivalent analyte reaction mechanism is present.
Embodiment 8
The method of any of Embodiments 6 to 7, wherein if the plot of σ1+σ2 versus analyte concentration is non-linear, it is concluded that a bivalent analyte reaction model is appropriate.
Embodiment 9
The method of any of Embodiments 1 to 8, wherein if a maximum exists that is greater than the R2 threshold in the plot of R2 versus analyte concentration, it is concluded that a bivalent analyte reaction mechanism is present.
Embodiment 10
The method of any of Embodiments 1 to 9, wherein if a maximum exists that is greater than the R2 threshold in the plot of R2 versus analyte concentration, it is concluded that a bivalent analyte reaction model is present.
Embodiment 11
The method of any of Embodiments 6 to 10, further comprising:

    
    
        plotting σ1*σ2 versus analyte concentration; and
        determining if the plot of σ1*σ2 versus analyte concentration is either linear or quadratic.
    
    


Embodiment 12
The method of Embodiment 11, wherein if the plot of σ1*σ2 versus analyte concentration is either linear or quadratic, it is concluded that a bivalent analyte reaction mechanism is not present.
Embodiment 13
The method of any of Embodiments 11 to 12, wherein if the plot of σ1*σ2 versus analyte concentration is either linear or quadratic, it is concluded that a bivalent analyte reaction model is not present.
Embodiment 14
The method of any of Embodiments 1 to 13, wherein the method is used in the medical or pharmaceutical fields for drug development, drug analysis, protein interactions, biomolecular interactions, and/or DNA interactions.
Embodiment 15
The method of any of Embodiments 1 to 13, wherein the method is used in the food or beverage industry for product quality control, product development, GMO testing, pesticide testing, and/or herbicide testing.
Embodiment 16
The method of any of Embodiments 1 to 13, wherein the method is used in veterinary medicine to treat animals, diagnose diseases, and/or to analyze biomolecular interactions.
Embodiment 17
The method of any of Embodiments 1 to 16, wherein the method is partially or completely implemented using a computing device, having one or more processors and memory, to increase measurement predictions and/or increase the efficiency of computing resources.
Embodiment 18
The method of any of Embodiments 1 to 17, wherein after the first batch of multiple SPR sensorgrams are recorded, a second batch of multiple SPR sensorgrams are recorded and the method steps of one or more of Embodiments 1 through 13 are conducted again to perfect the analysis (e.g., to better determine the “optimal concentration”, better identify whether a bivalent analyte mechanism is present, and/or better understand the applicability of a bivalent analyte model).
Embodiment 19
The method of Embodiment 18, wherein after the second batch of multiple SPR sensorgrams are recorded, a third batch of multiple SPR sensorgrams are recorded and the method steps of one or more of Embodiments 1 through 13 are conducted again to perfect the analysis.
Embodiment 19
The method of any of Embodiments 1 to 18, further comprising preparing the analyte for sensorgram recording prior to recording the sensorgram(s) (which can be performed once or before each sensorgram recording).
Embodiment 101
A surface plasmon resonance (SPR) system comprising:

    
    
        a machine for recording multiple SPR sensorgrams with an analyte, with the analyte having a different concentration for each sensorgram;
        a (non-transitory) computer-readable (or machine-readable) medium with computer executable instructions stored thereon that when executed by a processor perform a method for identifying a bivalent reaction mechanism or bivalent analyte model, the method comprising:
        
            inputting recordings of the multiple SPR sensorgrams (i.e., a first batch) with the analyte, with the analyte having a different concentration for each sensorgram;
            fitting each sensorgram with a single exponential function with exponents;
            determining the exponents for each sensorgram and R2 values for each sensorgram; and
            plotting R2 versus analyte concentration and determining if a maximum exists (i.e., determining if an optimal concentration exists).
        
        
    
    


Embodiment 102
The system of Embodiment 101, wherein the multiple SPR sensorgrams are conducted using equivalent sensor surfaces.
Embodiment 103
The system of any of Embodiments 101 to 102, wherein the single exponential function with exponents is




  
    
      
        
          
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or an equivalent function.

Embodiment 104
The system of any of Embodiments 101 to 103, wherein the method further comprises determining if the maximum R2 value is over a threshold.
Embodiment 105
The system of any of Embodiments 101 to 104, wherein the method further comprises determining if the plot of R2 versus analyte concentration takes the form of a quadratic function.
Embodiment 106
The system of any of Embodiments 101 to 105, wherein the method further comprises:

    
    
        fitting each sensorgram with a double exponential function,

R=D+Ee−σ1t+Fe−σ2t (association),

R=Ee−γ1t+Fe−γ2t (dissociation),

or an equivalent exponential function, with D, E, F, σ1, σ2, γ1, and γ2 being fitting parameters and with D=−(E+F); and

        plotting σ1+σ2 versus analyte concentration and determining if the plot of σ1+σ2 versus analyte concentration is non-linear.
    
    


Embodiment 107
The system of Embodiment 106, wherein if the plot of σ1+σ2 versus analyte concentration is non-linear, it is concluded that a bivalent analyte reaction mechanism is present.
Embodiment 108
The system of any of Embodiments 106 to 107, wherein if the plot of σ1+σ2 versus analyte concentration is non-linear, it is concluded that a bivalent analyte reaction model is appropriate.
Embodiment 109
The system of any of Embodiments 101 to 108, wherein if a maximum exists that is greater than the R2 threshold in the plot of R2 versus analyte concentration, it is concluded that a bivalent analyte reaction mechanism is present.
Embodiment 110
The system of any of Embodiments 101 to 109, wherein if a maximum exists in the plot of R2 versus analyte concentration that is greater than the R2 threshold, it is concluded that a bivalent analyte reaction model is appropriate.
Embodiment 111
The system of any of Embodiments 106 to 110, wherein the method further comprises:

    
    
        plotting σ1*σ2 versus analyte concentration; and
        determining if the plot of σ1*σ2 versus analyte concentration is either linear or quadratic.
    
    


Embodiment 112
The system of Embodiment 111, wherein if the plot of σ1*σ2 versus analyte concentration is either linear or quadratic, it is concluded that a bivalent analyte reaction mechanism is not present.
Embodiment 113
The system of any of Embodiments 111 to 112, wherein if the plot of σ1*σ2 versus analyte concentration is either linear or quadratic, it is concluded that a bivalent analyte reaction model is not appropriate.
Embodiment 114
The system of any of Embodiments 101 to 113, wherein the system is used in the medical or pharmaceutical fields for drug development, drug analysis, protein interactions, biomolecular interactions, and/or DNA interactions.
Embodiment 115
The system of any of Embodiments 101 to 113, wherein the system is used in the food or beverage industry for product quality control, product development, GMO testing, pesticide testing, and/or herbicide testing.
Embodiment 116
The system of any of Embodiments 101 to 113, wherein the system is used in veterinary medicine to treat animals, diagnose diseases, and/or analyze biomolecular interactions.
Embodiment 117
The system of any of Embodiments 101 to 116, wherein after the first batch of multiple SPR sensorgrams are recorded, a second batch of multiple SPR sensorgrams are recorded and the method steps of one or more of Embodiments 101 through 113 are conducted to perfect the analysis (e.g., to better determine the “optimal concentration”, better identify whether a bivalent analyte mechanism is present, or better understand the applicability of a bivalent analyte model).
Embodiment 118
The system of Embodiment 117, wherein after the second batch of multiple SPR sensorgrams are recorded, a third batch of multiple SPR sensorgrams are recorded and the method steps of one or more of Embodiments 101 through 113 are conducted again to perfect the analysis.
Embodiment 201
A (non-transitory) computer-readable (or machine-readable) medium with computer executable instructions stored thereon that when executed by a processor perform a method for identifying a bivalent reaction mechanism or bivalent analyte model in surface plasmon resonance (SPR) sensorgrams, the method comprising:

    
    
        inputting recordings of multiple SPR sensorgrams (i.e., a first batch) with an analyte, with the analyte having a different concentration for each sensorgram;
        fitting each sensorgram with a single exponential function with exponents;
        determining the exponents for each sensorgram and R2 values for each sensorgram; and
        plotting R2 versus analyte concentration and determining if a maximum exists (i.e., determining if an optimal concentration exists).
    
    


Embodiment 202
The computer-readable (or machine-readable) medium of Embodiment 201, wherein the multiple SPR sensorgrams were conducted using equivalent sensor surfaces.
Embodiment 203
The computer-readable (or machine-readable) medium of any of Embodiments 201 to 202, wherein the single exponential function with exponents is




  
    
      
        
          
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or an equivalent function.

Embodiment 204
The computer-readable (or machine-readable) medium of any of Embodiments 201 to 203, wherein the method further comprises determining if the maximum R2 value is over a threshold.
Embodiment 205
The computer-readable (or machine-readable) medium of any of Embodiments 201 to 204, wherein the method further comprises determining if the plot of R2 versus analyte concentration takes the form of a quadratic function.
Embodiment 206
The computer-readable (or machine-readable) medium of any of Embodiments 201 to 205, wherein the method further comprises:

    
    
        fitting each sensorgram with a double exponential function,

R=D+Ee−σ1t+Fe−σ2t (association),

R=Ee−γ1t+Fe−γ2t (dissociation),

or an equivalent exponential function, with D, E, F, σ1, σ2, γ1, and γ2 being fitting parameters and with D=−(E+F); and

        plotting σ1+σ2 versus analyte concentration and determining if the plot of σ1+σ2 versus analyte concentration is non-linear.
    
    


Embodiment 207
The computer-readable (or machine-readable) medium of Embodiment 206, wherein if the plot of σ1+σ2 versus analyte concentration is non-linear, it is concluded that a bivalent analyte reaction mechanism is present.
Embodiment 208
The computer-readable (or machine-readable) medium of any of Embodiments 206 to 207, wherein if the plot of σ1+σ2 versus analyte concentration is non-linear, it is concluded that a bivalent analyte reaction model is appropriate.
Embodiment 209
The computer-readable (or machine-readable) medium of any of Embodiments 201 to 208, wherein if a maximum exists that is greater than the R2 threshold in the plot of R2 versus analyte concentration, it is concluded that a bivalent analyte reaction mechanism is present.
Embodiment 210
The computer-readable (or machine-readable) medium of any of Embodiments 201 to 209, wherein if a maximum exists that is greater than the R2 threshold in the plot of R2 versus analyte concentration, it is concluded that a bivalent analyte reaction model is appropriate.
Embodiment 211
The computer-readable (or machine-readable) medium of any of Embodiments 206 to 210, wherein the method further comprises:

    
    
        plotting σ1*σ2 versus analyte concentration; and
        determining if the plot of σ1*σ2 versus analyte concentration is either linear or quadratic.
    
    


Embodiment 212
The computer-readable (or machine-readable) medium of Embodiment 211, wherein if the plot of σ1*σ2 versus analyte concentration is either linear or quadratic, it is concluded that a bivalent analyte reaction mechanism is not present.
Embodiment 213
The computer-readable (or machine-readable) medium of any of Embodiments 211 to 212, wherein if the plot of σ1*σ2 versus analyte concentration is either linear or quadratic, it is concluded that a bivalent analyte reaction model is not appropriate.
Embodiment 214
The computer-readable (or machine-readable) medium of any of Embodiments 201 to 213, wherein the method is used in the medical or pharmaceutical fields for drug development, drug analysis, protein interactions, biomolecular interactions, and/or DNA interactions.
Embodiment 215
The computer-readable (or machine-readable) medium of any of Embodiments 201 to 213, wherein the method is used in the food or beverage industry for product quality control, product development, GMO testing, pesticide testing, and/or herbicide testing.
Embodiment 216
The computer-readable (or machine-readable) medium of any of Embodiments 201 to 213, wherein the method is used in veterinary medicine to treat animals, diagnose diseases, and/or analyze biomolecular interactions.
Embodiment 217
The computer-readable (or machine-readable) medium of any of Embodiments 201 to 216, wherein the method increases measurement predictions or increases the efficiency of computing resources.
Embodiment 218
The computer-readable (or machine-readable) medium of any of Embodiments 201 to 217, wherein after the first batch of multiple surface plasmon resonance sensorgrams are inputted, a second batch of multiple surface plasmon resonance sensorgrams are inputted and the method steps of one or more of Embodiments 201 through 213 are conducted to perfect the analysis (to better determine the “optimal concentration”, better identify whether a bivalent analyte mechanism is present, or better understand the applicability of a bivalent analyte model).
Embodiment 219
The computer-readable (or machine-readable) medium of Embodiment 218, wherein after the second batch of multiple SPR sensorgrams are recorded, a third batch of multiple SPR sensorgrams are recorded and the method steps of one or more of Embodiments 201 through 213 are conducted again to perfect the analysis.
Embodiment 301
The method of any of Embodiments 4 to 19, the system of any of Embodiments 104 to 118, or the computer-readable (or machine-readable) medium of any of Embodiments 204 to 219, wherein the threshold is 0.900.
Embodiment 302
The method of any of Embodiments 4 to 19, the system of any of Embodiments 104 to 118, or the computer-readable (or machine-readable) medium of any of Embodiments 204 to 219, wherein the threshold is 0.910.
Embodiment 303
The method of any of Embodiments 4 to 19, the system of any of Embodiments 104 to 118, or the computer-readable (or machine-readable) medium of any of Embodiments 204 to 219, wherein the threshold is 0.920.
Embodiment 304
The method of any of Embodiments 4 to 19, the system of any of Embodiments 104 to 118, or the computer-readable (or machine-readable) medium of any of Embodiments 204 to 219, wherein the threshold is 0.930.
Embodiment 305
The method of any of Embodiments 4 to 19, the system of any of Embodiments 104 to 118, or the computer-readable (or machine-readable) medium of any of Embodiments 204 to 219, wherein the threshold is 0.940.
Embodiment 306
The method of any of Embodiments 4 to 19, the system of any of Embodiments 104 to 118, or the computer-readable (or machine-readable) medium of any of Embodiments 204 to 219, wherein the threshold is 0.950.
Embodiment 307
The method of any of Embodiments 4 to 19, the system of any of Embodiments 104 to 118, or the computer-readable (or machine-readable) medium of any of Embodiments 204 to 219, wherein the threshold is 0.960.
Embodiment 308
The method of any of Embodiments 4 to 19, the system of any of Embodiments 104 to 118, or the computer-readable (or machine-readable) medium of any of Embodiments 204 to 219, wherein the threshold is 0.970.
Embodiment 309
The method of any of Embodiments 4 to 19, the system of any of Embodiments 104 to 118, or the computer-readable (or machine-readable) medium of any of Embodiments 204 to 219, wherein the threshold is 0.975.
Embodiment 310
The method of any of Embodiments 4 to 19, the system of any of Embodiments 104 to 118, or the computer-readable (or machine-readable) medium of any of Embodiments 204 to 219, wherein the threshold is 0.980.
Embodiment 311
The method of any of Embodiments 4 to 19, the system of any of Embodiments 104 to 118, or the computer-readable (or machine-readable) medium of any of Embodiments 204 to 219, wherein the threshold is 0.985.
Embodiment 312
The method of any of Embodiments 4 to 19, the system of any of Embodiments 104 to 118, or the computer-readable (or machine-readable) medium of any of Embodiments 204 to 219, wherein the threshold is 0.990.
Embodiment 313
The method of any of Embodiments 4 to 19, the system of any of Embodiments 104 to 118, or the computer-readable (or machine-readable) medium of any of Embodiments 204 to 219, wherein the threshold is 0.995.
Embodiment 314
The method of any of Embodiments 4 to 19, the system of any of Embodiments 104 to 118, or the computer-readable (or machine-readable) medium of any of Embodiments 204 to 219, wherein the threshold is 0.998.
Embodiment 315
The method of any of Embodiments 4 to 19, the system of any of Embodiments 104 to 118, or the computer-readable (or machine-readable) medium of any of Embodiments 204 to 219, wherein the threshold is 0.999.
A greater understanding of the present invention and of its many advantages may be had from the following example, given by way of illustration. The following example is illustrative of some of the methods, applications, embodiments and variants of the present invention. It is, of course, not to be considered as limiting the invention. Numerous changes and modifications can be made with respect to the invention.
Example 1
The approaches to identifying interactions that fit a bivalent analyte model and the occurrence of a bivalent reaction mechanism discussed in this application were tested in proof of concept experiments, which included experimentally measured SPR sensorgrams. A BIACORE T200 instrument was used to record SPR sensorgrams. Sensor chip CM5 was used to immobilize recombinant ezrin proteins onto the sensor surface via standard amine coupling chemistry. Various concentrations (15.625 nM-500 nM) of anti-ezrin monoclonal antibody (ezrinAb) were passed through the ezrin immobilized sensor surface. FIG. 2 depicts the SPR sensorgrams for ezrinAb-ezrin binding. As shown in FIG. 2, the SPR association profiles did not reach an equilibrium state and, as a result, the simplest equilibrium data analysis method cannot be used. The lowest R2 value of fitting of both association and dissociation profiles (FIG. 2) was less than 0.75. This indicates that the interaction mechanism was not 1:1. The SPR sensorgrams were therefore fitted using the following double exponential functions:

R=D+Ee−σ1t+Fe−σ2t (association),

R=Ee−γ1t+Fe−γ2t (dissociation),

where D, E, F, σ1, σ2, γ1, and γ2 are fitting parameters with D=−(E+F). The lowest R2 value of the fitting of the SPR sensorgram was better than 0.97 (data not shown).

Table 1, below, gives on an analysis on how to use the fitting parameters derived from Equations (7) and (8) to determine appropriate mechanism or model. There are three commonly used models for the linear biphasic reaction, they are: Model 1 (the heterogeneous ligand model); Model 2 (the two-step conformational change model), and Model 3 (the bivalent ligand model). Of course, the focus of the application is Model 4 (the bivalent analyte model).







TABLE 1







Fitting parameters












Model
σ1σ2 vs C. 
σ1 and σ2 vs C
σ1 + σ2 vs C







1
Non-linear
Linear
Linear



2
Linear
Non-linear
Linear



3
Non-linear
Non-linear
Linear



4
Non-linear
Non-linear
Non-linear










As discussed in the related art, a “good” global fitting quality cannot guarantee the correct identification of the underlying mechanism. The behavior of the exponents (fitting parameters, Equation (7)) as a function of the analyte concentration should also be examined. The dependency of the sum of the exponents on ezrinAb concentration as shown in FIG. 3(a) shows that the underlying mechanism is none of the three models discussed in detail in Tiwari et al. (Rev. Sci. Jnstrnm. 86, 035001 (2015); reference 13 in the “References” section below), which is hereby incorporated herein by reference in its entirety. In addition, the product of the exponents should be either linear (two-step conformational change model) or quadratic (heterogeneous ligand model and bivalent ligand model) for the biphasic mechanism to be any of the three biphasic mechanisms. As shown in Tiwari et al., the quadratic dependency must have positive coefficients (coefficients of the quadratic, linear, and constant term in a quadratic equation). The dependency of the product of the exponents as shown in FIG. 3(b) therefore added another validation that the underlying mechanism is not any of the biphasic mechanisms as explained above. Notably, the biphasic models (two-step conformational change model, heterogeneous ligand model, and bivalent ligand model) are governed by coupled system of linear differential equations, unlike the bivalent analyte model presented in this report.
Finally, to correctly identify the underlying biphasic model, the signature of the bivalent analyte model explained above was utilized. The distribution of the R2 value obtained by fitting SPR association profiles at different analyte concentrations with a single exponential function (FIG. 2) is shown in FIG. 4.
The distribution of R2 values for the experimental data followed the theoretical model as predicted by Equations (4) and (6). Therefore, the underlying biphasic mechanism for ezrinAb-ezrin binding should be the bivalent analyte. The monoclonal anti-ezrin antibody is an IgG1 type antibody, which has two Fab portions for binding to ezrin. Therefore, the IgG antibody represents a good model for a bivalent analyte. From the fitting of the SPR association profiles (FIG. 2), the “optimal concentration” is determined to be 62.5 nM. Once the “optimal concentration” is determined, Equation (6) can be used to determine the ka1 and kd1 and hence the equilibrium dissociation constant (KD1) of the interaction (KD1=kd1/ka1) corresponding to the first phase of the interaction. The ka1, kd1, and KD1 values were determined to be 0.74×104 M−1 s−1, 0.92×10−3 s−1, and  ̃124 nM, respectively.
Experimental SPR sensorgrams were measured by using BIACORE T200 instrument available in the BIACORE Molecular Interaction Shared Resource (BMISR) facility at Georgetown University. The BMISR is supported by National Institutes of Health Grant No. P30CA51008.
It should be understood that the examples and embodiments described herein are for illustrative purposes only and that various modifications or changes in light thereof will be suggested to persons skilled in the art and are to be included within the spirit and purview of this application.
All patents, patent applications, provisional applications, and publications referred to or cited herein (including those in the “References” section) are incorporated by reference in their entirety, including all figures and tables, to the extent they are not inconsistent with the explicit teachings of this specification.
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## Claims

1. A method for identifying a bivalent reaction mechanism or bivalent analyte models in surface plasmon resonance (SPR) sensorgrams, the method comprising:
recording a first batch of multiple SPR sensorgrams with an analyte, with the analyte having a different concentration for each sensorgram;
fitting each sensorgram with a single exponential function with exponents;
determining the exponents for each sensorgram and R2 values for each sensorgram; and
plotting R2 versus analyte concentration and determining whether there is an optimal concentration,
the single exponential function with exponents being





  
    
      
        
          
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                  0
                
                2
              
              ⁢
              
                (
                
                  1
                  -
                  
                    e
                    
                      
                        -
                        
                          (
                          
                            4
                            ⁢
                            
                              
                            
                            ⁢
                            
                              k
                              
                                a
                                ⁢
                                
                                  
                                
                                ⁢
                                1
                              
                            
                            ⁢
                            
                              C
                              0
                            
                          
                          )
                        
                      
                      ⁢
                      t
                    
                  
                
                )
              
            
          
        
        ,
      
    
    
      
        
      
    
  



where B0 is an initial ligand concentration, ka1 is an association rate constant of a first ligand binding, kd1 is a dissociation rate constant of the first ligand unbinding, C0 is the optimal concentration and equals −kd1/2ka1, and t is time, and
the method further comprising determining if the plot of R2 versus analyte concentration takes the form of a quadratic function.

2. The method according to claim 1, further comprising determining if the optimal concentration R2 value is over a threshold and, if a maximum exists that is greater than the R2 threshold in the plot of R2 versus analyte concentration, a bivalent analyte reaction mechanism is indicated or a bivalent analyte reaction model is applied.

3. The method according to claim 1, further comprising:
fitting each sensorgram with a double exponential function,


R=D+Ee−σ1t+Fe−σ2t (association),



R=Ee−γ1t+Fe−γ2t (dissociation),


wherein D, E, F, σ1, σ2, γ1, and γ2 are fitting parameters and D=−(E+F); and
plotting σ1+σ2 versus analyte concentration and determining if the plot of σ1+σ2 versus analyte concentration is non-linear.

4. The method according to claim 3, wherein if the plot of σ1+σ2 versus analyte concentration is non-linear, a bivalent analyte reaction mechanism is indicated or a bivalent analyte reaction model is applied.

5. The method according to claim 3, further comprising:
plotting σ1*σ2 versus analyte concentration;
determining if the plot of σ1*σ2 versus analyte concentration is either linear or quadratic; and
if the plot of σ1*σ2 versus analyte concentration is either linear or quadratic, a bivalent analyte reaction mechanism is not indicated or a bivalent analyte reaction model is not applied.

6. A method for identifying a bivalent reaction mechanism or bivalent analyte models in surface plasmon resonance (SPR) sensorgrams, the method comprising:
recording a first batch of multiple SPR sensorgrams with an analyte, with the analyte having a different concentration for each sensorgram;
fitting each sensorgram with a single exponential function with exponents;
determining the exponents for each sensorgram and R2 values for each sensorgram; and
plotting R2 versus analyte concentration and determining whether there is an optimal concentration,
the single exponential function with exponents being





  
    
      
        
          
            Y
            ⁡
            
              (
              t
              )
            
          
          =
          
            
              
                
                  B
                  0
                
                2
              
              ⁢
              
                (
                
                  1
                  -
                  
                    e
                    
                      
                        -
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              
                                k
                                
                                  a
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  1
                                
                              
                              ⁢
                              
                                C
                                0
                              
                            
                            +
                            
                              k
                              
                                d
                                ⁢
                                
                                  
                                
                                ⁢
                                1
                              
                            
                          
                          )
                        
                      
                      ⁢
                      t
                    
                  
                
                )
              
            
            =
            
              
                
                  B
                  0
                
                2
              
              ⁢
              
                (
                
                  1
                  -
                  
                    e
                    
                      
                        -
                        
                          (
                          
                            4
                            ⁢
                            
                              
                            
                            ⁢
                            
                              k
                              
                                a
                                ⁢
                                
                                  
                                
                                ⁢
                                1
                              
                            
                            ⁢
                            
                              C
                              0
                            
                          
                          )
                        
                      
                      ⁢
                      t
                    
                  
                
                )
              
            
          
        
        ,
      
    
    
      
        
      
    
  



where B0 is an initial ligand concentration, ka1 is an association rate constant of a first ligand binding, kd1 is a dissociation rate constant of the first ligand unbinding, C0 is the optimal concentration and equals −kd1/2ka1, and t is time,
the method further comprising:
determining if the optimal concentration R2 value is over a threshold and, if a maximum exists that is greater than the R2 threshold in the plot of R2 versus analyte concentration, a bivalent analyte reaction mechanism is indicated or a bivalent analyte reaction model is applied;
if the optimal concentration R2 value is not over the threshold, recording a second batch of multiple SPR sensorgrams with an analyte, with the analyte having a different concentration for each sensorgram, and the concentrations being focused around a predicted optimum concentration determined from the first batch;
fitting each sensorgram of the second batch with a single exponential function with exponents;
determining the exponents for each sensorgram of the second batch and R2 values for each sensorgram of the second batch; and
plotting R2 versus analyte concentration of the second batch and determining whether there is an optimal concentration.

7. A non-transitory machine-readable medium with computer executable instructions stored thereon that when executed by a processor perform a method for identifying a bivalent reaction mechanism or bivalent analyte model in surface plasmon resonance (SPR) sensorgrams, the method comprising:
inputting recordings of a first batch of multiple SPR sensorgrams with an analyte, with the analyte having a different concentration for each sensorgram;
fitting each sensorgram with a single exponential function with exponents;
determining the exponents for each sensorgram and R2 values for each sensorgram; and
plotting R2 versus analyte concentration and determining whether there is an optimal concentration,
the single exponential function with exponents being





  
    
      
        
          
            Y
            ⁡
            
              (
              t
              )
            
          
          =
          
            
              
                
                  B
                  0
                
                2
              
              ⁢
              
                (
                
                  1
                  -
                  
                    e
                    
                      
                        -
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              
                                k
                                
                                  a
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  1
                                
                              
                              ⁢
                              
                                C
                                0
                              
                            
                            +
                            
                              k
                              
                                d
                                ⁢
                                
                                  
                                
                                ⁢
                                1
                              
                            
                          
                          )
                        
                      
                      ⁢
                      t
                    
                  
                
                )
              
            
            =
            
              
                
                  B
                  0
                
                2
              
              ⁢
              
                (
                
                  1
                  -
                  
                    e
                    
                      
                        -
                        
                          (
                          
                            4
                            ⁢
                            
                              
                            
                            ⁢
                            
                              k
                              
                                a
                                ⁢
                                
                                  
                                
                                ⁢
                                1
                              
                            
                            ⁢
                            
                              C
                              0
                            
                          
                          )
                        
                      
                      ⁢
                      t
                    
                  
                
                )
              
            
          
        
        ,
      
    
    
      
        
      
    
  



where B0 is an initial ligand concentration, ka1 is an association rate constant of a first ligand binding, kd1 is a dissociation rate constant of the first ligand unbinding, C0 is the optimal concentration and equals −kd1/2ka1, and t is time, and
the method further comprising determining if the plot of R2 versus analyte concentration takes the form of a quadratic function.

8. The machine-readable medium according to claim 7, wherein the method further comprises determining if the optimal concentration R2 value is over a threshold and, if a maximum exists that is greater than the R2 threshold in the plot of R2 versus analyte concentration, a bivalent analyte reaction mechanism is indicated or a bivalent analyte reaction model is applied.

9. The machine-readable medium according to claim 7, wherein the method further comprises:
fitting each sensorgram with a double exponential function,


R=D+Ee−σ1t+Fe−σ2t (association),



R=Ee−γ1t+Fe−γ2t (dissociation),


wherein D, E, F, σ1, σ2, γ1, and γ2 are fitting parameters and D=−(E+F); and
plotting σ1+σ2 versus analyte concentration and determining if the plot of σ1+σ2 versus analyte concentration is non-linear.

10. The machine-readable medium according to claim 9, wherein if the plot of σ1+σ2 versus analyte concentration is non-linear, a bivalent analyte reaction mechanism is indicated or a bivalent analyte reaction model is applied.

11. The machine-readable medium according to claim 9, wherein the method further comprises:
plotting σ1*σ2 versus analyte concentration;
determining if the plot of σ1*σ2 versus analyte concentration is either linear or quadratic; and
if the plot of σ1*σ2 versus analyte concentration is either linear or quadratic, a bivalent analyte reaction mechanism is not indicated or a bivalent analyte reaction model is not applied.

12. A non-transitory machine-readable medium with computer executable instructions stored thereon that when executed by a processor perform a method for identifying a bivalent reaction mechanism or bivalent analyte model in surface plasmon resonance (SPR) sensorgrams, the method comprising:
inputting recordings of a first batch of multiple SPR sensorgrams with an analyte, with the analyte having a different concentration for each sensorgram;
fitting each sensorgram with a single exponential function with exponents;
determining the exponents for each sensorgram and R2 values for each sensorgram; and
plotting R2 versus analyte concentration and determining whether there is an optimal concentration,
the single exponential function with exponents being





  
    
      
        
          
            Y
            ⁡
            
              (
              t
              )
            
          
          =
          
            
              
                
                  B
                  0
                
                2
              
              ⁢
              
                (
                
                  1
                  -
                  
                    e
                    
                      
                        -
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              
                                k
                                
                                  a
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  1
                                
                              
                              ⁢
                              
                                C
                                0
                              
                            
                            +
                            
                              k
                              
                                d
                                ⁢
                                
                                  
                                
                                ⁢
                                1
                              
                            
                          
                          )
                        
                      
                      ⁢
                      t
                    
                  
                
                )
              
            
            =
            
              
                
                  B
                  0
                
                2
              
              ⁢
              
                (
                
                  1
                  -
                  
                    e
                    
                      
                        -
                        
                          (
                          
                            4
                            ⁢
                            
                              
                            
                            ⁢
                            
                              k
                              
                                a
                                ⁢
                                
                                  
                                
                                ⁢
                                1
                              
                            
                            ⁢
                            
                              C
                              0
                            
                          
                          )
                        
                      
                      ⁢
                      t
                    
                  
                
                )
              
            
          
        
        ,
      
    
    
      
        
      
    
  



where B0 is an initial ligand concentration, ka1 is an association rate constant of a first ligand binding, kd1 is a dissociation rate constant of the first ligand unbinding, C0 is the optimal concentration and equals −kd1/2ka1, and t is time,
the method further comprising determining if the optimal concentration R2 value is over a threshold and, if a maximum exists that is greater than the R2 threshold in the plot of R2 versus analyte concentration, a bivalent analyte reaction mechanism is indicated or a bivalent analyte reaction model is applied, and
the threshold being 0.980.

13. A non-transitory machine-readable medium with computer executable instructions stored thereon that when executed by a processor perform a method for identifying a bivalent reaction mechanism or bivalent analyte model in surface plasmon resonance (SPR) sensorgrams, the method comprising:
inputting recordings of a first batch of multiple SPR sensorgrams with an analyte, with the analyte having a different concentration for each sensorgram;
fitting each sensorgram with a single exponential function with exponents;
determining the exponents for each sensorgram and R2 values for each sensorgram; and
plotting R2 versus analyte concentration and determining whether there is an optimal concentration,
the single exponential function with exponents being





  
    
      
        
          
            Y
            ⁡
            
              (
              t
              )
            
          
          =
          
            
              
                
                  B
                  0
                
                2
              
              ⁢
              
                (
                
                  1
                  -
                  
                    e
                    
                      
                        -
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              
                                k
                                
                                  a
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  1
                                
                              
                              ⁢
                              
                                C
                                0
                              
                            
                            +
                            
                              k
                              
                                d
                                ⁢
                                
                                  
                                
                                ⁢
                                1
                              
                            
                          
                          )
                        
                      
                      ⁢
                      t
                    
                  
                
                )
              
            
            =
            
              
                
                  B
                  0
                
                2
              
              ⁢
              
                (
                
                  1
                  -
                  
                    e
                    
                      
                        -
                        
                          (
                          
                            4
                            ⁢
                            
                              
                            
                            ⁢
                            
                              k
                              
                                a
                                ⁢
                                
                                  
                                
                                ⁢
                                1
                              
                            
                            ⁢
                            
                              C
                              0
                            
                          
                          )
                        
                      
                      ⁢
                      t
                    
                  
                
                )
              
            
          
        
        ,
      
    
    
      
        
      
    
  



where B0 is an initial ligand concentration, ka1 is an association rate constant of a first ligand binding, kd1 is a dissociation rate constant of the first ligand unbinding, C0 is the optimal concentration and equals −kd1/2ka1, and t is time,
the method further comprising:
determining if the optimal concentration R2 value is over a threshold and if a maximum exists that is greater than the R2 threshold in the plot of R2 versus analyte concentration, a bivalent analyte reaction mechanism is indicated or a bivalent analyte reaction model is applied;
if the optimal concentration R2 value is not over the threshold, recording a second batch of multiple SPR sensorgrams with an analyte, with the analyte having a different concentration for each sensorgram, and the concentrations being focused around a predicted optimum concentration determined from the first batch;
fitting each sensorgram of the second batch with a single exponential function with exponents;
determining the exponents for each sensorgram of the second batch and R2 values for each sensorgram of the second batch; and
plotting R2 versus analyte concentration of the second batch and determining whether there is an optimal concentration.