Estimation of kinetic rate constants from surface ...

Estimation of kinetic rate constants from surface plasmon resonance experiments Neil D. Evans ∗ David Lowe ∗∗ David Briggs ∗∗∗ Robert Higgins ∗∗∗∗ Andrew Bentall † Simon Ball † Daniel Mitchell ∗∗ Daniel Zehnder ∗∗ Michael J. Chappell ∗ ∗

School of Engineering, University of Warwick, Coventry CV4 7AL (e-mail: neil.evans@ warwick.ac.uk). ∗∗ Clinical Sciences Research Institute, Medical School Building, University of Warwick, Coventry CV4 7AL ∗∗∗ NHS Blood and Transplant, Birmingham B15 2SG ∗∗∗∗ Transplant Unit, University Hospitals Coventry and Warwickshire, Coventry CV2 2DX † University Hospital Birmingham, Birmingham B15 2TN Abstract: In order to characterise antibody binding characteristics it is necessary to determine reaction constants from quantitative measurements of the process. Surface plasmon resonance (SPR) provides convenient real-time measurement of the reaction that enables subsequent estimation of the reaction constants. Two models are considered that represent the binding reaction in the presence of transport effects. One of these models, the effective rate constant approximation, can be derived from the other applying a quasi-steady state assumption. Uniqueness of the reaction constants with respect to SPR measurements is considered via a structural identifiability analysis. It is shown that the effective rate constant model is unidentifiable, unless the analyte concentration is known, while the full model is structurally globally identifiable provided association and dissociation phases are considered. Both models provide comparable estimates for the unknown rate constants for a commercial anti-A monoclonal IgM experiment. Keywords: Biomedical systems, structural identifiability, parameter estimation, surface plasmon resonance, surface-volume reactions, binding affinity 1. INTRODUCTION Department of Health targets require an increased rate for all forms of transplantation. A strategy of increasing live donor transplantation will be most effective for renal failure because this can add more donor organs into the transplant pool. For a significant number of patients with preformed donor-specific antibody (DSA, HLA or ABO) live donor transplantation may not be an option because traditionally a positive crossmatch vetoes the transplant. Pretransplant antibody removal can now be used to overcome this barrier (Higgins et al., 2007), but these translants are immunologically very high risk and early rejection and dysfunction are a characteristic. In order to manage this risk a better understanding is needed of how much antibody a transplant can tolerate as well as of qualitative aspects, such as the type of antibody and how this is modulated by the transplant process and immunosuppression. Characterisation and identification of the rejection risk in individual patients will allow tailored immunosuppression and improved risk management (Higgins et al., 2009). Measurement of ABO-specific antibody levels by haemagglutination is notoriously irreproducible despite being the standard method for about 100 years (Aikawa et al., 2003). This is a significant problem in organ transplantation because of the rapid growth in ABO incompatible proce-

dures where treatment must be planned around base-line antibody levels and a safe maximum level at time of transplant must be defined. An ABO antibody detection and quantitation assay has been developed using the ProteOn XPR36 surface plasmon resonance (SPR) platform (BioRad) that allows up to six separate ligands to be analysed simultaneously by up to six analytes. Preliminary pilot studies have been undertaken to evaluate the feasibility of using this technology to address the issues of ABO antibody affinity and quantitative estimation. A number of authors have considered the problem of determining the rate constants for a binding reaction in which one of the reactants is immobilised on a sensor surface. One approach is to model the reaction as occuring on the sensor surface via a reacting boundary condition on a transport equation for the analyte (Myszka et al., 1998; Edwards et al., 1999). Another approach has been to include a receptor layer in the model, to allow for the dextran layer in which the receptor is immobilised, and to also consider analyte diffusion into the layer to reach binding sites (Edwards, 2001). Typically the full partial differential equations framework is used to consider the appropriateness of ordinary differential equation (ODE) based models that are more easily applied to experimental data to yield estimates for the binding constants (see, for example, (Edwards, 2001)). The simplest ODE model is

the rapid mixing model that considers the reaction, after a brief transient period, as the same as two reactants in a well-mixed volume. However, the measured interaction between analyte and receptor is a combination of the binding reaction and transport effects arising from the flow and diffusive processes (Myszka et al., 1998). Therefore, if the rapid mixing model is not appropriate then systematic errors in the estimates for the rate constants can arise (Chaiken et al., 1992). Myszka et al. (1998) propose a two-compartment model that includes transport effects while modelling the binding process using Langmuir-type mass action kinetics. Under a quasi-steady state (QSS) assumption this two-compartment model can be reduced to a single differential equation for the bound analyte. This QSS model is the effective rate constant approximation. Edwards (2001) derived the effective rate constant approximation directly from consideration of the fluid dynamics of the analyte in the flow and the receptor layer, and the subsequent binding. This model has been shown to provide a good approximation under certain conditions (Edwards, 2001). In order to estimate the (unknown) model parameters from SPR data it is necessary to include in the model an output structure, which corresponds to the function of the model variables that is to be compared with data. Before actually collecting experimental data it is necessary to analyse uniqueness of the unknown parameters with respect to this output structure, since estimates for unidentifiable parameters are effectively meaningless. Such a structural identifiability analysis assesses whether the model output contains enough information to determine all of the model parameters uniquely (Jacquez, 1996), and relates only to the structure of the model and output. For linear systems there are many well-established techniques for performing a structural identifiability analysis (see the tutorial by Godfrey and DiStefano III (1987) and other works in the same volume). For nonlinear systems, such as those involving binding kinetics, greater care has to be taken over choice of technique and computational tractability becomes a greater issue; suitable techniques for uncontrolled models are: those based on the uniqueness of a Taylor series expansion of the output (Pohjanpalo, 1978), or on the existence of a smooth transformation between models with identical outputs (Evans et al., 2002), or approaches based on differential algebra (Ljung and Glad, 1994; Saccomani et al., 2003), or on polynomial realisation theory (Nˇemcov´a, 2010). In this paper a differential equation for the output and its derivatives can be obtained that permits a direct analysis of the uniqueness of the parameters. This approach is similar in nature to that taken by Denis-Vidal et al. (2001). 2. COMPARTMENTAL MODELS Two models are applied to the problem of estimating the kinetic rate constants from SPR data: the first model is a two-compartment model proposed by Myszka et al. (1998) while the second model, the effective rate constant approximation, can be derived from the first via a QSS assumption (or directly from consideration of the fluid dynamics of the analyte in the flow and subsequent binding (Edwards, 2001)).

2.1 Full (two-compartment) model Since the SPR platform used permits multiple analytes (in this paper five dilutions of the same sample) across multiple immobilised receptors (A trisaccharide amine and linker) to be run in a single experiment the model proposed by Myszka et al. (1998) is extended to the following: ℎ𝑗 𝐶˙ 𝑖𝑗 = −𝑘𝑎𝑗 𝐶𝑖𝑗 (𝑅𝑖𝑗 − 𝐵𝑖𝑗 ) + 𝑘𝑑𝑗 𝐵𝑖𝑗 + 𝑘𝑀 𝑖𝑗 (𝐼𝑖 − 𝐶𝑖𝑗 ) 𝐵˙ 𝑖𝑗 = 𝑘𝑎𝑗 𝐶𝑖𝑗 (𝑅𝑖𝑗 − 𝐵𝑖𝑗 ) − 𝑘𝑑𝑗 𝐵𝑖𝑗

(1) for 𝑗 = 𝑎, 𝑙 (corresponding to amine and linker, respectively) and 𝑖 = 1, . . . , 5, where: 𝐶𝑖𝑗 (𝑡) is the (volume) concentration of analyte at the surface of the channel in ng/mm3 ; 𝐵𝑖𝑗 (𝑡) is the average bound (area) concentration in ng/mm2 ; ℎ𝑗 = 𝑉𝑗 /𝑆, where 𝑉𝑗 is the volume in contact with the surface from which binding takes place (receptor 𝑗) and 𝑆 is the surface area; 𝑘𝑎𝑗 and 𝑘𝑑𝑗 are the respective association and dissociation rate constants; 𝑘𝑀 𝑖𝑗 is the transport coefficient describing diffusive movement of analyte between the flow and the volume in contact with the surface; and 𝑅𝑖𝑗 is the maximum (area) density of bound analyte possible at the interaction site between flow 𝑖 and receptor 𝑗. Finally, 𝐼𝑖 (𝑡) is the inlet concentration of analyte in the flow channel 𝑖 and is given by { 0 𝑡 ∕∈ [𝑡𝑠 , 𝑡𝑓 ] 𝐼𝑖 (𝑡) = 𝑖 = 1, . . . , 5 (2) 𝑑𝑖 𝐶𝑇 𝑡 ∈ [𝑡𝑠 , 𝑡𝑓 ] where 𝑑𝑖 is the dilution factor for flow 𝑖, 𝐶𝑇 is the analyte sample concentration (in the flow), 𝑡𝑠 is the start time and 𝑡𝑓 the finish time for the association phase (the interval [𝑡𝑠 , 𝑡𝑓 ]). Therefore, the dissociation phase is for 𝑡 > 𝑡𝑓 . Note that, although the volume from which binding takes place is a theoretical construct to allow for mass-transport effects to be modelled, there seems to be no reason to expect it to be concentration/flow dependent, and so ℎ𝑗 = 𝑉𝑗 /𝑆 rather than ℎ𝑖𝑗 = 𝑉𝑖𝑗 /𝑆. Myszka et al. (1998) report that for the range of parameter values associated with Biacore experiments the solution to (1) is insensitive to the value of ℎ𝑗 . This remark suggests a lack of structural and/or numerical identifiability for ℎ𝑗 that is investigated in subsequent sections. The output structure for the model consists of the measurements of bound analyte and is given by 𝑦𝑖𝑗 (𝑡) = 𝛼𝐵𝑖𝑗 (𝑡), 𝑖 = 1, . . . , 5 𝑗 = 𝑎, 𝑙, (3) where 𝛼 is the conversion factor from the units of 𝐵𝑖𝑗 (𝑡) to the response units (RU) of the sensorgrams (1 RU = 10−3 ng/mm2 ). It is important to note that coupling between the differential equations is only via common parameters, and so these can be seen as distinct models being applied simultaneously (with some common parameters). 2.2 Effective rate constant approximation Applying a quasi-steady state assumption to (1) consists of solving 𝐶˙ 𝑖𝑗 (𝑡) = 0 for 𝐶𝑖𝑗 (𝑡) to yield 𝑘𝑑𝑗 𝐵𝑖𝑗 (𝑡) + 𝑘𝑀 𝑖𝑗 𝐼𝑖 (𝑡) 𝐶𝑖𝑗 (𝑡) = 𝑘𝑎𝑗 (𝑅𝑖𝑗 − 𝐵𝑖𝑗 (𝑡)) + 𝑘𝑀 𝑖𝑗 and then substituting this into the equation for 𝐵˙ 𝑖𝑗 (𝑡) gives 𝑘𝑎𝑗 𝐼𝑖 (𝑡)(𝑅𝑖𝑗 − 𝐵𝑖𝑗 (𝑡)) − 𝑘𝑑𝑗 𝐵𝑖𝑗 (𝑡) . 𝐵˙ 𝑖𝑗 (𝑡) = 1 + (𝑘𝑎𝑗 /𝑘𝑀 𝑖𝑗 )(𝑅𝑖𝑗 − 𝐵𝑖𝑗 (𝑡))

(4)

The inflows, 𝐼𝑖 (𝑡), and outputs, 𝑦𝑖𝑗 (𝑡) for the quasi-steady state version of the model are still given by (2) and (3), respectively. The initial conditions are 𝐵𝑖𝑗 (0) = 0 𝑖 = 1, . . . , 5 𝑗 = 𝑎, 𝑙. Equation (4) is the effective rate constant approximation derived by Edwards (2001), who showed that this equation gives a good approximation to a full fluid dynamics model up to 𝑂(Da2 ) where 𝑘𝑎 𝑅/𝑘𝑀 = Da ℎ𝑑 . Da is the Damk¨ ohler number, which is the ratio of the reaction velocity to diffusion velocity in the diffusive boundary layer, and ℎ𝑑 is a positive constant that incorporates effects of the receptor layer. Notice from (4), as discussed by Myszka et al. (1998), that the parameters ℎ𝑗 have been eliminated from the model. Therefore in the steady state situation these parameters have no bearing on the observation. 3. STRUCTURAL IDENTIFIABILITY In this section the uniqueness of the unknown parameters in a general nonlinear model is considered with respect to the outputs. More precisely, let 𝒑 ∈ Ω ⊂ ℝ𝑚 denote a vector comprising the unknown parameters in the model, which belongs to an open set of admissible vectors. To make the dependence of the model outputs on the unknown parameters more explicit they are written 𝑦𝑖 (𝑡, 𝒑), for 𝑖 = 1, . . . , 𝑟. Two parameter vectors 𝒑, 𝒑 ∈ Ω are indistinguishable, written 𝒑 ∼ 𝒑, if they give rise to identitical outputs: for all 𝑡 ≥ 0, 𝑖 = 1, . . . , 𝑟. 𝑦𝑖 (𝑡, 𝒑) = 𝑦𝑖 (𝑡, 𝒑) For generic 𝒑 ∈ Ω, the parameter 𝑝𝑖 is locally identifiable if there is a neighbourhood, 𝑁 , of 𝒑 such that 𝒑 ∈ 𝑁, 𝒑 ∼ 𝒑 implies that 𝑝𝑖 = 𝑝𝑖 . In particular, if 𝑁 = Ω in the above definition then 𝑝𝑖 is globally identifiable, otherwise it is nonuniquely (locally) identifiable. Notice that, for a given output, a locally identifiable parameter can take any of a distinct (countable) set of values. If there does not exist a suitable neighbourhood 𝑁 then 𝑝𝑖 is unidentifiable and, for a given output, can take an (uncountably) infinite set of values. A systems model is structurally globally identifiable (SGI) if all parameters are globally identifiable; it is structurally locally identifiable (SLI) if all parameters are locally identifiable and at least one is nonuniquely identifiable; and the model is structurally unidentifiable (SU) if at least one parameter is unidentifiable. Following a similar approach to that taken by Denis-Vidal et al. (2001), suppose that equations can be generated for the outputs that only involve 𝑦𝑖 (𝑡, 𝒑) and their derivatives. Then any indistinguishable parameter vector, 𝒑, must also satisfy the same equations with 𝒑 replaced by 𝒑. Therefore, for each 𝑦𝑖 (𝑡, 𝒑) and its derivatives, a polynomial of the following form can be generated: 𝑙 ∑ 𝑘=1

𝑐𝑘 (𝒑, 𝒑)𝜙𝑘 (𝑦𝑖 (𝑡, 𝒑), 𝑦𝑖′ (𝑡, 𝒑), 𝑦𝑖′′ (𝑡, 𝒑), . . . ) = 0

for all 𝑡 ≥ 0, where 𝜙𝑘 (𝑦𝑖 (𝑡, 𝒑), 𝑦𝑖′ (𝑡, 𝒑), 𝑦𝑖′′ (𝑡, 𝒑), . . . ) is a monomial in 𝑦𝑖 (𝑡, 𝒑) and its derivatives. If the 𝜙𝑘 (⋅) are linearly independent then it must be the case that 𝑘 = 1, . . . , 𝑙 𝑐𝑘 (𝒑, 𝒑) = 0 and so the relationship between 𝒑 and 𝒑 can be determined. If the only solution is 𝒑 = 𝒑 then the model is SGI, if there is a set of distinct solutions then the model is SLI and it is SU otherwise. 3.1 Full model For the full model (1) there are 27 unknown parameters (𝑘𝑎𝑗 , 𝑘𝑑𝑗 , 𝐶𝑇 , 𝑅𝑖𝑗 , 𝑘𝑀 𝑖𝑗 , and ℎ𝑗 for 𝑖 = 1, . . . , 5 and 𝑗 = 𝑎, 𝑙) for the 10 model outputs. In order to derive an equation for each output that only involves the output and its derivatives, first note from (3) that 1 𝐵𝑖𝑗 (𝑡, 𝒑) = 𝑦𝑖𝑗 (𝑡, 𝒑) 𝛼 and so from the second equation in (1) it can be seen that 𝑘𝑑𝑗 𝑦𝑖𝑗 + 𝑦˙ 𝑖𝑗 . 𝐶𝑖𝑗 = 𝛼𝑘𝑎𝑗 𝑅𝑖𝑗 − 𝑘𝑎𝑗 𝑦𝑖𝑗 From these relationships, in the association phase (𝐼𝑖 (𝑡) = 𝑑𝑖 𝐶𝑇 ), it can be seen that [ ( 1 2 𝑦¨𝑖𝑗 = (𝑘𝑎𝑗 𝐼𝑖 + 𝑘𝑑𝑗 )𝛼𝑘𝑀 𝑖𝑗 𝑦𝑖𝑗 𝛼ℎ𝑗 (𝛼𝑅𝑖𝑗 − 𝑦𝑖𝑗 ) ) ( − 𝑘𝑎𝑗 𝑦˙ 𝑖𝑗 + 𝛼𝑦𝑖𝑗 (𝑘𝑀 𝑖𝑗 + 2𝑘𝑎𝑗 𝑅𝑖𝑗 )𝑦˙ 𝑖𝑗 ) ( 2 − (2𝐼𝑖 𝑘𝑎𝑗 + 𝑘𝑑𝑗 )𝛼𝑘𝑀 𝑖𝑗 𝑅𝑖𝑗 + 𝛼 𝛼2 𝐼𝑖 𝑘𝑎𝑗 𝑘𝑀 𝑖𝑗 𝑅𝑖𝑗 ] ) 2 − 𝛼𝑅𝑖𝑗 (ℎ𝑗 𝑘𝑑𝑗 + 𝑘𝑀 𝑖𝑗 + 𝑘𝑎𝑗 𝑅𝑖𝑗 )𝑦˙ 𝑖𝑗 − ℎ𝑗 𝑦˙ 𝑖𝑗 , while in the dissociation phase (𝐼𝑖 (𝑡) = 0) it is seen that [ 1 3 𝑦¨𝑖𝑗 = 𝑘𝑑𝑗 (𝑘𝑑𝑗 − 𝑘𝑎𝑗 )𝑦𝑖𝑗 𝛼ℎ𝑗 (𝛼𝑅𝑖𝑗 − 𝑦𝑖𝑗 ) ( ) − 𝛼𝑦˙ 𝑖𝑗 𝛼𝑅𝑖𝑗 (ℎ𝑗 𝑘𝑑𝑗 + 𝑘𝑀 𝑖𝑗 + 𝑘𝑎𝑗 𝑅𝑖𝑗 ) + ℎ𝑗 𝑦˙ 𝑖𝑗 ( ) 2 𝛼𝑘𝑑𝑗 (𝑘𝑀 𝑖𝑗 + 2𝑘𝑎𝑗 𝑅𝑖𝑗 − 2𝑘𝑑𝑗 𝑅𝑖𝑗 ) − 𝑘𝑎𝑗 𝑦˙ 𝑖𝑗 + 𝑦𝑖𝑗 ( + 𝛼𝑦𝑖𝑗 𝛼𝑘𝑑𝑗 𝑅𝑖𝑗 (𝑘𝑑𝑗 𝑅𝑖𝑗 − 𝑘𝑀 𝑖𝑗 − 𝑘𝑎𝑗 𝑅𝑖𝑗 ) ] ) + (𝑘𝑀 𝑖𝑗 + 2𝑘𝑎𝑗 𝑅𝑖𝑗 )𝑦˙ 𝑖𝑗 . For any two parameter vectors, 𝒑 and 𝒑, to have identical outputs it must be the case that 𝑦𝑖𝑗 (𝑡, 𝒑) = 𝑦𝑖𝑗 (𝑡, 𝒑), 𝑦˙ 𝑖𝑗 (𝑡, 𝒑) = 𝑦˙ 𝑖𝑗 (𝑡, 𝒑), 𝑦¨𝑖𝑗 (𝑡, 𝒑) = 𝑦¨𝑖𝑗 (𝑡, 𝒑) and so using the above equation for 𝑦¨𝑖𝑗 in the association phase yields a polynomial of the following form 2 3 𝑐0 + 𝑐1 𝑦𝑖𝑗 + 𝑐2 𝑦𝑖𝑗 + 𝑐3 𝑦𝑖𝑗 + 𝑐4 𝑦˙ 𝑖𝑗 + 𝑐5 𝑦𝑖𝑗 𝑦˙ 𝑖𝑗

2 3 2 + 𝑐6 𝑦𝑖𝑗 𝑦˙ 𝑖𝑗 + 𝑐7 𝑦𝑖𝑗 𝑦˙ 𝑖𝑗 + 𝑐8 𝑦˙ 𝑖𝑗 = 0. The monomials in this polynomial are linearly independent (the Wronskian is non-zero at 𝑡 = 0) and so 𝑐𝑘 (𝒑, 𝒑) = 0 for 𝑘 = 0, . . . , 8. Solving these relationships between the components of 𝒑 and 𝒑 gives that the following parameters and combinations are unique (globally identifiable) with respect to the output in the association phase (𝐼𝑖 (𝑡) = 𝑑𝑖 𝐶𝑇 ): 𝑅𝑖𝑗 , 𝑘𝑑𝑗 , 𝑘𝑎𝑗 𝐶𝑇 , 𝑘𝑎𝑗 /ℎ𝑗 , and 𝑘𝑎𝑗 /𝑘𝑀 𝑖𝑗 for 𝑖 = 1, . . . , 5 and 𝑗 = 𝑎, 𝑙. Using an association experiment only results in the individual parameters 𝐶𝑇 ,

𝑘𝑎𝑗 , ℎ𝑗 , and 𝑘𝑀 𝑖𝑗 being unidentifiable, and in particular it would not be possible to estimate the unknown concentration, 𝐶𝑇 , or the binding affinity, 𝐴𝑗 = 𝑘𝑑𝑗 /𝑘𝑎𝑗 , reliably. However, if any one of these parameters were known a priori, then all of the remaining rate constants are globally identifiable and the model becomes SGI for association experiments. Using the above equation for 𝑦¨𝑖𝑗 in the dissociation phase, and the relationships between the components of 𝒑 and 𝒑 already determined, yields a polynomial of the form 2 3 4 𝑐9 𝑦𝑖𝑗 + 𝑐10 𝑦𝑖𝑗 + 𝑐11 𝑦𝑖𝑗 + 𝑐12 𝑦𝑖𝑗 = 0. Since this is a univariate polynomial that is identically zero, the coefficients must be equal to 0, that is 𝑐𝑘 (𝒑, 𝒑) = 0 for 𝑘 = 9, . . . , 12. Solving these relationships yields the additional information that ℎ𝑗 = ℎ𝑗 and so for experiments using the association and dissociation phases the following parameters are globally identifiable: 𝑅𝑖𝑗 , 𝑘𝑑𝑗 , 𝑘𝑎𝑗 , 𝐶𝑇 , ℎ𝑗 , and 𝑘𝑀 𝑖𝑗 for 𝑖 = 1, . . . , 5 and 𝑗 = 𝑎, 𝑙. The model is therefore SGI. 3.2 QSS model For the QSS model (4) there are 25 unknown parameters (𝑘𝑎𝑗 , 𝑘𝑑𝑗 , 𝐶𝑇 , 𝑅𝑖𝑗 , and 𝑘𝑀 𝑖𝑗 for 𝑖 = 1, . . . , 5 and 𝑗 = 𝑎, 𝑙) for the 10 model outputs. Since each output is a known scalar multiple of the corresponding model variable, any two parameter vectors, 𝒑 and 𝒑, that have identical outputs must satisfy the following: 𝑘𝑎𝑗 𝐼𝑖 (𝑅𝑖𝑗 − 𝛼1 𝑦𝑖𝑗 ) − 𝑘𝑑𝑗 𝛼1 𝑦𝑖𝑗 1 + (𝑘𝑎𝑗 /𝑘𝑀 𝑖𝑗 )(𝑅𝑖𝑗 − 𝛼1 𝑦𝑖𝑗 ) =

𝑘 𝑎𝑗 𝐼 𝑖 (𝑅𝑖𝑗 − 𝛼1 𝑦𝑖𝑗 ) − 𝑘 𝑑𝑗 𝛼1 𝑦𝑖𝑗

1 + (𝑘 𝑎𝑗 /𝑘 𝑀 𝑖𝑗 )(𝑅𝑖𝑗 − 𝛼1 𝑦𝑖𝑗 ) for 𝑖 = 1, . . . , 5 and 𝑗 = 𝑎, 𝑙. Rearranging this equation yields a polynomial of the form 2 𝑐0 (𝒑, 𝒑) + 𝑐1 (𝒑, 𝒑)𝑦𝑖𝑗 + 𝑐2 (𝒑, 𝒑)𝑦𝑖𝑗 =0 in the association phase (𝐼𝑖 (𝑡) = 𝑑𝑖 𝐶𝑇 ) and 2 =0 𝑐3 (𝒑, 𝒑)𝑦𝑖𝑗 + 𝑐4 (𝒑, 𝒑)𝑦𝑖𝑗

in the dissociation phase (𝐼𝑖 (𝑡) = 0). Both polynomials are univariate and so the coefficients are zero: 𝑐𝑘 (𝒑, 𝒑) = 0 for 𝑘 = 0, . . . , 4. Solving these relationships gives that the following parameters and combinations are globally identifiable: 𝑅𝑖𝑗 , 𝑘𝑑𝑗 , 𝑘𝑎𝑗 𝐶𝑇 , and 𝑘𝑎𝑗 /𝑘𝑀 𝑖𝑗 for 𝑖 = 1, . . . , 5 and 𝑗 = 𝑎, 𝑙. However, the individual parameters 𝐶𝑇 , 𝑘𝑎𝑗 , and 𝑘𝑀 𝑖𝑗 are all unidentifiable, and in particular it is not possible to estimate the unknown concentration, 𝐶𝑇 , or the binding affinity, 𝐴𝑗 = 𝑘𝑑𝑗 /𝑘𝑎𝑗 , reliably. If the sample concentration, 𝐶𝑇 , is known a priori, then all of the remaining rate constants are globally identifiable and the model becomes SGI. 4. PARAMETER ESTIMATION A ProteOn XPR36 SPR platform (Bio-Rad) was used in which A trisaccharide amine and linker were isolated onto

Parameter 𝑘𝑎𝑎 𝑘𝑎𝑙 𝑘𝑑𝑎 𝑘𝑑𝑙 𝑅1𝑎 𝑅2𝑎 𝑅3𝑎 𝑅4𝑎 𝑅5𝑎 𝑅1𝑙 𝑅2𝑙 𝑅3𝑙 𝑅4𝑙 𝑅5𝑙 𝑘𝑀 1𝑎 𝑘𝑀 2𝑎 𝑘𝑀 3𝑎 𝑘𝑀 4𝑎 𝑘𝑀 5𝑎 𝑘𝑀 1𝑙 𝑘𝑀 2𝑙 𝑘𝑀 3𝑙 𝑘𝑀 4𝑙 𝑘𝑀 5𝑙 𝐶𝑇 ℎ𝑎 ℎ𝑙

Value 3.48×10−2 3.92×10−2 6.80×10−4 9.24×10−4 1.20 1.14 6.02×10−1 5.47×10−1 5.45×10−1 8.99×10−1 8.52×10−1 4.27×10−1 3.93×10−1 3.94×10−1 3.09×10−3 5.48×10−3 6.85×10−3 1.11×10−2 1.38×10−2 2.20×10−3 3.92×10−3 4.59×10−3 7.14×10−3 7.79×10−3 6.38×10+1 1.00×10−3 1.00×10−3

SDLN 0.12 0.13 0.02 0.03 0.00 0.00 0.01 0.01 0.01 0.00 0.00 0.01 0.01 0.01 0.13 0.13 0.13 0.14 0.14 0.13 0.13 0.13 0.13 0.13 0.12 0.35 0.20

Value 3.61×10−2 4.09×10−2 6.90×10−4 9.30×10−4 1.20 1.14 6.10×10−1 5.54×10−1 5.51×10−1 9.00×10−1 8.54×10−1 4.31×10−1 3.97×10−1 4.00×10−1 3.26×10−3 5.81×10−3 7.12×10−3 1.16×10−3 1.49×10−3 2.33×10−3 4.15×10−3 4.83×10−3 7.51×10−3 8.15×10−3 60 -

SDLN 0.05 0.05 0.02 0.03 0.00 0.00 0.01 0.01 0.01 0.00 0.00 0.01 0.01 0.01 0.01 0.01 0.02 0.04 0.04 0.01 0.01 0.02 0.03 0.03 -

Table 1. Estimates for unknown model parameters (left full model, right QSS model): 𝑘𝑎𝑗 (mm3 /ng⋅s), 𝑘𝑑𝑗 (s−1 ), 𝑅𝑖𝑗 (ng/mm2 ), 𝑘𝑀 𝑖𝑗 (mm4 /ng.s2 ), 𝐶𝑇 (ng/mm3 ), and ℎ𝑗 (mm). separate lanes of a carboxylated SPR chip and five different dilutions (1:10, 1:20, 1:50, 1:75, and 1:100) of a commercially available anti-A monoclonal IgM sample passed down channels perpendicular to the dextra lanes. This experimental procedure permits quantitative responses to be obtained for 10 simultaneous binding reactions. The duration of the association phase was 120 s followed by a dissociation phase after the antibody stream was stopped with the total duration of the experiment being 794.7 s. The parameter estimation is performed using the software tool Facsimile for Windows 4 (MCPA Software, UK) for both models above using the data from the commercial typing reagent. The experiment consisted of five different dilutions corresponding to 𝑑1 = 0.1, 𝑑2 = 0.05, 𝑑3 = 0.02, 𝑑4 = 1/75 and 𝑑5 = 0.01. Although injection started at 𝑡 = 0 s there is a short delay of approximately 7.2 s before interaction between analyte and receptors, and so 𝑡𝑠 = 7.2 s and 𝑡𝑓 = 127.2 s. 4.1 Full model The fits of the model to the experimental data are presented in Figure 1. There seems to be a reasonable visual match with the data, but the peak that occurs between association and dissociation phases is missed. This suggests that perhaps the association phase needs to be extended closer to the equilibrium/saturation point. The estimated values of the model parameters are given in Table 1, where SDLN refers to the estimated standard deviation of the natural logarithm of the respective parameter estimate (since Facsimile works in terms of internal parameters

A Trisaccharide Amine

A Trisaccharide Amine 1400 Bound antibody concentration (RU)

Bound antibody concentration (RU)

1400 1200 1000 800 600 400 200 0 0

200

400 Time (s)

600

1200 1000 800 600 400 200 0 0

800

200

A Trisaccharide Linker

800

1200 Bound antibody concentration (RU)

Bound antibody concentration (RU)

600

A Trisaccharide Linker

1200 1000 800 600 400 200 0 0

400 Time (s)

200

400 Time (s)

600

1000 800 600 400 200 0 0

800

200

400 Time (s)

600

800

Fig. 1. Full model output (black lines) plotted with corresponding experimental data (grey lines).

Fig. 2. QSS model output (black lines) plotted with corresponding experimental data (grey lines).

that are the natural logarithms of the respective parameters) and so provides an estimate of the relevant confidence in the estimate.

The estimated correlation matrix for the fitted parameters shows high correlation between the association and dissociation constants: 𝑘𝑎𝑎 with 𝑘𝑑𝑎 (0.86), and 𝑘𝑎𝑙 with 𝑘𝑑𝑙 (0.88). This correlation shows that it is the binding affinity that is more accurately estimated from the data rather than the individual parameters. The calculated binding affinities (𝐴𝑗 = 𝑘𝑑𝑗 /𝑘𝑎𝑗 ) are:

Facsimile also estimates the correlation matrix for the fitted parameters and this matrix shows high correlation between the sample concentration, 𝐶𝑇 , and the association rate constants 𝑘𝑎𝑎 (-0.91) and 𝑘𝑎𝑙 (-0.91), the transport constants, 𝑘𝑀 𝑖𝑗 (less than -0.97), and ℎ𝑎 (0.93). In addition, the association constants are similarly highly correlated with each other (0.84), the transport constants, 𝑘𝑀 𝑖𝑗 (0.74–0.90), and ℎ𝑎 (-0.88). It is interesting that the association and dissociation constants, that give rise to the binding affinity, are only moderately correlated (approximately 0.5). The calculated binding affinities (𝐴𝑗 = 𝑘𝑑𝑗 /𝑘𝑎𝑗 ) are: 3

3

𝐴𝑎 = 1.95 × 10−2 ng/mm , 𝐴𝑙 = 2.36 × 10−2 ng/mm . 4.2 QSS model For the effective rate constant approximation model of (4) the analysis in the previous section showed that not all of the parameters are globally identifiable, but if the sample concentration, 𝐶𝑇 , is known a priori then the remaining unknown parameters are globally identifiable. The sample concentration was separately determined to be 60 ng/mm3 (0.06 mg/ml) and this was used in the parameter fitting. The fits of the model to the experimental data are presented in Figure 2. There seems to be a reasonable visual match with the data, but the peak that occurs between association and dissociation phases is again missed, which also suggests that perhaps the association phase needs to be extended closer to the equilibrium/saturation point. The estimated values of the model parameters are given in Table 1.

3

3

𝐴𝑎 = 1.91 × 10−2 ng/mm , 𝐴𝑙 = 2.27 × 10−2 ng/mm . 5. DISCUSSION Both of the models considered show reasonable correspondence between the simulated output and the experimental data, except that both do not seem to capture the dynamics at the point at which the experiment changes from the association to dissociation phases. It is noticable that for all of the flows the receptor is far from saturated, while there is significantly longer dissociation data than association data. Perhaps longer duration association data would assist in reducing this discrepancy. For both models considered the parameters are estimated to a high degree of confidence with SDLNs below 0.35 for the full model and 0.06 for the QSS model. The estimates for the common parameters are comparable, particularly those that were globally identifiable before determining the sample concentration 𝐶𝑇 (i.e., 𝑘𝑑𝑗 and 𝑅𝑖𝑗 , 𝑖 = 1, . . . , 5, and 𝑗 = 𝑎, 𝑙). The good comparison between the other parameters is due to the estimated sample concentration being close to the determined one (63.8 ng/mm3 compared with 60 ng/mm3 ). Both models provide comparable estimates for the binding affinities for the analyte for the two receptors, but again, this is because the sample concentration estimate is close to the determined value.

The QSS model proves to be easier to fit to the data, in being less sensitive to the initial estimates for the parameters, than the full model. Therefore, if the sample concentration is known, in which case the model becomes SGI, then the comparable parameter estimates for the two models means that it is slightly easier to fit the QSS and still obtain reasonable estimates. However, if the sample concentration is unknown then the QSS model is SU and so the full model must be used to obtain the final parameter estimates. The high estimated correlations for the full model show that, although SGI, certain combinations of parameters are more numerically identifiable than the individual parameters. Therefore, it is more that the combinations 𝑘𝑎𝑗 𝐶𝑇 and 𝑘𝑀 𝑖𝑗 𝐶𝑇 are being estimated from the data than the individual parameters. Myszka et al. (1998) report that for parameters relevant to the Biacore experiments the solution to the full model is insensitive to the value of ℎ𝑗 . It has already been seen that provided both association and dissociation phases are considered the full model is SGI. If only the association phase is considered it is necessary to know the sample concentration, 𝐶𝑇 , or one of the other unidentifiable parameters (ℎ𝑗 , 𝑘𝑎𝑗 , and 𝑘𝑀 𝑖𝑗 ) for the unknown parameters to be uniquely determined by the output. In particular, assigning an arbitrary value to ℎ𝑗 makes the model artificially SGI. The estimates for ℎ𝑗 for the full model have the greatest SDLN and hence greater uncertainty. However, though the SDLNs are large in comparison to the other estimated parameters the values are still relatively small. 6. CONCLUSIONS Two models typically used to estimate the kinetic rate constants in binding reactions from SPR experiments have been considered. The first model is a two-compartment system proposed by Myszka et al. (1998) that gives rise to the second model under a quasi-steady state assumption. The second model is the effective rate constant approximation, which has been previously shown to be a good approximation for low Damk¨ohler number (Edwards, 2001). The full two-compartment model has been shown to be structurally globally identifiable if both the association and dissociation phases are considered. However, the model is unidentifiable if only the association phase is considered, in which case certain parameters can not be uniquely determined from the output (and hence ultimately from sensorgrams). These unidentifiable parameters include the sample concentration and the association rate constant meaning that only a concentration corrected binding affinity, 𝑘𝑑 /(𝑘𝑎 𝐶𝑇 ), can be uniquely determined. The effective rate constant approximation has been shown to be structurally unidentifiable unless the sample concentration is known a priori. Although the dissociation rate constant is globally identifiable the corresponding association constant is unidentifiable. Therefore, the binding affinity can not be uniquely determined from the output. For mixed samples containing more than one analyte (for example, polyclonal samples) it would be necessary to know the respective concentrations in order to be able to

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