A unifying mathematical framework for experimental TCR-pMHC kinetic constants

Receptor binding and triggering are central in Immunology as T cells activated through their T cell receptors (TCR) by protein antigens orchestrate immune responses. In order to understand receptor-ligand interactions, many groups working with different experimental techniques and assays have generated a vast body of knowledge during the last decades. However, in recent years a type of assays, referred to as two-dimensional or membrane-to-membrane, has questioned our current understanding of the role of different kinetic constants (for instance, on- versus off-rate constants) on TCR-ligand interaction and subsequent T cell activation. Here we present a general mathematical framework that provides a unifying umbrella to relate fundamental and effective (or experimentally determined) kinetic constants, as well as describe and compare state-of-the-art experimental methods. Our framework is able to predict the correlations between functional output, such as 1/EC50, and effective kinetic constants for a range of different experimental assays (in two and three dimensions). Furthermore, our approach can be applied beyond Immunology, and serve as a “translation method” for the biochemical characterization of receptor-ligand interactions.


Description of the experimental systems
In this section we briefly describe the main 3D and 2D assays used by Zhu et al. [1] and Davis et al. [2], and identify the molecular states observed in the different experimental systems,as well as the associated effective kinetic constants.

3D assays
A) Surface plasmon resonance (SPR). Most procedures to analyze TCR-pMHC interactions in solution involve the use of SPR and either soluble TCR or pMHC as analyte (see [3] and references 3-7, 9-13, 15, 16, 18, 28, therein). In some publications, instead of using soluble TCRαβ heterodimers (without the transmembrane domain), an engineered TRBV-TRAV single chain, composed of the variable domains of TCRα and TCRβ chains, was used as analyte [4][5][6]. In the SPR technique a specific molecule is linked to a chip surface within a chamber and an analyte in solution is allowed to flow over that surface in a laminar flow.
In this technique, a laser light beam, propagating through a prism and reaching a metal-coated dielectric interface, is perpendicularly polarized to the plane of the interface. At a critical incidence angle, this beam generates an evanescent wave, by internal reflection at the metal-coated surface, and is able to excite a non-radiative surface plasmon wave, which causes a sharp decrease in the angle-dependent intensity of the reflected light. This critical angle increases proportionally to the number of analyte molecules that bind to the molecules linked to the surface. Washing the chamber with a buffer solution without the analyte leads to a progressive detachment of the analyte and a corresponding decrease of the SPR critical incidence angle. The rates of increase and decrease of the critical angle are proportional to the binding and unbinding rate constants, respectively, of the analyte. Typically the interaction of the linked molecule and the analyte is assumed to be well described by a simple Langmuir model, namely, where C is the bound complex of A and B.
B) Cells and fluorescent cell cytometer. Another technique to analyze TCR-pMHC interactions involves the use of OT-I T cells (expressing an ovalbumin peptide-specific transgenic TCR) [7], or the human lymphoblastoid T2 cell line transfected with either L d [8] or K b [9] human MHC heavy chain. The OT-I and T2 cells were incubated for some time with saturating concentrations of soluble pMHC-phycoerythrin tetramers and TCR, respectively, washed and resuspended. T2-pMHC cells covered with TCR were incubated with an excess of an antibody anti-variable (V) region of the TCR, and at different times cells were incubated with biotinylated antibody anti-TCR+streptavidin-phycoerythrin, and the corresponding decrease of mean fluorescence intensity (MFI) with time was recorded. In the case of OT-I T cells covered with pMHC-phycoerythrin, these were incubated with a high concentration of an antibody anti-V region of the TCR, and the decrease of MFI with time was recorded. In each case the kinetics of MFI change was used to estimate the unbinding rate constant of TCR:pMHC complexes. This experimental procedure assumes that unbinding is essentially an irreversible process. This, however, may not be the case, particularly in the case of OT-I T cells and pMHC tetramers, where due to the tetravalent nature of the ligand, rebinding of a bound molecule through a previously unbound second or third site can still occur.
C) Bulk FRET in solution (3D FRET). In this technique specific pMHC and TCR molecules, labeled with donor and acceptor fluorochromes, respectively, are mixed in solution for FRET analysis in a small reaction chamber of a stopped-flow instrument [10]. When the TCR and the pMHC are close enough (and properly positioned) or even bound, the donor and acceptor are at an adequate distance to allow the generation of a FRET signal. The association rate constant can be estimated from the time course of the donor quenching due to FRET and assuming a two-step process, with an initial formation of a transient complex followed by a bound complex, that can be described by the following reactions: where C is the bound complex of A and B. However, in a previous study by the same group using SPR, this two-step process was deemed incapable to explain the observed experimental results [11]. This experimental design cannot distinguish between bound molecules and those close enough and properly oriented. Therefore, the assumed twostep process should correspond to: (1) free chemical species, (2) encounter complexes, and (3) a combination of oriented and bound complexes. D) Single molecule fluorescent microscopy (SMFM) [12]. This assay is devised to measure the unbinding rate constant of TCR-pMHC complexes. It makes use of T cells attached to a poly-lysine-coated borosilicate slide and quantum-dot-tagged pMHC monomers added in solution. Using fluorescent microscopy, bound quantumdot-pMHC molecules were detected as single fluorescent spots on the T cell membrane. The distribution of the duration of individual pMHC binding events, starting and ending in a fixed time window, was used to calculate the average dwell-time and from it the dissociation rate constant.
A reported 2D-variant of this assay makes use of the duration of individual fluorescent spots of only those fluorochrome-tagged pMHC molecules showing slow diffusion, which characterizes TCR-bound pMHC molecules [13].

2D assays
A) Adhesion frequency assay [1]. This assay makes use of a red blood cell (RBC) decorated with univalent pMHC molecules or an RBC with an attached bead decorated with pMHC. With the help of a micropipette, a T cell was positioned to touch the RBC or the bead. After a given time, RBC elongation or bead displacement on T cell retraction, due to binding, may or may not happen. For a given contact time, many trials are performed and the frequency of binding events is recorded. This procedure is followed for a range of contact times. Assuming a simple Langmuir model and based on a probabilistic model for adhesion kinetics mediated by a small number of bonds, the corresponding affinity constant and dissociation rate constant can be derived by a best-fit procedure.
B) Thermal fluctuation assay [1]. This assay is based on the reduction/resumption of thermal position fluctuations of a pMHC-decorated bead touching a TCR-transgenic T cell held in a fixed position. The reduction of the bead's fluctuation is caused by bond formation between one or more bead pMHCs and T cell TCRs, restricting the movements of the bead. Time periods of restricted movement, t b , are considered durations of the bound state. Fitting the frequency distribution of many recorded t b periods to an exponential function allows one to derive an dissociation rate constant for the reaction.
C) FRET on cell membrane (2D FRET) [2]. This experimental setting makes use of TCR-transgenic T cells with the V-regions stained with a donor fluorochrome, MHC molecules complexed to peptides tagged with an acceptor fluorochrome, and total internal reflection microscopy to allow single molecule FRET. As is the case in the 3D assay, when the TCR and pMHC are bound or close to each other and oriented, the donor and acceptor dyes are at a suitable distance that leads to the generation of a FRET signal. The duration in single synapses of FRET signals that started and finished during an experimental recording was considered to correspond to a bound time period. Measuring the duration of many binding events and fitting their frequency distribution to an exponential function allows to obtain a dissociation rate constant for the reaction. In addition, after a long enough synaptic period (≤ 2.5 min), to safely assume a steady state has been reached, the local density of total TCRs, pMHCs and TCR-pMHC complexes was determined in a single synapse, considering average intensities of single-molecule fluorescence. This allowed to estimate the average local effective 2D affinity constant (K a ), and from this value and the dissociation rate constant, the association rate constant can be estimated.

Mathematical description
In the main text, the following set of coupled ordinary differential equations (ODEs) are derived from the general reaction scheme, in terms of the fundamental kinetic constants [see Eq. (1) and Eqs. (2)-(5) therein], In a similar fashion, we can obtain a specific set of coupled ODEs for every experimental model considered in Fig. 3 of the main text. Thus, • Pre-binding model A: • Pre-binding model B: • FRET model: • Single-step model: With these equations at hand, we can proceed to find a relationship between the fundamental kinetic rate constants of the general model and the effective kinetic constants in the coarse-grained experimental descriptions.

Effective kinetic constants in terms of fundamental ones 2.1 Local steady state balance
In order to derive the simplified equations of pre-binding models A and B, Eqs. (5)- (10), from the general model described by Eqs. (1)-(4), we introduce an additional hypothesis: due to experimental limitations, it might not be possible to discriminate between the different molecular states. In particular, when orientation prior to binding occurs fast enough, we cannot distinguish between encounter-non-oriented and encounter-oriented states. In mathematical terms, this means that, in a coarse-grained observation timescale, we can safely assume that 1 : for the pre-binding model A, and for the pre-binding model B.
We first carry out model reduction for the pre-binding model A. Defining RL = [RL * ] + [RL] and using Eq. (15) one gets: We add Eqs.
Equations (18)-(19) allow the identification of the effective rate constants in Eqs. (6)-(7) in terms of the fundamental rate constants. In particular, from the constants multiplying RL and [C] in Eqs. (19) and (7) we find, These values correspond to Eqs. (3)-(5) in Table 2 of the main text. Using Eqs. (17) and (5), we could determine k − * , so that In a similar way and for the pre-binding model B, we define C = [RL] + [C], and make use of Eq. (16) to obtain [RL] in terms of C . We add Eqs. (3)-(4) and substitute [RL] to obtain: As we have done above, Eqs. (22)-(23) allow us to identify the effective rate constants of Eqs. (9)-(10) in terms of the fundamental ones These equations correspond to Eqs. (6)- (8) in Table 2 of the main text.

Local steady state balance and perturbation analysis
The method introduced in the preceding section cannot be directly applied to the FRET model, as it involves the meta-states R , L and C , and the corresponding concentrations [ R ] and [ L ] enter non-linearly in the effective reactions. To solve this problem we proceed in two steps. First, we assume, a local steady state hypothesis: (1)-(2) to obtain an equation for the time evolution of R . In the same way as for R , a differential equation can be obtained for the remaining meta-states L and C , namely: We now need to obtain expressions for [RL * ] and [RL] in terms of the meta-states L and C . We make use of Eqs. (25) and (26) to find Eqn. (29) is quadratic and can be solved explicitly. It has two solutions that involve a square root due to the quadratic nature of the equation. Thus, the exact solution does not allow to obtain a linear relationship between the effective and fundamental kinetic rates. To overcome this limitation, we first rewrite Eq. (29) in terms of number concentrations instead of molar concentrations. Namely, we define n RL * , n R and n L as the absolute number of molecular species RL * , R and L , respectively. In general, n X = [X] N A V where V is the experimental volume and N A is Avogadro constant. Thus, Eq. (29) can be written as: Now, we can take advantage of the following relationship between the diffusion rate constants [14][15][16]: where N is the number of receptors, d = 2, 3 for 2D and 3D systems, respectively, and s and a are the approximate diameters of the receptor and the cell, respectively. As this ratio is small for the receptors and cells under consideration, we can perform a singular perturbation analysis of Eq. (31) in terms of a small

Note that
1 as s is about ten nanometers and a a few microns, so that s/a ∼ 10 −2 . The number of receptors is of the order of 10 4 in the 3D case and around 10 2 in the 2D case (the synapse) so, effectively, in both cases ∼ 10 −2 .
We now write Eq. (31) as an iterative relation for an unknown x ≡ n RL * for different orders i in the approximation. Namely, To lowest order we have, x 0 = n R n L ; that is The next order leads to For the sake of simplicity, we only make use of the lowest order, as higher corrections are O( ), and as mentioned above ∼ 10 −2 . We then substitute [RL] and [RL * ], as defined, respectively, by Eqs. (30) and (33), into Eqs. (27)-(28) to obtain: We can conclude that (encounter and oriented configurations, respectively) are assumed to be transient. This implies that the timescales associated to these states are short enough, so we can consider them to be slaved to the concentration of free species ([R] and [L]) and the bound complex ([C]). This approximation, often referred to as the quasi-steady state approximation (QSSA) is widely used, not only due to its simplicity, but as it has proven to be a good approximation [17], and it has served as a starting point for improved approximations [18]. In our case, the use of this approximation means dropping the time derivatives in Eqs. (2)-(3) and simplifying the slaved concentrations in terms of the remaining ones: [C], (36) Finally, substituting [RL * ] and [RL], as defined above, into Eqs. (1) and (4) leads to: [C].
If we replace the ligand concentration, [L], by the ligand number of molecules, n L = N A V [L], we obtain a relation for k + eff /(N A V ) instead of k + eff . We conclude with the expression of the effective rate constants in terms of the fundamental ones for the single-step model: