# A mathematical modelling tool for unravelling the antibody-mediated effects on CTLA-4 interactions

**Part of the following topical collections:**

## Abstract

### Background

Monoclonal antibodies blocking the Cytotoxic T-lymphocyte antigen 4 (CTLA-4) receptor have revolutionized the field of anti-cancer therapy for the last few years. The human T-cell-based immune responses are modulated by two contradicting signals. CTLA-4 provides a T cell inhibitory signal through its interaction with B7 ligands (B7–1 and B7–2), while CD28 provides a stimulatory signal when interacting with the same ligands. A previous theoretical model has focused on understanding the processes of costimulatory and inhibitory complex formations at the synapse. Nevertheless, the effects of monoclonal antibody (mAb)-mediation on these complexes are relatively unexplored. In this work, we expand on the previous model to develop a new mathematical framework for studying the effects of anti-CTLA-4 mAbs on the co-stimulatory (CD28/B7 ligands) and the co-inhibitory (CTLA-4/B7 ligands) complex formation at the immunological synapse. In particular, we focus on two promising anti-CTLA-4 mAbs, tremelimumab (from AstraZeneca) and ipilimumab (from Bristol-Myers Squibb), which are currently in clinical trials and the market, respectively, for targeting multiple tumors.

### Methods

The mathematical model in this work has been constructed based on ordinary differential equations and available experimental binding kinetics data for the anti-CTLA-4 antibodies from literature.

### Results

The numerical simulations from the current model are in agreement with a number of experimental data. Especially, the dose-curves for blocking the B7 ligand binding to CTLA-4 by ipilimumab are comparable with the results from a previous competitive binding assay by flow cytometry and ELISA. Our simulations predict the dose response and the relative efficacies of the two mAbs in blocking the inhibitory CTLA-4/B7 complexes.

### Conclusions

The results show that different factors, such as multivalent interactions, mobility of molecules and competition effects, could impact the effects of antibody-mediation. The results, in particular, describe that the competitive effects could impact the dose-dependent inhibition by the mAbs very significantly. We present this model as a useful tool that can easily be translated to study the effects of any anti-CTLA-4 antibodies on immunological synaptic complex formation, provided reliable biophysical data for mAbs are available.

## Keywords

CTLA-4 Immune checkpoints Ipilimumab Tremelimumab Antibody Mathematical modeling## Abbreviations

- APCs
Antigen-presenting cells

- CTLA-4
Cytotoxic T-lymphocyte antigen 4

- FDA
Food and drug administration

- mAbs
Monoclonal antibodies

- MHC
Major histocompatibility complex

- ODEs
Ordinary differential equations

- TCRs
T-cell receptors

## Background

Cancer cells are able to escape the immunological surveillance of T-cells, by overexpressing the inhibitory receptors that attenuate anti-tumor immune response [4, 5]. As a result, in principal, blocking the co-inhibitory receptor-ligand interactions (schematically shown in Fig. 1) should be able to facilitate the re-activation of T-cells, which in-turn will recognize and eliminate cancers. Monoclonal antibodies (mAbs) blocking inhibitory immune checkpoints have demonstrated exceptional therapeutic benefits in clinical trials [20, 21, 22], which is transforming human cancer treatment. Ipilimumab, a completely human IgG1 antibody from Bristol-Myers Squibb, became the first-in-class anti-CTLA-4 mAb to be approved by the US Food and Drug Administration (FDA) in 2011 for the treatment of metastatic melanoma [23, 24]. Several clinical trials showed that monotherapy with ipilimumab in metastasis melanoma patients increased the overall responsive rate by 10–20% [1, 25]. When combined with other immune checkpoint therapies, ipilimumab is able to offer much enhanced benefits [26, 27]. Another IgG2 anti-CTLA-4 mAb, Tremelimumab from AstraZeneca, is currently in phase III clinical trials [1, 5]. Similar to Ipilimumab, AstraZeneca’s tremelimumab binds specifically to CTLA-4 and blocks its interaction with the B7 ligands. Tremelimumab is also being tested in combination with other immune checkpoint mAbs for targeting multiple tumors [28, 29], including melanoma, colon cancer, and mesothelioma [28, 29]. The progress made by these two antibodies, from bench to final clinical trial phases or to the market, has boosted the interests towards developing more promising immune-checkpoint blocking inhibitors (both mAbs and, lately, small molecules).

Very recently, He et al. [1] investigated the binding profiles of both mAbs against CTLA-4 using surface plasmon resonance (SPR) experiments and reported their K_{D} values as 18.2 nM for ipilimumab/CTLA-4 complex and 5.89 nM for tremelimumab/CTLA-4 complex. These binding affinity values are higher than (or, in some cases, comparable to) those reported for B7–1 ligands with CTLA-4 [1]. However, it is important to note that different range of affinity values have been reported for the CTLA-4/B7–1 complexes. For example, an earlier study reported that a soluble B7–1 Ig fusion protein bound to CTLA-4 with a dissociation constant of ~ 12 nM [30]. However, this value was argued to be higher when compared with other protein-protein interactions occurring between T-cell surface and APCs [31]. The authors of this work [30] had described that the CTLA-4Ig in their experiments was not monomeric in solution, and possibly formed higher aggregates that might have possibly resulted in the high apparent K_{D} values for the interactions of CTLA-4Ig/B7Ig fusion protein. Another study [31] based on SPR experiments reported that, at 37 °C, soluble recombinant B7–1 bound to CTLA-4 with a K_{D} value of 0.42 μM. This indicates that affinity between B7–1 and CTLA-4 is relatively lower than that of ipilimumab/CTLA-4 and tremelimumab/CTLA-4 complexes. The association rate constant (k_{a}) of ipilimumab (3.83 × 10^{5}/Ms) and tremelimumab (3.08 × 10^{5}/Ms) are almost close to each other, however, their dissociation rate constants (k_{d}) were significantly different [1]. The k_{d} value for ipilimumab was 6.96 × 10^{− 3}/s, whereas for tremelimumab it was 1.8 × 10^{− 3}/s [1]. This clearly indicates that the tremelimumab is able to form much stable complex with CTLA-4 when compared to that of ipilimumab. Thus, understanding the effects of these two potent anti-CTLA-4 antibodies on the co-stimulatory and the co-inhibitory complex formation at the synapse will be useful to develop effective next-generation anti-CTLA-4 therapeutics. The ultimate objective of this work is to precisely study these effects using mathematical modelling and simulations.

Mathematical modelling and simulation remains a powerful tool to gain quantitative insights into the dynamics of complicated systems [32, 33, 34, 35, 36, 37, 38, 39, 40]. Especially, mathematical modelling is gaining more popularity in the field of cancer immunotherapy. For example, Kirschner et al. [38] developed a mathematical model, based on ordinary differential equations (ODEs) to simulate the dynamics between tumor cells, immune-effector cells and cytokine interleukin-2. This model was useful to explore the effects of adoptive cellular immunotherapy (ACI) on the model and found that a combination therapy with ACI and IL-2 could boost the immune system sufficiently to clear the tumor [38]. Krnoik et al. [37] developed an expanded mathematical model, from a previously published model, in order to simulate cellular immunotherapy in melanoma. This model was useful to understand the findings from clinical trials suggesting that patients with the smallest tumor load respond better for this kind treatment [37]. Mathematical models have also been useful to understand tumor growth, response to therapy and the interactions of immune cells with the cancer cell [39, 40, 41, 42]. Bidot et al. [43] developed a mathematical model for studying the kinetics of monoclonal T-cell-specific activation. This model attempted to account for the sequence of events starting from the TCR-MHC binding to T-cell activation and response [43]. Mathematical modelling has also been employed to study the complex formation between PD-1 receptor and its ligands [44]. Jansson et al. [33] developed a theoretical model for simulating the synaptic accumulation of the molecules involved in co-stimulation and inhibition of T-cells. This model, which was developed based on the Ordinary Differential Equations (ODEs) and rigorous biophysical and expression data from literature, explained the interactions of CTLA-4 and CD-28 with their B7-ligands in the context of a potentially dynamic synaptic microenvironment [33]. However, until now, there is no model that could predict the response to CTLA-4 blocking antibodies and the consequent effects on complex formation at immunological synapse.

In this work, we expand the model of Jansson et al. [33] by including the effect of an anti-CTLA-4 antibody. Our model is based on ODEs and additional biophysical data for antibody-CTLA-4 complexes [1]. Using this new model, we studied how the binding of mAbs (with different affinities) to CTLA-4 dynamically changed the interactions among co-inhibitory complexes (CTLA-4/B7–1 and CTLA-4/B7–2) and co-stimulatory complexes (CD28/B7–1 and CD28/B7–2) at the immunological synapse. The numerical simulations from the model have been validated by different experimental data reported earlier. This model should be a useful tool to predict the dose response of any anti-CTLA-4 antibodies and their impacts on synaptic complex formation processes.

## Methods

### Mathematical modelling assumptions

As mentioned above, the mathematical model in this work is based on the previous two-compartment model by Jansson et al. [33], which was constructed to simulate the CD28 and CTLA-4 complex formation with the B7-ligands at immunological synapses. In this study, we developed an expanded model to assess the changes in the co-stimulatory (CD28/B7 ligands) and the co-inhibitory (CTLA-4/B7 ligands) interactions upon binding of the antibody to the free CTLA-4 receptor sites. This model, as in Jansson’s model [33], involves two components that includes the synapse and the region outside of the synapse. It is assumed that CD28, B7–1 and B7–2 are primarily unbound and distributed uniformly over the surface. CTLA-4, on the other hand, is present intracellularly and gets injected into the synapse upon activation [45, 46]. The ‘free-diffusion’ model has been applied to control the mobility of the molecules, such that only the mobile molecules are able to diffuse into the synapse, while the immobile species outside of the synapse are ignored. It is understandable that the immobile species outside of the synapse are, anyway, not able to participate in any complex formation. On the other hand, the immobile molecules inside the synapse stay there and are involved in the protein-protein interactions. The model also assumes that the CTLA-4 receptors, once injected from the intracellular environment, stay within the synapse. In the case of antibody, unlike the membrane-bound ligands, it is in the free solution and is assumed to bind to free CTLA-4 monovalently. The model assumes that the binding site of B7–1, B7–2 and the antibody molecules on CTLA-4 overlap considerably [2], and, therefore, only one of them (B7–1 or B7–2 or the antibody) is able to bind with a CTLA-4 molecule. Moreover, in our model, the antibody is allowed to bind an unbound CTLA-4 monomer that is part of a dimer, where the other monomer can be bound with a B7 ligand. Due to the lack of parameters for the association and dissociation of complexes in the model, rate constants from similar complexes are used and reactions are modelled as parallel mass action reactions. Bivalent association and dissociation rate constants are employed for the binding/unbinding of B7 ligands to a CTLA-4 monomer that is part of a dimer, where the other monomer is bound to an antibody.

### Parameters

The parameters for the interactions of CTLA-4 and CD28 with the B7 ligands in the current model are almost similar to those in Jansson’s model [33], which are given in the supplementary information (Additional file 1: Table S1). However, the association and dissociation rate constants for the interactions of antibodies, tremelimumab and ipilimumab, with CTLA-4 receptor are collected from the literature [1] and employed in the current model. The reported SPR experiments [1] measured the association rate constants of ipilimumab and tremelimumab with CTLA-4 to be 3.83 × 10^{5}/Ms. and 3.08 × 10^{5}/Ms., respectively. Whereas, their dissociation rate constants were observed to be 6.96 × 10^{− 3}/s (ipilimumab) and 1.8 × 10^{− 3}/s [1] (tremelimumab). It is important to note that the rate constants for the interactions of antibodies are not converted to 2-D rates, as the antibodies are present in the solution. This approximation is reasonable, when the binding site is accessible [47].

### Antibody complex formation

The equations related to the modelling of rates of change for the complex formation between different species, such as CD28, CTLA-4, B7–1 and B7–2, are all similar to those of Jansson et al. [33]. However, a number of additional terms are employed in the current model in order to account for the increase or decrease in density of these species in response to the association and dissociation of antibodies to that of CTLA-4 monomers. Please note that the rate of change of the density of complexes for the newly added terms in the model are also written using the mass-action law.

### Rate of change of density of antibody/CTLA-4 complex

_{on}, in order to form the antibody/(CTLA-4)

_{2}complex (referred as Ac complex in the equation). The rate for dissociation of antibody from the Ac complex is k

_{off}. An unbound B7–1 associates with the Ac complex at the rate of α

_{44}to form a complex of Ac/B7–1 (or EAb

_{1}), whereas, an unbound B7–2 associates with the Ac complex at the rate of α

_{33}to form Ac/B7–2 (or Ab/CTLA4/B72). An antibody binds to the Ac complex at the rate of Kon to form the resultant Ab/Ac (or AcA) complex. The association of either of the B7–1 ligands or antibody to the Ac complex reduces the density of the latter. On the other hand, dissociation of B7–1 (rate = δ

_{44}), B7–2 (rate = δ

_{33}) and antibody (rate = k

_{off}) from the EAb1, Ab/CTLA4/B72, AcA complexes, respectively, increases the density. Hence, the rate of change in the density of the Ac complex can be written as follows,

### Modelling and simulations

Mathematical modelling and simulation procedures in this study were programmed using MATLAB software from MathWorks (https://www.mathworks.com/products/matlab.html). A stiff ordinary differential equation solver, ode15s, was used for solving the equations in the current model. The different components included in the model and their abbreviations are provided in the supplementary information (Additional file 3: Table S3). Each simulation was performed for 7 h (i.e., 25,200 s) unless otherwise stated. Initially, the simulations are performed with ‘0’ concentration of antibody, in order to reproduce the results from Jansson’s model [33]. Later, the simulations were performed with different concentrations of tremelimumab and ipilimumab and the effects on CTLA-4 and CD28 complex formation were analyzed.

## Results

### Simulation of antibody-free complex formation

### Binding of the antibodies to CTLA-4

_{D}values. As shown in Figs. 3, 50% of the CTLA-4 monomers were bound to antibody at the concentrations of 0.018 μM and 0.0058 μM of ipilimumab (Fig. 3a) and tremelimumab (Fig. 3b), respectively. These concentrations are approximately close to the K

_{D}values for these antibodies (ipilimumab – 18.2 nM; Tremelimumab – 5.89 nM), as reported by He et al. [1] based on their SPR experiments. Nevertheless, since the association and dissociation rate constants of the antibodies for the current model was obtained from the work of He et al. [1], it is expected that the model is able to achieve 50% complex formation at the concentration of the K

_{D}values of mAbs. Hence, this confirms that the model simulates the complex formation correctly.

### Competitive binding of the antibodies to CTLA-4 and model validation

The dose-curves for blocking B7–1 and B7–2 binding by ipilimumab (Fig. 4a) is comparable with the experimental dose curves reported by Keler et al. [50] for the same antibody. It is important to note that no values from this experimental work have been used in our simulations or in the construction of our model. In this previous work [50], the scientists at Medarex performed competitive binding assay by flow cytometry and ELISA to demonstrate the ability of ipilimumab (dubbed as 10D1 in the paper) [50] to block the interactions of CTLA-4 with B7–1 and B7–2, separately. The simulations based on our model was also carried out by having only one of the B7 ligands active at a time, so as to mimic the experimental set-up [50]. Although the overall trends in the dose-response curves for ipilimumab obtained from our model and the previous experiment [50] are in agreement, the percent inhibition predicted for each dose of the antibody by our model are higher than those observed in the experiments [50]. For example, our model predicts that, at 10 μM concentration of ipilimumab, ~ 89% of B7–1 and ~ 92% of B7–2 are blocked; the experiments, on the other hand, reported ~ 70% and ~ 90% inhibition of B7–1 and B7–2 binding [50], respectively. Nevertheless, it should be noted that the previous experiments [50] were carried out with a human CTLA-4 (hCTLA-4) expressing cell, which was constructed by a hCTLA-4/murineCD3 chimeric gene. Thus, taking into account this difference and other experimental conditions, in general, the predictions from our model and the experiments are in reasonable agreement. Particularly, Keler et al. [50] reported an IC_{50} value of ~ 1–3 μM for ipilimumab (or 10D1 as named in the paper) to block the B7 ligands, which is in excellent agreement with the values predicted by our model (IC_{50} = 1.11 μM for B7–2 blocking; and 3.5 μM for B7–1 blocking). This validates the ability of our model to simulate the competitive binding between antibodies and the B7 ligands reasonably well.

By comparing the dose-response curves for the two antibodies as predicted from our simulations, it is apparent that, at any given dose concentration, tremelimumab (Fig. 4b) is able to inhibit higher percentage of B7–1 and B7–2, when compared to that of ipilimumab (Fig. 4a). For instance, 1 μM of tremelimumab was found to inhibit ~ 54% of B7–1 binding and ~ 76% of B7–2 binding, which are much higher % inhibition than those obtained from 1 μM of ipilimumab (~ 25% of B7–1 inhibition; ~ 47% inhibition of B7–2) in our simulations. Such trends are in line with the affinity of these antibodies against CTLA-4 as reported by He et al [1]. It has been reported that both the antibodies have comparable association rate constants, however, the dissociation rate constants of ipilimumab is much higher than that of tremelimumab [1]. Hence, tremelimumab is able to block much higher amounts of B7 ligands from binding to CTLA-4, when matched with ipilimumab. In addition, it can be noted that higher percent inhibition of CTLA-4/B7–2 interactions than CTLA-4/B71 interactions can be achieved with low concentrations (≤ 10 μM) of the antibodies. This again accords with previous observation that the affinity of CTLA-4 to B7–1 is higher than its affinity to B7–2 [51]. In order to test this statement, we performed a simulation with our model, where only B7–1 and B7–2 were allowed to competitively bind with CTLA-4. To model this scenario, we again left the total initial concentrations of the antibodies and the CD28 receptor to 0, such that they do not have any effects on CTLA-4 binding to the B7 ligands. The result from this simulation is shown in Fig. 4c. Except for the first few minutes of the simulations, when the synapse was dominated by B7–2, the predominant amount of CTLA-4 remained in complex with B7–1 and the proportion of CTLA-4/B7–2 complex was meager.

### Effects of antibody-mediation on the overall complex formation

Comparison of the total numbers of CTLA-4/mAb, CTLA-4/B7 and CD28/B7 complexes formed at the end of simulations performed with the different concentrations (= 0 μM, 10 μM and 15 μM) of ipilimumab and tremelimumab

Interactions | Number of interacting CTLA-4/CD28 monomers | |||||
---|---|---|---|---|---|---|

Initial | Antibody =0 μM | Tremelimumab | Ipilimumab | |||

(10 μM) | (15 μM) | (10 μM) | (15 μM) | |||

CD28/B7–2 | 0 | 127 | 121 | 121 | 121 | 120 |

CD28/B7–1 | 0 | 54 | 83 | 84 | 77 | 81 |

CTLA-4/B7–2 | 0 | 8 | 17 | 12 | 40 | 32 |

CTLA-4/B7–1 | 0 | 790 | 20 | 11 | 129 | 60 |

CTLA-4/mAb | 0 | 0 | 763 | 778 | 631 | 707 |

From the dose curves, it is clear that both the antibodies have effectively inhibited the bivalent interactions of CTLA-4 and B7–1 even at very small concentrations. In fact, 50% of the B7–1 interactions with CTLA-4 were inhibited at concentrations < 5 μM in the case of both the antibodies. As much as 10 μM of either of the antibodies was sufficient to achieve ~ 90% (or more) inhibition of CTLA-4/B7–1 interactions (Fig. 6a). These percentage inhibition values for CTLA-4/B7–1 interactions shown by the antibodies in the fully competitive simulations are almost similar to those observed for B7–2 knockout-simulations (where B7–2 and CD28 were absent, in Fig. 4). Nevertheless, the inhibition of B7–2 interactions (with CTLA-4), in the full model simulations, required really very high dose of the antibodies. In fact, it can be seen that the CTLA-4/B7–2 inhibition was seen only at concentrations ≥20 μM for Tremelimumab and > 60 μM for ipilimumab (Fig. 6b). This indicates that at lower concentrations of the antibodies, there were actually some increase in the CTLA-4/B7–2 complexes, when compared to those seen in the untreated (or antibody-free) simulations. It should be noted that, in the antibody-free simulations, the CTLA-4/B7–2 complex dominated during the initial stages until the bivalent CTLA-4/B7–1 complex suppressed the monovalent CTLA-4/B7–2 interactions to become the dominant complex (see in Fig. 2). Whereas, in the antibody-mediated simulations, at low dose concentrations, the antibodies are more proactive in blocking the multivalent CTLA-4/B7–1 interactions, which relieves the suppression on the monovalent CTLA-4/B7–2 interactions. In addition, the actual numbers of CTLA-4/B7–2 complexes are in general much less than that of CTLA-4/B7–1 complex. For example, at the end of antibody-free simulations, there were only 8 CTLA-4/B7–2 complexes, when compared to 790 CTLA-4/B7–1 complexes. Nevertheless, higher antibody concentrations effectively blocked the CTLA-4/B7–2 interactions as well. This contradicts with the B7–1 knockout simulations (Fig. 4), where even the lower amounts of the antibodies inhibited predominant CTLA-4/B7–2 interactions. This suggests that the competitive effects could implicate the dose response predictions significantly. Finally, as indicated earlier, inhibition of CTLA-4/B7 interactions did not essentially translate to the more proportional increase in the CD28/B7 complex formation. Figure 6c compares the percentage of increase in the CD28/B7 complex seen at each dose concentration (from 0 μM to 1000 μM) of ipilimumab and tremelimumab. It can be seen that there were only a maximum of ~ 14% gain in the CD28/B7 complexes following the inhibition of CTLA-4/B7 interactions by the mAbs in this study.

### Effects of antibody-mediation on the overall complex formation

_{on}) and dissociation (k

_{off}) rate constants for tremelimumab on different interactions, such as CTLA-4/antibody complex (Fig. 7a), CTLA-4/B7–1 (Fig. 7b), CTLA-4/B7–2 (Fig. 7c), CD28/B7–1 (Fig. 7d) and CD28/B7–2 (Fig. 7e). As expected, the CTLA-4/antibody and the CTLA-4/B7 interactions were the most affected by these perturbations; indeed, the effects seen for CTLA-4/antibody complex were inverse to those of CTLA-4/B7 interactions. For example, the 50-fold reduction in the k

_{on}value led to the drop of ~ 60 numbers of monomer CTLA-4/antibody complexes, which was compensated by the increase in the total numbers of CTLA-4/B7 complexes (approximately + 48 for CTLA-4/B7–1 complex; + 12 for CTLA-4/B7–2 complex). On the other hand, 50-fold increase in the k

_{on}value resulted in a small gain of CTLA-4/antibody complexes, which again led to the drop in the total numbers of CTLA-4/B7 complexes (refer to Fig. 7a-c). Similar inverse effects in the CTLA-4/antibody and CTLA-4/B7 complexes were also seen for the changes in the k

_{off}values. Nevertheless, the CD28/B7 interactions were mostly insensitive to these perturbations. This highlights the fact that CD28 is not able to compete with the high-affinity and high-avidity interactions of CTLA-4/B7 and CTLA-4/antibody interactions.

The effects of varying the rate constants for association and dissociations for monovalent CD28/B7 complexes, monovalent CTLA-4/B7 complexes, bivalent CD28/B7–1 complex, and multivalent CTLA-4/B7 complexes were also tested (results provided in supplementary information, Additional file 4: Figure S1 and Additional file 5: Figure S2). These manipulated simulations described that the CTLA-4/antibody and CTLA-4/B7–1 interactions are mostly sensitive to the rate constant values for multivalent association and dissociation of CTLA-4/B7–1 complex; whereas, the CTLA-4/B7–2 interactions are predominantly affected by the rate constants for association and dissociation of bivalent CTLA-4/B7–2 interactions. The perturbations in the latter parameters also exhibited small effects on CTLA-4/antibody interactions. Nevertheless, none of the changes corresponding to CTLA-4/B7 interactions made any significant impacts on the CD28/B7 complexes, which were only sensitive to the rate constant values corresponding to their own association/dissociation. Although the model is sensitive to upto 50% variations in the selected kinetics parameters, the qualitative inference based on the original values remain the same.

## Discussions

Mathematical modeling and simulation remains a valuable tool to develop quantitative insights about the dynamic changes taking place within complex systems. It has particularly been employed in the field of cancer immunotherapy. Jansson et al. [33] developed a model for quantitative analysis of costimulatory complex formation, between CTLA4, CD28 and the B7- ligands, at the immunological synapse. However, there have been no study that modelled the effects of antibody-mediation on the complex formation at the synapse. In this study, we have taken one baby-step forward towards analyzing the effects of adding anti-CTLA-4 antibodies on the immunological balance between the co-stimulatory interactions (formed by CD28 and B7 ligands) and the co-inhibitory interactions (formed by CTLA-4 and B7 ligands) at the synapse, using a free-diffusion model. The study mainly focused on two promising CTLA-4 blocking antibodies, ipilimumab and tremelimumab, which are either in the market or in clinical trials, respectively. As acknowledged throughout the paper, this study is an extension of the Jansson’s model [33], where we included several new equations and parameters to account for the effects of antibody-mediation at the synapse. The model is able to reproduce the K_{D} values for the inhibition of CTLA-4 by the two antibodies. We also validated our model by showing a reasonable agreement between the dose-curves, for blocking the binding of B7–1 and B7–2 to CTLA-4 by ipilimumab, from our simulations against a previous experimental data from competitive binding assays by Keler et al. [50]. The study also helped to understand the relative efficacy of the two antibodies in CTLA-4 blockade. Although both tremelimumab and ipilimumab have similar K_{D} values, the former tends to show more effective inhibition of CTLA-4/B7 interactions, due to its much lower dissociation rate that that of ipilimumab. The modelling and simulations in this work have shown that different factors, such as multivalent interactions, mobility of molecules and competition effects, could impact the effects of antibody-mediation. The results, in particular, highlighted that the competitive effects played an important role in the dose-dependent inhibition of the B7 ligand interactions with CTLA-4 receptor by the antibodies. However, it is important to concede that, as in any case of mathematical modelling, the model in this work is also mainly dependent on the parameters employed to build it. However, it is known that the K_{D} values measured for the same systems under different experimental conditions could vary significantly. For example, different k_{on} and k_{off} rates for ipilimumab/CTLA-4 complex have been reported in the literature. Hence, in order to minimize the impacts from such variabilities, we have used the k_{on} and k_{off} rates for both ipilimumab/CTLA-4 and tremelimumab/CTLA-4 complexes from the same work of He et al. [1], which was very recently published.

Another important limitation of this model is that it is constructed based on the normal T-cell conditions, where both the co-stimulatory and co-inhibitory interactions at the synapse play important role in maintaining the much-needed immunological balance. However, a CTLA-4 blocking antibody (or any immune-checkpoint drug for that matter) is only administered in an abnormal micro-tumor environment, where the expression of the receptors and ligands will be different than those seen in the normal T-cells, thus shifting the balance more towards inhibitory interactions. But, unfortunately, comprehensive parameter data for simulating cancerous cells in the context of immunological synapse is not available in the literature. Hence, we made an informed choice of simulating the effects of antibody mediation in a normal T-cell environment, for which parameters are available and a preliminary model [33] (without antibody) was also published. Precisely, for this reason, we did not perform simulations by introducing the antibodies at various timescale (after reaching steady-state for instance). Instead, we only focused on simulating the competitive binding aspects of the antibodies to CTLA-4 and how it changes the co-stimulatory (CD28/B7) and the co-inhibitory (CTLA-4/B7) complex formations at the synapse, when compared to untreated (or antibody-free) simulation.

Despite the stated limitations, the numerical simulations performed with the current model are in agreement with different experiments, such as the dose curve for ipilimumab-mediated inhibition of B7 ligands. The model is able to predict the dose-dependent inhibition of CTLA-4/B7 interactions in an immunologically-relevant competitive environment, where both the B7-ligands and antibodies compete to bind with CTLA-4. In general, it is difficult (and not always practical) to measure the specific inhibition percentage of either B7–1 or B7–2 by the antibodies under such fully immunologically-relevant competitive binding environment. Most experiments measure the competitive binding of the anti-CTLA-4 antibodies only in the presence of either of the ligands and CTLA-4. Thus, this mathematical model could be a useful tool to gain some insights about the potencies of the antibodies to compete with both B7–1 and B7–2 to bind with the CTLA-4 receptor, at the dynamic immunological synapse. Although the simulations in this work were performed for only the two known antibodies, the model itself could serve as an easily transferable tool to study the effects of any anti-CTLA-4 antibodies on the co-stimulation by the CD28 pathway, provided the binding kinetics data for the query antibodies and CTLA-4 are available. Therefore, the results presented and the mathematical model will be useful for the research activity in the field of immune-checkpoints-targeted cancer therapy.

## Conclusion

In this work, we have developed an expanded mathematical modeling framework to quantitatively analyze the effects of anti-CTLA-4 antibody-mediation on the co-stimulatory and co-inhibitory complex formation at the immunological synapse. The numerical simulations performed using this model have been validated by different experimental data. The model predicted the dose curve for the B7-ligand blockade by ipilimumab, which was in a reasonable agreement with the experimental data obtained from competitive binding assays. Further, the model was also able to reproduce the K_{D} values for the binding of the antibodies against the CTLA-4 receptor. Our findings show that a number of significant factors, such as multivalent interactions, mobility of moleculesand competition effects contribute to the antibody-mediated interactions at the synapse. In particular, the competitive effects play a more predominant role. The simulations from our model show that in a less-competitive setting, the CTLA-4/B7–2 interactions are inhibited with much lower concentrations of antibodies, while the inhibition of B7–1 interactions required comparatively higher dose of antibodies. This is in concurrent with the previous findings that B7–1 is a preferred ligand for CTLA-4 and also has a higher affinity to CTLA-4 compared with B7–2. Nevertheless, our simulations show that the trend is reversed within a fully competitive and dynamic immunological synapse. In fact, the antibodies are more proactive in inhibiting the divalent CTLA-4/B7–1 interactions, which in turn relieves the suppression of CTLA-4/B7–2 complexes. As a result, the inhibition of CTLA-4/B7–2 in the full model required much higher concentrations of antibodies. Further, the inhibition of the CTLA-4/B7 interactions does not essentially lead to significant increase in the costimulatory CD28/B7 complexes. It is important to acknowledge that the model suffers from some of the important limitations, which are mainly caused due to lack of several parameters required to model a tumor microenvironment. Nevertheless, the current work represents an important first step towards understanding the antibody-mediated effects on synaptic complex formation. The model could also serve as an easily transferable predictive tool to study the effects of any anti-CTLA-4 antibodies on the co-stimulation by the CD28 pathway, provided the binding kinetics data for the query antibodies and CTLA-4 are available. Our natural next step will be expanding the current model by integrating it with the simulation of the main first signal (from TCR-MHC interactions) and also connecting to some downstream signaling processes, such as interleukin-2 activation pathway. Such an integrated mathematical model will be an excellent tool to guide immune-checkpoints research towards complete elimination of cancers.

## Notes

### Funding

The authors would like to acknowledge the funding support from Alberta Cancer Foundation (RES0025662) and Li Ka Shing Applied Virology Institute (Project ID:RES0028141) at the University of Alberta. The funding bodies had no role in this study in any way, including the study design, data collection and analysis, decision to publish and preparation of the manuscript.

### Availability of data and materials

All the equations and parameter data employed in this study are provided as supplementary information. And the complete MATLAB code for the present model can be accessed by contacting the corresponding author of this work.

### Authors’ contributions

AG and KB conceived and supervised the project. AG and TA contributed equally in relevant data collection, development of the model, testing and performing simulations. AG, TA , TC and KB analysed of the results and presentation. AG and TA wrote the manuscript. All authors read and approved the final manuscript.

### Ethics approval and consent to participate

Not applicable.

### Competing interests

The author declared that they have no competing interests.

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Supplementary material

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