 Research article
 Open Access
 Published:
A mathematical modelling tool for unravelling the antibodymediated effects on CTLA4 interactions
BMC Medical Informatics and Decision Making volume 18, Article number: 37 (2018)
Abstract
Background
Monoclonal antibodies blocking the Cytotoxic Tlymphocyte antigen 4 (CTLA4) receptor have revolutionized the field of anticancer therapy for the last few years. The human Tcellbased immune responses are modulated by two contradicting signals. CTLA4 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 antiCTLA4 mAbs on the costimulatory (CD28/B7 ligands) and the coinhibitory (CTLA4/B7 ligands) complex formation at the immunological synapse. In particular, we focus on two promising antiCTLA4 mAbs, tremelimumab (from AstraZeneca) and ipilimumab (from BristolMyers 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 antiCTLA4 antibodies from literature.
Results
The numerical simulations from the current model are in agreement with a number of experimental data. Especially, the dosecurves for blocking the B7 ligand binding to CTLA4 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 CTLA4/B7 complexes.
Conclusions
The results show that different factors, such as multivalent interactions, mobility of molecules and competition effects, could impact the effects of antibodymediation. The results, in particular, describe that the competitive effects could impact the dosedependent 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 antiCTLA4 antibodies on immunological synaptic complex formation, provided reliable biophysical data for mAbs are available.
Background
Blockade of immune checkpoints has recently been proven as a revolutionary strategy in the fight against cancers [1,2,3,4,5,6]. Tcells play a pivotal role in modulating the immune response against pathogens [7] and also in the prevention of autoimmunity [6]. According to the classic ‘twosignal’ model in immunology [8, 9], Tcellmediated immune responses are activated by two different proteinprotein interactions taking place at the immunological synapse [1,2,3,4,5, 10,11,12] (Fig. 1). The first signal is triggered when the Tcell receptors (TCRs) recognize the major histocompatibility complex (MHC) on the surface of antigenpresenting cells (APCs). A second signal comes from the binding of CD28, a coreceptor expressed on the Tcell surface, with its’ complementary B7 ligands, known as B7–1 (or CD80) and B7–2 (or CD86), on the surface of APCs. This second signal is required to activate and sustain the activity of Tcells [3, 5, 13, 14]; hence it is known as a costimulatory signal and CD28 is dubbed as a costimulatory receptor (Fig. 1). On the other hand, there are a number of negative signal stimuli (known as the inhibitory receptors) that act to inactivate the Tcells, including cytotoxic Tlymphocyteassociated protein4 (CTLA4 or CD152) and programmed death 1 (PD1 or CD279) [3]. Particularly, CTLA4, a CD28 homologue (with ~ 30% sequence identity) expressed on the Tcell surface, binds with the same set of B7ligands of CD28; albeit with higher affinity towards B7–1 and B7–2. Both CD28 and CTLA4 are transmembrane proteins of the immunoglobulin superfamily that exist as homodimers [15] .Nevertheless, CD28 is known to bind the ligands monovalently; whereas, CTLA4 is bivalent in nature that, infact, allows it to bind the B7 ligands with high avidity [15, 16]. Given such differences, under competitive environments, CTLA4 would be able to outcompete CD28 for ligandbinding [17]. The functionalities of CD28 and CTLA4 are interlinked and both costimulatory and coinhibitory signals are required for maintaining the immunological balance between selftolerance and defending against foreign entities and pathogens under normal conditions [4,5,6, 18, 19].
Cancer cells are able to escape the immunological surveillance of Tcells, by overexpressing the inhibitory receptors that attenuate antitumor immune response [4, 5]. As a result, in principal, blocking the coinhibitory receptorligand interactions (schematically shown in Fig. 1) should be able to facilitate the reactivation of Tcells, which inturn 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 BristolMyers Squibb, became the firstinclass antiCTLA4 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 antiCTLA4 mAb, Tremelimumab from AstraZeneca, is currently in phase III clinical trials [1, 5]. Similar to Ipilimumab, AstraZeneca’s tremelimumab binds specifically to CTLA4 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 immunecheckpoint blocking inhibitors (both mAbs and, lately, small molecules).
Very recently, He et al. [1] investigated the binding profiles of both mAbs against CTLA4 using surface plasmon resonance (SPR) experiments and reported their K_{D} values as 18.2 nM for ipilimumab/CTLA4 complex and 5.89 nM for tremelimumab/CTLA4 complex. These binding affinity values are higher than (or, in some cases, comparable to) those reported for B7–1 ligands with CTLA4 [1]. However, it is important to note that different range of affinity values have been reported for the CTLA4/B7–1 complexes. For example, an earlier study reported that a soluble B7–1 Ig fusion protein bound to CTLA4 with a dissociation constant of ~ 12 nM [30]. However, this value was argued to be higher when compared with other proteinprotein interactions occurring between Tcell surface and APCs [31]. The authors of this work [30] had described that the CTLA4Ig 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 CTLA4Ig/B7Ig fusion protein. Another study [31] based on SPR experiments reported that, at 37 °C, soluble recombinant B7–1 bound to CTLA4 with a K_{D} value of 0.42 μM. This indicates that affinity between B7–1 and CTLA4 is relatively lower than that of ipilimumab/CTLA4 and tremelimumab/CTLA4 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 CTLA4 when compared to that of ipilimumab. Thus, understanding the effects of these two potent antiCTLA4 antibodies on the costimulatory and the coinhibitory complex formation at the synapse will be useful to develop effective nextgeneration antiCTLA4 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, immuneeffector cells and cytokine interleukin2. 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 IL2 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 Tcellspecific activation. This model attempted to account for the sequence of events starting from the TCRMHC binding to Tcell activation and response [43]. Mathematical modelling has also been employed to study the complex formation between PD1 receptor and its ligands [44]. Jansson et al. [33] developed a theoretical model for simulating the synaptic accumulation of the molecules involved in costimulation and inhibition of Tcells. This model, which was developed based on the Ordinary Differential Equations (ODEs) and rigorous biophysical and expression data from literature, explained the interactions of CTLA4 and CD28 with their B7ligands in the context of a potentially dynamic synaptic microenvironment [33]. However, until now, there is no model that could predict the response to CTLA4 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 antiCTLA4 antibody. Our model is based on ODEs and additional biophysical data for antibodyCTLA4 complexes [1]. Using this new model, we studied how the binding of mAbs (with different affinities) to CTLA4 dynamically changed the interactions among coinhibitory complexes (CTLA4/B7–1 and CTLA4/B7–2) and costimulatory 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 antiCTLA4 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 twocompartment model by Jansson et al. [33], which was constructed to simulate the CD28 and CTLA4 complex formation with the B7ligands at immunological synapses. In this study, we developed an expanded model to assess the changes in the costimulatory (CD28/B7 ligands) and the coinhibitory (CTLA4/B7 ligands) interactions upon binding of the antibody to the free CTLA4 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. CTLA4, on the other hand, is present intracellularly and gets injected into the synapse upon activation [45, 46]. The ‘freediffusion’ 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 proteinprotein interactions. The model also assumes that the CTLA4 receptors, once injected from the intracellular environment, stay within the synapse. In the case of antibody, unlike the membranebound ligands, it is in the free solution and is assumed to bind to free CTLA4 monovalently. The model assumes that the binding site of B7–1, B7–2 and the antibody molecules on CTLA4 overlap considerably [2], and, therefore, only one of them (B7–1 or B7–2 or the antibody) is able to bind with a CTLA4 molecule. Moreover, in our model, the antibody is allowed to bind an unbound CTLA4 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 CTLA4 monomer that is part of a dimer, where the other monomer is bound to an antibody.
Parameters
The parameters for the interactions of CTLA4 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 CTLA4 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 CTLA4 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 2D 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, CTLA4, 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 CTLA4 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 massaction law.
Rate of change of density of antibody/CTLA4 complex
The antibody associates with a CTLA4 dimer at the rate of k_{on}, in order to form the antibody/(CTLA4)_{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,
The rate of change of density for the other antibodymediated complexes are given in supplementary information, Additional file 2: Table S2. The numbers of free antibody molecules is increased by the dissociation of mAb from any of the antibodyincluded complexes (such as, Ac, AcA, Ab/CTLA4/B72, EAb, CAb, DAb). On the other hand, association of free antibody to form any of the above complexes decreases the total number of free antibody molecules,and this rate of change in the antibody can be written as,
Density of mobile CD28, B71 and B72 molecules, present outside synapse is estimated by subtracting the total number of mobile molecules of CD28, B71,B72 that are part of complexes formed at synapse.
Similarly, the number of CTLA4 molecules present inside the cell is calculated by subtracting it from total number of CTLA4 molecules.
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 CTLA4 and CD28 complex formation were analyzed.
Results
Simulation of antibodyfree complex formation
Initially, the simulation based on the freediffusion model (Fig. 2a) was performed for ~ 7 h, without the inclusion of any antibody (i.e., 0 μM antibody concentration).This situation is ideally to reproduce the simulation from Jansson et al. [33].This simulation mimics an activated Tcell environment, where both CD28 and CTLA4 are present at the synapse and are competing for ligandbinding. During the initial hours (~ 15 min), the expression levels of B7–2 at synapse is much higher and hence, it remained the leading ligand for CTLA4 and CD28. Particularly, at about 6 min, the CTLA4/B72 interactions reached a peak with ~ 600 CTLA4 monomers were bound to B7–2, against ~ 140 CTLA4/B7–1 and 108 CD28/B7–2 complexes. The domination of CTLA4/B7–2 remained until ~ 17 min, however, this number (of CTLA4/B7–2 complex) dropped significantly to < 50 at ~ 1 h of simulation. Gradually, as more CTLA4 moved from the intracellular region to the synapse (see in Fig. 2b), the CTLA4/B7–1 complex replaced CTLA4/B7–2 and became the most populated complex after ~ 20 min. After reaching the steady state, it can be seen (in Fig. 2a) that the majority of the B7–1 was bound to CTLA4 (resulting ~ 790 CTLA4/B7–1 complexes), while only small numbers of B7–1 (~ 50) were engaged with the CD28 receptor. On the other hand, the amount of CD28/B7–2 complexes (~ 127) outnumbers the complexes formed by CTLA4 monomers and B7–2(~ 8). As Jansson et al. [33] noted, these behaviors in the simulation are in good agreement with previous findings that B7–1 is the preferred ligand for CTLA4 and B7–2 preferentially recruits CD28 at the synapse [48, 49]. Hence, 99% of CTLA4 are complexed with B7–1, while only 1% of the receptor is engaged with the B7–2 ligand. Overall, however, the percentage of CTLA4 complexed with the B7 ligands is much higher than that of CD28/B7 complexes (Fig. 2b). This trend is expected as CTLA4 has a higher affinity towards B7 ligands than that of CD28.
Binding of the antibodies to CTLA4
Initially, we performed two simulations with an in silico knockout model, where CD28 and the B7 ligands (both B7–1 and B7–2) were muted and only CTLA4 was allowed to bind with the antibodies, ipilimumab and tremelimumab, at different concentrations (0.002 μM to 0.018 μM). These simulations were carried out in order to confirm that, in a noncompetitive setting, 50% of CTLA4 had formed complex with the antibodies when the concentration of the antibodies equaled their respective K_{D} values. As shown in Figs. 3, 50% of the CTLA4 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 CTLA4 and model validation
Next, we performed the simulations to study how the antibodies, in the absence of CD28, competed with either B7–1 or B7–2 for binding to CTLA4. In order to perform these simulations, CD28 and either of the B7 ligands were muted in the model. The resulting dose response curves for the percent inhibition of B7–1 binding and B7–2 binding with CTLA4 at different concentrations of the antibodies, as predicted by the model, are shown in Fig. 4a and b, respectively. As shown in Fig. 4a and b, the model was able to simulate the dosedependent inhibition of CTLA4/B7 interactions by the competitive binding of ipilimumab and tremelimumab.
The dosecurves 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 CTLA4 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 setup [50]. Although the overall trends in the doseresponse 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 CTLA4 (hCTLA4) expressing cell, which was constructed by a hCTLA4/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 doseresponse 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 CTLA4 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 CTLA4, when matched with ipilimumab. In addition, it can be noted that higher percent inhibition of CTLA4/B7–2 interactions than CTLA4/B71 interactions can be achieved with low concentrations (≤ 10 μM) of the antibodies. This again accords with previous observation that the affinity of CTLA4 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 CTLA4. 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 CTLA4 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 CTLA4 remained in complex with B7–1 and the proportion of CTLA4/B7–2 complex was meager.
Effects of antibodymediation on the overall complex formation
Subsequently, we tested the effects of antibodymediation on the costimulatory (CD28/B7 ligands) and the coinhibitory (CTLA4/B7 ligands) complex formation at the synapse. Simulations were performed with a restraintfree competitive environment facilitated by the model, where all the species (such as CTLA4, B7–1, B7–2 and CD28) were present and a specific concentration of either of the antibodies was added. Initially, 10 μM concentration of ipilimumab (Fig. 5a) and tremelimumab (Fig. 5b) were added in the simulations to study their effects. The results are comparable to Fig. 2a, where the simulation performed without the antibodies (concentration = 0 μM) is provided. As expected, upon addition of 10 μM concentration of antibodies, the antibodybound CTLA4 complex outnumbered the other complex formations. Particularly, in the antibodyfree simulations, the B7–2bound CTLA4 complex (~ 600) was dominant during the initial hours; however, the presence of the antibody reduced the initial dominance of this complex (CTLA4/B7–2) by at least 75%.
In the case of simulation with ipilimumab (concentration = 10 μM), it can be seen that both the B7 ligands and ipilimumab contested for CTLA4 binding during the early hours. However, the antibody outcompeted the B7 ligands to form complexes with CTLA4. It can be seen in Fig. 5a that, at the end of the simulations, there were ~ 630 CTLA4/ipilimumab complexes against ~ 129 CTLA4/B7–1 complexes. This is, particularly, a significant reduction in the sheer dominance of CTLA4/B7–1 complexes (790 numbers) seen in the antibodyfree simulations. On the other hand, 10 μM concentration of tremelimumab was able to bind more effectively with CTLA4, thus remaining the most populated complex (~ 760 CTLA4/tremelimumab complexes) at the steady state of the simulations, as shown in Fig. 5b. As a result, there were only 20 complexes of CTLA4 and B7–1 at the end of simulations with tremelimumab (10 μM). In summary, the antibodymediation has significantly impacted the CTLA4/B7 ligand complex formation, which reduced from ~ 800 numbers (in antibodyfree simulations) to ~ 168 and ~ 42 in the simulations with 10 μM of ipilimumab and tremelimumab, respectively. Later, we increased the dose of the antibodies to 15 μM and performed simulations (Fig. 5cd). The higher dose of antibodies naturally increased the effects on CTLA4 binding and B7ligand blockade. In fact, predominant CTLA4 monomers were engaged in complexes with the antibodies, ipilimumab (~ 707) and tremelimumab (~ 780). And as a result, the inhibitory complex formation between CTLA4 and the B7ligands had significantly diminished (~ 23 with 15 μM of tremelimumab; ~ 92 with 15 μM of ipilimumab). However, the addition of antibody did not significantly impact the interactions of CD28 with the B7 ligands. While the amount of CD28/B7–2 complexes remained almost the same in antibodyfree and antibodyincluded simulations, the number of CD28/B71 complexes increased only slightly. The total numbers of different complexes (such as CTLA4/mAb, CTLA4/B7 and CD28/B7) at the end of simulations performed with different concentrations of ipilimumab and tremelimumab are summarized in Table 1.
Next, we simulated full doseresponse curves for the two antibodies (Fig. 6), ipilimumab and tremelimumab. The inhibition percentage of the B7 ligands (i.e., B7–1 and B7–2) at each concentration of the antibodies in this figure was calculated as follows,
From the dose curves, it is clear that both the antibodies have effectively inhibited the bivalent interactions of CTLA4 and B7–1 even at very small concentrations. In fact, 50% of the B7–1 interactions with CTLA4 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 CTLA4/B7–1 interactions (Fig. 6a). These percentage inhibition values for CTLA4/B7–1 interactions shown by the antibodies in the fully competitive simulations are almost similar to those observed for B7–2 knockoutsimulations (where B7–2 and CD28 were absent, in Fig. 4). Nevertheless, the inhibition of B7–2 interactions (with CTLA4), in the full model simulations, required really very high dose of the antibodies. In fact, it can be seen that the CTLA4/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 CTLA4/B7–2 complexes, when compared to those seen in the untreated (or antibodyfree) simulations. It should be noted that, in the antibodyfree simulations, the CTLA4/B7–2 complex dominated during the initial stages until the bivalent CTLA4/B7–1 complex suppressed the monovalent CTLA4/B7–2 interactions to become the dominant complex (see in Fig. 2). Whereas, in the antibodymediated simulations, at low dose concentrations, the antibodies are more proactive in blocking the multivalent CTLA4/B7–1 interactions, which relieves the suppression on the monovalent CTLA4/B7–2 interactions. In addition, the actual numbers of CTLA4/B7–2 complexes are in general much less than that of CTLA4/B7–1 complex. For example, at the end of antibodyfree simulations, there were only 8 CTLA4/B7–2 complexes, when compared to 790 CTLA4/B7–1 complexes. Nevertheless, higher antibody concentrations effectively blocked the CTLA4/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 CTLA4/B7–2 interactions. This suggests that the competitive effects could implicate the dose response predictions significantly. Finally, as indicated earlier, inhibition of CTLA4/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 CTLA4/B7 interactions by the mAbs in this study.
Effects of antibodymediation on the overall complex formation
It is important to acknowledge that any mathematical model is mainly dependent on the parameters employed to build it. Particularly, the sensitivity is much higher in the models that rely on biological parameters. For example, Jansson et al. [33] tested the dependence of their model on the affinity, mobility and expression levels of various species, by reducing each of these parameters by 10fold, and found that some of the simulated interactions are sensitive to these changes [33]. However, it is true even in the case of measured data, as the experimental conditions, such as stoichiometry and temperature, have been shown to affect the results. For instance, an earlier experimental study reported that the affinity between CTLA4 and B7–1 was 12 nM [30]; while another experimental study reported the affinity for the same complex to be 0.4 μM [31]. In the current study, we tested the sensitivity of our model towards the changes in the rate constants for association and dissociation of different species in the model (the parameters, P2P20 listed in Additional file 1: Table S1). To achieve this, we perturbed the values for each of these parameters in the range of − 50 to + 50% from their respective original values used in the model. Figure 7 compares the effects of changing the association (k_{on}) and dissociation (k_{off}) rate constants for tremelimumab on different interactions, such as CTLA4/antibody complex (Fig. 7a), CTLA4/B7–1 (Fig. 7b), CTLA4/B7–2 (Fig. 7c), CD28/B7–1 (Fig. 7d) and CD28/B7–2 (Fig. 7e). As expected, the CTLA4/antibody and the CTLA4/B7 interactions were the most affected by these perturbations; indeed, the effects seen for CTLA4/antibody complex were inverse to those of CTLA4/B7 interactions. For example, the 50fold reduction in the k_{on} value led to the drop of ~ 60 numbers of monomer CTLA4/antibody complexes, which was compensated by the increase in the total numbers of CTLA4/B7 complexes (approximately + 48 for CTLA4/B7–1 complex; + 12 for CTLA4/B7–2 complex). On the other hand, 50fold increase in the k_{on} value resulted in a small gain of CTLA4/antibody complexes, which again led to the drop in the total numbers of CTLA4/B7 complexes (refer to Fig. 7ac). Similar inverse effects in the CTLA4/antibody and CTLA4/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 highaffinity and highavidity interactions of CTLA4/B7 and CTLA4/antibody interactions.
The effects of varying the rate constants for association and dissociations for monovalent CD28/B7 complexes, monovalent CTLA4/B7 complexes, bivalent CD28/B7–1 complex, and multivalent CTLA4/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 CTLA4/antibody and CTLA4/B7–1 interactions are mostly sensitive to the rate constant values for multivalent association and dissociation of CTLA4/B7–1 complex; whereas, the CTLA4/B7–2 interactions are predominantly affected by the rate constants for association and dissociation of bivalent CTLA4/B7–2 interactions. The perturbations in the latter parameters also exhibited small effects on CTLA4/antibody interactions. Nevertheless, none of the changes corresponding to CTLA4/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 antibodymediation on the complex formation at the synapse. In this study, we have taken one babystep forward towards analyzing the effects of adding antiCTLA4 antibodies on the immunological balance between the costimulatory interactions (formed by CD28 and B7 ligands) and the coinhibitory interactions (formed by CTLA4 and B7 ligands) at the synapse, using a freediffusion model. The study mainly focused on two promising CTLA4 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 antibodymediation at the synapse. The model is able to reproduce the K_{D} values for the inhibition of CTLA4 by the two antibodies. We also validated our model by showing a reasonable agreement between the dosecurves, for blocking the binding of B7–1 and B7–2 to CTLA4 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 CTLA4 blockade. Although both tremelimumab and ipilimumab have similar K_{D} values, the former tends to show more effective inhibition of CTLA4/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 antibodymediation. The results, in particular, highlighted that the competitive effects played an important role in the dosedependent inhibition of the B7 ligand interactions with CTLA4 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/CTLA4 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/CTLA4 and tremelimumab/CTLA4 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 Tcell conditions, where both the costimulatory and coinhibitory interactions at the synapse play important role in maintaining the muchneeded immunological balance. However, a CTLA4 blocking antibody (or any immunecheckpoint drug for that matter) is only administered in an abnormal microtumor environment, where the expression of the receptors and ligands will be different than those seen in the normal Tcells, 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 Tcell 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 steadystate for instance). Instead, we only focused on simulating the competitive binding aspects of the antibodies to CTLA4 and how it changes the costimulatory (CD28/B7) and the coinhibitory (CTLA4/B7) complex formations at the synapse, when compared to untreated (or antibodyfree) 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 ipilimumabmediated inhibition of B7 ligands. The model is able to predict the dosedependent inhibition of CTLA4/B7 interactions in an immunologicallyrelevant competitive environment, where both the B7ligands and antibodies compete to bind with CTLA4. 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 immunologicallyrelevant competitive binding environment. Most experiments measure the competitive binding of the antiCTLA4 antibodies only in the presence of either of the ligands and CTLA4. 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 CTLA4 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 antiCTLA4 antibodies on the costimulation by the CD28 pathway, provided the binding kinetics data for the query antibodies and CTLA4 are available. Therefore, the results presented and the mathematical model will be useful for the research activity in the field of immunecheckpointstargeted cancer therapy.
Conclusion
In this work, we have developed an expanded mathematical modeling framework to quantitatively analyze the effects of antiCTLA4 antibodymediation on the costimulatory and coinhibitory 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 B7ligand 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 CTLA4 receptor. Our findings show that a number of significant factors, such as multivalent interactions, mobility of moleculesand competition effects contribute to the antibodymediated interactions at the synapse. In particular, the competitive effects play a more predominant role. The simulations from our model show that in a lesscompetitive setting, the CTLA4/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 CTLA4 and also has a higher affinity to CTLA4 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 CTLA4/B7–1 interactions, which in turn relieves the suppression of CTLA4/B7–2 complexes. As a result, the inhibition of CTLA4/B7–2 in the full model required much higher concentrations of antibodies. Further, the inhibition of the CTLA4/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 antibodymediated effects on synaptic complex formation. The model could also serve as an easily transferable predictive tool to study the effects of any antiCTLA4 antibodies on the costimulation by the CD28 pathway, provided the binding kinetics data for the query antibodies and CTLA4 are available. Our natural next step will be expanding the current model by integrating it with the simulation of the main first signal (from TCRMHC interactions) and also connecting to some downstream signaling processes, such as interleukin2 activation pathway. Such an integrated mathematical model will be an excellent tool to guide immunecheckpoints research towards complete elimination of cancers.
Abbreviations
 APCs:

Antigenpresenting cells
 CTLA4:

Cytotoxic Tlymphocyte antigen 4
 FDA:

Food and drug administration
 mAbs:

Monoclonal antibodies
 MHC:

Major histocompatibility complex
 ODEs:

Ordinary differential equations
 TCRs:

Tcell receptors
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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.
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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.
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Additional files
Additional file 1:
Table S1. Parameters used in the model. The parameters related to the interactions of CTLA4 and CD28 with the B7 ligands in the current model are provided in this file. (DOCX 26 kb)
Additional file 2:
Table S2. Equations related to the rate of change of density for the antibody included in different complexes. The rate of change of density for the different antibodymediated complexes are given in this file. Equations related to the rate of change of density for the antibody included in different complexes. The rate of change of density for the different antibodymediated complexes are given in this file. (DOCX 144 kb)
Additional file 3:
Table S3. Different components/species included in the model and their respective abbreviation used in the text. The different components included in our mathematical model, along with their corresponding abbreviations, are provided in this file. Different components/species included in the model and their respective abbreviation used in the text. The different components included in our mathematical model, along with their corresponding abbreviations, are provided in this file. (DOCX 13 kb)
Additional file 4:
Figure S1. Sensitivity of the CTLA4/antibody (a, d) and CTLA4/B7 (bc, ef) interactions towards the perturbations in the association and dissociation for the CTLA4/B7 and the CD28/B7 complexes. This file includes various figures corresponding to the sensitivity analyses performed to study the impacts of perturbations in the association and dissociation rates for the CTLA4/B7 and the CD28/B7 complexes. (DOCX 5236 kb)
Additional file 5:
Figure S2. Sensitivity of the CD28/B7–1 and CD28/B7–2 interactions towards the perturbations in the association and dissociation for the CTLA4/B7 and the CD28/B7 complexes. This file includes various figures corresponding to the sensitivity analyses performed to study the impacts of perturbations in the association and dissociation rates for the CTLA4/B7 and the CD28/B7 complexes. (DOCX 181 kb)
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Ganesan, A., Arulraj, T., Choulli, T. et al. A mathematical modelling tool for unravelling the antibodymediated effects on CTLA4 interactions. BMC Med Inform Decis Mak 18, 37 (2018). https://doi.org/10.1186/s129110180606x
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Keywords
 CTLA4
 Immune checkpoints
 Ipilimumab
 Tremelimumab
 Antibody
 Mathematical modeling