Modeling of B cell Synapse Formation by Monte Carlo Simulation Shows That Directed Transport of Receptor Molecules Is a Potential Formation Mechanism
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- Tsourkas, P.K. & Raychaudhuri, S. Cel. Mol. Bioeng. (2010) 3: 256. doi:10.1007/s12195-010-0123-1
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The formation of the protein segregation structure known as the “immunological synapse” in the contact region between B cells and antigen presenting cells appears to precede antigen (Ag) uptake by B cells. The mature B cell synapse consists of a central cluster of B cell receptor/Antigen (BCR/Ag) complexes surrounded by a ring of LFA-1/ICAM-1 complexes. In this study, we used an in silico model to investigate whether cytoskeletally driven transport of molecules toward the center of the contact zone is a potential mechanism of immunological synapse formation in B cells. We modeled directed transport by the cytoskeleton in an effective manner, by biasing the diffusion of molecules toward the center of the contact zone. Our results clearly show that biased diffusion of BCR/Ag complexes on the B cell surface is sufficient to produce patterns similar to experimentally observed immunological synapses. This is true even in the presence of significant membrane deformation as a result of receptor–ligand binding, which in previous work we showed had a detrimental effect on synapse formation at high antigen affinity values. Comparison of our model’s results to those of experiments shows that our model produces synapses for realistic length, time, and affinity scales. Our results also show that strong biased diffusion of free molecules has a negative effect on synapse formation by excluding BCR/Ag complexes from the center of the contact zone. However, synapses may still form provided the bias in diffusion of free molecules is an order-of-magnitude weaker than that of BCR/Ag complexes. We also show how diffusion trajectories obtained from single-molecule tracking experiments can generate insight into the mechanism of synapse formation.
KeywordsImmunological synapseB cell receptorCytoskeletonAgent-based simulationAntigen presenting cellReceptor–ligand dynamicsImmune responseAntigenLFA-1ICAM-1Computational modeling
The “immunological synapse” is a membrane–protein segregation structure that forms during contact between a lymphocyte and an antigen presenting cell (APC) during recognition of antigen by the lymphocyte. The function and formation mechanisms of the immunological synapse are among the least understood aspects of lymphocyte activation. Originally observed to form during antigen recognition by T cells,17,20,24,35 immunological synapses have also been observed during contact between B cells and APCs.2 During synapse formation in B cells, the B cell receptors (BCR) bind antigen (Ag) on the APC surface and the resultant BCR/Ag complexes cluster at the center of the contact zone, while the integrin Lymphocyte function-associated antigen-1 (LFA-1) on the B cell surface binds its ligand Intercellular adhesion molecule-1 (ICAM-1). The LFA-1/ICAM-1 complexes surround the central cluster of BCR/Ag complexes, resulting in a concentric pattern.2,6,8 It is widely believed that the immunological synapse modulates intracellular signaling and immune cell response.7,11,15,22
A considerable modeling effort has been expended to understand the formation mechanisms of the immunological synapse, particularly for the T cell synapse,4,9,10,16,22,23,25,26,33 and to a lesser extent the B cell synapse,15,18,31,32 although many aspects of synapse formation still remain unresolved. In T cells, differences in equilibrium bond length between TCR/MHCp and LFA-1/ICAM-1 complexes are thought to be sufficient to generate immunological synapses (the so-called “topographic model”).25,26 However, experimental studies involving T cells show that receptor accumulation at the T cell-APC interface is also driven by the cell’s cytoskeleton.5,13,19,34
Even though immunological synapse patterns observed in B cells resemble the canonical T cell synapse pattern, the significantly larger size of the BCR (~25 nm1) compared to the TCR/MHCp bond (~15 nm4) means that differences in equilibrium bond length between BCR/Antigen and LFA-1/ICAM-1 complexes (equilibrium bond length ~40 nm4) are less likely to account for synapse formation in B cells. In addition, B cells recognize antigen over a considerably wider range of affinity values (KA = 106–1011 M−1) than do T cells (KA = 106–108 M−1). Other significant differences between B cells and T cells include receptor valency (the BCR is bivalent, as compared to the TCR, which is monovalent), and that the density of B cell receptors on the B cell surface is at least an order of magnitude greater than that of T cell receptors on the T cell surface. Lastly, the number of experimental studies that focuses on exploring the molecular basis of B cell synapse formation is small compared to such studies for T cells, making it even more difficult to reach a consensus on the formation mechanism of immunological synapses in B cells.
We have previously studied B cell synapse formation using Monte Carlo simulations.31,32 Our modeling work indicated that it is difficult for high affinity antigens (KA ≥ 108 M−1) to cluster into a synapse by purely passive mechanisms, such as a difference in bond length between the BCR/Ag and LFA-1/ICAM-1 complexes, in the occurrence of significant membrane deformation.31 We also showed that BCR molecules undergo sub-diffusive motion upon binding antigen, making it difficult for them to cluster in a synapse pattern within biologically realistic time-scales.32 It thus seems likely that synapse formation in B cells is driven by an active, signaling-driven mechanism that involves the actin cytoskeleton.28,29 A recent study of B cell activation also showed that BCR accumulation at the synapse is compromised when the cytoplasmic signaling domain of BCR molecules has been truncated so as to inhibit their ability to signal and attach to the actin cytoskeleton.15
In this study we use a Monte Carlo simulation procedure to investigate whether cytoskeletally driven transport of receptor molecules is a potential mechanism of synapse formation in B cells. Cytoskeletally driven transport toward the center of the contact zone is simulated in a computationally efficient manner, by biasing the random diffusion of membrane bound molecules toward the center of the contact zone. In a series of in silico parametric experiments, we vary the strength of the bias in diffusion of the various membrane-bound species so as to generate both qualitative and quantitative insight into the nature of the synapse formation mechanism.
Our results indicate that a bias in the diffusion of BCR/Ag complexes toward the center of the cell–cell contact zone is a sufficient mechanism of synapse formation in situations where mechanisms that depend on differences in bond properties between BCR/Ag and LFA-1/ICAM-1 fail to generate synapses. However, our results also show that a strong bias in diffusion of LFA-1/ICAM-1 complexes and free molecules has a detrimental effect on synapse formation, although synapse formation can still occur if the bias in diffusion of these species is weak compared to that of BCR/Ag complexes. We also show how molecular diffusion trajectories obtained from single-molecule tracking experiments can be used to test for the presence of directed transport. If such transport is present, we show that the mean distance of receptor–ligand complexes from the center of the contact zone can be used to determine precisely which species are affected by the transport mechanism, as well as the relative strength of directed transport of the various species. Specifically, we show that if cytoskeletally driven transport affects BCR/Ag complexes more strongly than LFA-1/ICAM-1 complexes, or free receptor molecules, the respective mean distance from the center differs by an order of magnitude. Recently, Tolar et al.28,29 have studied receptor diffusion using single-molecule tracking during the course of B cell synapse formation. Our results can thus be used synergistically with parallel single molecule tracking experiments such as those in Tolar et al.28,29 to elucidate the mechanism of synapse formation in B cells.
We use a Monte Carlo procedure similar to our previous work.31,32 Receptor and ligand molecules are randomly sampled to undergo reaction or diffusion according to specific probabilities. A distinguishing feature of our method is a mapping between the probabilistic parameters of the Monte Carlo simulation and their physical counterparts. This makes it possible to compare our model’s results to those of physical experiments to within an order of magnitude.
Only one molecule can occupy a node, so we choose a nodal spacing of 10 nm, which is approximately equal to a membrane protein’s exclusion radius (resulting in N = 300 nodes). The exception are BCR molecules, which being bivalent, have a width of ~25 nm1 and thus occupy three nodes, with either a horizontal or vertical orientation on the lattice. For the radius of B lymphocytes we use RB = 6 μm and z0 = 40 nm.
At the start of a simulation run, molecules are uniformly distributed over the two surfaces at random. The molecular species simulated are BCR and LFA-1 on the B cell surface, and their ligands, antigen and ICAM-1, on the bilayer surface. At each time step in the simulation, the molecules are individually sampled at random to undergo either diffusion or reaction events, determined by means of a coin toss with probability 0.5.
If a molecule has been selected to undergo reaction, we first check the facing node on the opposite surface for a binding partner. If that is the case, a random number trial with probability pon(i) is performed to determine if the two molecules will form a receptor–ligand complex. BCR molecules are able to bind two antigen molecules, one on each end node (but not the middle node). Thus, if a BCR molecule is selected for a reaction, an additional coin toss is performed to pick one of the end nodes, and the bilayer surface opposite the chosen node is checked for a free antigen molecule. Sometimes a BCR molecule may have bound an antigen molecule on one Fab domain and have the other Fab domain free, forming a BCR/Ag complex. If the free Fab domain is selected, the reaction proceeds as described above, which may result in a second antigen molecule binding to the BCR/Ag complex (forming a BCR/Ag2 complex). If the Fab domain with the bound antigen is selected, the BCR/Ag complex may dissociate into its component molecules with probability poff(i). Three reversible reactions are thus possible: LFA-1 + ICAM-1 ↔ LFA-1/ICAM-1, BCR + Ag ↔ BCR/Ag, and BCR/Ag + Ag ↔ BCR/Ag2. The binding and dissociation probabilities for the two reactions involving antigen are assumed to be the same and thus the subscript i refers to the BCR/Ag reactions when i = BA and the LFA-1/ICAM-1 reaction when i = LI. The overall sampling rate for reaction or dissociation is thus 0.5 × pon(i) or 0.5 × poff(i).
The quantity PA(i)(z) defined in Eq. (4) is analogous to the overall receptor–ligand affinity, and consists of both the intrinsic affinity PA(i)max and the bond stiffness κi. Varying ponmax and poffmin while keeping the ratio PA(i)max constant changes the time scale of the simulation, but not the equilibrium behavior.
If a molecule has been selected to undergo diffusion, a random number trial with probability pdiff(i) is used to determine whether the diffusion move will occur successfully. The overall sampling rate for diffusion is thus 0.5 × pdiff(i). Although the probability of diffusion can be different for all seven species present in the simulation (free BCR, free antigen, free LFA-1, free ICAM-1, BCR/Ag complexes, BCR/Ag2 complexes, LFA-1/ICAM-1 complexes), we assume for simplicity that free molecules in the bilayer (antigen, ICAM-1) diffuse with the same probability (denoted as pdiff(FB)), molecules on the cell surface diffuse with probability pdiff(FC), and receptor–ligand complexes with probability pdiff(C).
If the trial with probability pdiff(i) is successful, one of the four neighboring nodes is selected at random for the molecule to diffuse to. Because two molecules are not allowed to occupy the same node, the molecule will only move if the target node is unoccupied. For the case of BCR molecules, three nodes need to be free for the molecule to diffuse in the direction transverse to its length, while only one free node is needed in order for it to diffuse along its length. In the case of complexes, the target nodes on both surfaces need to be free (two nodes for monomeric LFA-1/ICAM-1 complexes, two or four for BCR/Ag complexes, and three or five for BCR/Ag2 complexes).
Because of the intricacies and computational cost involved in explicitly modeling the cytoskeleton, cytoskeletally driven motion toward the center of the contact zone is simulated in a computationally efficient, indirect manner by biasing the diffusion of molecules toward the center (see Appendix for details). We define a biasing factor η that we multiply pdiff(i) by if the target node is closer to the center of the simulation domain than the molecule’s current location. The case η = 1 corresponds to purely random motion, with biased motion toward the center increasing with larger η.
Sampling and Time Step Size
Experimentally measured parameter values and their probabilistic counterparts
Measured or estimated value
Same as meas. value
Same as est. value
Same as meas. value
Same as est. value
Dfree molecules(B cell)
~10−12 m4/J sa
Same as est. value
Same as meas. value
5 × 10−20 J25
Same as est. value
The diffusion coefficient of free receptor molecules in a cell membrane is in the range of ~0.01–0.1 μm2/s,14 while we estimate that it is an order of magnitude greater for molecules in a lipid bilayer. Since free antigen and ICAM-1 on the bilayer are the fastest diffusing species, we set pdiff(FB) = 1 and accordingly pdiff(FC) = 0.1. Due to the difficulty of experimentally measuring the diffusion coefficient of receptor–ligand complexes, we have not been able to find measured values for this quantity. B cell synapse experiments, however, show that a significant loss in mobility occurs upon antigen binding.28,29 Consequently, we use a value of pdiff(C) = 0.01 in our simulations. In this case, the bias in diffusion will affect free molecules and receptor–ligand complexes differently, with slow diffusing complexes being proportionately more affected (see Appendix). We carry out in silico experiments in which we vary the ηi for free BCR, BCR/Ag complexes, free LFA-1, and LFA-1/ICAM-1 complexes. Free antigen and ICAM-1 molecules, here modeled as being on artificial lipid bilayer, are always assumed to have ηi = 1.
Parameter and Time Step Mapping
Because some of the parameters of our model are probabilistic in nature and therefore dimensionless, it is necessary to map them onto physical quantities in order to physically interpret the results. Two such mappings are necessary: One which maps the probabilistic affinity PAmax to the association constant KA and one which maps the size of our model’s time step to physical time by relating pdiff to the physical diffusion coefficient D.
Biased Diffusion of BCR/Ag Complexes Results in Synapse Formation with Membrane Deformation
Synapses Can Only Form When the Diffusion Bias of All Membrane Bound Species Is Smaller Than That of BCR/Ag Complexes
Single Molecule Trajectories Can Be Used to Test for the Relative Strength of Diffusion Bias of the Various Membrane-Bound Species
Mean Distance From the Center Can Be Used to test for Cytoskeletally Driven Directed Transport of Molecules
In Fig. 7b, the mean distance from the center of BCR/Ag complexes increases as the diffusion bias factor of free BCR, free LFA-1, and LFA-1/ICAM-1 complexes increases, and the mean distance from the center of free BCR and free LFA-1 molecules decreases. This is in spite of the fact that the diffusion bias factor of BCR/Ag complexes is fixed at ηBA = 1.1. The mean distance from the center of LFA-1/ICAM-1 complexes decreases, albeit weakly, as the effect of increasing diffusion bias (ηLI) is offset by increasing crowding at the center of the contact zone by free BCR and LFA-1 molecules. We also note that when the diffusion bias of free BCR and LFA-1 is strong, the mean distance from the center of these molecules is comparable to that of BCR/Ag complexes, while when the bias of free BCR and LFA-1 molecules is weak, the mean distance from the center is approximately an order of magnitude greater than that of BCR/Ag complexes. These data suggest that it is possible to obtain insight into the nature of the cytoskeletal transport mechanism by calculating the mean distance from the center of the contact region of the various species involved in B cell synapse formation.
In this study, we used a kinetic Monte Carlo simulation method to investigate whether directed transport of molecules toward the center of the B cell/bilayer interface is a potential mechanism of immunological synapse formation in B cells. A distinguishing feature of our method is the development of a mapping between probabilistic parameters of the Monte Carlo simulation and their physical counterparts, thereby allowing quantitative comparison of our model’s results to those of biological experiments.27,31,32 Significantly, the length and time scales of synapse formation in our model matched experimentally observed length and time scales of B cell synapse formation. The synapses formed in our model are of the order of ~1–1.5 μm in diameter, which is comparable to the diameter of physiological B cell synapses, while the formation time of ~100 s predicted by our model matches relatively well with the experimentally observed synapse formation time of 1–2 min. In addition, our model also reproduced the existence of a threshold value of BCR/Ag affinity for synapse formation of KA = 106 M−1. As in synapse formation experiments,6,8,15 our model did not produce synapses below this critical value of BCR/Ag affinity, thereby matching the experimentally observed affinity range of synapse formation.
We modeled directed transport of receptors in an implicit manner, by biasing the diffusion of molecules toward the center of the B cell/bilayer interface. This approach, rather than an explicit simulation of molecular attachment to the cytoskeleton, was chosen based on computational efficiency considerations. In the Appendix, we show that biased diffusion is mathematically equivalent to explicit simulation of attachment and detachment to the cytoskeleton. Our results reveal that biased diffusion of BCR/Ag complexes is sufficient to produce synapse patterns similar to those observed in biological experiments, even for high affinity antigens. In the absence of such a transport mechanism, BCR molecules show sub-diffusive behavior in the early phase of synapse formation, which inhibits rapid clustering of BCR/Ag complexes, especially for high affinity antigens.32 In the late phase of synapse formation, significant membrane deformation makes it difficult for synapses to form for high affinity antigens in the absence of a transport mechanism.31 These results are thus all the more important in light of the fact that BCR can bind antigens with high affinity (up to 1010 M−1), and that there is experimental evidence to suggest that significant membrane deformation does indeed occur during B cell synapse formation.2,6,8,15
Our results lie in contrast to theoretical studies of T cell synapse formation where it was shown that a difference in equilibrium bond length between TCR/MHCp and LFA1/ICAM1 complexes was enough to segregate the two receptor–ligand pairs into an immunological synapse pattern.25,26 However, it has been shown during T-cell synapse formation experiments that the time needed for an antibody-coated bead to traverse half a cell circumference is approximately 6 min (for a ~10 μm diameter T cell).34 Assuming a diffusion coefficient of the order of ~0.1 μm2/s for receptors on a cell surface,14 a simple calculation (time~distance2/diffusion coefficient) shows that the time needed for receptor to traverse half a cell circumference by pure diffusion is much larger than the time indicated in the work of Wülfing and Davis.34 Thus, it seems likely that active transport of receptors assists the bond length difference-mediated synapse formation mechanism in T cells. In B cells, however, the absence of a difference in equilibrium bond length between BCR/Ag and LFA-1/ICAM1 makes these species’ spontaneous segregation into a synapse pattern difficult. Moreover, in previous work, we also showed that canonical synapse formation by purely passive mechanisms based on differences in the properties of the BCR/Ag and LFA-1/ICAM-1 bonds (e.g., equilibrium bond length, bond stiffness, affinity) is not possible, especially for high affinity antigens on a cell membrane that undergoes deformation.31 However, the results of this study show that directed transport of BCR/Ag complexes to the center of the cell–cell interface is capable of generating canonical synapse patterns even in situations where mechanisms based on differences in bond properties between BCR/Ag and LFA-1/ICAM-1 fail to do so. Importantly, recent B cell synapse formation experiments show impaired synapse formation in signaling-deficient B cells.15 The likely explanation for this is that signaling-deficient B cell receptors cannot generate the signal needed for cytoskeleton-mediated transport of B cell receptors to the synapse.
Our results also indicate that biased diffusion of the other membrane-bound species involved in B cell synapse formation (free BCR, free LFA-1, LFA-1/ICAM-1 complexes) hinders synapse formation unless the bias in the diffusion of BCR/Antigen complexes is significantly stronger than that of the other species. A similar, differential transport mechanism has been shown to be effective in experimental studies of T cell synapse formation.19
We show that it is possible to gain insight into the mechanism of synapse formation by obtaining the trajectories of individual antigen molecules over the course of synapse formation, as well as by calculating the mean distance from the center of the various species. The trajectories can potentially reveal the presence or absence of cytoskeletal transport, while the mean distance from the center can reveal whether cytoskeletal transport affects the various species differently. We believe that an iterative process of computational modeling and physical experimentation can lead to a full understanding of the mechanism of immunological synapse formation in B cells.
PT and SR are supported from NIH Grant AI074022.
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