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MTF and MMH conceived the idea of the paper. MTF designed and performed the numerical experiments. MTF and MMH analyzed the data. MTF and MMH wrote the paper.

The authors have declared that no competing interests exist.

The interaction of T cells and antigen-presenting cells is central to adaptive immunity and involves the formation of immunological synapses in many cases. The surface molecules of the cells form a characteristic spatial pattern whose formation mechanisms and function are largely unknown. We perform computer simulations of recent experiments on geometrically repatterned immunological synapses and explain the emerging structure as well as the formation dynamics. Only the combination of in vitro experiments and computer simulations has the potential to pinpoint the kind of interactions involved. The presented simulations make clear predictions for the structure of the immunological synapse and elucidate the role of a self-organizing attraction between complexes of T cell receptor and peptide–MHC molecule, versus a centrally directed motion of these complexes.

Adaptive immunity is a response of the immune system that involves the activation of lymphocytes and that is most effective in defending against virus-infected cells, cancer cells, fungi, and intracellular bacteria. Central to this response is the interaction between a T cell and an antigen-presenting cell, and in particular the communication of information mediated by the T cell receptor and co-receptors. The contact zone between the cells is a highly organized interface, which is termed the immunological synapse, where both the spatial and the temporal organization of the bound receptors contribute to the generated activation signal on antigen recognition. Although a considerable amount of experimental and theoretical studies have dealt with the immunological synapse, the mechanisms that control its formation are still under discussion. In 2005, Mossman et al. conducted ingenious experiments using nanometer-scale structures to geometrically repattern the immunological synapse. These experiments are reproduced by Figge and Meyer-Hermann applying computer simulations, based on an agent-based model approach, to uncover the emerging structures as well as the underlying formation mechanisms. Clear predictions for the structure of proposed geometrically repatterned immunological synapses are obtained that will further elucidate the role of the involved formation mechanisms.

The recognition of pathogens by the T cells of the immune system relies on antigen-presenting cells (APCs) that process pathogen-derived molecules and present them with major histocompatibility complex (MHC) molecules. The surface of APCs is scanned by T cells that bind to peptide–MHC (pMHC) complexes with their specific T cell receptors (TCRs). This interaction can initiate the dynamic formation of an immunological synapse (IS), which is an adhesive junction with a nanometer scale gap between the two cells [

The prototypical IS matures within minutes into a well-organized structure with a characteristic bull's-eye pattern that may remain stable for hours [

Recently, K. D. Mossman et al. [

The chromium barriers (black) are implemented in the synthetic bilayer APC and confine the free movement of TCR–pMHC (green) and LFA-1-ICAM-1 (red), as indicated by the crossed arrow. The TCR–pMHC complexes interact with each other via the cytoskeleton of the T cell.

Various aspects of the IS have been successfully analyzed by in silico experiments [

A different model approach is adopted in the present paper, where we focus on the high potential of geometrical repatterning to uncover the nature of the interaction mechanisms underlying the formation and geometry of the ISs. This is achieved by performing a comparative study of in silico experiments that are based on a generic cellular automaton. In this agent-based approach, receptor–ligand complexes are treated as discrete entities that evolve into a pattern by moving due to thermally induced stochastic motion and according to their mutual interactions.

The experimental basis for these models is given by the observation that, due to large differences in the length of TCR–pMHC complexes (∼15 nm) and LFA-1–ICAM-1 complexes (∼45 nm), elastic membrane forces will drive their segregation [

In contrast to previous agent-based models [

It should be noted that in the present model we consider receptor–ligand complexes to move as multimeric units by neglecting the individual unbinding and rebinding of receptors and ligands. As a consequence, we neglect the possibility that receptor–ligand complexes might cross the imposed barriers, which is supported by the experimental observation that stable microclusters are formed and that individual TCRs do not percolate over the barriers [

The comparison with existing in vitro experiments on geometrically repatterned ISs reveals that three interaction mechanisms are essential during the synapse formation: (i) adhesion between neighboring TCR–pMHC complexes, (ii) repulsive short-range interactions between TCR–pMHC and LFA-1–ICAM-1 complexes, and (iii) either a centrally directed motion of TCR–pMHC complexes or a long-range attractive interaction between them. To determine the relevant type of TCR–pMHC aggregation mechanism, we propose novel experiments on geometrically repatterned ISs and make quantitative predictions for the occurrence of a pattern transition.

The in silico experiments are performed for the same geometrically repatterned ISs that were recently studied in in vitro experiments by K. D. Mossman et al. [^{−2} and 100 μm^{−2}, respectively. The receptor–ligand complexes perform random moves within the interface region and interact among each other. The same diffusion constant, D = 0.06 μm^{2}/s, is chosen for TCR–pMHC and LFA-1–ICAM-1, which corresponds to a typical value for these membrane-anchored macromolecules [_{i} and the relative interaction strength w_{i} with I = rep, att, and dir, respectively. The details of the cellular automaton are summarized in

In

Both simulations take adhesion between TCR–pMHC complexes (α = 1), diffusion of TCR–pMHC and LFA-1–ICAM-1 (D = 0.06 μm^{2}/s, and short-range repulsion between TCR–pMHC and LFA-1–ICAM-1 (L_{rep} = 0.1R, w_{rep} = −1) into account.

(A–H) TCR–pMHC aggregation due to long-range attraction (L_{att} = R, w_{att} = 0.14, w_{dir} = 0).

(I–P) TCR–pMHC aggregation due to centrally directed motion (L_{dir} = R, w_{dir} = 3, w_{att} = 0). The IS formation is shown after 30 s in (E) and (M), after 2 min in (F) and (N), after 5 min in (G) and (O), and after 10 min in (H) and (P).

It cannot be excluded that other types of interactions are present, e.g., adhesive forces between LFA-1–ICAM-1 complexes; however, the comparison with the experimentally observed geometrically repatterned ISs indicates that the included mechanisms are sufficient, are all required, and seem to be the most important ones. In addition, depending on the precise interaction parameters, a rich variety of IS patterns is observed. In _{att} has a strong impact on the IS pattern. This can be seen in _{att}, starting from the same simulation parameters as in _{att} < R/2. In the context of immature T cells (thymocytes), multifocal synapse patterns have been attributed to the reduced density of TCRs and thermal fluctuations [

(A–D) Same parameters as in _{att} = R, (B) L_{att} = 0.43R, (C) L_{att} = 0.29R, and (D) L_{att} = 0.15R.

(E–H) Same parameters as in _{att} = 0) for the geometrically repatterned ISs.

(I–L) IS pattern formation in the absence of long-range attraction between TCR–pMHCs (w_{att} = 0) and short-range repulsion between TCR–pMHC and LFA-1–ICAM-1 (w_{rep} = 0), and with strong TCR–pMHC adhesion: α = 5.

(M–P) Same parameters as in _{rep} = 0).

In

An interesting aspect can be observed for the geometrically repatterned ISs in

The formation of the bull's-eye pattern is observed if long- and short-range interactions are omitted while adhesion between pairs of TCR–pMHC complexes is increased in strength (see

We finally show in

To reproduce the experimentally observed geometrically repatterned ISs by in silico experiments, three relevant interaction mechanisms play an important role: (i) adhesion between neighboring TCR–pMHC complexes, (ii) repulsive short-range interactions between TCR–pMHC and LFA-1–ICAM-1 complexes, and (iii) either a centrally directed motion of TCR–pMHC complexes mediated by aggregation proteins, or a long-range attractive interaction between TCR–pMHC pairs mediated by the cytoskeleton. To answer the question by which aggregation mechanism TCR–pMHCs accumulate at the center of the IS, we propose a conclusive procedure that makes once again use of the high potential of geometrical repatterning experiments. The difference between an attractive long-range interaction and a directed motion of TCR–pMHC can be made visible by realizing that the former interaction depends in a crucial way on the distribution of TCR–pMHC complexes, whereas the latter mechanism is governed by the distribution of proteins. It thus follows that the two mechanisms can be distinguished if the number of TCR–pMHC complexes is geometrically confined in such a way that these mechanisms give rise to clearly distinguishable IS patterns. We propose experiments where the freedom of TCR–pMHC movement is geometrically confined by a barrier that subdivides the IS into an inner and an outer region, respectively, with an inner TCR–pMHC number, N_{i}, and an outer TCR–pMHC number, N_{o}. Varying the size of the inner compartment is accompanied by a change in the ratio N_{o}/N_{i} of the TCR–pMHC numbers in the outer to the inner region. In the presence of directed TCR–pMHC motion, the resulting IS pattern will not change qualitatively as a function of N_{o}/N_{i}. However, we expect that in the presence of an attractive long-range interaction between TCR–pMHCs, the c-SMAC will only form if N_{o} << N_{i}, whereas for N_{o} >> N_{i} the TCR–pMHCs of the inner region will be attracted towards the geometric boundary. To prevent the blurring of the desired effect by the repulsion between TCR–pMHC and LFA-1–ICAM-1 that is acting across the barrier, as has been discussed in connection with _{rep}.

In

The parameters are the same as in

(A–D) and (I–L) TCR–pMHC aggregation due to long-range attractive interaction (L_{att} = R, w_{att} = 0.14, w_{dir} = 0).

(E–H) and (M–P) TCR–pMHC aggregation due to centrally directed motion (L_{dir} = R, w_{dir} = 3, w_{att} = 0). The radius of the circular geometry and the simulation time are (A) and (E) r = 0.36R after 30 min, (B) and (F) r = 0.43R after 30 min, (C) and (G) r = 0.43R after 60 min, (D) and (H) r = 0.50R after 30 min. The side length of the quadratic geometry and the simulation time are (I) and (M) s = 0.57R after 30 min, (J) and (N) s = 0.71R after 30 min, (K) and (O) s = 0.71R after 60 min, (L) and (P) s = 0.85R after 30 min.

A quantitative estimate for the occurrence of the pattern transition is obtained as follows. Assuming the initial random distribution of TCR–pMHCs to be homogeneous, the ratio n_{r} = N_{o}/N_{i} is directly related to the areas of the outer and inner regions, respectively, A_{o} and A_{i}. The area of the outer region may be expressed in terms of the total interface area A = πR^{2}. We then estimate:
_{i} = πr^{2} for the circular geometry and A_{i} = s^{2} for the quadratic geometry. The pattern transition takes place at a critical value of the ratio, n_{r} = n_{c}, where n_{c} may depend on geometrical constraints, diffusion, and adhesion, as well as on effects of the repulsive interaction. The pattern transition is found to occur at the critical extensions r_{c} ≈ 0.43R and s_{c} ≈ 0.78R, respectively, for the circular and quadratic geometry. This implies s_{c}/r_{c} ≈ π^{1/2} and, thus, that the pattern transition occurs for these geometries at approximately the same critical area for the inner region: A_{i} ≈ 0.2A. Inserting this value for A_{i} into the expression for n_{r} yields a quantitative estimate for the critical ratio:

It should be noted that, in principle, the IS may be formed by a combination of the long-range attraction and the directed motion of TCR–pMHC. In this case, the transition is expected to be shifted to a larger ratio n_{c} > 4 and thus to a smaller critical value for the area of the inner region, A_{i} < 0.2A. The quantitative estimate for the critical ratio, N_{o}/N_{i} ≈ A_{o}/A_{i} ≈ 4, may still serve as a guideline for the experimental realization of the pattern transition in the IS formation.

We conclude by once again emphasizing the high potential of geometrical repatterning of ISs with respect to gaining new insight into the underlying mechanisms that govern IS formation. The computer simulations are performed in the classical spirit of an interdisciplinary approach [

A cellular automaton is used to perform in silico experiments on the formation of geometrically repatterned ISs. To keep the number of involved parameters as small as possible, a minimal phenomenological model is considered where the cell–cell interface is represented by a square lattice of circular geometry with radius R and N sites. To simulate a T cell with a diameter of approximately 10 μm, the lattice constant is set to a = 70 nm and the radius is set to R = 70a, which gives rise to a lattice of circular geometry with roughly N = 15 × 10^{3} sites. Each site has four nearest-neighbors and four (diagonal) next-nearest-neighbors, and can be either empty or occupied by one of the N_{TM} and N_{LI} complexes of TCR–pMHC and LFA-1–ICAM-1 in the system, respectively. The number of receptor–ligand complexes relative to the number of sites that are not excluded by the presence of barriers, are in all simulations fixed around 0.2 and 0.47, respectively, for TCR–pMHC and LFA-1–ICAM-1 [

Initially all complexes of TCR–pMHC and of LFA-1–ICAM-1 are distributed randomly on the lattice, i.e., we neglect the recruitment of TCR and LFA-1 from the backside of the T cell since it is known that the cell–cell interface is fully developed during the first 30 s of synapse formation [_{TM} + N_{LI} sites per time step at random and change the system configuration locally due to the random motion of receptor–ligand complexes and due to their interactions among each other. This procedure is represented by the flowchart in

See the text for details.

If a chosen site is occupied, the receptor–ligand complex is allowed to move randomly with a probability p_{s}. In the case of LFA-1–ICAM-1, this move is performed if the neighbor site, which is randomly chosen from the eight nearest-neighbor and next-nearest-neighbor sites, is empty. In this procedure, the probability p_{s} for moving to one of the four next-nearest-neighbors is reduced by a factor 2^{−1/2}. In the case of TCR–pMHC, whether or not the move is performed depends in addition on adhesive forces between TCR–pMHC complexes at the four nearest-neighbor sites. Adhesive forces are taken into account by an adhesive factor that reduces the probability p_{s} for the move. In the model, the adhesive factor is given by f_{α}(N_{nn}) = 1/(1 + N_{nn})^{α}, where N_{nn} is the actual number of nearest TCR–pMHC-neighbors (0 ≤ N_{nn} ≤ 4), and the parameter α is a measure for the strength of the adhesive force. In all simulations presented in this paper we have chosen p_{s} = 1, from which we estimate the time step for a freely moving membrane-anchored macromolecule with diffusion constant D = 0.06 μm^{2}/s to be τ = a^{2}/(4D) = 0.02 s.

Furthermore, a randomly chosen receptor–ligand complex may undergo interactions with other receptor–ligand complexes and move according to these interactions with probability p_{i}. In the case of TCR–pMHC, this move is again subjected to adhesive forces due to its nearest-neighbor TCR–pMHC complexes. In all simulations presented in this paper we have chosen p_{i} = 0.3, which implies a general dominance of the number of randomly induced moves over the number of moves that are induced by interactions. In other words, the ratio p_{i}/p_{s} is comparable to the ratio of the potential to the kinetic energy, and p_{i}/p_{s} < 1 has been chosen in the spirit of a fluidity model for the plasma membrane.

Different types of interactions between receptor–ligand complexes are considered. The first type of interaction is related to elastic membrane forces that arise due to the different lengths of TCR–pMHC and LFA-1–ICAM-1 when they are close together. This repulsive interaction of weight w_{rep} is responsible for the segregation of TCR–pMHC and LFA-1–ICAM-1 driving them away from each other if the distance between them is less than the length L_{rep}. The distance is related to the region of membrane distortion and is typically on the order of several lattice sites, L_{rep} = 0.1R << R. The second type of interaction gives rise to the aggregation of TCR–pMHC at the c-SMAC of the IS. Two possibilities for the origin of this interaction are considered, which are referred to as model A and model B in _{att} (model A); (ii) a centrally directed motion of TCR–pMHC mediated by aggregation proteins that enhance the TCR–pMHC accumulation at a specific point. The interaction range is defined by L_{dir} (model B). In the simulations presented here we either use the interaction of type (i) or (ii) with, respectively, L_{att} = R or L_{dir} = R, if not stated otherwise.

The precise functional dependence of the involved forces is not known and depends on numerous complicated factors, e.g., the time-dependent changes of the membrane under the formation of the IS that have not been monitored in experiments. Thus, we apply the following intuitive rule: if the randomly chosen lattice site is occupied by a TCR–pMHC complex, we calculate the unit vectors in the direction of all LFA-1–ICAM-1 complexes that are less than the distance L_{rep} apart, sum them up, and give this direction a weight w_{rep} < 0 that is related to the strength of the repulsive force. Next, in the case of interaction type (i), we calculate the unit vectors in the direction of all TCR–pMHC complexes that are less than the distance L_{att} apart, sum them up, and give this direction a weight w_{att} > 0 (model A). In the case of interaction type (ii), we calculate the unit vector in the direction of the center of the lattice and give this direction a weight w_{dir} > 0 (model B). In both cases, the two computed vectors are added and the resulting vector is normalized. The latter vector points in the direction of one of its eight neighboring lattice sites, in which the TCR–pMHC complex moves with a probability subjected to the adhesive factor f_{α}(N_{nn}).

We proceed in a corresponding manner if the randomly chosen lattice site is occupied by an LFA-1–ICAM-1 complex; however, in this case we only have to account for the repulsive force due to membrane distortions by surrounding TCR–pMHC complexes.

In the present in silico experiments, strict barriers are imposed, i.e., receptor–ligand complexes are not allowed to cross the barriers. Related to this issue, at this stage we do not account for the unbinding and rebinding of receptor and ligand, which might induce a small portion of barrier crossing. Furthermore, the recruitment of TCR and LFA-1 from the backside of the T cell during the first few seconds of the IS formation could be taken into account. This would give rise to a time-dependent increase of the TCR–pMHC and LFA-1–ICAM-1 densities at the cell–cell interface that is accompanied by a pattern inversion from an outer TCR–pMHC ring into an outer LFA-1–ICAM-1 ring, as is experimentally observed during early IS formation [

antigen-presenting cell

central supramolecular activation cluster

intercellular adhesion molecule-1

immunological synapse

leukocyte function–associated antigen-1

major histocompability complex

peptide–major histocompability complex

peripheral supramolecular activation cluster

T cell receptor