A Java High Performance Tool For Topological Data Analysis

REFERENCES:

[1] MASSIMO BERNASCHI AND FILIPPO CASTIGLIONE. Design and implementation of an immune system simulator. Computers in Biology and Medicine, 31(5):303–331, 2001.

[2] NIELS K JERNE. Towards a network theory of the immune system. In Annales d’immunologie, 125, pages 373–389, 1974.

[3] MATTEO RUCCO, FILIPPO CASTIGLIONE, EMANUELA MERELLI, AND MARCO PETTINI. Characterisation of the idiotypic immune network through persistent entropy. In proceedings of European Conference on Complex Systems, Lucca, 2014, Springer, Complexity.

[4] EMANUELA MERELLI, MATTEO RUCCO, PETER SLOOT, AND LUCA TESEI. Topological Characterization of Complex Systems: Using Persistent Entropy. Entropy, 17(10):6872–6892, 2015.

Temporal evolution of the Persistent Entropy. The peaks correspond to a double injection of antigen. The plateau corresponds to the immune memory.

Idiotypic Network during the simulation

Histogram of antibodies. The most frequent antibodies are directly involved in the dynamics of whole system. If they are removed the persistent entropy is flattened.

Example of topological graph obtained from antibodies' network. The nodes are simplices involved in persistent topological holes. An edge exists between two nodes if they belong to the same hole. The thickness of the edge is proportional to the frequency of appearance.

In this work, we performed a preliminary application of TDA to a model of the mammal immune system, the so-called idiotypic network and simulated with C-ImmSim that is the agent-based simulator simulation of the immune system [1]. The computational ABM employed is “discrete” in mathematical terms because it represents biological entities as individuals in a heterogeneous population of cells and molecules. In particular, the major classes of cells of the lymphoid lineage and some of the myeloid lineage. Moreover the model accounts for various interleukins and messengers. This “discreteness” confers the model the character of being “easily scalable” in terms of introducing new biological complexity, at variance with corresponding equation based models.The model is stochastic, meaning it can naturally display biological “controlled” variability: for example, it is possible to separate the sorting of repertoire specificities from the random occurrences (encounters, binding, cell death, cell replacement, diffusion, cell division) during the running of the response. In other words, each repertoire expresses a private specificity, and by repeating runs with random events, the impact of different repertoires can be compared and their variations statistically determined, at the same time increasing the significance of results. The ABM model, is a polyclonal model, since all lymphocytes are equipped with a receptor represented as a binary string. This allows for a number of immunological features such as expressed and potential repertoire definition, specific recognition/binding, antigen’s peptides presentation, specific clonal memory, hypermutation, etc.

Idiotypic Network , historically is the former model among the models of the mammal immune system. Accordingly to Jerne [2], the Idiotypic Network can be described by three main states: virgin, activation and memory. In the virgin state there are not antibodies but only non-specific T-cells that start their activities after the activation signal sent by B-cells that recognize the presence of pathogens. Then, the Idiotypic Network proliferates antibodies and reaches the activation state, which is not a steady state. After the activation, the Idiotypic Network performs the immunization during which the antibodies play a dual role. In fact, an antibody can be seen both as a self protein (a protein of the organism) or a non-self protein (a pathogen to be suppressed). After the immunization the Idiotypic Network reaches the memory state. This state represents a steady condition in which there is only a selection of antibodies. All the transitions and states of Idiotypic Network are characterized by the fact that antibodies perform basically two actions: elicitation and suppression.

In the rest of this page we report about the application of Clique Weight Rank Persistent Homology, by jHoles, to the Idiotypic Network simulated with C-ImmSim. In our configuration a simulation has a lifespan of 2190 ticks, where a tick=8 hours, and a repertoire of at most 10^12 antibodies and antigen volume equal to V = 10mL. The results are the average over 100 runs. In the simulator each idiotype (both antigens and antibodies) is represented with a bit-string, in our case of 12 bit length. An idiotype interacts with each other if and only if their Hamming distance is 11__<__d(Aj;Ak)__<__12. The pair-wise distances are stored in a matrix, the so-called Affinity matrix.We defined a weight function for the idiotypic network, the coexistence function between antibodies:

Where [Abi(t)] is the concentration of the i-th antibody Abl at tick t. The meaning of equation is that for lower values of affinity the concentration must be more significant because the match between antibodies is less probable.

For each simulation we computed the coexistence function and we used the weighted idiotypic network as input for the persistent homology computation. In this work we used the clique weight rank persistent homology (CWRPH) algorithm implemented in jHoles. The output of the persistent homology has been used for computing both the persistent entropy and for identifying the persistent holes and their generators, namely the persistent antibodies that govern the evolution of the idiotypic network during the virgin state, the activation and the immune memory. Persistent entropy is able to recognize the activation of the immune system: the peaks in the charts point out the immune activation that is following by a transient that represents the immune response. During the immune response the antibodies play a dual role: they can simultaneously elicit and suppress each other. After this transient there is a plateau that represents the persistent immune network activation corresponding to the immune memory.

Persistent entropy is directly computed from the result of persistent homology: the Betti numbers.Persistent entropy can be thought as complexity measures for simplicial complexes. The reason is evident in the mathematical definition: persistent entropy depends by the topological noise and by the persistent topological features.

The analysis of the generators of the homological classes allows to identify the real number of antibodies that have been used: 203 instead of 4096. The analysis of the persistent Betti numbers reveals that there is a subset of antibodies arranged in a 1-dimensional hole that is present both in the activation state and in the memory state. This 1-dimensional hole is formed by the antibodies: Ab1; Ab2; Ab7; Ab13. This hole is formed by the most active antibodies, see the histogram. The removal of this 1-dimensional hole from the barcodes will flatten the entropy, that means this cycle is formed by the most specialized antibodies for the antigen that has been injected [3,4].

jHoles